Options

Learning Objectives 2

The Five Pillars 3

Nobel Prize winner and former Univ. of Chicago professor, Merton Miller, published a paper called the “The History of Finance”

Miller identified five “pillars on which the field of finance rests” These include

1. Miller-Modigliani Propositions• Merton Miller 1990 • Franco Modigliani 1985

2. Capital Asset Pricing Model• William Sharpe 1990

3. Efficient Market Hypothesis• (Eugene Fama, Paul Samuelson, …)

4. Modern Portfolio Theory• Harry Markowitz 1990

5. Options • Myron Scholes and Robert Merton 1997

Objectives 4

Options vs Forwards

Forward Long

Obligation to buy and take delivery of asset for $K at time T

Short Obligation to sell and deliver an

asset for $K at time T

Option Call

Long Right to buy an asset at price $K

at time T Short

Obligation to sell an asset at price $K at time T

Put Long

Right to sell an asset at price $K at time T

Short Obligation to buy an asset at

price $K at time T

5

Options vs Forwards6

-$15

-$10

-$5

$0

$5

$10

$15

$75 $80 $85 $90 $95 $100 $105 $110

Profi

t

ST

Opt 1

Fwd

Strike

-$15

-$10

-$5

$0

$5

$10

$15

$75 $80 $85 $90 $95 $100 $105 $110

Profi

t

ST

Opt 1

Fwd

Strike

Options vs Forwards7

-$15

-$10

-$5

$0

$5

$10

$15

$70 $75 $80 $85 $90 $95 $100 $105 $110

Profi

t

ST

Opt 1

Fwd

Total

Strike

-$15

-$10

-$5

$0

$5

$10

$15

$70 $75 $80 $85 $90 $95 $100 $105 $110

Profi

t

ST

Opt 1

Fwd

Total

Strike

Basic Options8

-$4

-$2

$0

$2

$4

$6

$8

$10

$30 $35 $40 $45 $50 $55 $60

-$6

-$4

-$2

$0

$2

$4

$6

$8

$10

$30 $35 $40 $45 $50 $55 $60

-$20

-$15

-$10

-$5

$0

$5

$10

$30 $35 $40 $45 $50 $55 $60

-$15

-$10

-$5

$0

$5

$10

$30 $35 $40 $45 $50 $55 $60

Short Call: Obligation to sell an asset at price $K at time T if buyer chooses to exercise.

Short Put: Obligation to buy an asset at price $K at time T if buyer chooses to exercise.

Long Put: Right to sell an asset at price $K at time T

Long Call: Right to buy an asset at price $K at time T

Options

Types European

Can be exercised only at time T Without dividends With dividends

American Can be exercised anytime 0 < t ≤ T

Binary European Cash or nothing Asset or nothing

Other Asian

Underlying Assets Equity Exchange Rates Commodity futures Interest Rates

9

-$15

-$10

-$5

$0

$5

$10

$30 $35 $40 $45 $50 $55 $60PT

ST-$20

-$15

-$10

-$5

$0

$5

$10

$30 $35 $40 $45 $50 $55 $60PT

ST

-$6

-$4

-$2

$0

$2

$4

$6

$8

$10

$30 $35 $40 $45 $50 $55 $60

PT

ST-$4

-$2

$0

$2

$4

$6

$8

$10

$30 $35 $40 $45 $50 $55 $60

CT

ST

Basic Options – Value at Expiry 10

Long put

PT = max(K – ST , 0)Long call

CT = max(ST-K , 0)

Short call

-CT = min(K-ST , 0)

Short put

-PT = min(ST –K , 0)

Forward and Option Resolution

Forward Buyer (Long) Buy asset at $K at time T

Option Buyer Sell the option

“Sell to close” Exercise

Put: Sell asset for $K at time T

Call: Buy asset at $K at time T Let option expire

Premium is lost

Forward Seller (Short) Sell asset at $K at time T

Option Seller Buy the option back

“Buy to cover” Get exercised

Put: Buy asset at $K at time T Call: Sell asset at $K at time T

Option expires Keep premium

11

Put – Call Parity 12

Portfolio of one share of stock, S, one long put, P, one short call, CSame strike, K, and time to expiry T

