5.1

Option pricing:pre-analytics

Lecture 5

5.2

Notation• c : European call

option price

• p : European put option price

• S0 : Stock price today

• X : Strike price

• T : Life of option

• : Volatility of stock price

• C : American Call option price

• P : American Put option price

• ST :Stock price at time T

• D : Present value of dividends during option’s life

• r : Risk-free rate for maturity T with cont comp

5.3Calls: An Arbitrage

Opportunity?

• Suppose that

c = 3 S0 = 20

T = 1 r = 10%

X = 18 D = 0

• Is there an arbitrage opportunity?

5.4

Lower Bound for European Call Option Prices; No Dividends

c + Xe -rT S0

in fact, at maturity:if S>X, identical resultif S<X, superior result

consider 2 portfolios:a. buy a call and a ZCB worth X at Tb. buy the stock

5.5

Lower Bound for European Call Option Prices; No Dividends

c + Xe -rT S0

c S0 -Xe -rT

c 20 -18*0.9048

c

5.6

arbitrageif c<3.71

TODAY

• sell the stock at 20, and with the proceeds:

• buy the call at 3 and a ZCB maturing in 1 year time for 16,28 today (i.e. 18=X)

• invest the diff=Y=20-3-16,28=0.72 in a ZCB maturing in 1 year

AT MATURITY

• if S<18, you have 18+0.79 (so better off than keeping the stock

• if S>18, you get:

- S-X from the call (=S-18)

- X from the ZCB (=18)

- 0.79 from the residual

• sum up the terms and you will be happier than having kept the stock

5.7Puts: An Arbitrage

Opportunity?

• Suppose that

p = 1 S0 = 37 T = 0.5 r =5% X = 40 D = 0

• Is there an arbitrage opportunity?

5.8Lower Bound for European Put Prices; No Dividends

p + S0 Xe -rT

p Xe -rT - S0

consider 2 portfolios:a. buy a put and a stock b. buy the ZCB worth X at maturity

5.9

Put-Call Parity; No Dividends

• Consider the following 2 portfolios:

- Portfolio A: European call on a stock + PV of the strike price in cash

- Portfolio B: European put on the stock + the stock

• Both are worth MAX(ST , X ) at the maturity of the options

• They must therefore be worth the same today

- This means that

c + Xe -rT = p + S0

5.10

Arbitrage Opportunities

• Suppose that

c = 3 S0 = 31

T = 0.25 r = 10%

X =30 D = 0

• What are the arbitrage possibilities when

p = 2.25 ? p = 1 ?

5.11arbitrage

• c + Xe -rT = p + S0

3+30*0.97531=p+31

3+29.2593=p+31

p=1.2593

• if p=1, then

• TODAY:- sell the call at 3 and buy the put at 1. You have 2 euro remaining.

- buy the stock at 31, by borrowing 30.23 and using 0.77 from the options proceeds

- invest 1.23 (out of the original 2) in a ZCB

• in 3 months time:- if S=33, for example, put=0, call=3. So you are at -3.

- sell the stock at 33 and repay your debt. So you are at +2

- so you are at -1, but in reality you still have 1.23*exp(0.25*10%)

5.12The Impact of Dividends on Lower Bounds to Option Prices

rTXeDSc 0

0SXeDp rT

5.13Effect of Variables on Option

Pricing

c p C PVariable

S0

XTrD

+ + –+

+ + + ++ + + ++ – + –

–– – +

– + – +

5.14

American vs European Options

An American option is worth at least as much as the corresponding European option

C cP p

5.15

The Black-ScholesModel

5.16

The Stock Price Assumption

• Consider a stock whose price is S

• In a short period of time of length t the change in then stock price is assumed to be normal with mean Sdt and standard deviation

is expected return and is volatility

S t

5.17

The Lognormal Distribution

E S S e

S S e e

TT

TT T

( )

( ) ( )

0

02 2 2

1

var

5.18

The Concepts Underlying Black-Scholes

• The option price & the stock price depend on the same underlying source of uncertainty

• We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty

• The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate

• This leads to the Black-Scholes differential equation

5.19

1 of 3: The Derivation of theBlack-Scholes Differential Equation

shares :ƒ

+

derivative :1

of consisting portfolio a upset e W

ƒƒ

½ƒƒ

ƒ

)(ƒ

½ƒƒ

ƒ

),(ƒ

222

2

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2

S

zSS

tSSt

SS

toSS

tt

SS

tSf

zStSS

5.20

The value of the portfolio is given by

ƒƒ

The change in its value in time is given by

ƒƒ

SS

t

SS

2 of 3: The Derivation of theBlack-Scholes Differential Equation

• there are no stochastic terms inside

5.21

3 of 3: The Derivation of theBlack-Scholes Differential Equation

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½ƒƒ

:equation aldifferenti Scholes-Black

get the toequations in these and ƒfor substitute We

Hence rate. free-risk thebemust portfolio on thereturn The

2

222 r

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5.22

Risk-Neutral Valuation

• The variable does not appear in the Black-Scholes equation

• The equation is independent of all variables affected by risk preference

• The solution to the differential equation is therefore the same in a risk-free world as it is in the real world

• This leads to the principle of risk-neutral valuation

5.23

Applying Risk-Neutral Valuation

1. Assume that the expected return from the stock price is the risk-free rate

2. Calculate the expected payoff from the option

3. Discount at the risk-free rate

5.24

The Black-Scholes Formulas

c S N d X e N d

p X e N d S N d

dS X r T

T

dS X r T

Td T

rT

rT

0 1 2

2 0 1

10

20

1

2 2

2 2

where

( ) ( )

( ) ( )

ln( / ) ( / )

ln( / ) ( / )

5.25

Implied Volatility

• The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price

• The is a one-to-one correspondence between prices and implied volatilities

• Traders and brokers often quote implied volatilities rather than dollar prices

5.26

Dividends

• European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into Black-Scholes

• Only dividends with ex-dividend dates during life of option should be included

• The “dividend” should be the expected reduction in the stock price expected

5.27European Options on StocksPaying Continuous Dividends

continued

We can value European options by reducing the stock price to S0e–q T and then behaving as though there is no dividend

5.28

Formulas for European Options

T

TqrXSd

T

TqrXSd

dNeSdNXep

dNXedNeScqTrT

rTqT

)2/2()/ln(

)2/2()/ln( where

)()(

)()(

02

01

102

210

5.29

The Foreign Interest Rate

• We denote the foreign interest rate by rf

• When a European company buys one unit of the foreign currency it has an investment of S0 euro

• The return from investing at the foreign rate is rf S0 euro

• This shows that the foreign currency provides a “dividend yield” at rate rf

5.30

Valuing European Currency Options

• A foreign currency is an asset that provides a continuous “dividend yield” equal to rf

• We can use the formula for an option on a stock paying a continuous dividend yield :

Set S0 = current exchange rate

Set q = rƒ

5.31Formulas for European Currency Options

c S e N d Xe N d

p Xe N d S e N d

dS X r r

fT

T

dS X r r

fT

T

r T rT

rT r T

f

f

0 1 2

2 0 1

1

0

2

0

2 2

2 2

( ) ( )

( ) ( )

ln( / ) ( / )

ln( / ) ( / )

where