### given the simulated sample paths from the previous question 1 find prices of europea 3951665

Question 2

Given the simulated sample paths from the previous question, 1. Find prices of European call options for strike prices, X = \$40, \$45, \$50 (Note that the payoff of each sample path is max{4) — X, 0} i = 1,2, … , 100 (6) Then take the discounted average of these. ) 2. Find the continuous time equivalents for these calls, i.e. Black Scholes. Recall that the formula for this is C = SoAr(di) — Xe—riAi(d2) (7)

where

ln(S0/X) + (r + o-2/2)T d1 = ; d2 = dl — a-VT” (8) aff

What do you notice? 3. Given the same simulated sample paths calculate the price of put options for strikes X = \$40, \$45, \$50.

4. Next, use put-call parity, i.e. P = C — So + Xe—rT derive the prices, P, given the simulated call prices from item 1.

Question 3

(9)

We now wish to connect the Binomial model to the continuous time equivalent provided by Black Scholes Merton. To that end, we need to introduce a measure of volatility as measured by the standard deviation, a. To that end, consider the inputs r = 0.045, a = 0.24, T = 1/4 and So = \$45 and strikes X = \$40, \$45, \$50. For n = 1,2, … , 100 set

1. u = exp(a7./,) and d= 1/u. 2. R = (1+ r1n)T 3. q = (R— d)/(u — d) 4. q’ = uq/R 5. a which is the smallest positive integer greater than (ln(So/X) + n ln(d))/ ln(d/u)

(10)

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