K

0KSS K)(S0SΠ

KS

K 0SKS

CPSΠKS

CPSΠ

TT

TTT

T

TT

TTTT

T

TTTT

t

tTrtt

0Tr

00

000Tr

0000

Tr0

T

SKeCP

SKeCP

CPSKe

CPSΠ

eKΠ

KΠ

*

*

*

*

Put – Call Parity and Forwards at Expiry 13

-$15

-$10

-$5

$0

$5

$10

$15

$30 $35 $40 $45 $50 $55 $60

Put

Forward

Call

TTT

TrTT

TrTTT

fPCeKSf

eKSPC*

*

Long call = Long put + long forwardLong forward = Long call + short put

TTT

TTT

PCffPC

TTT

TTT

CPffCP

Long put = Long call + short forwardShort forward = Long put + short call

-$15

-$10

-$5

$0

$5

$10

$15

$30 $35 $40 $45 $50 $55 $60

Call

Forward

Put

TTT fCP TTT fPC

Basic Options – Profit at Expiry 14

-$6

-$4

-$2

$0

$2

$4

$6

$8

$10

$30 $35 $40 $45 $50 $55 $60

-$4

-$2

$0

$2

$4

$6

$8

$10

$30 $35 $40 $45 $50 $55 $60

-$20

-$15

-$10

-$5

$0

$5

$10

$30 $35 $40 $45 $50 $55 $60

-$15

-$10

-$5

$0

$5

$10

$30 $35 $40 $45 $50 $55 $60

Long put

PT = max(K – ST , 0)-P0

Long call

CT = max(ST-K , 0)-C0

Short call

CT = min(K-ST , 0)+C0

Short put

PT = min(ST –K , 0)+P0

Option Market Participants

Hedgers Reduce or eliminate a natural

exposure to financial risk

Speculators Take on risk in pursuit of profit

Arbitrageurs Risk free profits from mispricing

between asset and derivatives priced relative to the asset

Market Maker Profit from derivative’s bid – ask

spread plus commission Likely will “lay-off” or hedge

positions created in their ‘book’

15

Speculate - needs work

Buy a put Speculate on declining asset

value Sell a put

Believe asset will not fall below K, so receive premium

If the buyer exercises the put (S<K) the put seller must buy the asset for K

Seller may have a strategy to buy asset at K anyway but reduce cost by via the premium

Buy a call Speculate on increasing asset

value Sell a call

Believe asset will not rise above K, so collect a call premium

If the buyer exercises the call (S>K) the call seller must sell the asset at K If the call seller owns the

asset, it is covered call Seller may have a strategy to sell

at K anyway and enhance return with a call premium

04/10/2023

16

Topic 16 UNDER CONSTRUCTION

Hedge needs work

Buy a put Protect current value of a long

position in an asset Protective Put

Buy a call Protect current value of a short

position in an asset

Protective Put 18

-$15

-$10

-$5

$0

$5

$10

$15

$85 $90 $95 $100 $105 $110 $115 $120

Profi

t

ST

Opt 1

Fwd

Total

Strike

Asset Info Option 1 Option 2 Forward r* 4.00% Call or Put P Call or Put Strike, K 107.00$ s 15.00% Strike, K 107.00$ Strike, K Long / Sht L

S0 100.00$ Long / Sht L Long / Sht Num 1T 0.50 Number 1 Number

Premium 7.203$ Premium

Covered Call19

-$15

-$10

-$5

$0

$5

$10

$15

$85 $90 $95 $100 $105 $110 $115 $120

Profi

t

ST

Opt 1

Fwd

Total

Strike

Asset Info Option 1 Option 2 Forward r* 4.00% Call or Put C Call or Put Strike, K 107.00$ s 15.00% Strike, K 107.00$ Strike, K Long / Sht L

S0 100.00$ Long / Sht S Long / Sht Num 1T 0.50 Number 1 Number

Premium 2.322$ Premium

Speculate on Volatility 20

-$15

-$10

-$5

$0

$5

$10

$15

$85 $90 $95 $100 $105 $110 $115 $120

Profi

t

ST

Opt 1

Opt 2

Total

Strike

Asset Info Option 1 Option 2 Forward r* 4.00% Call or Put C Call or Put P Strike, Ks 15.00% Strike, K 107.00$ Strike, K 107.00$ Long / Sht

