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MATH 361: Financial Mathematics for Actuaries I
Albert Cohen
Actuarial Sciences ProgramDepartment of Mathematics
Department of Statistics and ProbabilityC336 Wells Hall
Michigan State UniversityEast Lansing MI
[email protected]@stt.msu.edu
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 1 / 161
Course Information
Syllabus to be posted on class page in first week of classes
Homework assignments will posted there as well
Page can be found at https://www.math.msu.edu/classpages/
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 2 / 161
Course Information
Many examples within these slides are used with kind permission ofProf. Dmitry Kramkov, Dept. of Mathematics, Carnegie MellonUniversity.
Book for course: Financial Mathematics: A Comprehensive Treatment(Chapman and Hall/CRC Financial Mathematics Series) 1st Edition.Can be found in MSU bookstores now
Some examples here will be similar to those practice questionspublicly released by the SOA. Please note the SOA owns thecopyright to these questions.
This book will be our reference, and some questions for assignmentswill be chosen from it. Copyright for all questions used from this bookbelongs to Chapman and Hall/CRC Press .
From time to time, we will also follow the format of Marcel Finan’s ADiscussion of Financial Economics in Actuarial Models: A Preparationfor the Actuarial Exam MFE/3F. Some proofs from there will bereferenced as well. Please find these notes here
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 3 / 161
What are financial securities?
Traded Securities - price given by market.
For example:
StocksCommodities
Non-Traded Securities - price remains to be computed.
Is this always true?
We will focus on pricing non-traded securities.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 4 / 161
What are financial securities?
Traded Securities - price given by market.
For example:
StocksCommodities
Non-Traded Securities - price remains to be computed.
Is this always true?
We will focus on pricing non-traded securities.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 4 / 161
What are financial securities?
Traded Securities - price given by market.
For example:
StocksCommodities
Non-Traded Securities - price remains to be computed.
Is this always true?
We will focus on pricing non-traded securities.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 4 / 161
What are financial securities?
Traded Securities - price given by market.
For example:
StocksCommodities
Non-Traded Securities - price remains to be computed.
Is this always true?
We will focus on pricing non-traded securities.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 4 / 161
How does one fairly price non-traded securities?
By eliminating all unfair prices
Unfair prices arise from Arbitrage Strategies
Start with zero capitalEnd with non-zero wealth
We will search for arbitrage-free strategies to replicate the payoff of anon-traded security
This replication is at the heart of the engineering of financial products
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 5 / 161
How does one fairly price non-traded securities?
By eliminating all unfair prices
Unfair prices arise from Arbitrage Strategies
Start with zero capitalEnd with non-zero wealth
We will search for arbitrage-free strategies to replicate the payoff of anon-traded security
This replication is at the heart of the engineering of financial products
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 5 / 161
How does one fairly price non-traded securities?
By eliminating all unfair prices
Unfair prices arise from Arbitrage Strategies
Start with zero capitalEnd with non-zero wealth
We will search for arbitrage-free strategies to replicate the payoff of anon-traded security
This replication is at the heart of the engineering of financial products
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 5 / 161
How does one fairly price non-traded securities?
By eliminating all unfair prices
Unfair prices arise from Arbitrage Strategies
Start with zero capitalEnd with non-zero wealth
We will search for arbitrage-free strategies to replicate the payoff of anon-traded security
This replication is at the heart of the engineering of financial products
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 5 / 161
How does one fairly price non-traded securities?
By eliminating all unfair prices
Unfair prices arise from Arbitrage Strategies
Start with zero capitalEnd with non-zero wealth
We will search for arbitrage-free strategies to replicate the payoff of anon-traded security
This replication is at the heart of the engineering of financial products
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 5 / 161
More Questions
Existence - Does such a fair price always exist?
If not, what is needed of our financial model to guarantee at least onearbitrage-free price?
Uniqueness - are there conditions where exactly one arbitrage-freeprice exists?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 6 / 161
And What About...
Does the replicating strategy and price computed reflect uncertaintyin the market?
Mathematically, if P is a probabilty measure attached to a series ofprice movements in underlying asset, is P used in computing theprice?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 7 / 161
And What About...
Does the replicating strategy and price computed reflect uncertaintyin the market?
Mathematically, if P is a probabilty measure attached to a series ofprice movements in underlying asset, is P used in computing theprice?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 7 / 161
Notation
Forward Contract:
A financial instrument whose initial value is zero, and whose finalvalue is derived from another asset. Namely, the difference of thefinal asset price and forward price:
V (0) = 0,V (T ) = S(T )− F (1)
Value at end of term can be negative - buyer accepts this in exchangefor no premium up front
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 8 / 161
Notation
Forward Contract:
A financial instrument whose initial value is zero, and whose finalvalue is derived from another asset. Namely, the difference of thefinal asset price and forward price:
V (0) = 0,V (T ) = S(T )− F (1)
Value at end of term can be negative - buyer accepts this in exchangefor no premium up front
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 8 / 161
Notation
Forward Contract:
A financial instrument whose initial value is zero, and whose finalvalue is derived from another asset. Namely, the difference of thefinal asset price and forward price:
V (0) = 0,V (T ) = S(T )− F (1)
Value at end of term can be negative - buyer accepts this in exchangefor no premium up front
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 8 / 161
Notation
Interest Rate:
The rate r at which money grows. Also used to discount the valuetoday of one unit of currency one unit of time from the present
V (0) =1
1 + r,V (1) = 1 (2)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 9 / 161
Notation
Interest Rate:
The rate r at which money grows. Also used to discount the valuetoday of one unit of currency one unit of time from the present
V (0) =1
1 + r,V (1) = 1 (2)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 9 / 161
An Example of Replication
Forward Exchange Rate: There are two currencies, foreign anddomestic:
SBA = 4 is the spot exchange rate - one unit of B is worth SB
A of Atoday (time 0)
rA = 0.1 is the domestic borrow/lend rate
rB = 0.2 is the foreign borrow/lend rate
Compute the forward exchange rate FBA . This is the value of one unit
of B in terms of A at time 1.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 10 / 161
An Example of Replication
Forward Exchange Rate: There are two currencies, foreign anddomestic:
SBA = 4 is the spot exchange rate - one unit of B is worth SB
A of Atoday (time 0)
rA = 0.1 is the domestic borrow/lend rate
rB = 0.2 is the foreign borrow/lend rate
Compute the forward exchange rate FBA . This is the value of one unit
of B in terms of A at time 1.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 10 / 161
An Example of Replication
Forward Exchange Rate: There are two currencies, foreign anddomestic:
SBA = 4 is the spot exchange rate - one unit of B is worth SB
A of Atoday (time 0)
rA = 0.1 is the domestic borrow/lend rate
rB = 0.2 is the foreign borrow/lend rate
Compute the forward exchange rate FBA . This is the value of one unit
of B in terms of A at time 1.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 10 / 161
An Example of Replication
Forward Exchange Rate: There are two currencies, foreign anddomestic:
SBA = 4 is the spot exchange rate - one unit of B is worth SB
A of Atoday (time 0)
rA = 0.1 is the domestic borrow/lend rate
rB = 0.2 is the foreign borrow/lend rate
Compute the forward exchange rate FBA . This is the value of one unit
of B in terms of A at time 1.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 10 / 161
An Example of Replication
Forward Exchange Rate: There are two currencies, foreign anddomestic:
SBA = 4 is the spot exchange rate - one unit of B is worth SB
A of Atoday (time 0)
rA = 0.1 is the domestic borrow/lend rate
rB = 0.2 is the foreign borrow/lend rate
Compute the forward exchange rate FBA . This is the value of one unit
of B in terms of A at time 1.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 10 / 161
An Example of Replication: Solution
At time 1, we deliver 1 unit of B in exchange for FBA units of domestic
currency A.
This is a forward contract - we pay nothing up front to achieve this.
Initially borrow some amount foreign currency B, in foreign market togrow to one unit of B at time 1. This is achieved by the initial
amountSBA
1+rB(valued in domestic currency)
Invest the amountFBA
1+rAin domestic market (valued in domestic
currency)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 11 / 161
An Example of Replication: Solution
At time 1, we deliver 1 unit of B in exchange for FBA units of domestic
currency A.
This is a forward contract - we pay nothing up front to achieve this.
Initially borrow some amount foreign currency B, in foreign market togrow to one unit of B at time 1. This is achieved by the initial
amountSBA
1+rB(valued in domestic currency)
Invest the amountFBA
1+rAin domestic market (valued in domestic
currency)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 11 / 161
An Example of Replication: Solution
At time 1, we deliver 1 unit of B in exchange for FBA units of domestic
currency A.
This is a forward contract - we pay nothing up front to achieve this.
Initially borrow some amount foreign currency B, in foreign market togrow to one unit of B at time 1. This is achieved by the initial
amountSBA
1+rB(valued in domestic currency)
Invest the amountFBA
1+rAin domestic market (valued in domestic
currency)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 11 / 161
An Example of Replication: Solution
At time 1, we deliver 1 unit of B in exchange for FBA units of domestic
currency A.
This is a forward contract - we pay nothing up front to achieve this.
Initially borrow some amount foreign currency B, in foreign market togrow to one unit of B at time 1. This is achieved by the initial
amountSBA
1+rB(valued in domestic currency)
Invest the amountFBA
1+rAin domestic market (valued in domestic
currency)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 11 / 161
An Example of Replication: Solution
This results in the initial value
V (0) =FBA
1 + rA−
SBA
1 + rB(3)
Since the initial value is 0, this means
FBA = SB
A
1 + rA
1 + rB= 3.667 (4)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 12 / 161
An Example of Replication: Solution
This results in the initial value
V (0) =FBA
1 + rA−
SBA
1 + rB(3)
Since the initial value is 0, this means
FBA = SB
A
1 + rA
1 + rB= 3.667 (4)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 12 / 161
Discrete Probability Space
Let us define an event as a point ω in the set of all possible outcomes Ω.This includes the events ”The stock doubled in price over two tradingperiods” or ”the average stock price over ten years was 10 dollars”.
In our initial case, we will consider the simple binary spaceΩ = H,T for a one-period asset evolution. So, given an initialvalue S0, we have the final value S1(ω), with
S1(H) = uS0,S1(T ) = dS0 (5)
with d < 1 < u. Hence, a stock increases or decreases in price,according to the flip of a coin.
Let P be the probability measure associated with these events:
P[H] = p = 1− P[T ] (6)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 13 / 161
Discrete Probability Space
Let us define an event as a point ω in the set of all possible outcomes Ω.This includes the events ”The stock doubled in price over two tradingperiods” or ”the average stock price over ten years was 10 dollars”.
In our initial case, we will consider the simple binary spaceΩ = H,T for a one-period asset evolution. So, given an initialvalue S0, we have the final value S1(ω), with
S1(H) = uS0,S1(T ) = dS0 (5)
with d < 1 < u. Hence, a stock increases or decreases in price,according to the flip of a coin.
Let P be the probability measure associated with these events:
P[H] = p = 1− P[T ] (6)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 13 / 161
Discrete Probability Space
Let us define an event as a point ω in the set of all possible outcomes Ω.This includes the events ”The stock doubled in price over two tradingperiods” or ”the average stock price over ten years was 10 dollars”.
In our initial case, we will consider the simple binary spaceΩ = H,T for a one-period asset evolution. So, given an initialvalue S0, we have the final value S1(ω), with
S1(H) = uS0,S1(T ) = dS0 (5)
with d < 1 < u. Hence, a stock increases or decreases in price,according to the flip of a coin.
Let P be the probability measure associated with these events:
P[H] = p = 1− P[T ] (6)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 13 / 161
Discrete Probability Space
Let us define an event as a point ω in the set of all possible outcomes Ω.This includes the events ”The stock doubled in price over two tradingperiods” or ”the average stock price over ten years was 10 dollars”.
In our initial case, we will consider the simple binary spaceΩ = H,T for a one-period asset evolution. So, given an initialvalue S0, we have the final value S1(ω), with
S1(H) = uS0,S1(T ) = dS0 (5)
with d < 1 < u. Hence, a stock increases or decreases in price,according to the flip of a coin.
Let P be the probability measure associated with these events:
P[H] = p = 1− P[T ] (6)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 13 / 161
Discrete Probability Space
Let us define an event as a point ω in the set of all possible outcomes Ω.This includes the events ”The stock doubled in price over two tradingperiods” or ”the average stock price over ten years was 10 dollars”.
In our initial case, we will consider the simple binary spaceΩ = H,T for a one-period asset evolution. So, given an initialvalue S0, we have the final value S1(ω), with
S1(H) = uS0,S1(T ) = dS0 (5)
with d < 1 < u. Hence, a stock increases or decreases in price,according to the flip of a coin.
Let P be the probability measure associated with these events:
P[H] = p = 1− P[T ] (6)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 13 / 161
Arbitrage
Assume that S0(1 + r) > uS0
Where is the risk involved with investing in the asset S ?
Assume that S0(1 + r) < dS0
Why would anyone hold a bank account (zero-coupon bond)?
Lemma Arbitrage free ⇒ d < 1 + r < u
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 14 / 161
Arbitrage
Assume that S0(1 + r) > uS0
Where is the risk involved with investing in the asset S ?
Assume that S0(1 + r) < dS0
Why would anyone hold a bank account (zero-coupon bond)?
Lemma Arbitrage free ⇒ d < 1 + r < u
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 14 / 161
Arbitrage
Assume that S0(1 + r) > uS0
Where is the risk involved with investing in the asset S ?
Assume that S0(1 + r) < dS0
Why would anyone hold a bank account (zero-coupon bond)?
Lemma Arbitrage free ⇒ d < 1 + r < u
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 14 / 161
Arbitrage
Assume that S0(1 + r) > uS0
Where is the risk involved with investing in the asset S ?
Assume that S0(1 + r) < dS0
Why would anyone hold a bank account (zero-coupon bond)?
Lemma Arbitrage free ⇒ d < 1 + r < u
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 14 / 161
Arbitrage
Assume that S0(1 + r) > uS0
Where is the risk involved with investing in the asset S ?
Assume that S0(1 + r) < dS0
Why would anyone hold a bank account (zero-coupon bond)?
Lemma Arbitrage free ⇒ d < 1 + r < u
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 14 / 161
Derivative Pricing
Let S1(ω) be the price of an underlying asset at time 1. Define thefollowing instruments:
Zero-Coupon Bond : V B0 = 1
1+r ,VB1 (ω) = 1
Forward Contract : V F0 = 0,V F
1 = S1(ω)− F
Call Option : V C1 (ω) = max(S1(ω)− K , 0)
Put Option : V P1 (ω) = max(K − S1(ω), 0)
In both the Call and Put option, K is known as the Strike.Once again, a Forward Contract is a deal that is locked in at time 0 forinitial price 0, but requires at time 1 the buyer to purchase the asset forprice F .
What is the value V0 of the above put and call options?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 15 / 161
Derivative Pricing
Let S1(ω) be the price of an underlying asset at time 1. Define thefollowing instruments:
Zero-Coupon Bond : V B0 = 1
1+r ,VB1 (ω) = 1
Forward Contract : V F0 = 0,V F
1 = S1(ω)− F
Call Option : V C1 (ω) = max(S1(ω)− K , 0)
Put Option : V P1 (ω) = max(K − S1(ω), 0)
In both the Call and Put option, K is known as the Strike.Once again, a Forward Contract is a deal that is locked in at time 0 forinitial price 0, but requires at time 1 the buyer to purchase the asset forprice F .
What is the value V0 of the above put and call options?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 15 / 161
Derivative Pricing
Let S1(ω) be the price of an underlying asset at time 1. Define thefollowing instruments:
Zero-Coupon Bond : V B0 = 1
1+r ,VB1 (ω) = 1
Forward Contract : V F0 = 0,V F
1 = S1(ω)− F
Call Option : V C1 (ω) = max(S1(ω)− K , 0)
Put Option : V P1 (ω) = max(K − S1(ω), 0)
In both the Call and Put option, K is known as the Strike.
Once again, a Forward Contract is a deal that is locked in at time 0 forinitial price 0, but requires at time 1 the buyer to purchase the asset forprice F .
What is the value V0 of the above put and call options?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 15 / 161
Derivative Pricing
Let S1(ω) be the price of an underlying asset at time 1. Define thefollowing instruments:
Zero-Coupon Bond : V B0 = 1
1+r ,VB1 (ω) = 1
Forward Contract : V F0 = 0,V F
1 = S1(ω)− F
Call Option : V C1 (ω) = max(S1(ω)− K , 0)
Put Option : V P1 (ω) = max(K − S1(ω), 0)
In both the Call and Put option, K is known as the Strike.Once again, a Forward Contract is a deal that is locked in at time 0 forinitial price 0, but requires at time 1 the buyer to purchase the asset forprice F .
What is the value V0 of the above put and call options?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 15 / 161
Derivative Pricing
Let S1(ω) be the price of an underlying asset at time 1. Define thefollowing instruments:
Zero-Coupon Bond : V B0 = 1
1+r ,VB1 (ω) = 1
Forward Contract : V F0 = 0,V F
1 = S1(ω)− F
Call Option : V C1 (ω) = max(S1(ω)− K , 0)
Put Option : V P1 (ω) = max(K − S1(ω), 0)
In both the Call and Put option, K is known as the Strike.Once again, a Forward Contract is a deal that is locked in at time 0 forinitial price 0, but requires at time 1 the buyer to purchase the asset forprice F .
What is the value V0 of the above put and call options?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 15 / 161
Put-Call Parity
Can we replicate a forward contract using zero coupon bonds and put andcall options?
