Question

Section B)

A bank has written a call option on one stock and a put option on another stock. For the first option the stock price is 50, the strike price is 51, the volatility is 28% per annum, and the time to maturity is 9 months. For the second option the stock price is 20, the strike price is 19, and the volatility is 25% per annum, and the time to maturity is 1 year. Neither stock pays a dividend. The risk-free rate is 6% per annum, and the correlation between stock price returns is 0.4.

1. 1) Please derive an approximate linear relationship between the change in the portfolio value and the change in the underlying stocks, and then estimate the 10-day 99% VaR based on this relation.

2. 2) Using C/C++ or Java or Matlab to calculate the 10-day 99% Monte Carlo Simulation based VaR for the portfolio. Set the number of simulation to 5000.

3. 3) What else data is required to calculate the 10-day 99% Historical based VaR for the portfolio?

Answer 1

From DerivaGem Model, Value of the first option position -5.413; Delta of the first option position -0.589 Value of the second option position -1.014; Delta of the second option position -0.284 Linear model to the change in the portfolio value to proportional change (X1) in the first stock price and the proportional change (X2) in the second stock price is given by: Change in Price = (-0.589 x $50)X1 + (-0.284 x $20) X2 Change in Price -$29.45X1-$5.68X2 Option-1 Option-2 Annual volatility percentage #Active working days in a year 28% 25% 252 252 [28%)SQRT(252); 25%/SQRT(252); SQRT-->Square root] Daily volatility percentage 0.0176 0.0157 Using linear equation framed in (1) and daily volatility percentage calculated, One-day variance of change in price shall be: One-day variance of change in price (29.457 x0.0176)+(5.68 x0.0157)-(2x29.45 x0.176x5.68x0.0157 x0.4) where 0.4=correlation between stock price returns.

One-day variance of change in price (867.30 x0.00031)(32.26 x 0.00025)-0.03698 One-day variance of change in price 0.26866+0.00795-0.03698 ay variance of change in price0.276610.03698 One-day variance of change in price 0.2396 One-day standard deviation of change in price One-day 99% VaR value from the table # days for which 99% VaR is calculated Therefore, 10-day 99% VaR [Square root of variance SQRT(0.2396) 0.4895] 0.4895 2.33 10 .61 [0.4895 x 2.33 x SQRT(10); SORT ->Square root]