S0 100.00$ Long / Sht L Long / Sht L Num T 0.50 Number 1 Number 1

Premium 2.322$ Premium 7.203$

Bull Spread – Madoff’s Core ‘Strategy’ 21

-$15

-$10

-$5

$0

$5

$10

$15

$75 $80 $85 $90 $95 $100 $105 $110

Profi

t

ST

Opt 1

Opt 2

Fwd

Total

Strike

Asset Info Option 1 Option 2 Forward r* 4.00% Call or Put C Call or Put P Strike, K 91.00$ s 15.00% Strike, K 95.00$ Strike, K 87.00$ Long / Sht L

S0 91.00$ Long / Sht S Long / Sht L Num 1T 0.25 Number 1 Number 1

Premium 1.504$ Premium 0.897$

Arbitrage & Market Making needs work

Arbitrage Example

Market Maker Buy and sell options May also buy and sell underlying

and forwards, futures to hedge exposure from market making

See big hedge eqn

22

Black-Scholes-Merton Equation 23

SΔV1Π

dSΔdVdΠ

dwσSdtμSdS

dwSσSV

dtSσSV

21

tV

SμSV

dV

*

222

2*

Portfolio, P, is defined as a short position in a derivative, V, and a long position of D shares of equity S (following GBM). The goal is to define the derivative and its price, V.

Use differential form and substitute dV and dS from Topic 14

dwSσΔdtμSΔdwSσSV

dtSσSV

21

SμSV

tV

dΠ *222

2*

Black-Scholes-Merton Equation 24

dtSσSV

21

tV

dΠ 222

2

The differential portfolio value, dP, no longer contains randomness, dw, nor the expected return on the equity , m *

This result is quite remarkable, however actually due to the definition of D above. The price of the derivative, V, will be defined such that the portfolio profit over dt equal to the risk free rate of return.

The number of equity shares, D , is now defined as then substitute into last equation S

VΔ

The two dw terms and the two m * term cancel each other

dwSσSV

dtμSSV

dwSσSV

dtSσSV

21

μSSV

tV

dΠ *222

2*

dtrΠdΠ *

Black-Scholes-Merton Equation 25

Substitute

dtrΠdtSσSV

21

tV

dΠ *222

2

Now set the two expressions for dP equal

SSV

V Π

**222

2

*222

2

*222

2

rSSV

rVSσSV

21

tV

dtrSSV

VdtSσSV

21

tV

dtrΠdtSσSV

21

tV

26

Black-Scholes-Merton Equation

“Underlying” asset price, S, is modeled with GBM

Derivative price is defined by a partial differential equation

Each derivative type has initial and boundary conditions Derivative defined to fully hedge portfolio, P, over time dt

This is ‘delta hedging’ where the portfolio is hedged i.e., portfolio returns the risk free rate relative to changes in the price of the underlying equity

The equation is deterministic – no random term dw or z·√dt And no expected return rate on the asset, m*

Derivatives are priced in a ‘risk neutral’ world Expected risk and return are priced into the equity

r* is the continuously compounded risk free rate of return and remains constant The derivative is priced relative to the equity

Derivative exercise is “European” (only at expiry) The equity pays no dividends during the option's life Commissions, spreads, and taxes are ignored

VrrSSV

SσSV

21

tV **22

2

2

-$15

-$10

-$5

$0

$5

$10

$15

$30 $35 $40 $45 $50 $55 $60

Put

Forward

Call

-$15

-$10

-$5

$0

$5

$10

$15

$30 $35 $40 $45 $50 $55 $60

Call

Forward

Put

Put – Call Parity and Forwards before Expiry 27

ttt

trtt

trttt

fPCeKSf

eKSPC*

*

Long call = Long put + long forwardLong forward = Long call + short put

ttt

ttt

PCffPC

ttt

ttt

CPffCP

Long put = Long call + short forwardShort forward = Long put + short call

ttt fCP ttt fPC

Claims on Capital via Put – Call Parity28

K is the book value of debt at T

ET = max(ST – K, 0)

The equity holders have a payoff equivalent to a call with strike price K on the value of the firm with

Dt = K·e-r*·T – max (K-ST , 0)