Yes: The final value of a replicating strategy X has value
V C1 − V P
1 + (K − F ) = S1 − F = X1(ω) (7)
This is achieved (replicated) by
Purchasing one call option
Selling one put option
Purchasing K − F zero coupon bonds with value 1 at maturity.
all at time 0.Since this strategy must have zero initial value, we obtain
V C0 − V P
0 =F − K
1 + r(8)
Question: How would this change in a multi-period model?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 16 / 161
Put-Call Parity
Can we replicate a forward contract using zero coupon bonds and put andcall options?Yes: The final value of a replicating strategy X has value
V C1 − V P
1 + (K − F ) = S1 − F = X1(ω) (7)
This is achieved (replicated) by
Purchasing one call option
Selling one put option
Purchasing K − F zero coupon bonds with value 1 at maturity.
all at time 0.Since this strategy must have zero initial value, we obtain
V C0 − V P
0 =F − K
1 + r(8)
Question: How would this change in a multi-period model?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 16 / 161
Put-Call Parity
Can we replicate a forward contract using zero coupon bonds and put andcall options?Yes: The final value of a replicating strategy X has value
V C1 − V P
1 + (K − F ) = S1 − F = X1(ω) (7)
This is achieved (replicated) by
Purchasing one call option
Selling one put option
Purchasing K − F zero coupon bonds with value 1 at maturity.
all at time 0.Since this strategy must have zero initial value, we obtain
V C0 − V P
0 =F − K
1 + r(8)
Question: How would this change in a multi-period model?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 16 / 161
Put-Call Parity
Can we replicate a forward contract using zero coupon bonds and put andcall options?Yes: The final value of a replicating strategy X has value
V C1 − V P
1 + (K − F ) = S1 − F = X1(ω) (7)
This is achieved (replicated) by
Purchasing one call option
Selling one put option
Purchasing K − F zero coupon bonds with value 1 at maturity.
all at time 0.
Since this strategy must have zero initial value, we obtain
V C0 − V P
0 =F − K
1 + r(8)
Question: How would this change in a multi-period model?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 16 / 161
Put-Call Parity
Can we replicate a forward contract using zero coupon bonds and put andcall options?Yes: The final value of a replicating strategy X has value
V C1 − V P
1 + (K − F ) = S1 − F = X1(ω) (7)
This is achieved (replicated) by
Purchasing one call option
Selling one put option
Purchasing K − F zero coupon bonds with value 1 at maturity.
all at time 0.Since this strategy must have zero initial value, we obtain
V C0 − V P
0 =F − K
1 + r(8)
Question: How would this change in a multi-period model?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 16 / 161
Put-Call Parity
Can we replicate a forward contract using zero coupon bonds and put andcall options?Yes: The final value of a replicating strategy X has value
V C1 − V P
1 + (K − F ) = S1 − F = X1(ω) (7)
This is achieved (replicated) by
Purchasing one call option
Selling one put option
Purchasing K − F zero coupon bonds with value 1 at maturity.
all at time 0.Since this strategy must have zero initial value, we obtain
V C0 − V P
0 =F − K
1 + r(8)
Question: How would this change in a multi-period model?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 16 / 161
General Derivative Pricing -One period model
If we begin with some initial capital X0, then we end with X1(ω). To pricea derivative, we need to match
X1(ω) = V1(ω) ∀ ω ∈ Ω (9)
to have X0 = V0, the price of the derivative we seek.
A strategy by the pair (X0,∆0) wherein
X0 is the initial capital
∆0 is the initial number of shares (units of underlying asset.)
What does the sign of ∆0 indicate?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 17 / 161
Replicating Strategy
Initial holding in bond (bank account) is X0 −∆0S0
Value of portfolio at maturity is
X1(ω) = (X0 −∆0S0)(1 + r) + ∆0S1(ω) (10)
Pathwise, we compute
V1(H) = (X0 −∆0S0)(1 + r) + ∆0uS0
V1(T ) = (X0 −∆0S0)(1 + r) + ∆0dS0
Algebra yields
∆0 =V1(H)− V1(T )
(u − d)S0(11)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 18 / 161
Replicating Strategy
Initial holding in bond (bank account) is X0 −∆0S0
Value of portfolio at maturity is
X1(ω) = (X0 −∆0S0)(1 + r) + ∆0S1(ω) (10)
Pathwise, we compute
V1(H) = (X0 −∆0S0)(1 + r) + ∆0uS0
V1(T ) = (X0 −∆0S0)(1 + r) + ∆0dS0
Algebra yields
∆0 =V1(H)− V1(T )
(u − d)S0(11)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 18 / 161
Replicating Strategy
Initial holding in bond (bank account) is X0 −∆0S0
Value of portfolio at maturity is
X1(ω) = (X0 −∆0S0)(1 + r) + ∆0S1(ω) (10)
Pathwise, we compute
V1(H) = (X0 −∆0S0)(1 + r) + ∆0uS0
V1(T ) = (X0 −∆0S0)(1 + r) + ∆0dS0
Algebra yields
∆0 =V1(H)− V1(T )
(u − d)S0(11)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 18 / 161
Replicating Strategy
Initial holding in bond (bank account) is X0 −∆0S0
Value of portfolio at maturity is
X1(ω) = (X0 −∆0S0)(1 + r) + ∆0S1(ω) (10)
Pathwise, we compute
V1(H) = (X0 −∆0S0)(1 + r) + ∆0uS0
V1(T ) = (X0 −∆0S0)(1 + r) + ∆0dS0
Algebra yields
∆0 =V1(H)− V1(T )
(u − d)S0(11)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 18 / 161
Risk Neutral Probability
Let us assume the existence of a pair (p, q) of positive numbers, and usethese to multiply our pricing equation(s):
pV1(H) = p(X0 −∆0S0)(1 + r) + p∆0uS0
qV1(T ) = q(X0 −∆0S0)(1 + r) + q∆0dS0
Addition yields
X0(1 + r) + ∆0S0(pu + qd − (1 + r)) = pV1(H) + qV1(T ) (12)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 19 / 161
Risk Neutral Probability
Let us assume the existence of a pair (p, q) of positive numbers, and usethese to multiply our pricing equation(s):
pV1(H) = p(X0 −∆0S0)(1 + r) + p∆0uS0
qV1(T ) = q(X0 −∆0S0)(1 + r) + q∆0dS0
Addition yields
X0(1 + r) + ∆0S0(pu + qd − (1 + r)) = pV1(H) + qV1(T ) (12)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 19 / 161
If we constrain
0 = pu + qd − (1 + r)
1 = p + q
0 ≤ p
0 ≤ q
then we have a risk neutral probability P where
V0 = X0 =1
1 + rE[V1] =
pV1(H) + qV1(T )
1 + r
p = P[X1(ω) = H] =1 + r − d
u − d
q = P[X1(ω) = T ] =u − (1 + r)
u − d
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 20 / 161
If we constrain
0 = pu + qd − (1 + r)
1 = p + q
0 ≤ p
0 ≤ q
then we have a risk neutral probability P where
V0 = X0 =1
1 + rE[V1] =
pV1(H) + qV1(T )
1 + r
p = P[X1(ω) = H] =1 + r − d
u − d
q = P[X1(ω) = T ] =u − (1 + r)
u − d
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 20 / 161
Example: Pricing a forward contract
Consider the case of a stock with
S0 = 400
u = 1.25
d = 0.75
r = 0.05
Then the forward price is computed via
0 =1
1 + rE[S1 − F ]⇒ F = E[S1] (13)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 21 / 161
Example: Pricing a forward contract
Consider the case of a stock with
S0 = 400
u = 1.25
d = 0.75
r = 0.05
Then the forward price is computed via
0 =1
1 + rE[S1 − F ]⇒ F = E[S1] (13)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 21 / 161
This leads to the explicit price
F = puS0 + qdS0
= (p)(1.25)(400) + (1− p)(0.75)(400)
= 500p + 300− 300p = 300 + 200p
= 300 + 200 · 1 + 0.05− 0.75
1.25− 0.75= 300 + 200 · 3
5
= 420
Homework Question: What is the price of a call option in the caseabove,with strike K = 375?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 22 / 161
This leads to the explicit price
F = puS0 + qdS0
= (p)(1.25)(400) + (1− p)(0.75)(400)
= 500p + 300− 300p = 300 + 200p
= 300 + 200 · 1 + 0.05− 0.75
1.25− 0.75= 300 + 200 · 3
5
= 420
Homework Question: What is the price of a call option in the caseabove,with strike K = 375?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 22 / 161
General one period risk neutral measure
We define a finite set of outcomes Ω ≡ ω1, ω2, ..., ωn and anysubcollection of outcomes A ∈ F1 := 2Ω an event.
Furthermore, we define a probability measure P, not necessarily thephysical measure P to be risk neutral if
P[ω] > 0 ∀ ω ∈ ΩX0 = 1
1+r E[X1]
for all strategies X .
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 23 / 161
General one period risk neutral measure
The measure is indifferent to investing in a zero-coupon bond, or arisky asset X
The same initial capital X0 in both cases produces the same”‘average”’ return after one period.
Not the physical measure attached by observation, experts, etc..
In fact, physical measure has no impact on pricing
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 24 / 161
General one period risk neutral measure
The measure is indifferent to investing in a zero-coupon bond, or arisky asset X
The same initial capital X0 in both cases produces the same”‘average”’ return after one period.
Not the physical measure attached by observation, experts, etc..
In fact, physical measure has no impact on pricing
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 24 / 161
General one period risk neutral measure
The measure is indifferent to investing in a zero-coupon bond, or arisky asset X
The same initial capital X0 in both cases produces the same”‘average”’ return after one period.
Not the physical measure attached by observation, experts, etc..
In fact, physical measure has no impact on pricing
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 24 / 161
General one period risk neutral measure
The measure is indifferent to investing in a zero-coupon bond, or arisky asset X
The same initial capital X0 in both cases produces the same”‘average”’ return after one period.
Not the physical measure attached by observation, experts, etc..
In fact, physical measure has no impact on pricing
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 24 / 161
Example: Risk Neutral measure for trinomial case
Assume that Ω = ω1, ω2, ω3 with
S1(ω1) = uS0
S1(ω2) = S0
S1(ω3) = dS0
Given a payoff V1(ω) to replicate, are we assured that a replicatingstrategy exists?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 25 / 161
Example: Risk Neutral measure for trinomial case
Assume that Ω = ω1, ω2, ω3 with
S1(ω1) = uS0
S1(ω2) = S0
S1(ω3) = dS0
Given a payoff V1(ω) to replicate, are we assured that a replicatingstrategy exists?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 25 / 161
Example: Risk Neutral measure for trinomial case
Homework:
Try our first example with
(S0, u, d , r) = (400, 1.25.0.75, 0.05)
V digital1 (ω) = 1S1(ω)>450(ω).
Now, assume you are observe the price on the market to be
V digital0 =
1
1 + rE[V digital
1 ] = 0.25. (14)
Use this extra information to price a call option with strike K = 420.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 26 / 161
Example: Risk Neutral measure for trinomial case
Homework:
Try our first example with
(S0, u, d , r) = (400, 1.25.0.75, 0.05)
V digital1 (ω) = 1S1(ω)>450(ω).
Now, assume you are observe the price on the market to be
V digital0 =
1
1 + rE[V digital
1 ] = 0.25. (14)
Use this extra information to price a call option with strike K = 420.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 26 / 161
Solution: Risk Neutral measure for trinomial case
The above scenario is reduced to finding the risk-neutral measure(p1, p2, p3). This can be done by finding the rref of the matrix M:
M =
1 1 1 1500 400 300 420
1 0 0 0.25(1.05)
(15)
which results in
rref (M) =
1 0 0 0.26250 1 0 0.6750 0 1 0.0625
. (16)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 27 / 161
Solution: Risk Neutral measure for trinomial case
It follows that (p1, p2, p3) = (0.2625, 0.675, 0.0625), and so
V C0 =
1
1.05E[(S1 − 420)+ | S0 = 400]
=0.2625
1.05× (500− 420) = 20.
(17)
Could we perhaps find a set of digital options as a basis setV d1
1 (ω),V d21 (ω),V d3
1 (ω)
= 1A1(ω), 1A2(ω), 1A3(ω) (18)
with A1,A2,A3 ∈ F1 to span all possible payoffs at time 1?
How about (A1,A2,A3) = (ω1 , ω2 , ω3) ?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 28 / 161
Solution: Risk Neutral measure for trinomial case
It follows that (p1, p2, p3) = (0.2625, 0.675, 0.0625), and so
V C0 =
1
1.05E[(S1 − 420)+ | S0 = 400]
=0.2625
1.05× (500− 420) = 20.
(17)
Could we perhaps find a set of digital options as a basis setV d1
1 (ω),V d21 (ω),V d3
1 (ω)
= 1A1(ω), 1A2(ω), 1A3(ω) (18)
with A1,A2,A3 ∈ F1 to span all possible payoffs at time 1?
How about (A1,A2,A3) = (ω1 , ω2 , ω3) ?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 28 / 161
Exchange one stock for another
Assume now an economy with two stocks, X and Y . Assume that
(X0,Y0, r) = (100, 100, 0.01) (19)
and
(X1(ω),Y1(ω)) =
(110, 105) : ω = ω1
(100, 100) : ω = ω2
(80, 95) : ω = ω3.
Consider two contracts, V and W , with payoffs
V1(ω) = max Y1(ω)− X1(ω), 0W1(ω) = Y1(ω)− X1(ω).
(20)
Price V0 and W0.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 29 / 161
Exchange one stock for another
In this case, our matrix M is such that
M =
1 1 1 1110 100 80 101105 100 95 101
(21)
which results in
rref (M) =
1 0 0 310
0 1 0 610
0 0 1 110
. (22)
It follows that
W0 =E[Y1]− E[X1]
1.01= Y0 − X0 = 0
V0 =1
1.01· (15p3) =
1.5
1.01= 1.49.
(23)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 30 / 161
Existence of Risk Neutral measure
Let P be a probability measure on a finite space Ω. The following areequivalent:
P is a risk neutral measure
For all traded securities S i , S i0 = 1
1+r E[S i
1
]Proof: Homework (Hint: One direction is much easier than others. Also,strategies are linear in the underlying asset.)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 31 / 161
Existence of Risk Neutral measure
Let P be a probability measure on a finite space Ω. The following areequivalent:
P is a risk neutral measure
For all traded securities S i , S i0 = 1
1+r E[S i
1
]Proof: Homework (Hint: One direction is much easier than others. Also,strategies are linear in the underlying asset.)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 31 / 161
Existence of Risk Neutral measure
Let P be a probability measure on a finite space Ω. The following areequivalent:
P is a risk neutral measure
For all traded securities S i , S i0 = 1
1+r E[S i
1
]
Proof: Homework (Hint: One direction is much easier than others. Also,strategies are linear in the underlying asset.)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 31 / 161
Existence of Risk Neutral measure
Let P be a probability measure on a finite space Ω. The following areequivalent:
P is a risk neutral measure
For all traded securities S i , S i0 = 1
1+r E[S i
1
]Proof: Homework (Hint: One direction is much easier than others. Also,strategies are linear in the underlying asset.)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 31 / 161
Complete Markets
A market is complete if it is arbitrage free and every non-traded asset canbe replicated.
Fundamental Theorem of Asset Pricing 1: A market is arbitrage freeiff there exists a risk neutral measure
Fundamental Theorem of Asset Pricing 2: A market is complete iffthere exists exactly one risk neutral measure
Proof(s): We will go over these in detail later!
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 32 / 161
Complete Markets
A market is complete if it is arbitrage free and every non-traded asset canbe replicated.
Fundamental Theorem of Asset Pricing 1: A market is arbitrage freeiff there exists a risk neutral measure
Fundamental Theorem of Asset Pricing 2: A market is complete iffthere exists exactly one risk neutral measure
Proof(s): We will go over these in detail later!
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 32 / 161
Complete Markets
A market is complete if it is arbitrage free and every non-traded asset canbe replicated.
Fundamental Theorem of Asset Pricing 1: A market is arbitrage freeiff there exists a risk neutral measure
Fundamental Theorem of Asset Pricing 2: A market is complete iffthere exists exactly one risk neutral measure
Proof(s): We will go over these in detail later!
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 32 / 161
Optimal Investment for a Strictly Risk Averse Investor
Assume a complete market, with a unique risk-neutral measure P.
Characterize an investor by her pair (x ,U) of initial capital x ∈ X andutility function U : X → R+.
Assume U ′(x) > 0.
Assume U ′′(x) < 0.
Define the Radon-Nikodym derivative of P to P as the randomvariable
Z (ω) :=P(ω)
P(ω). (24)
Note that Z is used to map expectations under P to expectationsunder P: For any random variable X , it follows that
E[X ] = E[ZX ]. (25)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 33 / 161
Optimal Investment for a Strictly Risk Averse Investor
A strictly risk-averse investor now wishes to maximize her expected utilityof a portfolio at time 1, given initial capital at time 0:
u(x) := maxX1∈Ax
E[U(X1)]
Ax := all portfolio values at time 1 with initial capital x .(26)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 34 / 161
Optimal Investment for a Strictly Risk Averse Investor
Theorem
Define X1 via the relationship
U ′(
X1
):= λZ (27)
where λ sets X1 as a strategy with an average return of r under P:
E[X1] = x(1 + r). (28)
Then X1 is the optimal strategy.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 35 / 161
Optimal Investment for a Strictly Risk Averse Investor
Proof.
