fec financial engineering club. an intro to options
TRANSCRIPT
FEC FINANCIAL ENGINEERING CLUB
AN INTRO TO OPTIONS
AGENDA
What are options?
Bounds on prices
Spread strategies
Greeks
OPTIONS CONTRACTS
An option contract is a right to buy (call option) or sell (put option) an underlying security at a pre-specified date in the future and at a pre-specified price. Date is called the maturity or expiration date Pre-specified price is called the strike price
Ex) AAPL is currently at (about) $509.00
You want to buy a call option with a strike of $505.00 whose expiration is March 21, 2014.
This means you can ‘exercise’ your option to buy AAPL at $505.00 at the maturity date (European-style option) or before (American-style option)
OPTION VALUE
What is the value of such an option? Depends on many things—most importantly, the underlying (AAPL) price.
Suppose this call option expired today and AAPL was at $509.00. How much would you be willing to pay for it? Right to buy AAPL (worth $509.00) for $505.00
CallIntrinsic = (S-K)+ = max{S-K,0}, S is the price of the underlying todayK is the strike price
INTRINSIC VALUE
The intrinsic value is the value of an option if it expires right now.
, ,
S is the price of the underlying todayK is the strike price
For AAPL at 509.00, what is the intrinsic value of?
Call(K=520)? $0.00 Out of the money (trading lingo)
Put(K=520)? $11.00 In the money
Call(K=500)? $9.00 In the money
Put(K=500)? $0.00 Out of the money
INTRINSIC VALUE
Intrinsic Value of a Call Option (Green)
Intrinsic Value of a Put Option (Green)
TIME VALUE
However, if there is time left until expiration, the stock price (at time t) St, could change and thus the value of the option would change. This component of price that changes over time is known as time value.
Time value is the value associated with the likelihood that the option will become in the money (valuable) by a favorable move in the underlying price
Some determinants of time value: How volatile is the underlying stock What is your borrowing rate, are there any dividends from the underlying
stock How much time is left until maturity
BUYING VS SELLING OPTIONS
If you buy an option you have the option to exercise it:
Long Call option: You may pay K to receive the stock. Long Put option: You may sell the stock for K.
BUYING VS SELLING OPTIONS
When you sell an option, you give the buyer the right to exercise the option:
Short Call option: Buyer may buy the stock from you for K.
Short Put option: Buyer may sell the stock to you for K.
When will the buyer buy the stock from you?
In the same situations you would exercise the call option—when S > K. They would not exercise when S < K.
BUYING VS SELLING OPTIONS
Same logic applies for short puts.
When you sell (called writing) an option and it is exercised by the buyer, it is said to be assigned against you.
OPTION CONTRACT STYLES
European—Option may be exercised at maturity only.
American—Option may be exercised at any time preceding maturity.
Others—Asian, Bermudan, Barrier options.
For this lecture, we will discuss the simplest and most common cases—European and American options.
COMMON SENSE BOUNDS
Without making any assumptions about the returns of the underlying or imposing any model, what can we say about option prices?
Denote Current price of underlying
Strike price on option
current date
Expiration date
Value of European call option
Value of American call option
Value of European put option
Value of American put option
COMMON SENSE BOUNDS
Options cannot have negative value:
Why?
COMMON SENSE BOUNDS
American options are at least as valuable as European options:
Why?
You can exercise American options in more (potentially more profitable) situations.
COMMON SENSE BOUNDS
Call options never worth more than underlying stock; puts never worth more than exercise price:
Why?
If P > K, sell the put for P and invest K of the proceeds. In the worst scenario, the stock will be worthless and you will pay K and receive the stock. However, you earned P > K.
COMMON SENSE BOUNDS
American options are worth at least their intrinsic value:
COMMON SENSE BOUNDS
Calls with lower strikes are more valuable. Puts with higher strikes are more valuable:
If
Why?
The strike price is what you pay to exercise a call—smaller K means larger payoff. Converse is true for puts.
COMMON SENSE BOUNDS
Options are worth more when there is additional time until maturity:
If
Why?
PUT-CALL PARITY
What happens if I sell a put at strike price K, buy an identical call, and lend PV(K)?
Has the same payoff as a long position in the underlying! Therefore costs of our combined position must equal that of the underlying.
PUT-CALL PARITY
PUT-CALL PARITY
−𝑆𝑡=−𝑐 (𝑆 ,𝐾 , 𝑡 ,𝑇1 )+𝑝 (𝑆 ,𝐾 , 𝑡 ,𝑇1 )−𝑃𝑉 (𝐾 )Cost to be long underlying
Cost to be long call option
Cost to be short put
Cash outflow from lending PV(K)
This, and other bounds may be used in the Black-Scholes PDE (next lecture)
SPREAD STRATEGIES
SPREADS
Spread strategies are multi-legged option positions
What is a leg? A position using one type of options contract
Example: What if we buy a call option and buy an identical put option (same strike, time until maturity, etc)? One leg is the call option One leg is the put option What does our position look like?
SPREADS
When S > K, what happens?
We exercise our long call option(s)
When S < K, what happens?
