1 16-Option Valuation. 2 Pricing Options Simple example of no arbitrage pricing: Stock with known price: S 0 =$3 Consider a derivative contract on S:

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16-Option Valuation

2

Pricing Options

Simple example of no arbitrage pricing: Stock with known price: S0=$3 Consider a derivative contract on S:

Payoff is 2*ST, where ST is the value of the stock at time T Assume stock pays no dividends What is price of derivative contract?

Answer is simple to find since payoff is a linear function of the payoff on A. Can “replicate” derivative by buying 2 shares of A

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Binomial Option Pricing:Call Option on Dell

The current price is S0 = $60. After six months, the stock price will either grow to

$66 or fall to $54. Pick what ever probabilities you want.

The annual risk-free interest rate is 1%. Assume yield curve is flat

What is the value of a call option with a strike price of $65 that expires after 6 months?

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Stock Price Tree Option Price Tree

60

66

54

?

1

0

Find value of a corresponding call option with X=65:

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Claim: we can use the stock along with a risk-free bond to replicate the option

Replicating portfolio: Position of shares of the stock

If is positive, that means you “own” the stock If is negative, that means you are “short” the stock

Position of $B in bonds (B=present value) If B is positive, that means you “own” the bond If B is negative, that means you are “short” the bond

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Strategy: If we know that holding shares of stock and $B in bonds will replicate the payoffs of the option, then we know the cost of the option is S0 + B

Example: Suppose the stock is currently $60, and we find that holding 1 share of stock and shorting $55 in bonds will give us the exact same payoffs as the option (in either state).

Then we know the price of the option is ________.60-55 = 5

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We want to find and B such that

and 54 are the payoffs from holding shares of the stock B(1.01)1/2 is the payoff from holding $B of the bond

Mathematically possible Two equations and two unknowns

1/ 2

1/ 2

66 (1.01) 1

54 (1.01) 0

B

B

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Shortcut to finding :

Subscripts: H – the state in which the stock price is high L – the state in which the stock price is low

1/ 2

1/ 2

66 (1.01) 1

54 (1.01) 0

B

B

1 0 1

66 54 12H L

H L

C C

S S

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Once we know , it is easy to find B

So if we buy 1/12 shares of stock Short $4.48 of the bond Then we have a portfolio that replicates the option

1/ 2

1/ 2

154 (1.01) 0

12

1 544.48

12 (1.01)

B

B

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Binomial Option Pricing:Call Option on Dell Do we know how to price the replicating portfolio?

Yes:

We know the price of the stock is $60 1/12 shares of the stock will cost $5

When we short $4.49 of the bond we get $4.48

Total cost of replicating portfolio is 5.00 - 4.48 = 0.52

This is the price of the option.

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Binomial Option Pricing:Put Option On Dell

Stock Price Tree Option Price Tree

60

66

54

?

0

11

Find value of a corresponding put option with X=65:

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Binomial Option Pricing:Put Option on Dell

We want to find and B such that

and 54 are the payoffs from holding shares of the stock B(1.01)1/2 is the payoff from holding B shares of the bond

Mathematically possible Two equations and two unknowns

1/ 2

1/ 2

66 (1.01) 0

54 (1.01) 11

B

B

13


Shortcut to finding :

Subscripts: H – the state in which the stock price is high L – the state in which the stock price is low

1/ 2

1/ 2

66 (1.01) 0

54 (1.01) 11

B

B

0 11 11

66 54 12H L

H L

P P

S S

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Once we know , it is easy to find B

So if we short 11/12 shares of stock buy $60.20 of the bond Then we have a portfolio that replicates the option

1/ 2

1/ 2

1166 (1.01) 0

12

11 6660.20

12 (1.01)

B

B

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Do we know how to price the replicating portfolio? Yes:

The price of the stock is $60 When we short 11/12 shares of the stock we

will get $55.00

To buy $60.20 of the bond This will cost $60.20

Total cost of replicating portfolio is 60.20 - 55.00 = 5.20

This is the price of the option.

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Binomial Option Pricing:Call Option on Dell You can assume can be a fraction – that is you can buy a

fraction of a share of stock.