The debt holders have a payoff equivalent to the value of the firm with a short put with strike price K

TTT

TTr

TT

DEVPeKCS

*

-$45

-$30

-$15

$0

$15

$30

$45

$60

$0 $10 $20 $30 $40 $50 $60 $70

K

Put

Debt

-$45

-$30

-$15

$0

$15

$30

$45

$60

$0 $10 $20 $30 $40 $50 $60 $70

Debt

Equity

Value

Option Value Components 29

In the moneyOut of the money

Intrinsic Value

Time Value

At expiry

Prior to expiry

K

St

Value of a forward with contract price K

0

0.01

0.02

0.03

0.04

0.05

$10 $20 $30 $40 $50 $60 $70 $80 $90

Call Option Price30

)d(N~K)d(N~eS

)d(N~K)d(N~SE

K)d(N~)d(N~

SE)d(N~

0)d(N~KKS|SE)d(N~

0KSPr

KS|KEKS|SEKSPr

0,KSmaxECE

21Tr

0

21T

2

1T2

2TT2

T

TTTT

TT

*

yprobabilit neutral riskrP

nexpectatio neutral riskE

2

T

dN~

KSPr

2

T

d-N~ KSPr

KST KST

K

Call Option Price31

Tσ

Tσ.5rKS

lnd

Tσ

Tσ.5rKS

lnd

2*0

2

2*0

1

r* is the expected risk-free rate of return (continuously compounded)

)d(N~Ke)d(N~S

)d(N~K)d(N~eSeC

2Tr

10

21Tr

0Tr

0*

**

This formula is also the solution to the B-S PDE for the European call option initial and boundary conditions

Call Option Price 32

If ST > K, then CT = KIf ST ≤ K, then CT = 0

776.14$34867.45$e

.38892-N~45$e

dN~KeC

06.

106.

2Tr

0

*

If ST > K, then CT = ST

If ST ≤ K, then CT = 0

003.17$42509.40$ .18892-N~40$ dN~SC 100

A European call option is a portfolio of long an “asset or nothing” call option and short a “cash or nothing” call option

Recall the “asset or nothing” and “cash or nothing” binary call options from Topic 13

In both examples, the information was as follows: S0=$40, K=$45, r*=6%, s=20%, and T=1 year.

23.2$776.14$003.17$)d(N~Ke)d(N~S C 2

Tr100

*

0

0.01

0.02

0.03

0.04

0.05

$10 $20 $30 $40 $50 $60 $70 $80 $90

Comparing the Two Binary Options

cash or nothing call option asset or nothing call option

33

10

Tr

0

1Tr

0

2

1T2

TTTT

dN~S

CEe C

dN~eS

dN~dN~

SEdN~

KS|SEKSPrCE

T*

*

2Tr

0

T

2

TT

dN~KeC

KS|KEK

dN~K

KSPrKCE

*

0

T

dN~KSPr

2

T

dN~KSPr

KST KST

Binary Call Option Value 34

$0

$10

$20

$30

$40

$50

$60

$30 $35 $40 $45 $50 $55 $60

(T-t)=1.0

(T-t)=0.5

(T-t)=0.25

(T-t)=0.0

$0

$10

$20

$30

$40

$50

$60

$30 $35 $40 $45 $50 $55 $60

(T-t)=1.0

(T-t)=0.5

(T-t)=0.25

(T-t)=0.0

cash or nothing call optionasset or nothing call option

European Call Option Value 35

$0

$2

$4

$6

$8

$10

$12

$14

$16

$18

$20

$30 $35 $40 $45 $50 $55 $60

(T-t)=1.0

(T-t)=0.5

(T-t)=0.25

(T-t)=0.0

Option portfolio: long ‘asset or nothing call option’ and ‘short a cash or nothing call option’ – equivalent to a European call option