Assume X1 to be an arbitrary strategy with initial capital x . Then for
f (y) := E[U(yX1 + (1− y)X1)] (29)
it follows that
f ′(0) = E[U ′(X1)
(X1 − X1
)]= E
[λZ(
X1 − X1
)]= λE
[(X1 − X1
)]= 0
f ′′(y) = E[
U ′′(yX1 + (1− y)X1)(
X1 − X1
)2]< 0
(30)
and so f attains its maximum at y = 0. We conclude thatE[U(X1)] < E[U(X1)] for any admissible strategy X1.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 36 / 161
Optimal Investment: Example
Assume an investor and economy defined by
U(x) = ln (x)
(S0, u, d , p, q, r) = (400, 1.25.0.75, 0.5, 0.5, 0.05).(31)
It follows that
(p, q) =
(3
5,
2
5
)(Z (H),Z (T )) =
(6
5,
4
5
).
(32)
Since U ′(x) = 1x , we have
X1(ω) =1
λ
1
Z (ω)
x = X0 =1
1 + rE[X1] =
1
1 + r
(p · 1
λ
p
p+ q · 1
λ
q
q
)=
1
λ
1
1 + r.
(33)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 37 / 161
Optimal Investment: Example
Combining the previous results, we see that
X1(ω) =x(1 + r)
Z (ω)
u(x) = p ln X1(H) + (1− p) ln X1(T )
= p ln
(x(1 + r)
Z (H)
)+ (1− p) ln
(x(1 + r)
Z (T )
)= ln
((1 + r)
Z (H)pZ (T )1−p x
)= ln (1.0717x) > ln (1.05x).
(34)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 38 / 161
Optimal Investment: Example
In terms of her actual strategy, we see that
π0 :=∆0S0
X0=
S0
x
X1(H)− X1(T )
S1(H)− S1(T )=
1 + r
u − d
(1
Z (H)− 1
Z (T )
)=
1 + r
u − d
(p
p− 1− p
1− p
)=
1.05
0.5
(5
6− 5
4
)= −0.875.
(35)
Therefore, the optimal strategy is to sell a stock portfolio worth 87.5% ofher initial wealth x and invest the proceeds into a safe bank account.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 39 / 161
Optimal Investment: Example
In fact, since π0 = 1+ru−d
(pp −
1−p1−p
), we see that qualitatively, her optimal
strategy involves
π0 =
> 0 : p > p= 0 : p = p< 0 : p < p.
This links with her strategy via
1 + r1(ω) :=X1(ω)
X0= (1− π0)(1 + r) + π0
S1(ω)
S0(36)
and so for our specific case where (r , u, d , π0) = (0.05, 1.25, 0.75,−0.875),we have
1 + r1(H) = (1− π0)(1 + r) + π0u = 0.875
1 + r1(T ) = (1− π0)(1 + r) + π0d = 1.3125.(37)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 40 / 161
Optimal Investment: U(x) =√x
Consider now the same set-up as before, only that the utility functionchanges to U(x) =
√x .
It follows that
U ′(X1) =1
2
1√X1
⇒ X1 =1
4λ2
1
Z 2
(38)
Solving for λ returns
x(1 + r) = E[X1]
= E[Z X1]
= E[
Z1
4λ2
1
Z 2
]=
1
4λ2E[
1
Z
].
(39)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 41 / 161
Optimal Investment: U(x) =√x
Combining the results above, we see that
X1 =x(1 + r)
Z 2E[
1Z
]⇒ u(x) = E
[√X1
]= E
[√x(1 + r)
Z 2E[
1Z
]]
=√
x(1 + r)
√E[
1
Z
].
(40)
Question: Is it true for all (p, p) ∈ (0, 1)× (0, 1) that√E[
1
Z
]=
√p2
p+
(1− p)2
1− p≥ 1? (41)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 42 / 161
Optimal Betting at the Omega Horse Track!
Imagine our investor with U(x) = ln (x) visits a horse track.
There are three horses: ω1, ω2 and ω3.
She can bet on any of the horses to come in 1st .
The payoff is 1 per whole bet made.
She observes the price of each bet with payoff 1 right before the raceto be
(B10 ,B
20 ,B
30 ) = (0.5, 0.3, 0.2). (42)
Symbolically,B i
1(ω) = 1ωi(ω). (43)
Our investor feels the physical probabilities of each horse winning is
(p1, p2, p3) = (0.6, 0.35, 0.05). (44)
How should she bet if the race is about to start?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 43 / 161
Optimal Betting at the Omega Horse Track!
In this setting, we can assume r = 0.
This directly implies that
(p1, p2, p3) = (0.5, 0.3, 0.2). (45)
Our Radon-Nikodym derivative of P to P is now
(Z (ω1),Z (ω2),Z (ω3)) =
(0.5
0.6,
0.3
0.35,
0.2
0.05
). (46)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 44 / 161
Optimal Betting at the Omega Horse Track!
Her optimal strategy X1 reflects her betting strategy, and satisfies
X1(ω) =X0
Z (ω)
∴
(X1(ω1)
X0,
X1(ω2)
X0,
X3(ω1)
X0
)=
(6
5,
7
6,
1
4
).
(47)
So, per dollar of wealth, she buys 65 of a bet for Horse 1 to win, 7
6 ofa bet for Horse 2 to win, and 1
4 of a bet for Horse 3 to win.
The total price (per dollar of wealth) is thus
6
5· 0.5 +
7
6· 0.3 +
1
4· 0.2 = 0.6 + 0.35 + 0.05 = 1. (48)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 45 / 161
Dividends
What about dividends? How do they affect the risk neutral pricing ofexchange and non-exchange traded assets? What if they are paid atdiscrete times? Continuously paid?
Recall that if dividends are paid continuously at rate δ, then 1 share attime 0 will accumulate to eδT shares upon reinvestment of dividends intothe stock until time T .
It follows that to deliver one share of stock S with initial price S0 at timeT , only e−δT shares are needed. Correspondingly,
Fprepaid = e−δTS0
F = erT e−δTS0 = e(r−δ)TS0.(49)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 46 / 161
Dividends
What about dividends? How do they affect the risk neutral pricing ofexchange and non-exchange traded assets? What if they are paid atdiscrete times? Continuously paid?
Recall that if dividends are paid continuously at rate δ, then 1 share attime 0 will accumulate to eδT shares upon reinvestment of dividends intothe stock until time T .
It follows that to deliver one share of stock S with initial price S0 at timeT , only e−δT shares are needed. Correspondingly,
Fprepaid = e−δTS0
F = erT e−δTS0 = e(r−δ)TS0.(49)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 46 / 161
Dividends
What about dividends? How do they affect the risk neutral pricing ofexchange and non-exchange traded assets? What if they are paid atdiscrete times? Continuously paid?
Recall that if dividends are paid continuously at rate δ, then 1 share attime 0 will accumulate to eδT shares upon reinvestment of dividends intothe stock until time T .
It follows that to deliver one share of stock S with initial price S0 at timeT , only e−δT shares are needed. Correspondingly,
Fprepaid = e−δTS0
F = erT e−δTS0 = e(r−δ)TS0.(49)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 46 / 161
Binomial Option Pricing w/ cts Dividends and Interest
Over a period of length h, interest increases the value of a bond by afactor erh and dividends the value of a stock by a factor of eδh.
Once again, we compute pathwise,
V1(H) = (X0 −∆0S0)erh + ∆0eδhuS0
V1(T ) = (X0 −∆0S0)erh + ∆0eδhdS0
and this results in the modified quantities
∆0 = e−δhV1(H)− V1(T )
(u − d)S0
p =e(r−δ)h − d
u − d
q =u − e(r−δ)h
u − d
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 47 / 161
Binomial Option Pricing w/ cts Dividends and Interest
Over a period of length h, interest increases the value of a bond by afactor erh and dividends the value of a stock by a factor of eδh.
Once again, we compute pathwise,
V1(H) = (X0 −∆0S0)erh + ∆0eδhuS0
V1(T ) = (X0 −∆0S0)erh + ∆0eδhdS0
and this results in the modified quantities
∆0 = e−δhV1(H)− V1(T )
(u − d)S0
p =e(r−δ)h − d
u − d
q =u − e(r−δ)h
u − dAlbert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 47 / 161
Binomial Models w/ cts Dividends and Interest
For σ, the annualized standard deviation of continuously compoundedstock return, the following models hold:
Futures - Cox (1979)
u = eσ√h
d = e−σ√h.
General Stock Model
u = e(r−δ)h+σ√h
d = e(r−δ)h−σ√h.
Currencies with rf the foreign interest rate, which acts as a dividend:
u = e(r−rf )h+σ√h
d = e(r−rf )h−σ√h.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 48 / 161
1- and 2-period pricing
Consider the case r = 0.10, δ = 0.05, h = 0.01, σ = 0.1,S0 = 10 = K .
Now price two digital options, using the
1 General Stock Model
2 Futures-Cox Model
with respective payoffs
V K1 (ω) := 1S1≥K(ω)
V K2 (ω) := 1S2≥K(ω).
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 49 / 161
1- and 2-period pricing
Consider the case r = 0.10, δ = 0.05, h = 0.01, σ = 0.1,S0 = 10 = K .
Now price two digital options, using the
1 General Stock Model
2 Futures-Cox Model
with respective payoffs
V K1 (ω) := 1S1≥K(ω)
V K2 (ω) := 1S2≥K(ω).
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 49 / 161
Calibration Exercise
Assume table below of realized gains & losses over a ten-period cycle.
Use the adjusted values (r , δ, h,S0,K ) = (0.02, 0, 0.10, 10, 10).
Calculate binary options from last slide using these assumptions.
Period Return
1 S1S0
= 1.05
2 S2S1
= 1.02
3 S3S2
= 0.98
4 S4S3
= 1.01
5 S5S4
= 1.02
6 S6S5
= 0.99
7 S7S6
= 1.03
8 S8S7
= 1.05
9 S9S8
= 0.96
10 S10S9
= 0.97
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 50 / 161
Calibration Exercise: Linear Approximation
We would like to compute σ for the logarithm of returns ln(
SiSi−1
).
Assume the returns per period are all independent.
Q: Can we use a linear (simple) return model instead of a compoundreturn model as an approximation?
If so, then for our observed simple return rate values:
Calculate the sample variance σ2∗.
Estimate that σ ≈ σ∗√h
.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 51 / 161
Calibration Exercise: Linear Approximation
Note that if SiSi−1
= 1 + γ for γ 1, then
ln
(Si
Si−1
)≈ γ =
Si − Si−1
Si−1. (50)
Approximation: Convert our previous table, using simple interest.
Over small time periods h, define linear return values for i th period:
Xih :=Si − Si−1
Si−1. (51)
In other words, for simple rate of return Xi for period i :
Si = Si−1 · (1 + Xih). (52)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 52 / 161
Calibration Exercise: Linear Approximation
Our returns table now looks like
Period Return
1 S1−S0S0
= 0.05
2 S2−S1S1
= 0.02
3 S3−S2S2
= −0.02
4 S4−S3S3
= 0.01
5 S5−S4S4
= 0.02
6 S6−S5S5
= −0.01
7 S7−S6S6
= 0.03
8 S8−S7S7
= 0.05
9 S9−S8S8
= −0.04
10 S10−S9S9
= −0.03
sample standard deviation σ∗ = 0.0319estimated return deviation σ ≈ 0.1001
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 53 / 161
Calibration Exercise: Linear Approximation
We estimate, therefore, that under the Futures-Cox model
(u, d) = (e0.0319, e−0.0319) = (1.0324, 0.9686)
p =e0.002 − e−0.0319
e0.0319 − e−0.0319= 0.5234.
(53)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 54 / 161
Calibration Exercise: Linear Approximation
For the one-period digital option:
V0 = e−rhE0[1S1≥10] = e−0.002 · p = 0.5224. (54)
For the two-period digital option:
V0 = e−2rhE0[1S2≥10] = e−0.004 ·[p2 + 2pq
]= 0.7698. (55)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 55 / 161
Calibration Exercise: Linear Approximation
For the one-period digital option:
V0 = e−rhE0[1S1≥10] = e−0.002 · p = 0.5224. (54)
For the two-period digital option:
V0 = e−2rhE0[1S2≥10] = e−0.004 ·[p2 + 2pq
]= 0.7698. (55)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 55 / 161
Calibration Exercise: No Approximation
Without the linear approximation, we can directly estimate
σY√
h = 0.03172
(u, d) = (e0.03172, e−0.03172) = (1.0322, 0.9688)
p =e0.002 − e−0.03172
e0.03172 − e−0.03172= 0.5246.
(56)
For the one-period digital option:
V0 = e−rhE0[1S1≥10] = e−0.002 · p = 0.5236. (57)
For the two-period digital option:
V0 = e−2rhE0[1S2≥10] = e−0.004 ·[p2 + 2pq
]= 0.7721. (58)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 56 / 161
Calibration Exercise: No Approximation
Without the linear approximation, we can directly estimate
σY√
h = 0.03172
(u, d) = (e0.03172, e−0.03172) = (1.0322, 0.9688)
p =e0.002 − e−0.03172
e0.03172 − e−0.03172= 0.5246.
(56)
For the one-period digital option:
V0 = e−rhE0[1S1≥10] = e−0.002 · p = 0.5236. (57)
For the two-period digital option:
V0 = e−2rhE0[1S2≥10] = e−0.004 ·[p2 + 2pq
]= 0.7721. (58)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 56 / 161
1- and 2-period pricing
We can solve for 2-period problems
on a case-by-case basis, or
by developing a general theory for multi-period asset pricing.
In the latter method, we need a general framework to carry out ourcomputations
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 57 / 161
1- and 2-period pricing
We can solve for 2-period problems
on a case-by-case basis, or
by developing a general theory for multi-period asset pricing.
In the latter method, we need a general framework to carry out ourcomputations
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 57 / 161
1- and 2-period pricing
We can solve for 2-period problems
on a case-by-case basis, or
by developing a general theory for multi-period asset pricing.
In the latter method, we need a general framework to carry out ourcomputations
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 57 / 161
Risk Neutral Pricing Formula
Assume now that we have the ”regular assumptions” on our coin flipspace, and that at time N we are asked to deliver a path dependentderivative value VN . Then for times 0 ≤ n ≤ N, the value of thisderivative is computed via
Vn = e−rhEn [Vn+1] (59)
and so
X0 = E0 [XN ]
Xn :=Vn
enh.
(60)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 58 / 161
Risk Neutral Pricing Formula
Assume now that we have the ”regular assumptions” on our coin flipspace, and that at time N we are asked to deliver a path dependentderivative value VN . Then for times 0 ≤ n ≤ N, the value of thisderivative is computed via
Vn = e−rhEn [Vn+1] (59)
and so
X0 = E0 [XN ]
Xn :=Vn
enh.
(60)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 58 / 161
Risk Neutral Pricing Formula
Assume now that we have the ”regular assumptions” on our coin flipspace, and that at time N we are asked to deliver a path dependentderivative value VN . Then for times 0 ≤ n ≤ N, the value of thisderivative is computed via
Vn = e−rhEn [Vn+1] (59)
and so
X0 = E0 [XN ]
Xn :=Vn
enh.
(60)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 58 / 161
Risk Neutral Pricing Formula
Assume now that we have the ”regular assumptions” on our coin flipspace, and that at time N we are asked to deliver a path dependentderivative value VN . Then for times 0 ≤ n ≤ N, the value of thisderivative is computed via
Vn = e−rhEn [Vn+1] (59)
and so
X0 = E0 [XN ]
Xn :=Vn
enh.
(60)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 58 / 161
Computational Complexity
Consider the case
p = q =1
2
S0 = 4, u =4
3, d =
3
4
(61)
but now with term n = 3.
There are 23 = 8 paths to consider.
However, there are 3 + 1 = 4 unique final values of S3 to consider.
In the general term N, there would be 2N paths to generate SN , butonly N + 1 distinct values.
At any node n units of time into the asset’s evolution, there are n + 1distinct values.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 59 / 161
Computational Complexity
At each value s for Sn, we know that Sn+1 = 43 s or Sn+1 = 3
4 s.
Using multi-period risk-neutral pricing, we can generate forvn(s) := Vn(Sn(ω1, ..., ωn)) on the node (event) Sn(ω1, ..., ωn) = s:
vn(s) = e−rh[pvn+1
(4
3s)
+ qvn+1
(3
4s)]. (62)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 60 / 161
An Example:
Assume r , δ, and h are such that
p =1
2= q, e−rh =
9
10
S0 = 4, u = 2, d =1
2V3 := max 10− S3, 0 .
(63)
It follows thatv3(32) = 0
v3(8) = 2
v3(2) = 8
v3(0.50) = 9.50.
(64)
Compute V0.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 61 / 161
Markov Processes
If we use the above approach for a more exotic option, say a lookbackoption that pays the maximum over the term of a stock, then we findthis approach lacking.
There is not enough information in the tree or the distinct values forS3 as stated. We need more.
Consider our general multi-period binomial model under P.
Definition We say that a process X is adapted if it depends only on thesequence of flips ω := (ω1, ..., ωn)
Definition We say that an adapted process X is Markov if for every0 ≤ n ≤ N − 1 and every function f (x) there exists another function g(x)such that
En [f (Xn+1)] = g(Xn). (65)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 62 / 161
Markov Processes
This notion of Markovity is essential to our state-dependent pricingalgorithm.
Indeed, our stock process evolves from time n to time n + 1, usingonly the information in Sn.
We can in fact say that for every f (s) there exists a g(s) such that
g(s) = En [f (Sn+1) | Sn = s] . (66)
In fact, that g depends on f :
g(s) = e−rh[pf(4
3s)
+ qf(3
4s)]. (67)
So, for any f (s) := VN(s), we can work our recursive algorithmbackwards to find the gn(s) := Vn(s) for all 0 ≤ n ≤ N − 1
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 63 / 161
Markov Processes
Some more thoughts on Markovity:
Consider the example of a Lookback Option.