This assumption does not make the answers “unrealistic”

Suppose you could replicate the payoff of one option by buying 4.25 shares of the stock and shorting $25 in bonds.

Then 17 shares of stock and shorting $100 in bonds would replicate the payoff of four options.

The price of four options would be _______________________________________________

Or rather, the price of one option would be _______________________________________________

the price of 17 shares of stock minus $100.

the price of 4.25 shares of stock less $25.

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Insights on Option Pricing

The value of a derivative Does not depend on the investor’s risk-

preferences. Does not depend on the investor’s assessments

of the probability of low and high returns. To value any derivative, just find a replicating

portfolio. The procedures outlined above apply to any

derivative with any payoff function

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Multi-Period Binomial Model

A shortcoming of the binomial model is that future stock prices can only take two possible values and that stock prices change only once during the period.

We can generalize our binomial model by cutting time into smaller pieces and modeling what prices can do over those sub-periods.

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Two-period Binomial Modelfor Stock Prices

70

56

84

70

77

63

Time=0 Time=3 mo Time=6 mo

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Price a call option with strike of 80

Suppose the stock price in 3 months is $77.

Use method above to price call option at this point = B= Price of call is

$77

$84

$70

0.2857

-19.95 [Solve 0.2857(70) + B(1.01)1/4 = 0] for B

0.2857(77) – 19.95= 2.05

Stock Call

4

0

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Suppose the stock price in 3 months is $63.

What is the value of the call option struck at 80?

$63

$70

$56

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In 3 months the call value will be either 2.05 (if the stock price is at $77) 0 (if the stock price is at $63)

If you buy the call option now and were to “sell it” in three months, what would be payoff? 2.05 (if the stock price is at $77) 0 (if the stock price is at $63)

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The payoff trees over the next three months:

Use method above to price call option at this point = B= Price of call is

Stock Option

70

77

63

2.05

0.00

?

2.05/14= 0.1464

-9.20 [Solve 0.1464(63) + B(1.01)1/4 = 0] for B

(0.1464)70-9.20 =1.048

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To model stock prices, pick up and down movements to match expected return (e.g. from the CAPM) estimated volatility

Involves solving two equations and two unknowns. Reference: Hull “Options, Futures, and Other

Derivatives” page 213-214

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What Did Black-Scholes Do?

Showed how to replicate an option assuming the stock price has a “log-normal” distribution, not just two possible outcomes.

Actually, the simple binomial approach was developed after Black and Scholes solved the more complex problem.

If we cut time into infinitely small pieces, the binomial model converges to the Black-Scholes solution.

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Delta Hedging Delta Hedging: the practice of replicating an option by using just

the stock and the bond.

When would you do it? When the option doesn’t exist You want to “tailor” the risk of an option position.

How do you do it? Figure out the delta (position in stock) and B (the position in

bonds). Delta and B will change as other factors change, such as the

stock price and time to maturity. Requires heavy portfolio rebalancing.

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Black-Scholes Formula

The Black-Scholes Formula for European options on stocks paying no dividends is:

where

1 2

hTc S N d e X N d

2

1

ln 2S X T hd

T

Tdd 12

S = current stock priceX = strike priceh = “continuously compounded” risk-free rateT = time until option expires = standard deviation of stock return (not price)

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is a “continuous time” discounting factor In discrete time the discounting factor would be

Example: suppose the 1-year risk-free rate is 10% and the continuously compounded rate is 9.53%

What is the PV of $100 received 1-year from now?

100/1.10 =90.91

100e-.0953=90.91

1(1 )Tr

hTe

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N(d) is the cumulative distribution function for a standard normal random variable - use Excel normsdist function (not normdist).

0

0.2

0.4

0.6

0.8

1

-4 -3 -2 -1 0 1 2 3 4

Cum

ulat

ive

Dis

trib

utio

n F

unct

ion

d

N(d)

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The price of a European put is given by:

1 2

rTp S N d e X N d

Documents

1 16-Option Valuation. 2 Pricing Options Simple example of no arbitrage pricing: Stock with known price: S 0 =$3 Consider a derivative contract on S: