option pricing ba 543 aoyang long. agenda binomial pricing model black—scholes model
TRANSCRIPT
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