Here, the payoff is dependent on the realized maximumMn := max0≤i≤nSi of the asset.
Mn is not Markov by itself, but the two-factor process (Mn, Sn) is.Why?
Let’s generate the tree!
Homework Can you think of any other processes that are not Markov?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 64 / 161
Call Options on Zero-Coupon Bonds
Assume an economy where
One period is one year
The one year short term interest rate from time n to time n + 1 is rn.
The rate evolves via a stochastic process:
r0 = 0.02
rn+1 = Xrn
P[X = 2k ] =1
3for k ∈ −1, 0, 1 .
(68)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 65 / 161
Call Options on Zero-Coupon Bonds
Consider now a zero-coupon bond that matures in 3−years withcommon face and redemption value F = 100.
Also consider a call option on this bond that expires in 2−years withstrike K = 97.
Denote Bn and Cn as the bond and call option values, respectively.
Note that we iterate backwards from the values
B3(r) = 100
C2(r) = max B2(r)− 97, 0 .(69)
Compute (B0,C0).
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 66 / 161
Call Options on Zero-Coupon Bonds
Our general recursive formula is
Bn(r) =1
1 + rE[Bn+1(rn+1) | rn = r ]
Cn(r) =1
1 + rE[Cn+1(rn+1) | rn = r ].
(70)
Iterating backwards, we see that at t = 2,
B2(r) =1
1 + r
1
3
1∑k=−1
B3(2k r). (71)
At time t = 2, we have that
r2 ∈ 0.08, 0.04, 0.02, 0.01, 0.005 . (72)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 67 / 161
Call Options on Zero-Coupon Bonds
Our associated Bond and Call Option values at time 2:
r2 B2 C2
0.08 92.59 00.04 96.15 00.02 98.04 1.040.01 99.01 2.01
0.005 99.50 2.50
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 68 / 161
Call Options on Zero-Coupon Bonds
Our associated Bond and Call Option values at time 1:
r1 B1 C1
0.04 91.92 0.330.02 95.82 1.000.01 97.87 1.83
Our associated Bond and Call Option values at time 0:
r0 B0 C0
0.02 93.34 1.03
Question: What if the delivery time of the option is changed to 3?Symbolically, what if
C3(r) = max B3(r)− 97, 0? (73)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 69 / 161
Capital Structure Model
As an analyst for an investments firm, you are tasked with advisingwhether a company’s stock and/or bonds are over/under-priced.
You receive a quarterly report from this company on it’s return onassets, and have compiled a table for the last ten quarters below.
Today, just after the last quarter’s report was issued, you see that inbillions of USD, the value of the company’s assets is 10.
There are presently one billions shares of this company that are beingtraded.
The company does not pay any dividends.
Six months from now, the company is required to pay off a billionzero-coupon bonds. Each bond has a face value of 9.5.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 70 / 161
Capital Structure Model
Assume Miller-Modigliani holds with At = Bt + St , where the assetsof a company equal the sum of its share and bond price.
Presently, the market values are (B0, S0, r) = (9, 1, 0.02).
The Merton model for corporate bond pricing asserts that atredemption time T ,
Bt = e−r(T−t)E [min AT ,F]St = e−r(T−t)E [max AT − F , 0] .
(74)
With all of this information, your job now is to issue a Buy or Sell onthe stock and the bond issued by this company.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 71 / 161
Capital Structure Model
Table of return on assets for Company X, with h = 0.25.
Period Return on Assets
1 A1−A0A0
= 0.05
2 A2−A1A1
= 0.02
3 A3−A2A2
= −0.02
4 A4−A3A3
= 0.01
5 A5−A4A4
= 0.02
6 A6−A5A5
= −0.01
7 A7−A6A6
= 0.03
8 A8−A7A7
= 0.05
9 A9−A8A8
= −0.04
10 A10−A9A9
= −0.03
sample standard deviation σ∗ = 0.0319estimated return deviation σ ≈ 0.0638
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 72 / 161
Capital Structure Model
We scale all of our calculation in terms of billions ($, shares, bonds).
Using the Futures- Cox model, we have
(u, d) = (e0.0319, e−0.0319) = (1.0324, 0.9686)
p =e0.005 − e−0.0319
e0.0319 − e−0.0319= 0.5706.
(75)
Using this model, the only time the payoff of the bond is less than theface is on the path ω = TT .
The price of the bond and stock are thus modeled to be
B0 = e−0.02·(2·0.25)[p2 · 9.5 + 2pq · 9.5 + q2 · 9.38
]= 9.38 > 9.00
S0 = 10− 9.38 = 0.62 < 1.00.
(76)
It follows that,according to our model, one should Buy the bond as itis underpriced and one should Sell the stock as it is overpriced.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 73 / 161
The Interview Process
Consider the following scenario:
After graduating, you go on the job market, and have 4 possible jobinterviews with 4 different companies.
So sure of your prospects that you know that each company will makean offer, with an identically, independently distributed probabilityattached to the 4 possible salary offers:
P [Salary Offer=50, 000] = 0.1
P [Salary Offer=70, 000] = 0.3
P [Salary Offer=80, 000] = 0.4
P [Salary Offer=100, 000] = 0.2.
(77)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 74 / 161
The Interview Process
Questions:
How should you interview?
Specifically, when should you accept an offer and cancel theremaining interviews?
How does your strategy change if you can interview as many times asyou like, but the distribution of offers remains the same as above?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 75 / 161
The Interview Process
Questions:
How should you interview?
Specifically, when should you accept an offer and cancel theremaining interviews?
How does your strategy change if you can interview as many times asyou like, but the distribution of offers remains the same as above?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 75 / 161
The Interview Process
Questions:
How should you interview?
Specifically, when should you accept an offer and cancel theremaining interviews?
How does your strategy change if you can interview as many times asyou like, but the distribution of offers remains the same as above?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 75 / 161
The Interview Process: Strategy
Some more thoughts...
At any time the student will know only one offer, which she can eitheraccept or reject.
Of course, if the student rejects the first three offers, than she has toaccept the last one.
So, compute the maximal expected salary for the student after thegraduation and the corresponding optimal strategy.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 76 / 161
The Interview Process: Optimal Strategy
The solution process Xk4k=1 follows an Optimal Stopping Strategy:
Xk(s) = max
s, E[Xk+1 | kth offer = s
]. (78)
At time 4, the value of this game is X4(s) = s, with s being the salaryoffered.
At time 3, the conditional expected value of this game is
E[X4 | 3rd offer = s
]= E[X4]
= 0.1× 50, 000 + 0.3× 70, 000
+ 0.4× 80, 000 + 0.2× 100, 000
= 78, 000.
(79)
Hence, one should accept an offer of 80, 000 or 100, 000, and rejectthe other two.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 77 / 161
The Interview Process: Optimal Strategy
This strategy leads to a valuation:
X3(50, 000) = 78, 000
X3(70, 000) = 78, 000
X3(80, 000) = 80, 000
X3(100, 000) = 100, 000.
(80)
At time 2, similar reasoning using E2[X3] leads to the valuation
X2(50, 000) = 83, 200
X2(70, 000) = 83, 200
X2(80, 000) = 83, 200
X2(100, 000) = 100, 000.
(81)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 78 / 161
The Interview Process: Optimal Strategy
At time 1,X1(50, 000) = 86, 560
X1(70, 000) = 86, 560
X1(80, 000) = 86, 560
X1(100, 000) = 100, 000.
(82)
Finally, at time 0, the value of this optimal strategy is
E0[X1] = E[X1] = 0.8× 86, 560 + 0.2× 100, 000 = 89, 248. (83)
So, the optimal strategy is, for the first two interviews, accept only anoffer of 100, 000. If after the third interview, and offer of 80, 000 or100, 000 is made, then accept. Otherwise continue to the last interviewwhere you should accept whatever is offered.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 79 / 161
Review
Let’s review the basic contracts we can write:
Forward Contract Initial Value is 0, because both buyer and sellermay have to pay a balance at maturity.
(European) Put/Call Option Initial Value is > 0, because both onlyseller must pay balance at maturity.
(European) ”Exotic” Option Initial Value is > 0, because both onlyseller must pay balance at maturity.
During the term of the contract, can the value of the contract ever fallbelow the intrinsic value of the payoff? Symbolically, does it ever occurthat
vn(s) < g(s) (84)
where g(s) is of the form of g(S) := max S − K , 0, in the case of a Calloption, for example.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 80 / 161
Review
Let’s review the basic contracts we can write:
Forward Contract Initial Value is 0, because both buyer and sellermay have to pay a balance at maturity.
(European) Put/Call Option Initial Value is > 0, because both onlyseller must pay balance at maturity.
(European) ”Exotic” Option Initial Value is > 0, because both onlyseller must pay balance at maturity.
During the term of the contract, can the value of the contract ever fallbelow the intrinsic value of the payoff? Symbolically, does it ever occurthat
vn(s) < g(s) (84)
where g(s) is of the form of g(S) := max S − K , 0, in the case of a Calloption, for example.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 80 / 161
Review
Let’s review the basic contracts we can write:
Forward Contract Initial Value is 0, because both buyer and sellermay have to pay a balance at maturity.
(European) Put/Call Option Initial Value is > 0, because both onlyseller must pay balance at maturity.
(European) ”Exotic” Option Initial Value is > 0, because both onlyseller must pay balance at maturity.
During the term of the contract, can the value of the contract ever fallbelow the intrinsic value of the payoff? Symbolically, does it ever occurthat
vn(s) < g(s) (84)
where g(s) is of the form of g(S) := max S − K , 0, in the case of a Calloption, for example.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 80 / 161
Review
Let’s review the basic contracts we can write:
Forward Contract Initial Value is 0, because both buyer and sellermay have to pay a balance at maturity.
(European) Put/Call Option Initial Value is > 0, because both onlyseller must pay balance at maturity.
(European) ”Exotic” Option Initial Value is > 0, because both onlyseller must pay balance at maturity.
During the term of the contract, can the value of the contract ever fallbelow the intrinsic value of the payoff? Symbolically, does it ever occurthat
vn(s) < g(s) (84)
where g(s) is of the form of g(S) := max S − K , 0, in the case of a Calloption, for example.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 80 / 161
Review
Let’s review the basic contracts we can write:
Forward Contract Initial Value is 0, because both buyer and sellermay have to pay a balance at maturity.
(European) Put/Call Option Initial Value is > 0, because both onlyseller must pay balance at maturity.
(European) ”Exotic” Option Initial Value is > 0, because both onlyseller must pay balance at maturity.
During the term of the contract, can the value of the contract ever fallbelow the intrinsic value of the payoff? Symbolically, does it ever occurthat
vn(s) < g(s) (84)
where g(s) is of the form of g(S) := max S − K , 0, in the case of a Calloption, for example.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 80 / 161
Review
Let’s review the basic contracts we can write:
Forward Contract Initial Value is 0, because both buyer and sellermay have to pay a balance at maturity.
(European) Put/Call Option Initial Value is > 0, because both onlyseller must pay balance at maturity.
(European) ”Exotic” Option Initial Value is > 0, because both onlyseller must pay balance at maturity.
During the term of the contract, can the value of the contract ever fallbelow the intrinsic value of the payoff? Symbolically, does it ever occurthat
vn(s) < g(s) (84)
where g(s) is of the form of g(S) := max S − K , 0, in the case of a Calloption, for example.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 80 / 161
Review
Let’s review the basic contracts we can write:
Forward Contract Initial Value is 0, because both buyer and sellermay have to pay a balance at maturity.
(European) Put/Call Option Initial Value is > 0, because both onlyseller must pay balance at maturity.
(European) ”Exotic” Option Initial Value is > 0, because both onlyseller must pay balance at maturity.
During the term of the contract, can the value of the contract ever fallbelow the intrinsic value of the payoff? Symbolically, does it ever occurthat
vn(s) < g(s) (84)
where g(s) is of the form of g(S) := max S − K , 0, in the case of a Calloption, for example.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 80 / 161
Early Exercise
If σ = 0, and so uncertainty vanishes, then an investor would seek toexercise early if
rK > δS . (85)
If σ > 0, then the situation involves deeper analysis.
Whether solving a free boundary problem or analyzing a binomialtree, it is likely that a computer will be involved in helping theinvestor to determine the optimal exercise time.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 81 / 161
For Freedom! (we must charge extra...)
What happens if we write a contract that allows the purchaser to exercisethe contract whenever she feels it to be in her advantage? By allowing thisextra freedom, we must
Charge more than we would for a European contract that is exercisedonly at the term N.
Hedge our replicating strategy X differently, to allow for thepossibility of early exercise.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 82 / 161
For Freedom! (we must charge extra...)
What happens if we write a contract that allows the purchaser to exercisethe contract whenever she feels it to be in her advantage? By allowing thisextra freedom, we must
Charge more than we would for a European contract that is exercisedonly at the term N.
Hedge our replicating strategy X differently, to allow for thepossibility of early exercise.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 82 / 161
For Freedom! (we must charge extra...)
What happens if we write a contract that allows the purchaser to exercisethe contract whenever she feels it to be in her advantage? By allowing thisextra freedom, we must
Charge more than we would for a European contract that is exercisedonly at the term N.
Hedge our replicating strategy X differently, to allow for thepossibility of early exercise.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 82 / 161
American Options
In the end, the option v is valued after the nth value of the stockSn(ω) = s is revealed via the recursive formula along each path ω:
vn(Sn(ω)) = max
g(Sn(ω)), e−rhE[v(Sn+1(ω)) | Sn(ω)
]τ∗(ω) = inf k ∈ 0, 1, ..,N | vk(Sk(ω)) = g(Sk(ω)) .
(86)
Here, τ∗ is the optimal exercise time.
In the Binomial case, we reduce to
vn(s) = max
g(s), e−rh [pvn+1(us) + qvn+1(ds)]
τ∗ = inf k | vk(s) = g(s) .(87)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 83 / 161
American Options
Some examples:
”American Bond:” g(s) = 1
”American Digital Option:” g(s) = 16≤s≤10
”American Square Option:” g(s) = s2.
Does an investor exercise any of these options early? Consider again thesetting
p =1
2= q, e−rh =
9
10
S0 = 4, u = 2, d =1
2.
(88)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 84 / 161
American Square Options
Consider the American Square Option. We know via Jensen’s Inequalitythat
e−rhE[g(Sn+1) | Sn
]= e−rhE
[S2n+1 | Sn
]≥ e−rh
(E[Sn+1 | Sn
]2)= e−rh
(erhSn
)2
= erhS2n > S2
n = g(Sn).
(89)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 85 / 161
American Square Options
It follows that vN(s) = s2 and
vN−1(s) = max
g(s), e−rhE[vN(SN) | SN−1 = s]
= max
s2, e−rhE[S2N | SN−1 = s]
= e−rhE[S2
N | SN−1 = s]
= e−rhE[vN(SN) | SN−1 = s].
(90)
In words, with one period to go, don’t exercise yet!!
The American and European option values coincide. Keep going.
How about with two periods left before expiration?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 86 / 161
American Options
Let’s return to the previous European Put example, where
p =1
2= q, e−rh =
9
10
S0 = 4, u = 2, d =1
2V3 := max 10− S3, 0 .
(91)
It follows that S3(ω) ∈
12 , 2, 8, 32
.
Use this to compute v3(s) and the American Put recursively.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 87 / 161
Matching Interest Rates to Market Conditions
Consider again a series of coin flips (ω1, ..., ωn) where at time n, theinterest rate from n to n + 1 is modeled via
rn = rn(ω1, ..., ωn) (92)
and a stochastic volatility σ at time n via
σn =1
2ln
(rn(ω1, ..., ωn−1, ωn = H)
rn(ω1, ..., ωn−1, ωn = T )
). (93)
Keep in mind that we will build a recombining binomial tree for thismodel. So, for example,
r2(H,T ) = r2(T ,H). (94)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 88 / 161
Matching Interest Rates to Market Conditions
Futhermore, we can define the yield rate y(t,T , r(t)) for a zero-couponbond B(t,T , r(t)) via
B(t,T , r(t)) =1
(1 + y(t,T , r(t)))T−t(95)
and the corresponding yield rate volatility
σn =1
2ln
(y(1, n, r1(H))
y(1, n, r1(T ))
). (96)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 89 / 161
Matching Interest Rates to Market Conditions
We can in fact see that
y(1, 2, r1(H)) = r1(H)
y(1, 2, r1(T )) = r1(T ).(97)
But, for example, it is not clear how to obtain
(y(1, 3, r1(H)), y(1, 3, r1(T ))) . (98)
Furthermore, how can we match to market conditions and update ourestimates for interest rates rn?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 90 / 161
Matching Interest Rates to Market Conditions
One way forward is the Black Derman Toy Model 1
Consider a market with the following observations:
Maturity Yield to Maturity y(0,T , r) Yield Volatility σT1 0.100 N.A.2 0.110 0.1903 0.120 0.1804 0.125 0.1505 0.130 0.140
(For Daily US Treasury Real Yield Curve Rates click here )
1Black, Fischer, Emanuel Derman, and William Toy. ”A one-factor modelof interest rates and its application to treasury bond options.” FinancialAnalysts Journal 46.1 (1990): 33-39.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 91 / 161
Matching Interest Rates to Market Conditions
In this setting, we match market conditions to a model where at eachtime, an interest rate moves up or down with a ”risk-neutral”probability of 1
2 .
It follows that we have from time t = 0 to t = 1, with an initial rater0 = r ,
1
1 + y(0, 1, r)=
1
1 + r
⇒ y(0, 1, r) = r .
(99)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 92 / 161
Matching Interest Rates to Market Conditions
From time t = 0 to t = 2, again with our initial rate r0 = r ,
Connecting our observed two-year yield with yearly interest ratesreturns
1
(1 + y(0, 2, r))2=
1
1 + r
(1
2
1
1 + r1(H)+
1
2
1
1 + r1(T )
)⇒ 1
1.112=
1
1.10
(1
2
1
1 + r1(H)+
1
2
1
1 + r1(T )
).
(100)
Also, connecting our one-year yields with yearly interest rates leads to
σ1 =1
2ln
(r1(H)
r1(T )
)=
1
2ln
(y(1, 2, r1(H))
y(1, 2, r1(T ))
)= σ2 = 0.190.
(101)
Solution leads to the pair
(r1(H), r1(T )) = (0.1432, 0.0979). (102)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 93 / 161
Matching Interest Rates to Market Conditions
From time t = 0 to t = 3, again with our initial rate r0 = r , we try toestimate the matching (r2(H,H), r2(H,T ), r2(T ,T )).
Connecting our observed two-year yield with yearly interest ratesreturns
1
(1 + y(0, 3, r))3=
1
1 + r
(1
4
1
1 + r1(H)
1
1 + r2(H,H)
)+
1
1 + r
(1
4
1
1 + r1(H)
1
1 + r2(H,T )
)+
1
1 + r
(1
4
1
1 + r1(T )
1
1 + r2(T ,H)
)+
1
1 + r
(1
4
1
1 + r1(T )
1
1 + r2(T ,T )
).
(103)
We can now substitute our values(r , r1(H), r1(T ), y(0, 3, r)) = (0.10, 0.1432, 0.0979, 0.120).
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 94 / 161
Matching Interest Rates to Market Conditions
We also know that σ2 6= σ2(ω1, ω2) and so
σ2 =1
2ln
(r2(H,H)
r2(H,T )
)σ2 =
1
2ln
(r2(T ,H)
r2(T ,T )
)⇒ r2(H,T ) = r2(T ,H) =
√r2(H,H)r2(T ,T ).
(104)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 95 / 161
Matching Interest Rates to Market Conditions
Finally, we have one more matching condition via
0.180 = σ3 =1
2ln
(y(1, 3, r1(H))
y(1, 3, r1(T ))
)(105)
where
1
(1 + y(1, 3, r1(H))2=
1
1 + r1(H)
(1
2
1
1 + r2(H,H)+
1
2
1
1 + r2(H,T )
)1
(1 + y(1, 3, r1(T ))2=
1
1 + r1(T )
(1
2
1
1 + r2(T ,H)+
1
2
1
1 + r2(T ,T )
).
(106)Now solve for (r2(H,H), r2(H,T ), r2(T ,T ))!!HW1: Price a bond that matures in two years, with the aboveobservations for term structure, and with F = 100 and coupon rate 10%.HW2: What about r3, r4, r5? Can we compute them?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 96 / 161
Pricing a Two-year Coupon Bond Using Market Conditions
Consider the previous observations for term structure, and a bond withF = 100 and coupon rate 10%. In this setting, we have
Time Interest Rate Value
0 r0 0.1001 r1(H) 0.14321 r1(T ) 0.09792 r2(H,H) 0.19422 r2(H,T ) 0.13772 r2(T ,H) 0.13772 r2(T ,T ) 0.0976
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 97 / 161
Pricing a Two-year Coupon Bond Using Market Conditions
We can decompose the two-year coupon bond into two componentzero-coupon bonds.
The first is B(1), which has face 10, maturity T = 1, and initial priceB(1)(0,T , r) at t = 0.
The first is B(2), which has face 110, maturity T = 2, and initial priceB(2)(0,T , r) at t = 0.
The total coupon bond price is thus
B(0, 2, r) = B(1)(0, 1, r) + B(2)(0, 2, r). (107)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 98 / 161
Pricing a Two-year Coupon Bond Using Market Conditions
Our component bonds thus have prices
Time Interest Rate B1(t,T , r)
0 r0 = 0.10 B(1)(0, 1, r0) = 9.09
1 r1(H) = 0.1432 B(1)(1, 1, r1(H)) = 10
1 r1(T ) = 0.0979 B(1)(1, 1, r1(T )) = 10
Time Interest Rate B2(t,T , r)
0 r0 = 0.10 B(2)(0, 2, r0) = 89.28
1 r1(H) = 0.1432 B(2)(1, 2, r1(H)) = 96.22
1 r1(T ) = 0.0979 B(2)(1, 2, r1(T )) = 100.19
2 r2(H,H) = 0.1942 B(2)(2, 2, r2(H,H)) = 110
2 r2(H,T ) = 0.1377 B(2)(2, 2, r2(H,T )) = 110
2 r2(T ,H) = 0.1377 B(2)(2, 2, r2(T ,H)) = 110
2 r2(T ,T ) = 0.0976 B(2)(2, 2, r2(T ,T )) = 110
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 99 / 161
Pricing a Two-year Coupon Bond Using Market Conditions
Finally, we have our coupon bond with price
Time Interest Rate B(t,T , r)
0 r0 = 0.10 B(0, 2, r0) = 98.371 r1(H) = 0.1432 B(1, 2, r1(H)) = 106.221 r1(T ) = 0.0979 B(1, 2, r1(T )) = 110.192 r2(H,H) = 0.1942 B(2, 2, r2(H,H)) = 1102 r2(H,T ) = 0.1377 B(2, 2, r2(H,T )) = 1102 r2(T ,H) = 0.1377 B(2, 2, r2(T ,H)) = 1102 r2(T ,T ) = 0.0976 B(2, 2, r2(T ,T )) = 110
Note: Note that with our coupons, the yield yc(0, 2, r) = 0.1095, which isobtained by solving
98.37 =10
1 + yc(0, 2, r)+
110
(1 + yc(0, 2, r))2. (108)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 100 / 161
Pricing a 1-year Call Option on our 2-year Coupon Bond
Consider the previous term structure, and a European Call Option on thetwo year bond with K = 97 and expiry of 1 year.
Compute the initial Call price C0(r).
Compute the initial number of bonds to hold to replicate this option.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 101 / 161
Pricing a 1-year Call Option on our 2-year Coupon Bond
In this case, we look at the price of the bond minus the accrued interest:
Time Interest Rate B(t,T , r)
0 r0 = 0.10 B(0, 2, r0) = 98.371 r1(H) = 0.1432 B(1, 2, r1(H)) = 96.221 r1(T ) = 0.0979 B(1, 2, r1(T )) = 100.192 r2(H,H) = 0.1942 B(2, 2, r2(H,H)) = 1002 r2(H,T ) = 0.1377 B(2, 2, r2(H,T )) = 1002 r2(T ,H) = 0.1377 B(2, 2, r2(T ,H)) = 1002 r2(T ,T ) = 0.0976 B(2, 2, r2(T ,T )) = 100
Pricing and hedging is accomplished via
C0(0.10) =1
1.10
(1
2· 0 +
1
23.19
)= 1.45
∆0(0.10) =0− 3.19
96.22− 100.19= 0.804.
(109)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 102 / 161
Asian Options
In times of high volatility or frequent trading, a company may want toprotect against large price movements over an entire time period, using anaverage. For example, if a company is looking at foreign exchange marketsor markets that may be subject to stock pinning due to large actors.
As an input, the average of an asset is used as an input against a strike,instead of the spot price.
There are two possibilities for the input in the discrete case: h = TN and
Arithmetic Average: IA(T ) := 1N
∑Nk=1 Skh .
Geometric Average: IG (T ) :=(
ΠNk=1Skh
) 1N
.
HW: Is there an ordering for IA, IG that is independent of T ?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 103 / 161
Asian Options
In times of high volatility or frequent trading, a company may want toprotect against large price movements over an entire time period, using anaverage. For example, if a company is looking at foreign exchange marketsor markets that may be subject to stock pinning due to large actors.
As an input, the average of an asset is used as an input against a strike,instead of the spot price.
There are two possibilities for the input in the discrete case: h = TN and
Arithmetic Average: IA(T ) := 1N
∑Nk=1 Skh .
Geometric Average: IG (T ) :=(
ΠNk=1Skh
) 1N
.
HW: Is there an ordering for IA, IG that is independent of T ?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 103 / 161
Asian Options
In times of high volatility or frequent trading, a company may want toprotect against large price movements over an entire time period, using anaverage. For example, if a company is looking at foreign exchange marketsor markets that may be subject to stock pinning due to large actors.
As an input, the average of an asset is used as an input against a strike,instead of the spot price.
There are two possibilities for the input in the discrete case: h = TN and
Arithmetic Average: IA(T ) := 1N
∑Nk=1 Skh .
Geometric Average: IG (T ) :=(
ΠNk=1Skh
) 1N
.
HW: Is there an ordering for IA, IG that is independent of T ?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 103 / 161
Asian Options
In times of high volatility or frequent trading, a company may want toprotect against large price movements over an entire time period, using anaverage. For example, if a company is looking at foreign exchange marketsor markets that may be subject to stock pinning due to large actors.
As an input, the average of an asset is used as an input against a strike,instead of the spot price.
There are two possibilities for the input in the discrete case: h = TN and
Arithmetic Average: IA(T ) := 1N
∑Nk=1 Skh .
Geometric Average: IG (T ) :=(
ΠNk=1Skh
) 1N
.
HW: Is there an ordering for IA, IG that is independent of T ?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 103 / 161
Asian Options: An Example:
Notice that these are path-dependent options, unlike the put and calloptions that we have studied until now. Assume r , δ, and h are such that
S0 = 4, u = 2, d =1
2, e−rh =
9
10g(I ) = max I − 2.5, 0 .
(110)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 104 / 161
Asian Options: An Example:
Consider an arithmetic average with N = 2. Then
v2(HH) = max
8 + 16
2− 2.5, 0
= 9.5
v2(HT ) = max
8 + 4
2− 2.5, 0
= 3.5
v2(TH) = max
2 + 4
2− 2.5, 0
= 0.5
v2(TT ) = max
2 + 1
2− 2.5, 0
= 0.
(111)
Compute v0, assuming a European structure. How about an Americanstructure?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 105 / 161
Asian Options: An Example:
Consider an arithmetic average with N = 2. Then
v2(HH) = max
8 + 16
2− 2.5, 0
= 9.5
v2(HT ) = max
8 + 4
2− 2.5, 0
= 3.5
v2(TH) = max
2 + 4
2− 2.5, 0
= 0.5
v2(TT ) = max
2 + 1
2− 2.5, 0
= 0.
(111)
Compute v0, assuming a European structure.
How about an Americanstructure?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 105 / 161
Asian Options: An Example:
Consider an arithmetic average with N = 2. Then
v2(HH) = max
8 + 16
2− 2.5, 0
= 9.5
v2(HT ) = max
8 + 4
2− 2.5, 0
= 3.5
v2(TH) = max
2 + 4
2− 2.5, 0
= 0.5
v2(TT ) = max
2 + 1
2− 2.5, 0
= 0.
(111)
Compute v0, assuming a European structure. How about an Americanstructure?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 105 / 161
Lognormality
Analyzing returns, we assume:
A probability space(
Ω,F ,P).
Our asset St(ω) has an associated return over any period (t, t + u)defined as
rt,u(ω) := ln
(St+u(ω)
St(ω)
). (112)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 106 / 161
Lognormality
Partition the interval [t,T ] into n intervals of length h = T−tn , then:
The return over the entire period can be taken as the sum of thereturns over each interval:
rt,T−t(ω) = ln
(ST (ω)
St(ω)
)=
n∑k=1
rtk ,h(ω)
tk = t + kh.
(113)
We model the returns as being independent and possessing a binomialdistribution.
Employing the Central Limit Theorem, it can be shown that asn→∞, this distribution approaches normality.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 107 / 161
Binomial Tree and Discrete Dividends
Another issue encountered in elementary credit and investment theoryis the case of different compounding and deposit periods.
This also occurs in the financial setting where a dividend is not paidcontinuously, but rather at specific times.
It follows that the dividend can be modeled as delivered in the middleof a binomial period, at time τ(ω) < T .
This view is due to Schroder and can be summarized as viewing theinherent value of St(ω) as the sum of a prepaid forward PF and thepresent value of the upcoming dividend payment D:
PFt(ω) = St(ω)− De−r(τ(ω)−t)
u = erh+σ√h
d = erh−σ√h.
(114)
Now, the random process that we model as having up and downmoves is PF instead of S .
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 108 / 161
Back to the Continuous Time Case
Consider the case of a security whose binomial evolution is modeled as anup or down movement at the end of each day. Over the period of oneyear, this amounts to a tree with depth 365. If the tree is not recombining,then this amounts to 2365 branches. Clearly, this is too large to evaluatereasonably, and so an alternative is sought.
Whatever the alternative, the concept of replication must hold. This isthe reasoning behind the famous Black-Scholes-Merton PDE approach.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 109 / 161
Back to the Continuous Time Case
Consider the case of a security whose binomial evolution is modeled as anup or down movement at the end of each day. Over the period of oneyear, this amounts to a tree with depth 365. If the tree is not recombining,then this amounts to 2365 branches. Clearly, this is too large to evaluatereasonably, and so an alternative is sought.
Whatever the alternative, the concept of replication must hold. This isthe reasoning behind the famous Black-Scholes-Merton PDE approach.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 109 / 161
Monte Carlo Techniques
Our model for asset evolution is
⇒ St = S0e(α−δ− 12σ2)t+σ
√tZ
Z ∼ N(0, 1).(115)
Consider now the possibility of simulating the stock evolution bysimulating the random variable Z , or in fact an i.i.d. sequence
Z (i)
ni=1
.
For a European option with time expiry T , we can simulate the expirytime payoff mulitple times:
V(
S (i),T)
= G(
S (i))
= G(
S0e(α−δ− 12σ2)T+σ
√TZ (i)
). (116)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 110 / 161
Monte Carlo Techniques
Our model for asset evolution is
⇒ St = S0e(α−δ− 12σ2)t+σ
√tZ
Z ∼ N(0, 1).(115)
Consider now the possibility of simulating the stock evolution bysimulating the random variable Z , or in fact an i.i.d. sequence
Z (i)
ni=1
.
For a European option with time expiry T , we can simulate the expirytime payoff mulitple times:
V(
S (i),T)
= G(
S (i))
= G(
S0e(α−δ− 12σ2)T+σ
√TZ (i)
). (116)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 110 / 161
Monte Carlo Techniques
Our model for asset evolution is
⇒ St = S0e(α−δ− 12σ2)t+σ
√tZ
Z ∼ N(0, 1).(115)
Consider now the possibility of simulating the stock evolution bysimulating the random variable Z , or in fact an i.i.d. sequence
Z (i)
ni=1
.
For a European option with time expiry T , we can simulate the expirytime payoff mulitple times:
V(
S (i),T)
= G(
S (i))
= G(
S0e(α−δ− 12σ2)T+σ
√TZ (i)
). (116)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 110 / 161
Monte Carlo Techniques
If we sample uniformly from our simulated values
V(
S (i),T)n
i=1we
can appeal to a sampling-convergence theorem with the appoximation
V (S , 0) = e−rT1
n
n∑i=1
V(
S (i),T). (117)
The challenge now is to simulate our lognormally distributed assetevolution.
One can simulate the value ST directly by one random variable Z , or amultiple of them to simulate the path of the evolution until T .
The latter method is necessary for Asian options.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 111 / 161
Monte Carlo Techniques
If we sample uniformly from our simulated values
V(
S (i),T)n
i=1we
can appeal to a sampling-convergence theorem with the appoximation
V (S , 0) = e−rT1
n
n∑i=1
V(
S (i),T). (117)
The challenge now is to simulate our lognormally distributed assetevolution.
One can simulate the value ST directly by one random variable Z , or amultiple of them to simulate the path of the evolution until T .
The latter method is necessary for Asian options.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 111 / 161
Monte Carlo Techniques
If we sample uniformly from our simulated values
V(
S (i),T)n
i=1we
can appeal to a sampling-convergence theorem with the appoximation
V (S , 0) = e−rT1
n
n∑i=1
V(
S (i),T). (117)
The challenge now is to simulate our lognormally distributed assetevolution.
One can simulate the value ST directly by one random variable Z , or amultiple of them to simulate the path of the evolution until T .
The latter method is necessary for Asian options.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 111 / 161
Monte Carlo Techniques
If we sample uniformly from our simulated values
V(
S (i),T)n
i=1we
can appeal to a sampling-convergence theorem with the appoximation
V (S , 0) = e−rT1
n
n∑i=1
V(
S (i),T). (117)
The challenge now is to simulate our lognormally distributed assetevolution.
One can simulate the value ST directly by one random variable Z , or amultiple of them to simulate the path of the evolution until T .
The latter method is necessary for Asian options.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 111 / 161
Monte Carlo Techniques
There are multiple ways to simulate Z . One way is to find a randomnumber U taken from a uniform distribution U[0, 1].
It follows that one can now map U → Z via inversion of the Nornal cdf N:
Z = N−1(U). (118)
It can be shown that in a sample, the standard deviation of the sampleaverage σsample is related to the standard deviation of an individual drawvia
σsample =σdraw√
n. (119)
If σdraw = σ, then we can see that we must increase our sample size by22k if we wish to cut our σsample by a factor of 2k .
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 112 / 161
Monte Carlo Techniques
There are multiple ways to simulate Z . One way is to find a randomnumber U taken from a uniform distribution U[0, 1].
It follows that one can now map U → Z via inversion of the Nornal cdf N:
Z = N−1(U). (118)
It can be shown that in a sample, the standard deviation of the sampleaverage σsample is related to the standard deviation of an individual drawvia
σsample =σdraw√
n. (119)
If σdraw = σ, then we can see that we must increase our sample size by22k if we wish to cut our σsample by a factor of 2k .
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 112 / 161
Monte Carlo Techniques
There are multiple ways to simulate Z . One way is to find a randomnumber U taken from a uniform distribution U[0, 1].
It follows that one can now map U → Z via inversion of the Nornal cdf N:
Z = N−1(U). (118)
It can be shown that in a sample, the standard deviation of the sampleaverage σsample is related to the standard deviation of an individual drawvia
σsample =σdraw√
n. (119)
If σdraw = σ, then we can see that we must increase our sample size by22k if we wish to cut our σsample by a factor of 2k .
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 112 / 161
Black Scholes Pricing using Underlying Asset
In the next course, we will derive the following solutions to theBlack-Scholes PDE:
V C (S , t) = e−r(T−t)E [(ST − K )+ | St = S ]
= Se−δ(T−t)N(d1)− Ke−r(T−t)N(d2)
V P(S , t) = e−r(T−t)E [(K − ST )+ | St = S ]
= Ke−r(T−t)N(−d2)− Se−δ(T−t)N(−d1)
d1 =ln(
SK
)+ (r − δ + 1
2σ2)(T − t)
σ√
T − t
d2 = d1 − σ√
T − t
N(x) =1√2π
∫ x
−∞e−
z2
2 dz .
(120)
Notice that V C (S , t)− V P(S , t) = Se−δ(T−t) − Ke−r(T−t).Question: What underlying model of stock evolution leads to this value?How can we support such a probability measure?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 113 / 161
Lognormal Random Variables
We say that Y ∼ LN(µ, σ) is Lognormal if ln(Y ) ∼ N(µ, σ2).
As sums of normal random variables remain normal, products of lognormalrandom variables remain lognormal.
Recall that the moment-generating function ofX ∼ N(µ, σ2) ∼ µ+ σN(0, 1) is
MX (t) = E[etX ] = eµt+ 12σ2t2
(121)
If Y = eµ+σZ , then, it can be seen that
E[Y n] = E[enX ] = eµn+ 12σ2n2
(122)
and
fY (y) =1
σ√
2πyexp
(− (ln(y)− µ)2
2σ2
)(123)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 114 / 161
Lognormal Random Variables
We say that Y ∼ LN(µ, σ) is Lognormal if ln(Y ) ∼ N(µ, σ2).
As sums of normal random variables remain normal, products of lognormalrandom variables remain lognormal.
Recall that the moment-generating function ofX ∼ N(µ, σ2) ∼ µ+ σN(0, 1) is
MX (t) = E[etX ] = eµt+ 12σ2t2
(121)
If Y = eµ+σZ , then, it can be seen that
E[Y n] = E[enX ] = eµn+ 12σ2n2
(122)
and
fY (y) =1
σ√
2πyexp
(− (ln(y)− µ)2
2σ2
)(123)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 114 / 161
Stock Evolution and Lognormal Random Variables
One application of lognormal distributions is their use in modeling theevolution of asset prices S . If we assume a physical measure P with α theexpected return on the stock under the physical measure, then
ln
(St
S0
)= N
((α− δ − 1
2σ2)t, σ2t
)⇒ St = S0e(α−δ− 1
2σ2)t+σ
√tZ
(124)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 115 / 161
Stock Evolution and Lognormal Random Variables
We can use the previous facts to show
E[St ] = S0e(α−δ)t
P[St > K ] = N
(ln S0
K + (α− δ − 0.5σ2)t
σ√
t
).
(125)
Note that under the risk-neutral measure P, we exchange α with r , therisk-free rate:
E[St ] = S0e(r−δ)t
P[St > K ] = N(d2).(126)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 116 / 161
Stock Evolution and Lognormal Random Variables
We can use the previous facts to show
E[St ] = S0e(α−δ)t
P[St > K ] = N
(ln S0
K + (α− δ − 0.5σ2)t
σ√
t
).
(125)
Note that under the risk-neutral measure P, we exchange α with r , therisk-free rate:
E[St ] = S0e(r−δ)t
P[St > K ] = N(d2).(126)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 116 / 161
Stock Evolution and Lognormal Random Variables
Risk managers are also interested in Conditional Tail Expectations (CTE’s)of random variables:
CTEX (k) := E[X | X > k] =E[X 1X>k
]P[X > k]
. (127)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 117 / 161
Stock Evolution and Lognormal Random Variables
In our case,
E[St | St > K ] =
E
[S0e(α−δ− 1
2σ2)t+σ
√tZ1
S0e(α−δ− 1
2 σ2)t+σ√tZ>K
]
P[S0e(α−δ− 1
2σ2)t+σ
√tZ > K
]
= S0e(α−δ)t
N
(ln
S0K
+(α−δ+0.5σ2)t
σ√t
)
N
(ln
S0K
+(α−δ−0.5σ2)t
σ√t
)
⇒ E[St | St > K ] = S0e(r−δ)t N(d1)
N(d2)(128)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 118 / 161
Stock Evolution and Lognormal Random Variables
In fact, we can use this CTE framework to solve for the European Calloption price in the Black-Scholes framework, where P0[A] = P[A | S0 = S ]and
V C (S , 0) := e−rT E[(ST − K )+ | S0 = S
]= e−rT E0
[ST − K | ST > K
]· P0[ST > K ]
= e−rT E0
[ST | ST > K
]· P0[ST > K ]− Ke−rT P0[ST > K ]
= e−rTSe(r−δ)T N(d1)
N(d2)· N(d2)− Ke−rTN(d2)
= Se−δTN(d1)− Ke−rTN(d2).(129)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 119 / 161
Black Scholes Analysis: Option Greeks
For any option price V (S , t), define its various sensitivities as follows:
∆ =∂V
∂S
Γ =∂∆
∂S=∂2V
∂S2
ν =∂V
∂σ
Θ =∂V
∂t
ρ =∂V
∂r
Ψ =∂V
∂δ.
(130)
These are known accordingly as the Option Greeks.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 120 / 161
Black Scholes Analysis: Option Greeks
Straightforward partial differentiation leads to
∆C = e−δ(T−t)N(d1)
∆P = −e−δ(T−t)N(−d1)
ΓC = ΓP =e−δ(T−t)N ′(d1)
σS√
T − t
νC = νP = Se−δ(T−t)√
T − tN ′(d1)
(131)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 121 / 161
Black Scholes Analysis: Option Greeks
as well as..
ρC = (T − t)Ke−r(T−t)N(d2)
ρP = −(T − t)Ke−r(T−t)N(−d2)
ΨC = −(T − t)Se−δ(T−t)N(d1)
ΨP = (T − t)Se−δ(T−t)N(−d1).
(132)
What do the signs of the Greeks tell us?
HW: Compute Θ for puts and calls.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 122 / 161
Black Scholes Analysis: Option Greeks
as well as..
ρC = (T − t)Ke−r(T−t)N(d2)
ρP = −(T − t)Ke−r(T−t)N(−d2)
ΨC = −(T − t)Se−δ(T−t)N(d1)
ΨP = (T − t)Se−δ(T−t)N(−d1).
(132)
What do the signs of the Greeks tell us?
HW: Compute Θ for puts and calls.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 122 / 161
Option Elasticity
Define
Ω(S , t) := limε→0
V (S+ε,t)−V (s,t)V (S,t)
S+ε−SS
=S
V (S , t)limε→0
V (S + ε, t)− V (s, t)
S + ε− S
=∆ · S
V (S , t).
(133)
Consequently,
ΩC (S , t) =∆C · S
V C (S , t)=
Se−δ(T−t)
Se−δ(T−t) − Ke−r(T−t)N(d2)≥ 1
ΩP(S , t) =∆P · S
V P(S , t)≤ 0.
(134)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 123 / 161
Option Elasticity
Theorem
The volatility of an option is the option elasticity times the volatility of thestock:
σoption = σstock× | Ω | . (135)
The proof comes from Finan: Consider the strategy of hedging a portfolioof shorting an option and purchasing ∆ = ∂V
∂S shares.The initial and final values of this portfolio are
Initally: V (S(t), t)−∆(S(t), t) · S(t)
Finally: V (S(T ),T )−∆(S(t), t) · S(T )
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 124 / 161
Option Elasticity
Proof.
If this portfolio is self-financing and arbitrage-free requirement, then
er(T−t)(
V (S(t), t)−∆(S(t), t) ·S(t))
= V (S(T ),T )−∆(S(t), t) ·S(T ).
(136)It follows that for κ := er(T−t),
V (S(T ),T )− V (S(t), t)
V (S(t), t)= κ− 1 +
[S(T )− S(t)
S(t)+ 1− κ
]Ω
⇒ Var
[V (S(T ),T )− V (S(t), t)
V (S(t), t)
]= Ω2Var
[S(T )− S(t)
S(t)
]⇒ σoption = σstock× | Ω | .
(137)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 125 / 161
Option Elasticity
If γ is the expected rate of return on an option with value V , α is theexpected rate of return on the underlying stock, and r is of course the riskfree rate, then the following equation holds:
γ · V (S , t) = α ·∆(S , t) · S + r ·(
V (S , t)−∆(S , t) · S). (138)
In terms of elasticity Ω, this reduces to
Risk Premium(Option) := γ − r = (α− r)Ω. (139)
Furthermore, we have the Sharpe Ratio for an asset as the ratio of riskpremium to volatility:
Sharpe(Stock) =(α− r)
σ=
(α− r)Ω
σΩ= Sharpe(Call). (140)
HW Sharpe Ratio for a put? How about elasticity for a portfolio ofoptions? Now read about Calendar Spreads, Implied Volatility, andPerpetual American Options.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 126 / 161
Option Elasticity
If γ is the expected rate of return on an option with value V , α is theexpected rate of return on the underlying stock, and r is of course the riskfree rate, then the following equation holds:
γ · V (S , t) = α ·∆(S , t) · S + r ·(
V (S , t)−∆(S , t) · S). (138)
In terms of elasticity Ω, this reduces to
Risk Premium(Option) := γ − r = (α− r)Ω. (139)
Furthermore, we have the Sharpe Ratio for an asset as the ratio of riskpremium to volatility:
Sharpe(Stock) =(α− r)
σ=
(α− r)Ω
σΩ= Sharpe(Call). (140)
HW Sharpe Ratio for a put? How about elasticity for a portfolio ofoptions? Now read about Calendar Spreads, Implied Volatility, andPerpetual American Options.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 126 / 161
Option Elasticity
If γ is the expected rate of return on an option with value V , α is theexpected rate of return on the underlying stock, and r is of course the riskfree rate, then the following equation holds:
γ · V (S , t) = α ·∆(S , t) · S + r ·(
V (S , t)−∆(S , t) · S). (138)
In terms of elasticity Ω, this reduces to
Risk Premium(Option) := γ − r = (α− r)Ω. (139)
Furthermore, we have the Sharpe Ratio for an asset as the ratio of riskpremium to volatility:
Sharpe(Stock) =(α− r)
σ=
(α− r)Ω
σΩ= Sharpe(Call). (140)
HW Sharpe Ratio for a put? How about elasticity for a portfolio ofoptions? Now read about Calendar Spreads, Implied Volatility, andPerpetual American Options.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 126 / 161
Example: Hedging
Under a standard framework, assume you write a 4− yr European Calloption a non-dividend paying stock with the following:
S0 = 10 = K
σ = 0.2
r = 0.02.
(141)
Calculate the initial number of shares of the stock for your hedgingprogram.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 127 / 161
Example: Hedging
Recall
∆C = e−δ(T−t)N(d1)
d1 =ln(
SK
)+ (r − δ + 1
2σ2)(T − t)
σ√
T − t
d2 = d1 − σ√
T − t.
(142)
It follows that
∆C = N(
0.4)
= 0.6554. (143)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 128 / 161
Example: Risk Analysis
Assume that an option is written on an asset S with the followinginformation:
The expected rate of return on the underlying asset is 0.10.
The expected rate of return on a riskless asset is 0.05.
The volatility on the underlying asset is 0.20.
V (S , t) = e−0.05(10−t)(
S2eS)
Compute Ω(S , t) and the Sharpe Ratio for this option.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 129 / 161
Example: Risk Analysis
By definition,
Ω(S , t) =∆ · S
V (S , t)=
S · ∂V (S,t)∂S
V (S , t)
=S d
dS (S2eS)
(S2eS)=
S · (2SeS + S2eS)
S2eS
= 2 + S .
(144)
Furthermore, since Ω = 2 + S ≥ 2, we have
Sharpe =0.10− 0.05
0.20= 0.25. (145)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 130 / 161
Example: Risk Analysis
By definition,
Ω(S , t) =∆ · S
V (S , t)=
S · ∂V (S,t)∂S
V (S , t)
=S d
dS (S2eS)
(S2eS)=
S · (2SeS + S2eS)
S2eS
= 2 + S .
(144)
Furthermore, since Ω = 2 + S ≥ 2, we have
Sharpe =0.10− 0.05
0.20= 0.25. (145)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 130 / 161
Example: Black Scholes Pricing
Consider a portfolio of options on a non-dividend paying stock S thatconsists of a put and a call, both with strike K = 5 = S0. What is the Γfor this option as well as the option value at time 0 if the time toexpiration is T = 4, r = 0.02, σ = 0.2.
In this case,
V = V C + V P
Γ =∂2
∂S2
(V C + V P
)= 2ΓC .
(146)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 131 / 161
Example: Black Scholes Pricing
Consider a portfolio of options on a non-dividend paying stock S thatconsists of a put and a call, both with strike K = 5 = S0. What is the Γfor this option as well as the option value at time 0 if the time toexpiration is T = 4, r = 0.02, σ = 0.2.
In this case,
V = V C + V P
Γ =∂2
∂S2
(V C + V P
)= 2ΓC .
(146)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 131 / 161
Example: Black Scholes Pricing
Consequently, d1 = 0.4 and d2 = 0.4− 0.2√
4 = 0, and so
V (5, 0) = V C (5, 0) + V P(5, 0)
= 5(
N(d1) + e−4rN(−d2)− e−4rN(d2)− N(−d1))
= 5(
N(0.4) + e−4rN(0)− e−4rN(0)− N(−0.4))
= 1.5542
Γ(5, 0) =2N ′(0.4)
0.2 · 5 ·√
4= N ′(0.4) =
1√2π
e−0.5·(0.4)2= 0.4322.
(147)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 132 / 161
Market Making
On a periodic basis, a Market Maker, services the option buyer byrebalancing the portfolio designed to replicate the payoff written into theoption contract.Define
Vi = Option Value i periods from inception
∆i = Delta required i periods from inception
∴ Pi = ∆iSi − Vi
(148)
Rebalancing at time i requires an extra (∆i+1 −∆i ) shares.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 133 / 161
Market Making
On a periodic basis, a Market Maker, services the option buyer byrebalancing the portfolio designed to replicate the payoff written into theoption contract.Define
Vi = Option Value i periods from inception
∆i = Delta required i periods from inception
∴ Pi = ∆iSi − Vi = Cost of Strategy
(149)
Rebalancing at time i requires an extra (∆i+1 −∆i ) shares.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 134 / 161
Market Making
Define
∂Si = Si+1 − Si
∂Pi = Pi+1 − Pi
∂∆i = ∆i+1 −∆i
(150)
Then
∂Pi = Net Cash Flow = ∆i∂Si − ∂Vi − rPi
= ∆i∂Si − ∂Vi − r(
∆iSi − Vi
) (151)
Under what conditions is the Net Flow = 0?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 135 / 161
Market Making
For a continuous rate r , we can see that if ∆ := ∂V∂S , Pt = ∆tSt − Vt ,
dV := V (St + dSt , t + dt)− V (St , t)
≈ Θdt + ∆ · dSt +1
2Γ · (dSt)
2
⇒ dPt = ∆tdSt − dVt − rPtdt
≈ ∆tdSt −(
Θdt + ∆ · dSt +1
2Γ · (dSt)
2
)− r (∆tSt − Vt) dt
≈ −
(Θdt + r(∆St − V (St , t))dt +
1
2Γ · [dSt ]
2
).
(152)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 136 / 161
Market Making
If dt is small, but not infinitessimally small, then on a periodic basis giventhe evolution of St , the periodic jump in value from St → St + dSt may beknown exactly and correspond to a non-zero jump in Market Maker profitdPt .
If dSt · dSt = σ2S2t dt, then if we sample continuously and enforce a zero
net-flow, we retain the BSM PDE for all relevant (S , t):
∂V
∂t+ r(
S∂V
∂S− V
)+
1
2σ2S2∂
2V
∂S2= 0
V (S ,T ) = G (S) for final time payoff G (S).
(153)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 137 / 161
Market Making
If dt is small, but not infinitessimally small, then on a periodic basis giventhe evolution of St , the periodic jump in value from St → St + dSt may beknown exactly and correspond to a non-zero jump in Market Maker profitdPt .
If dSt · dSt = σ2S2t dt, then if we sample continuously and enforce a zero
net-flow, we retain the BSM PDE for all relevant (S , t):
∂V
∂t+ r(
S∂V
∂S− V
)+
1
2σ2S2∂
2V
∂S2= 0
V (S ,T ) = G (S) for final time payoff G (S).
(153)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 137 / 161
Note: Delta-Gamma Neutrality vs Bond Immunization
In an actuarial analysis of cashflow, a company may wish to immunizeits portfolio. This refers to the relationship between a non-zero valuefor the second derivative with respect to interest rate of the(deterministic) cashflow present value and the subsequent possibilityof a negative PV.
