Financial Simulation - Google and Jp Morgan

Solution 1:

We selected two stocks: GOOG and JPM. Google is a leading player in the tech industry and JP Morgan is one of leading US banks. Hence, they demonstrate fairly high correlation in terms of daily returns for the past three years. JPM is traded on NYSE and GOOG on NASDAQ and both are large cap companies. We select three years of adjusted closing prices for our model.  By definition an exchange option gives the holder a right to purchase one stock when the price of another hits a certain strike price. The formula for simulation using the Black Scholes Model is given by

[pic 1]

The price of the exchange option is derived using the following formula where Wt is the standard Weiner process:

[pic 2]

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We calibrate the model using three year historical data:

After this we can generate correlated random variate by

[pic 5]

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Where Y1, Y2 and Y3 are random variates generated by following the standard normal distribution.

Now we choose the maturity of the exchange option to be 1 year and set q1 as 1, so the q2 as S1/S2 given by:

[pic 7]

We set GOOG as S1 and JPM as S2.

We simulated 10,000 paths and got the price of the exchange option to be $73.3826

99% Confidence Interval is $[70.5004, 76.2648].

Solution 2:

We use the CEV model for the stock evolution of GOOG based on three years of adjusted closing prices.

The price of the CEV model is evolved as:

[pic 8]

The µ and σ can be calibrated using

[pic 9]

[pic 10]

after we choose a specific alpha by observing the distribution of the log-return.

We calibrate the model using three-year data and get

The Compound Option under reference is a call option on a call option, so the payoff can be given by

[pic 11]

Where K1 is the strike price of the compound option and K2 is the strike price of the underlying call option, T1 is the maturity of the compound option and S(T1) is the underlying stock price at time T1 of the underlying call option.

The price of the underlying call option at T1 is derived by simulating stock price after T1 and obtaining the average payoff at T1. Hence, we would need a nested for loop to simulate this, one to determine option price at T1 and the other used to determine the compounded option price.

We get the simulated compounded option price to be $4.0597

99% Confidence Interval:  $[3.5571, 4.5622]

Solution 3:

The control variable is the underlying stock price or underlying option price. It can be shown that the estimator is still unbiased and the variance reduction will be achieved.

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Variance reduction is achieved by

[pic 14]

We construct b as

[pic 15]

It will give

[pic 16]

We set the control variate to be the underlying stock price as in question 1 and the underlying call option price for question 2, we conclude that the variance is reduced in both cases and the 99% confidence interval is $[71.0051, 75.7601] for solution 1 and $[3.8179, 4.3014] for solution 2. The tightness is reduced from 5.76 to 4.76 in solution 1 and from 1.0051 to 0.4835 in solution 2.

We chose S1 in question 1 to be the control variate because we can show from the above derivation that the estimator would be unbiased after subtracting its difference from its mean. Also, they are iid variables to some extent in the sense that they are identically distributed by are not completely independent. This is the very same reason for choosing the call option price as the control variate.        

Appendix:

clear;

clc;

%% Solution 1

S01 = 823.56; %CMP of Google

S02 = 84.4; %CMP of JPM

mug = 0.129938; %Google mu

...