Question

1. What is the price of an American-style call option assuming a 4% annual risk-free rate, a strike price = $150, and 3 years to maturity.  In each year the price can either rise by a factor of 1.3 or fall by a factor of 0.9.  The current price of the underlying asset is $100 and it pays no dividends.
2. Why is the price in part a different than you would get from inputting a 10% drift and 20% volatility into the Black Scholes equation?
3. What would be the price of an American-style put option on the same stock with the same maturity as in part a above?

Answer 1

a) We value the call using 3-step binomial option pricing with step size t=1

The probability of up-move is given by

Probability of down-move is

where r is the risk-free rate

D is the down-move factor

U is the up-move factor

Substituting, we get the probability of up-move = (e^(0.04*1) - 0.9 )/(1.3-0.9) = 0.3520

Probability of down-move = 0.6480

Current stock price

Strike price

The call value at each node is calculated above the node (in black) by discounting the expected call value in the following nodes by the risk-free rate

Hence, American style call option value = $3.146

b)

Black Scholes model gives the price of the call option as

S is the current stock price

S=$100

where q is the dividend yield = 0

K = Strike price = $150

r = risk-free rate = 0.04

Volatility s = 0.2

T=3

Substituting we get, d1 = (ln(100/150)+((0.04+0.2*0.2/2)*3)/(0.2*(3^0.5)) = -0.65

N(d1)= 0.2575

d2 = -0.65 - (0.2*(3^0.5)) = -0.996

N(d2) = 0.1596

Subsituting N(d1) and N(d2) in call option c formula

c= 100*0.2575 - 150*e^(-0.04*3)*0.1596

c= $4.517

The call option value using Black-Scholes is $4.517

This value is different from the value in part a, as we have used binomial option pricing in part a and we take certain time-period length or steps in a binomial model ( here we have taken 3 time intervals). As the step size in binomial pricing becomes infinite, the option value comes closer to the one calculated through Black-Scholes model.

c.)

Since put-call partiy do not work for American option, we calculate the value of American-put of similar strike and marturity as in Part a

We execise the option early ( since, American option), when the payoff at a node is greater than the discounted expected .payoff in subsequent nodes

Here, the option is exercised early at t=2 node

This is because (0.648*44.7*e^(-0.04) + 0*0 )=27.82 < 33

Similarly [0.648*77.1*e^(-0.04) + 0.3520*44.7*e^(-0.04)] = 63.1 < 69

Also, option is exercised eary at the bottom node at t=1 since

[0.648*69*e^(-0.04) + 0.3520*33*e^(-0.04)] = 54.12 < 60

Hence, the price of American put on the same stock with the same maturity as in part a. above is $44.30