2A simple example A stock is currently priced at $40 per share. In 1 month, the stock price maygo up by 25%, orgo down by 12.5%.
3A simple example Stock price dynamics: t = now t = now + 1 month up state$40x(1+.25) = $50$40$40x(1-.125) = $35down state
4Call option A call option on this stock has a strike price of $45 t=0 Stock Price=$50;Call Value=$5Stock Price=$40;Call Value=$cStock Price=$35;Call Value=$0
5A replicating portfolio Consider a portfolio containing D shares of the stock and $B invested in risk-free bonds.The present value (price) of this portfolio is DS + B = $40 D + B
6Portfolio value t=0 t=1 up state down state $50 D + (1+r/12)B $40 D + Bdown state
7A replicating portfolio This portfolio will replicate the option if we can find a D and a B such thatUp state$50 D + (1+r/12) B = $5andDown state$35 D + (1+r/12) B = $0Portfolio payoff=Option payoff
8The replicating portfolio Solution:D = 1/3B = -35/(3(1+r/12)).Eg, if r = 5%, then the portfolio contains1/3 share of stock (current value $40/3 = $13.33)partially financed by borrowing $35/(3x ) = $11.62
10The replicating portfolio Since the the replicating portfolio has the same payoff in all states as the call, the two must also have the same price.The present value (price) of the replicating portfolio is $ $11.62 = $1.71.Therefore, c = $1.71
12An observation about DAs the time interval shrinks toward zero, delta becomes the derivative.
13Put option What about a put option with a strike price of $45 t=0 t=1 Stock Price=$50;Put Value=$0Stock Price=$40;Put Value=$pStock Price=$35;Put Value=$10
14A replicating portfolio up state$50 D + (1+r/12)B$35 D + (1+r/12)B$40 D + Bdown state
15A replicating portfolio This portfolio will replicate the put if we can find a D and a B such thatUp state$50 D + (1+r/12) B = $0andDown state$35 D + (1+r/12) B = $10Portfolio payoff=Option payoff
16The replicating portfolio Solution:D = -2/3B = 100/(3(1+r/12)).Eg, if r = 5%, then the portfolio containsshort 2/3 share of stock (current value $40x2/3 = $26.66)lending $100/(3x ) = $33.19.
17Two PeriodsSuppose two price changes are possible during the life of the optionAt each change point, the stock may go up by Ru% or down by Rd%
18Two-Period Stock Price Dynamics For example, suppose that in each of two periods, a stocks price may rise by 3.25% or fall by 2.5%The stock is currently trading at $47At the end of two periods it may be worth as much as $50.10 or as little as $44.68
23Estimating Ru and RdAccording to Rendleman and Barter you can estimate Ru and Rd from the mean and standard deviation of a stock’s returns
24Estimating Ru and RdIn these formulas, t is the option’s time to expiration (expressed in years) and n is the number of intervals t is carved into
25For ExampleConsider a call option with 4 months to run (t = .333 yrs) andn = 2 (the 2-period version of the binomial model)
26For ExampleIf the stock’s expected annual return is 14% and its volatility is 23%, then
27For ExampleThe price of a call with an exercise price of $105 on a stock priced at $108.25
28Anders ConsultingFocusing on the Nov and Jan options, how do Black-Scholes prices compare with the market prices listed in case Exhibit 2?Hints:The risk-free rate was 7.6% and the expected return on stocks was 14%.Historical Estimates: sIBM = .24 & sPepsico = .38