Option PricingBA 543

Aoyang Long

Agenda• Binomial pricing model

• Black—Scholes model

Binomial Option Pricing Model

• Interest rate =8%

• Price0 = (60%*$ 80+40%*$ 55)/(1+8%) = $ 64.81

?

$80

$55

60%

40%

t0 t1

Binomial Option Pricing Model

?

$80

$55

60%

40%

t0 t1 Stock Price = $ 80

Stock Price = $ 55

Call option payoff

$ 10 $ 0

• Interest rate =8%

• Exercise price= $70

• Value of call = (60%*$ 10) / (1+8%) = $ 5.56

Multiple Periodst0 t1 t2 t3 t4

90

80

70

60

50

Price0

60%

40% 60%

40%

60%

40%

60%

40%

60%

40%

60%

40%

60%

40%

60%

40%

60%

40%

60%

40%

How many path for a stock price of $80?

Pascal’s Triangle

Each number in the triangle is the sum of the two directly above it.

Lognormal Distribution

$20 $10 $0 ($10) ($20)0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Multiple Period

Lognormal Distribution

$20 $10 $0 ($10) ($20)0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Multiple Period

Black—Scholes Model• The trick is to set up an option equivalent by combing common stock

investment and borrowing. The net cost of buying the option equivalent must equal the value of the option.

-- Black and Scholes• Assumptions

- European call option only- Underlying assets does not pay dividends until expiration date- Both the interest rate and the variance of the return on the stock are

constant- Stock prices are continuous ( no sudden jump)

Black—Scholes Model

d1=log [P/PV(X)] /σ√t+σ√t2d2=d1-σ√tN(d) = cumulative normal probability function X = exercise pricet = number of periods to exercise dateS = current stock priceσ= standard deviation per period of (continuously compounded) rate of return on stock

Black—Scholes Model• Example• S = 55• X = 55• r = 4% per year• t = 0.5 year = 182.5 days• σ = 40.69%

• Black-Scholes Calculator

Summary• Binomial pricing model:

- discrete model- both European and American call- slow

• Black—Scholes model: - continuous model- European call- quick