This is similar to the case of market maker with a non-zero Gamma.In the market makers cash flow, a move of dS in the stockcorresponds to a move 1
2 Γ(dS)2 in the portfolio value.
In order to protect against large swings in the stock causing non-lineareffects in the portfolio value, the market maker may choose to offsetpositions in her present holdings to maintain Gamma Neutrality or shewish to maintain Delta Neutrality, although this is only a linear effect.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 138 / 161
Option Greeks and Analysis - Some Final Comments
It is important to note the similarities between Market Making andActuarial Reserving. In engineering the portfolio to replicate thepayoff written into the contract, the market maker requires capital.
The idea of Black Scholes Merton pricing is that the portfolio shouldbe self-financing.
One should consider how this compares with the capital required byinsurers to maintain solvency as well as the possibility of obtainingreinsurance.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 139 / 161
Exam Practice
Consider an economy where :
The current exchange rate is x0 = 0.011 dollaryen .
A four-year dollar-denominated European put option on yen with astrike price of 0.008$ sells for 0.0005$.
The continuously compounded risk-free interest rate on dollars is 3%.
The continuously compounded risk-free interest rate on yen is 1.5%.
Compute the price of a 4−year dollar-denominated European call optionon yens with a strike price of 0.008$.
ANSWER: By put call parity, and the Black Scholes formula, with theasset S as the exchange rate, and the foreign risk-free rate rf = δ,
V C (x0, 0) = V P(x0, 0) + x0e−rf T − Ke−rT
= 0.0005 + 0.011e−0.015·4 − 0.008e−0.03·4
= 0.003764.
(154)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 140 / 161
Exam Practice
Consider an economy where :
The current exchange rate is x0 = 0.011 dollaryen .
A four-year dollar-denominated European put option on yen with astrike price of 0.008$ sells for 0.0005$.
The continuously compounded risk-free interest rate on dollars is 3%.
The continuously compounded risk-free interest rate on yen is 1.5%.
Compute the price of a 4−year dollar-denominated European call optionon yens with a strike price of 0.008$.ANSWER: By put call parity, and the Black Scholes formula, with theasset S as the exchange rate, and the foreign risk-free rate rf = δ,
V C (x0, 0) = V P(x0, 0) + x0e−rf T − Ke−rT
= 0.0005 + 0.011e−0.015·4 − 0.008e−0.03·4
= 0.003764.
(154)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 140 / 161
Exam Practice
An investor purchases a 1−year, 50− strike European Call option ona non-dividend paying stock by borrowing at the risk-free rate r .
The investor paid V C (S0, 0) = 10.
Six months later, the investor finds out that the Call option hasincreased in value by one: V C (S0.05, 0.5) = 11.
Assume (σ, r) = (0.2, 0.02).
Should she close out her position after 6 months?
ANSWER: Simply put, her profit if she closes out after 6 months is
11− 10e0.02 12 = 0.8995. (155)
So, yes, she should liquidate her position.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 141 / 161
Exam Practice
An investor purchases a 1−year, 50− strike European Call option ona non-dividend paying stock by borrowing at the risk-free rate r .
The investor paid V C (S0, 0) = 10.
Six months later, the investor finds out that the Call option hasincreased in value by one: V C (S0.05, 0.5) = 11.
Assume (σ, r) = (0.2, 0.02).
Should she close out her position after 6 months?
ANSWER: Simply put, her profit if she closes out after 6 months is
11− 10e0.02 12 = 0.8995. (155)
So, yes, she should liquidate her position.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 141 / 161
Exam Practice
Consider a 1−year at the money European Call option on anon-dividend paying stock.
You are told that ∆C = 0.65, and the economy bears a 1% rate.
Can you estimate the volatility σ?
ANSWER: By definition,
∆C = e−δTN(d1) = N( r + 1
2σ2
σ
)= N
(0.01 + 12σ
2
σ
)= 0.65
⇒0.01 + 1
2σ2
σ= 0.385
⇒ σ ∈ 0.0269, 0.7431 .
(156)
More information is needed to choose from the two roots computed above.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 142 / 161
Exam Practice
Consider a 1−year at the money European Call option on anon-dividend paying stock.
You are told that ∆C = 0.65, and the economy bears a 1% rate.
Can you estimate the volatility σ?
ANSWER: By definition,
∆C = e−δTN(d1) = N( r + 1
2σ2
σ
)= N
(0.01 + 12σ
2
σ
)= 0.65
⇒0.01 + 1
2σ2
σ= 0.385
⇒ σ ∈ 0.0269, 0.7431 .
(156)
More information is needed to choose from the two roots computed above.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 142 / 161
Exam Practice
Consider a 1−year at the money European Call option on anon-dividend paying stock.
You are told that ∆C = 0.65, and the economy bears a 1% rate.
Can you estimate the volatility σ?
ANSWER: By definition,
∆C = e−δTN(d1) = N( r + 1
2σ2
σ
)= N
(0.01 + 12σ
2
σ
)= 0.65
⇒0.01 + 1
2σ2
σ= 0.385
⇒ σ ∈ 0.0269, 0.7431 .
(156)
More information is needed to choose from the two roots computed above.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 142 / 161
Exam Practice
Consider a 1−year at the money European Call option on anon-dividend paying stock.
You are told that ∆C = 0.65, and the economy bears a 1% rate.
Can you estimate the volatility σ?
ANSWER: By definition,
∆C = e−δTN(d1) = N( r + 1
2σ2
σ
)= N
(0.01 + 12σ
2
σ
)= 0.65
⇒0.01 + 1
2σ2
σ= 0.385
⇒ σ ∈ 0.0269, 0.7431 .
(156)
More information is needed to choose from the two roots computed above.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 142 / 161
Exam Pointers
When reviewing the material for the exam, consider the followingmilestones and examples:
The definition of the Black-Scholes pricing formulae for Europeanputs and calls.
What are the Greeks? Given a specific option, could you compute theGreeks?
What is the Option Elasticity? How is it useful? How about theSharpe ratio of an option? Can you compute the Elasticity andSharpe ration of a given option?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 143 / 161
Exam Pointers
What is Delta Hedging?
If the Delta and Gamma values of an option are known, can youcalculate the change in option value given a small change in theunderlying asset value?
How does this correspond the Market Maker’s profit?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 144 / 161
Probability Spaces - Introduction
We define the finite set of outcomes, the Sample Space, as Ω and anysubcollection of outcomes A ⊂ Ω an event.
How does this relate to the case of 2 consecutive coin flips
Ω ≡ HH,HT ,TH,TTSet of events includes statements like at least one head
= HH,HT ,TH
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 145 / 161
Probability Spaces - Introduction
We define the finite set of outcomes, the Sample Space, as Ω and anysubcollection of outcomes A ⊂ Ω an event.
How does this relate to the case of 2 consecutive coin flips
Ω ≡ HH,HT ,TH,TTSet of events includes statements like at least one head
= HH,HT ,TH
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 145 / 161
Probability Spaces - Introduction
We define the finite set of outcomes, the Sample Space, as Ω and anysubcollection of outcomes A ⊂ Ω an event.
How does this relate to the case of 2 consecutive coin flips
Ω ≡ HH,HT ,TH,TT
Set of events includes statements like at least one head
= HH,HT ,TH
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 145 / 161
Probability Spaces - Introduction
We define the finite set of outcomes, the Sample Space, as Ω and anysubcollection of outcomes A ⊂ Ω an event.
How does this relate to the case of 2 consecutive coin flips
Ω ≡ HH,HT ,TH,TTSet of events includes statements like at least one head
= HH,HT ,TH
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 145 / 161
Probability Spaces - Introduction
We define the finite set of outcomes, the Sample Space, as Ω and anysubcollection of outcomes A ⊂ Ω an event.
How does this relate to the case of 2 consecutive coin flips
Ω ≡ HH,HT ,TH,TTSet of events includes statements like at least one head
= HH,HT ,TH
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 145 / 161
Probability Spaces - Introduction
We define, ∀A,B ⊆ Ω
Ac = ω ∈ Ω : ω /∈ AA ∩ B = ω ∈ Ω : ω ∈ A and ω ∈ BA ∪ B = ω ∈ Ω : ω ∈ A or ω ∈ Bφ as the Empty Set
A,B to be Mutually Exclusive or Disjoint if A ∩ B = φ
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 146 / 161
Probability Spaces - Introduction
We define, ∀A,B ⊆ Ω
Ac = ω ∈ Ω : ω /∈ A
A ∩ B = ω ∈ Ω : ω ∈ A and ω ∈ BA ∪ B = ω ∈ Ω : ω ∈ A or ω ∈ Bφ as the Empty Set
A,B to be Mutually Exclusive or Disjoint if A ∩ B = φ
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 146 / 161
Probability Spaces - Introduction
We define, ∀A,B ⊆ Ω
Ac = ω ∈ Ω : ω /∈ AA ∩ B = ω ∈ Ω : ω ∈ A and ω ∈ B
A ∪ B = ω ∈ Ω : ω ∈ A or ω ∈ Bφ as the Empty Set
A,B to be Mutually Exclusive or Disjoint if A ∩ B = φ
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 146 / 161
Probability Spaces - Introduction
We define, ∀A,B ⊆ Ω
Ac = ω ∈ Ω : ω /∈ AA ∩ B = ω ∈ Ω : ω ∈ A and ω ∈ BA ∪ B = ω ∈ Ω : ω ∈ A or ω ∈ B
φ as the Empty Set
A,B to be Mutually Exclusive or Disjoint if A ∩ B = φ
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 146 / 161
Probability Spaces - Introduction
We define, ∀A,B ⊆ Ω
Ac = ω ∈ Ω : ω /∈ AA ∩ B = ω ∈ Ω : ω ∈ A and ω ∈ BA ∪ B = ω ∈ Ω : ω ∈ A or ω ∈ Bφ as the Empty Set
A,B to be Mutually Exclusive or Disjoint if A ∩ B = φ
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 146 / 161
Probability Spaces - Introduction
We define, ∀A,B ⊆ Ω
Ac = ω ∈ Ω : ω /∈ AA ∩ B = ω ∈ Ω : ω ∈ A and ω ∈ BA ∪ B = ω ∈ Ω : ω ∈ A or ω ∈ Bφ as the Empty Set
A,B to be Mutually Exclusive or Disjoint if A ∩ B = φ
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 146 / 161
σ−algebras
Given a non-empty set Ω of outcomes, a σ−algebra F is a collection ofsubsets of Ω that satisfies
∅ ∈ FA ∈ F ⇒ Ac ∈ FA1,A2,A3, .... ∈ F ⇒ ∪∞n=1An ∈ F
Some Examples
F0 = ∅,ΩF1 = ∅,Ω, HH,HT, TT ,THF2 = ∅,Ω, HH, HT, TT, TH, ....
F2 is completed by taking all unions of ∅,Ω, HH, HT, TT, TH.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 147 / 161
σ−algebras
Given a non-empty set Ω of outcomes, a σ−algebra F is a collection ofsubsets of Ω that satisfies
∅ ∈ F
A ∈ F ⇒ Ac ∈ FA1,A2,A3, .... ∈ F ⇒ ∪∞n=1An ∈ F
Some Examples
F0 = ∅,ΩF1 = ∅,Ω, HH,HT, TT ,THF2 = ∅,Ω, HH, HT, TT, TH, ....
F2 is completed by taking all unions of ∅,Ω, HH, HT, TT, TH.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 147 / 161
σ−algebras
Given a non-empty set Ω of outcomes, a σ−algebra F is a collection ofsubsets of Ω that satisfies
∅ ∈ FA ∈ F ⇒ Ac ∈ F
A1,A2,A3, .... ∈ F ⇒ ∪∞n=1An ∈ F
Some Examples
F0 = ∅,ΩF1 = ∅,Ω, HH,HT, TT ,THF2 = ∅,Ω, HH, HT, TT, TH, ....
F2 is completed by taking all unions of ∅,Ω, HH, HT, TT, TH.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 147 / 161
σ−algebras
Given a non-empty set Ω of outcomes, a σ−algebra F is a collection ofsubsets of Ω that satisfies
∅ ∈ FA ∈ F ⇒ Ac ∈ FA1,A2,A3, .... ∈ F ⇒ ∪∞n=1An ∈ F
Some Examples
F0 = ∅,ΩF1 = ∅,Ω, HH,HT, TT ,THF2 = ∅,Ω, HH, HT, TT, TH, ....
F2 is completed by taking all unions of ∅,Ω, HH, HT, TT, TH.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 147 / 161
σ−algebras
Given a non-empty set Ω of outcomes, a σ−algebra F is a collection ofsubsets of Ω that satisfies
∅ ∈ FA ∈ F ⇒ Ac ∈ FA1,A2,A3, .... ∈ F ⇒ ∪∞n=1An ∈ F
Some Examples
F0 = ∅,ΩF1 = ∅,Ω, HH,HT, TT ,THF2 = ∅,Ω, HH, HT, TT, TH, ....
F2 is completed by taking all unions of ∅,Ω, HH, HT, TT, TH.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 147 / 161
σ−algebras
Given a non-empty set Ω of outcomes, a σ−algebra F is a collection ofsubsets of Ω that satisfies
∅ ∈ FA ∈ F ⇒ Ac ∈ FA1,A2,A3, .... ∈ F ⇒ ∪∞n=1An ∈ F
Some Examples
F0 = ∅,Ω
F1 = ∅,Ω, HH,HT, TT ,THF2 = ∅,Ω, HH, HT, TT, TH, ....
F2 is completed by taking all unions of ∅,Ω, HH, HT, TT, TH.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 147 / 161
σ−algebras
Given a non-empty set Ω of outcomes, a σ−algebra F is a collection ofsubsets of Ω that satisfies
∅ ∈ FA ∈ F ⇒ Ac ∈ FA1,A2,A3, .... ∈ F ⇒ ∪∞n=1An ∈ F
Some Examples
F0 = ∅,ΩF1 = ∅,Ω, HH,HT, TT ,TH
F2 = ∅,Ω, HH, HT, TT, TH, ....
F2 is completed by taking all unions of ∅,Ω, HH, HT, TT, TH.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 147 / 161
σ−algebras
Given a non-empty set Ω of outcomes, a σ−algebra F is a collection ofsubsets of Ω that satisfies
∅ ∈ FA ∈ F ⇒ Ac ∈ FA1,A2,A3, .... ∈ F ⇒ ∪∞n=1An ∈ F
Some Examples
F0 = ∅,ΩF1 = ∅,Ω, HH,HT, TT ,THF2 = ∅,Ω, HH, HT, TT, TH, ....
F2 is completed by taking all unions of ∅,Ω, HH, HT, TT, TH.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 147 / 161
σ−algebras
Given a non-empty set Ω of outcomes, a σ−algebra F is a collection ofsubsets of Ω that satisfies
∅ ∈ FA ∈ F ⇒ Ac ∈ FA1,A2,A3, .... ∈ F ⇒ ∪∞n=1An ∈ F
Some Examples
F0 = ∅,ΩF1 = ∅,Ω, HH,HT, TT ,THF2 = ∅,Ω, HH, HT, TT, TH, ....
F2 is completed by taking all unions of ∅,Ω, HH, HT, TT, TH.
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 147 / 161
Power Sets
As a useful example, how many subsets are there of a set containingn elements?
Answer: 2n
Proof:
Consider strings of length n where the elements are either 0 or 1....
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 148 / 161
Power Sets
As a useful example, how many subsets are there of a set containingn elements?
Answer: 2n
Proof:
Consider strings of length n where the elements are either 0 or 1....
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 148 / 161
Power Sets
As a useful example, how many subsets are there of a set containingn elements?
Answer: 2n
Proof:
Consider strings of length n where the elements are either 0 or 1....
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 148 / 161
Power Sets
As a useful example, how many subsets are there of a set containingn elements?
Answer: 2n
Proof:
Consider strings of length n where the elements are either 0 or 1....
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 148 / 161
Notice that F0 ⊂ F1 ⊂ F2.
Correspondingly, given an Ω, we define a
Filtration as ..
a sequence of σ−algebras F0,F1,F2, ...,Fn, ... such that
F0 ⊂ F1 ⊂ F2 ⊂ ... ⊂ Fn ⊂ ...and F = σ(Ω) as the σ−algebra of all subsets of Ω.
Given a pair (Ω,F), we define a Random Variable X (ω) as a mappingX : Ω→ R
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 149 / 161
Notice that F0 ⊂ F1 ⊂ F2.
Correspondingly, given an Ω, we define a
Filtration as ..
a sequence of σ−algebras F0,F1,F2, ...,Fn, ... such that
F0 ⊂ F1 ⊂ F2 ⊂ ... ⊂ Fn ⊂ ...and F = σ(Ω) as the σ−algebra of all subsets of Ω.
Given a pair (Ω,F), we define a Random Variable X (ω) as a mappingX : Ω→ R
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 149 / 161
Notice that F0 ⊂ F1 ⊂ F2.
Correspondingly, given an Ω, we define a
Filtration as ..
a sequence of σ−algebras F0,F1,F2, ...,Fn, ... such that
F0 ⊂ F1 ⊂ F2 ⊂ ... ⊂ Fn ⊂ ...and F = σ(Ω) as the σ−algebra of all subsets of Ω.
Given a pair (Ω,F), we define a Random Variable X (ω) as a mappingX : Ω→ R
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 149 / 161
Notice that F0 ⊂ F1 ⊂ F2.
Correspondingly, given an Ω, we define a
Filtration as ..
a sequence of σ−algebras F0,F1,F2, ...,Fn, ... such that
F0 ⊂ F1 ⊂ F2 ⊂ ... ⊂ Fn ⊂ ...
and F = σ(Ω) as the σ−algebra of all subsets of Ω.