European Call Option and Forward Value 36

-$20

-$15

-$10

-$5

$0

$5

$10

$15

$20

$30 $35 $40 $45 $50 $55 $60

(T-t)=1.0

(T-t)=0.5

(T-t)=0.25

(T-t)=0.0

(T-t)=0.0

)d(N~Ke)d(N~S C 2Tr

100

*

KeSF Tr0

*

$30 $35 $40 $45 $50 $55 $60

-$20

-$15

-$10

-$5

$0

$5

$10

$15

$20

(T-t)=1.0

(T-t)=0.5

(T-t)=0.25

(T-t)=0.0

(T-t)=0.0

European Put Option and Short Forward Value 37

)d(N~Ke)d(N~S- P 2Tr

100

*

KeSF Tr0

*

Put Option Price38

22

11

102Tr

102Tr

0Tr

2Tr

10

0Tr

00

dN~1dN~dN~1dN~

dN~SdN~eK

dN~1SdN~1eK

SKedN~eKdN~S

SKeCP

*

*

**

*

From put – call parity

This formula is also the solution to the B-S PDE for the European put option initial and boundary conditions

Call & Put Price Example 39

38892.-1.0.2

1.0.04$50$40

lnd

18892.1.0.2

1.0.08$50$40

lnd

2

1

23.2$ 34867.e45$42508.40$

dN~eKdN~S C0.106.

2Tr

100

2Tr

100 dN~eKdN~S C

61.4$ 57492.40$65133.e45$

dN~SdN~eKP0.106.

102Tr

0

61.4$00.40$38.42$23.2$

SKeCP 0Tr

00

Current stock price, S0 = $40.00Expected (continuously compounded) rate of return, m* = 16.00 %Annual volatility, s = 20%

Strike price, K: $45.00Risk free (continuously compounded) rate of return, r*: 6%Time to expiry, T = 1.0 years

Stock Price PDF at T = 1 yr 40

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

$10 $20 $30 $40 $50 $60 $70 $800

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

$10 $20 $30 $40 $50 $60 $70 $80

Option Pricing41

If this variable increases

The call price

The put price

Stock price, S Increases Decreases

Exercise price, K Decreases Increases

Volatility of asset, s Increases Increases

Time to expiry, T-t Increases Either

Risk free interest rate, r

Increases Decreases

Dividend payout Decreases Increases

)d(N~Ke)d(N~S C 2Tr

100

*

)d(N~Ke)d(N~S- P 2Tr

100

*

Tσ

Tσ.5rKS

lnd

Tσ

Tσ.5rKS

lnd

2*0

2

2*0

1

Effect of s on Call Value

04/10/2023Topic 16 UNDER CONSTRUCTION

42

$-

$5

$10

$15

$20

$25

$30

$20 $30 $40 $50 $60 $70 $80

Current Stock Price

Va

lue

of

Ca

ll

0% 12% 24% 36%

Implied Volatility 43

Supporting Derivations 44

Z)(N~KSPr

Tσ

T2

σμ

KS

ln

Tσ

T2

σμ

KS

ln- Z

KS

Tσ

TμSS

lnZ

T

2*0

2*0

T

0

T

d2 supporting derivation

Tσ

T2

σr

KS

ln

Tσ

Tσ2

σr

KS

ln

TσTσ

Tσ

T2

σr

KS

lnd

Tσdd

2*t

22

*t

2

2*t

1

21

d1 supporting derivation

)d(N~KSPr

Tσ

T2

σr

KS

lnd

2T

2*t

2

From Topic 13 and the basic equity price process

For options pricing

Option Price Simulation 45

Current stock price, S0 = $40.00Expected (continuously compounded) rate of return, m * = 16.00 %Annual volatility, s = 20%

Strike price, K: $45.00Risk free (continuously compounded) rate of return, r*: 6%Time to expiry, T = 1.0 years

Ē[ST] = $42.473

Ē[ST |ST > K ] = $51.781

C0 =$2.227P0 = $4.606

From simulation with 10,000 runs

Ē[ST] = $42.435Ē[ST |ST > K ] = $51.788

r=3.922%r*= 5.94%s=20.051%

CT =$2.347P0 = $4.912

C0 =$2.210P0 = $4.626

Option Price Simulation

Do a path dependent option Histograms and plots

46

Delta Hedging: Calls47

SΔC

SSC

C

SSC

C 0

C

dtrSSC

CdΠ

SΔC SΔVΠ

*

C

2')tT(r

1'

t1t

t

2)tT(r

1tt

dN~eKdN~SdN~SC

dN~eKdN~S C

Short DC calls for each share of stock held for a delta neutral portfolio

So what is DC exactly for a European call ?