We exercise our short put options(s)
Profit/Loss Diagram is:
LONG STRADDLE
This is known as a long Straddle position. One of the simpler spread positions
When would one want to trade a straddle?
A ) When volatility is high ?
B) When volatility is low?
C) When we are certain the underlying will increase?
LONG STRADDLE
This is known as a long Straddle position. One of the simpler spread positions
When would one want to trade a straddle?
A ) When volatility is high ?
B) When volatility is low?
C) When we are certain the underlying will increase?
MORE SPREAD STRATEGIES
Underlying is 37
Strategy: Long call (Strike = 40); Long a put (Strike = 35). The call is worth $3. The put is worth $1.
Underlying Long Call Long Put Long Strangle0 5
10 15 20 25 30 35 40 45 50 55 60 65 70 75
STRANGLE
Underlying
Long Call
Long Put
Long Strangle
0 -3 34 315 -3 29 26
10 -3 24 2115 -3 19 1620 -3 14 1125 -3 9 630 -3 4 135 -3 -1 -440 -3 -1 -445 2 -1 150 7 -1 655 12 -1 1160 17 -1 1665 22 -1 2170 27 -1 2675 32 -1 31
MORE SPREAD STRATEGIES
From a payoff standpoint (ignore costs), would you prefer to be long Position 1: two call options (K = 35) , or Position 2: one call option (K = 30) and another call option (K = 40)
MORE SPREAD STRATEGIES
Underlying
Long Call (K=35)
Position 1
0 0 05 0 0
10 0 015 0 020 0 025 0 030 0 035 0 040 5 1045 10 2050 15 3055 20 4060 25 5065 30 6070 35 7075 40 80
Underlying
Long Call (K = 30)
Long Call (K = 40)
Position 2
0 0 0 05 0 0 0
10 0 0 015 0 0 020 0 0 025 0 0 030 0 0 035 5 0 540 10 0 1045 15 5 2050 20 10 3055 25 15 4060 30 20 5065 35 25 6070 40 30 7075 45 35 80
MORE SPREAD STRATEGIES
GENERAL APPROACH TO SPREADS
Options can replicate any risk profile at maturity with exclusively puts or calls.
That is, you can construct a position like this:
GENERAL APPROACHES TO SPREADS
38
50
51
52
37
REPLICATION WITH CALLS
Evaluate positions from left to right
38
50
51
52
37
1) Slope must be 10—buy 10 Calls at 37
2) Slope from 38 to 50 must be 0—sell 10 Calls at 38 to get flat
3) Slope from 50 to 51 must be -5—sell 5 Calls at 50
4) Slope from 51 to 52 must be -3—buy 2 Calls at 51
5) Slope after 52 is 0—buy 3 Calls at 52 to get flat
+10 Calls(37)
-10 Calls(38)
-5 Calls(50)
+2 Calls(51)
+3 Calls(52)
REPLICATION WITH PUTS
Evaluate positions from right to left
38
50
51
52
37
5) Slope must be 0—buy 10 Puts at 37 to get flat
4) Slope from 38 to 37 must be 10—sell 10 Puts at 38
3) Slope from 50 to 38 must be 0—sell 5 Puts at 50 to get flat
2) Slope from 51 to 50 must be -5—buy 2 Puts at 51
1) Slope from 52 to 51 is -3—buy 3 Puts at 52
+3 Put(52)
+2 Put(51)
-5 Put(50)
-10 Put(38)
+10 Put(37)
GREEKS
GREEKS
Recall that there are five drivers of an option’s price: Price of the underlying Volatility of the returns on the underlying Interest rates Strike price Time until maturity
What is the risk of an option? How does the price of an option change as the underlying factors change? These are the greeks.
DELTA
The sensitivity of an option with respect to a change in the underlying’s price.
Ex) Suppose that the underlying is at 60. A call option with strike has a delta of .5 (usually quoted as 50). What happens if underlying moves to 65?
Option price increases by .5*(65-50) = .5*15 = 7.50
DELTA-HEDGING
In general, for an option with a delta of , its price will move by
Many traders like to be delta—neutral. That is, they prefer to be immune to the risk of the underlying price trading.
Delta-hedging is the process of offsetting the delta of your portfolio to 0, by selling the underlying. You may want to do this in the trading competition.
DELTA-HEDGING
Ex) Suppose you are long 20 call options with 0.3 years until maturity and strike is $40. The risk-free interest rate is 0 and the expected standard deviation of returns over the next 0.3 years is 0.2. The underlying is at $41.
Delta is .61 We want to trade X shares of the underlying so that
What is ?
DELTA-HEDGING
Ex) Suppose you are long 20 call options with 0.3 years until maturity and strike is $40. The risk-free interest rate is 0 and the expected standard deviation of returns over the next 0.3 years is 0.2. The underlying is at $41.
Delta is .61 We want to trade X shares of the underlying so that
Now, the underlying decreases to $39.00. What happens?
Sell 1220 Shares
DELTA-HEDGING
;
;
You make money even though your calls become less valuable
Now = 0.43 Repeat the process:
Why did delta change? Answer: Gamma
Sell 860 Shares
NEXT LECTURE
Continuous models for option valuation Stochastic calculus Black-Scholes-Merton More greeks
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