Question

Problem 1: Evaluation of a known integral using various quadratures: In this problem, we are going to compute the price of a European call option with 3 month expiry, strike 12, and implied vol 20, Assume the underlying is 10 now and the interest rate is 4%.

1. Use Black-Scholes formula to compute the price of the call analytically.

2. Calculate the price of the call numerically using the following 3 quadrature methods:

(a) Left Riemann rule

(b) Midpoint rule

(c) Gauss nodes of your choice (say explicitly why you made that choice) with the number of nodes N = 5, 10, 50, 100 and compute the calculation error as a function of N for each of the methods.

3. Estimate the experimental rate of convergence of each method and compare it with the known theoretical estimate.

4. Which method is your favorite and why

Answer 1

1) & 4) I will be explaining you the Black-Scholes model since it gives very good accuracy to value plain vanilla options and requires much less time and computations.

Black Scholes Formula =

The function N(x) is the cumulative probability distribution function for a standardized normal distribution.  The variables c and p are the European call and European put price, is the stock price at time zero, K is the strike price, r is the continuously compounded risk-free rate, is the stock price volatility, and T is the time to maturity of the option.  

Lets value the call option given in the problem:

T = 3 months = 3/12 = 0.25

K = 12

= 20%

=10

r = 4%

Lets calculate d1 and d2 from above formula:

= = -1.67322

= = -1.77322

Now, lets calculate and using excel's Normdist function,

= N(-1.67322) = 0.047142

= N(-1.77322) = 0.038096

Now we can calculate the price of Call option from above formula:

C = 10 * 0.047142 - 12 * * 0.038096 = 0.0188