Given a pair (Ω,F), we define a Random Variable X (ω) as a mappingX : Ω→ R
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 149 / 161
Notice that F0 ⊂ F1 ⊂ F2.
Correspondingly, given an Ω, we define a
Filtration as ..
a sequence of σ−algebras F0,F1,F2, ...,Fn, ... such that
F0 ⊂ F1 ⊂ F2 ⊂ ... ⊂ Fn ⊂ ...and F = σ(Ω) as the σ−algebra of all subsets of Ω.
Given a pair (Ω,F), we define a Random Variable X (ω) as a mappingX : Ω→ R
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 149 / 161
Notice that F0 ⊂ F1 ⊂ F2.
Correspondingly, given an Ω, we define a
Filtration as ..
a sequence of σ−algebras F0,F1,F2, ...,Fn, ... such that
F0 ⊂ F1 ⊂ F2 ⊂ ... ⊂ Fn ⊂ ...and F = σ(Ω) as the σ−algebra of all subsets of Ω.
Given a pair (Ω,F), we define a Random Variable X (ω) as a mappingX : Ω→ R
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 149 / 161
Given a pair (Ω,F), we define a Probability Space as the triple(Ω,F ,P), where
P : F → [0, 1]
P[∅] = 0
For any countable disjoint sets A1,A2, ... ∈ FP [∪∞n=1An] =
∑∞n=1 P[An]
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 150 / 161
And so
P[A] :=∑
ω∈A P[ω]
E[X ] :=∑
ω X (ω)P[ω] =∑n
k=1 xkP[X (ω) = xk]
with Variance := E[(X − E[X ])2
]Some useful properties:
P[Ac ] = 1− P[A]
P[A] ≤ 1
P[A ∪ B] = P[A] + P[B]− P[A ∩ B]
P[A ∪ B ∪ C ] =P[A] + P[B] + P[C ]− P[A∩B]− P[A∩ C ]− P[B ∩ C ] + P[A∩B ∩ C ]
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 151 / 161
And so
P[A] :=∑
ω∈A P[ω]
E[X ] :=∑
ω X (ω)P[ω] =∑n
k=1 xkP[X (ω) = xk]
with Variance := E[(X − E[X ])2
]Some useful properties:
P[Ac ] = 1− P[A]
P[A] ≤ 1
P[A ∪ B] = P[A] + P[B]− P[A ∩ B]
P[A ∪ B ∪ C ] =P[A] + P[B] + P[C ]− P[A∩B]− P[A∩ C ]− P[B ∩ C ] + P[A∩B ∩ C ]
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 151 / 161
And so
P[A] :=∑
ω∈A P[ω]
E[X ] :=∑
ω X (ω)P[ω] =∑n
k=1 xkP[X (ω) = xk]
with Variance := E[(X − E[X ])2
]Some useful properties:
P[Ac ] = 1− P[A]
P[A] ≤ 1
P[A ∪ B] = P[A] + P[B]− P[A ∩ B]
P[A ∪ B ∪ C ] =P[A] + P[B] + P[C ]− P[A∩B]− P[A∩ C ]− P[B ∩ C ] + P[A∩B ∩ C ]
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 151 / 161
And so
P[A] :=∑
ω∈A P[ω]
E[X ] :=∑
ω X (ω)P[ω] =∑n
k=1 xkP[X (ω) = xk]
with Variance := E[(X − E[X ])2
]Some useful properties:
P[Ac ] = 1− P[A]
P[A] ≤ 1
P[A ∪ B] = P[A] + P[B]− P[A ∩ B]
P[A ∪ B ∪ C ] =P[A] + P[B] + P[C ]− P[A∩B]− P[A∩ C ]− P[B ∩ C ] + P[A∩B ∩ C ]
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 151 / 161
Example
Consider the case where two dice are rolled separately. What is the SampleSpace Ω here? How about the probability that the dots on the faces of thepair add up to 3 or 4 or 5?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 152 / 161
How Should We Count?
Think about the following problems:
How many pairings of aliens and humans, (aliens, human), can wehave if we can choose from 8 aliens and 9 people?
How many different strings of length 5 can we expect to find of 0′s
and 1′s ?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 153 / 161
Examples
From a group of 3 aliens and 5 humans, how many alien-peoplecouncils can be formed with 2 of each on the board?
What if two of the humans refuse to serve together?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 154 / 161
Examples
From a group of 3 aliens and 5 humans, how many alien-peoplecouncils can be formed with 2 of each on the board?
What if two of the humans refuse to serve together?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 154 / 161
Permutations
In general, we are interested in the number of different ways, orCombinations of ways r objects could be grouped when selected from apool of n total objects. Notationally, for r ≤ n we define this as
(nr
)and
the formula can be shown to be(n
r
)=
n!
(n − r)!r !
n! = 1 · 2 · ... · n(n
r
)=
(n − 1
r − 1
)+
(n − 1
r
) (157)
If order matters when selecting the r objects, then we define the numberof Permutations
Pk,n = r ! ·(
n
r
)=
n!
(n − r)!(158)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 155 / 161
Permutations
In general, we are interested in the number of different ways, orCombinations of ways r objects could be grouped when selected from apool of n total objects. Notationally, for r ≤ n we define this as
(nr
)and
the formula can be shown to be(n
r
)=
n!
(n − r)!r !
n! = 1 · 2 · ... · n
(n
r
)=
(n − 1
r − 1
)+
(n − 1
r
) (157)
If order matters when selecting the r objects, then we define the numberof Permutations
Pk,n = r ! ·(
n
r
)=
n!
(n − r)!(158)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 155 / 161
Permutations
In general, we are interested in the number of different ways, orCombinations of ways r objects could be grouped when selected from apool of n total objects. Notationally, for r ≤ n we define this as
(nr
)and
the formula can be shown to be(n
r
)=
n!
(n − r)!r !
n! = 1 · 2 · ... · n(n
r
)=
(n − 1
r − 1
)+
(n − 1
r
) (157)
If order matters when selecting the r objects, then we define the numberof Permutations
Pk,n = r ! ·(
n
r
)=
n!
(n − r)!(158)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 155 / 161
What if another event has already occured?
Consider the case where two events in our σ−field are under consideration.In fact, we know that B has already happened. How does that affect thechances of A happening?
For example, if you know your friend has one boy, and the chance of a boyor girl is equal at 0.5, then what is the chance all three of his children areboys?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 156 / 161
What if another event has already occured?
Consider the case where two events in our σ−field are under consideration.In fact, we know that B has already happened. How does that affect thechances of A happening?
For example, if you know your friend has one boy, and the chance of a boyor girl is equal at 0.5, then what is the chance all three of his children areboys?
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 156 / 161
What if another event has already occured?
In symbols, we seek
P [A | B] ≡ P [A ∩ B]
P [B]
=P [ all are boys ∩ first child is a boy]
P [ first child is a boy]
=P [ all are boys]
P [ first child is a boy]
=12 ·
12 ·
12
12
=1
4
(159)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 157 / 161
What if another event has already occured?
In symbols, we seek
P [A | B] ≡ P [A ∩ B]
P [B]
=P [ all are boys ∩ first child is a boy]
P [ first child is a boy]
=P [ all are boys]
P [ first child is a boy]
=12 ·
12 ·
12
12
=1
4
(159)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 157 / 161
What if another event has already occured?
In symbols, we seek
P [A | B] ≡ P [A ∩ B]
P [B]
=P [ all are boys ∩ first child is a boy]
P [ first child is a boy]
=P [ all are boys]
P [ first child is a boy]
=12 ·
12 ·
12
12
=1
4
(159)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 157 / 161
What if another event has already occured?
In symbols, we seek
P [A | B] ≡ P [A ∩ B]
P [B]
=P [ all are boys ∩ first child is a boy]
P [ first child is a boy]
=P [ all are boys]
P [ first child is a boy]
=12 ·
12 ·
12
12
=1
4
(159)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 157 / 161
What if another event has already occured?
In symbols, we seek
P [A | B] ≡ P [A ∩ B]
P [B]
=P [ all are boys ∩ first child is a boy]
P [ first child is a boy]
=P [ all are boys]
P [ first child is a boy]
=12 ·
12 ·
12
12
=1
4
(159)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 157 / 161
What if another event has already occured?
Now, what if you know your friend has at least one boy. Then what is thechance all three of his children are boys? This is also known as TheBoy-Girl Paradox.
P [A | C ] ≡ P [A ∩ C ]
P [C ]
=P [ all are boys ∩ at least one child is a boy]
P [at least one child is a boy]
=P [ all are boys]
P [ at least one child is a boy]
=P [ all are boys]
1− P [all girls]
=12 ·
12 ·
12
1− 18
=1
7
(160)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 158 / 161
What if another event has already occured?
Now, what if you know your friend has at least one boy. Then what is thechance all three of his children are boys? This is also known as TheBoy-Girl Paradox.
P [A | C ] ≡ P [A ∩ C ]
P [C ]
=P [ all are boys ∩ at least one child is a boy]
P [at least one child is a boy]
=P [ all are boys]
P [ at least one child is a boy]
=P [ all are boys]
1− P [all girls]
=12 ·
12 ·
12
1− 18
=1
7
(160)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 158 / 161
What if another event has already occured?
Now, what if you know your friend has at least one boy. Then what is thechance all three of his children are boys? This is also known as TheBoy-Girl Paradox.
P [A | C ] ≡ P [A ∩ C ]
P [C ]
=P [ all are boys ∩ at least one child is a boy]
P [at least one child is a boy]
=P [ all are boys]
P [ at least one child is a boy]
=P [ all are boys]
1− P [all girls]
=12 ·
12 ·
12
1− 18
=1
7
(160)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 158 / 161
What if another event has already occured?
Now, what if you know your friend has at least one boy. Then what is thechance all three of his children are boys? This is also known as TheBoy-Girl Paradox.
P [A | C ] ≡ P [A ∩ C ]
P [C ]
=P [ all are boys ∩ at least one child is a boy]
P [at least one child is a boy]
=P [ all are boys]
P [ at least one child is a boy]
=P [ all are boys]
1− P [all girls]
=12 ·
12 ·
12
1− 18
=1
7
(160)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 158 / 161
What if another event has already occured?
Now, what if you know your friend has at least one boy. Then what is thechance all three of his children are boys? This is also known as TheBoy-Girl Paradox.
P [A | C ] ≡ P [A ∩ C ]
P [C ]
=P [ all are boys ∩ at least one child is a boy]
P [at least one child is a boy]
=P [ all are boys]
P [ at least one child is a boy]
=P [ all are boys]
1− P [all girls]
=12 ·
12 ·
12
1− 18
=1
7
(160)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 158 / 161
What if another event has already occured?
Now, what if you know your friend has at least one boy. Then what is thechance all three of his children are boys? This is also known as TheBoy-Girl Paradox.
P [A | C ] ≡ P [A ∩ C ]
P [C ]
=P [ all are boys ∩ at least one child is a boy]
P [at least one child is a boy]
=P [ all are boys]
P [ at least one child is a boy]
=P [ all are boys]
1− P [all girls]
=12 ·
12 ·
12
1− 18
=1
7
(160)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 158 / 161
What if another event has already occured?
Now, what if you know your friend has at least one boy. Then what is thechance all three of his children are boys? This is also known as TheBoy-Girl Paradox.
P [A | C ] ≡ P [A ∩ C ]
P [C ]
=P [ all are boys ∩ at least one child is a boy]
P [at least one child is a boy]
=P [ all are boys]
P [ at least one child is a boy]
=P [ all are boys]
1− P [all girls]
=12 ·
12 ·
12
1− 18
=1
7
(160)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 158 / 161
Total Probability
Notice from our definition of conditional probabiltity that
P[A ∩ B] = P[A | B] · P[B] (161)
We can expand on this idea: For our sample space Ω, assume we have aset A1, ...,An where the members are
mutually exclusive - Ai ∩ Aj = φ for all i 6= j
exhaustive - A1 ∪ A2 ∪ ... ∪ An = Ω .
Then
P[B] = P[B ∩ A1] + P[B ∩ A2] + ...+ P[B ∩ An]
= P[B | A1]P[A1] + P[B | A2]P[A2] + ...+ P[B | An]P[An](162)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 159 / 161
Total Probability
Notice from our definition of conditional probabiltity that
P[A ∩ B] = P[A | B] · P[B] (161)
We can expand on this idea: For our sample space Ω, assume we have aset A1, ...,An where the members are
mutually exclusive - Ai ∩ Aj = φ for all i 6= j
exhaustive - A1 ∪ A2 ∪ ... ∪ An = Ω .
Then
P[B] = P[B ∩ A1] + P[B ∩ A2] + ...+ P[B ∩ An]
= P[B | A1]P[A1] + P[B | A2]P[A2] + ...+ P[B | An]P[An](162)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 159 / 161
Total Probability
Notice from our definition of conditional probabiltity that
P[A ∩ B] = P[A | B] · P[B] (161)
We can expand on this idea: For our sample space Ω, assume we have aset A1, ...,An where the members are
mutually exclusive - Ai ∩ Aj = φ for all i 6= j
exhaustive - A1 ∪ A2 ∪ ... ∪ An = Ω .
Then
P[B] = P[B ∩ A1] + P[B ∩ A2] + ...+ P[B ∩ An]
= P[B | A1]P[A1] + P[B | A2]P[A2] + ...+ P[B | An]P[An](162)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 159 / 161
Total Probability
Notice from our definition of conditional probabiltity that
P[A ∩ B] = P[A | B] · P[B] (161)
We can expand on this idea: For our sample space Ω, assume we have aset A1, ...,An where the members are
mutually exclusive - Ai ∩ Aj = φ for all i 6= j
exhaustive - A1 ∪ A2 ∪ ... ∪ An = Ω .
Then
P[B] = P[B ∩ A1] + P[B ∩ A2] + ...+ P[B ∩ An]
= P[B | A1]P[A1] + P[B | A2]P[A2] + ...+ P[B | An]P[An](162)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 159 / 161
Bayes Theorem
For our sample space Ω, assume we have an exhaustive set of eventsA1, ...,An with prior probabilities P[Ai ] for i = 1, .., n. Then for anyother event B in our σ−field where P[B] > 0, the posterior probabilty ofAj given that B has occured is
P [Aj | B] =P [Aj ∩ B]
P [B]=
P [B ∩ Aj ]
P [B]
=P[B | Aj ] · P[Aj ]
P[B | A1]P[A1] + ...+ P[B | An]P[An]
(163)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 160 / 161
Bayes Theorem
For our sample space Ω, assume we have an exhaustive set of eventsA1, ...,An with prior probabilities P[Ai ] for i = 1, .., n. Then for anyother event B in our σ−field where P[B] > 0, the posterior probabilty ofAj given that B has occured is
P [Aj | B] =P [Aj ∩ B]
P [B]=
P [B ∩ Aj ]
P [B]
=P[B | Aj ] · P[Aj ]
P[B | A1]P[A1] + ...+ P[B | An]P[An]
(163)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 160 / 161
Bayes Theorem
For our sample space Ω, assume we have an exhaustive set of eventsA1, ...,An with prior probabilities P[Ai ] for i = 1, .., n. Then for anyother event B in our σ−field where P[B] > 0, the posterior probabilty ofAj given that B has occured is
P [Aj | B] =P [Aj ∩ B]
P [B]=
P [B ∩ Aj ]
P [B]
=P[B | Aj ] · P[Aj ]
P[B | A1]P[A1] + ...+ P[B | An]P[An]
(163)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 160 / 161
Bayes Theorem
For our sample space Ω, assume we have an exhaustive set of eventsA1, ...,An with prior probabilities P[Ai ] for i = 1, .., n. Then for anyother event B in our σ−field where P[B] > 0, the posterior probabilty ofAj given that B has occured is
P [Aj | B] =P [Aj ∩ B]
P [B]=
P [B ∩ Aj ]
P [B]
=P[B | Aj ] · P[Aj ]
P[B | A1]P[A1] + ...+ P[B | An]P[An]
(163)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 160 / 161
Independent Events
For our sample space Ω, assume we have two events A,B. We say that Aand B are independent if P[A | B] = P[A] and dependent ifP[A | B] 6= P[A]
In other words, A and B are independent if and only ifP[A ∩ B] = P[A] · P[B].Generally speaking, for any collection of events A1, ...,An we have forany subcollection Ai1 , ...,Ain ⊆ A1, ...,An
P[Ai1 ∩ ... ∩ Ain ] = P[Ai1 ] · .. · ...P[Ain ] (164)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 161 / 161
Independent Events
For our sample space Ω, assume we have two events A,B. We say that Aand B are independent if P[A | B] = P[A] and dependent ifP[A | B] 6= P[A]In other words, A and B are independent if and only ifP[A ∩ B] = P[A] · P[B].
Generally speaking, for any collection of events A1, ...,An we have forany subcollection Ai1 , ...,Ain ⊆ A1, ...,An
P[Ai1 ∩ ... ∩ Ain ] = P[Ai1 ] · .. · ...P[Ain ] (164)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 161 / 161
Independent Events
For our sample space Ω, assume we have two events A,B. We say that Aand B are independent if P[A | B] = P[A] and dependent ifP[A | B] 6= P[A]In other words, A and B are independent if and only ifP[A ∩ B] = P[A] · P[B].Generally speaking, for any collection of events A1, ...,An we have forany subcollection Ai1 , ...,Ain ⊆ A1, ...,An
P[Ai1 ∩ ... ∩ Ain ] = P[Ai1 ] · .. · ...P[Ain ] (164)
Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring 2016 161 / 161