In: Finance
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
Substituting 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 parity do not work for an American option, we calculate the value of American-put of similar strike and maturity as in Part a
We exercise 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