KtTσedN

K

eSdNStTσ

1dN~

eKS

dNdN

)reference needs( eKS

dN~dN~

StTσdN

Sd

dNdN~

StTσdN

Sd

dNdN~

tTr1

tTrt1

t2

'

tTrt12

tTrt1

'2

'

t

2

t

222

'

t

1

t

111

'

**

*

*

Delta Hedging: Calls48

1

111

tTr1Tr1

1

2'Tr

1'

t1t

t

dN~ Tσ

dNTσ

dNdN~

KtTσedN

eKtTσ

dNdN~

dN~eKdN~SdN~SC

**

*

SdN~C

dN~SC

SΔC

1

1

C

Short calls for each share of stock held for a delta neutral portfolio

1C dN~SC

Δ

Delta Hedging: Puts 49

Complete similar to calls

SΔP

SSP

P

SSP

P 0

P

dtrSSP

PdΠ

SΔPΠ

*

P

Short DC calls for each share of stock held for a delta neutral portfolio

2')tT(r

1'

t1t

t

2)tT(r

1tt

dN~eKdN~SdN~SP

dN~eKdN~S P

Delta For Call50

-$6

-$3

$0

$3

$6

$9

$36 $38 $40 $42 $44 $46 $48 $50 $52

(T-t)=0.25

Slope

(T-t)=0.0

Delta For Put 51

-$6

-$3

$0

$3

$6

$9

$36 $38 $40 $42 $44 $46 $48 $50 $52

(T-t)=0.25

(T-t)=0.0

Slope

Strike

European Equity w Dividend Options 52

)d(N~Ke)d(N~eS C 2tTr

1tTd

tt

**

t-Tσ

t-Tσ.5drKS

lnd

t-Tσ

t-Tσ.5drKS

lnd

2**0

2

2**0

1

Dividends are assumed continuously paid at rate q*

)d(N~Ke)d(N~eS- P 2tTr

1tTd

tt

**

Discrete dividends are assumed

)d(N~Ke)d(N~edPVS C 2tTr

1

nd

ii

tΔritt

*i

*

)d(N~Ke)d(N~edPVS- P 2tTr

1

nd

ii

tΔritt

*i

*

t-Tσ

t-Tσ.5rKS

lnd

t-Tσ

t-Tσ.5rKS

lnd

2*0

2

2*0

1

European Options: FX 53

)d(N~Ke)d(N~eS C 2tTr

1tTc

tt

**

Tσ

Tσ.5crKS

lnd

Tσ

Tσ.5crKS

lnd

2**t

2

2**t

1

Options on a foreign exchange rate expressed as Commodity currency / terms currency Risk free interest rate on the commodity currency c*

Risk free interest rate on the terms currency r*

Rates are continuously compounded expected annual rates

)d(N~Ke)d(N~eS- P 2tTr

2tTc

tt

**

Other Options

Commodity FX

54

Then the domestic currency value of a call option into the foreign currency is

The value of a put option has value

where : S0 is the current spot rate

K is the strike priceN is the cumulative normal distribution functionrd is domestic risk free simple interest rate

rf is foreign risk free simple interest rate

T is the time to maturity (calculated according to the appropriate day count convention)and σ is the volatility of the FX rate.

American Equity Options 55

Put – Call Parity

04/10/2023

56

For European options

K=$20, T=5/12, S=$19, r=10%, C=$1.50

What is P?

P = C - S + Ke-rt = 1.50 – 19 + 20e-.1*5/12

= $1.68

Now if American options, what is P?

S - K C - P S - Ke-rt

19 - 20 1.50 - P 19 - 20e-rt

-1 1.50 - P 19 – 19.184

-1 1.50 - P -.184

-1 1.50 - P -.184

2.50 P 1.68

Option Variants

Warrants Incentive Stock Options

57

Essential Points 58

Appendix: Probability and Expectation Summary59

2

1TTT

2

T

2

1TTT

2

T

dN~dN~

SEKS|SE

dN~KSPr

zN~zN~

SEKS|SE

zN~KSPr

2

T

2

0

T

d-N~KSPr

z-N~zN~

KSPr

Essential Concepts 60