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Monte Carlo Methods

When analytical formulae for complex derivative instruments are not available,

Monte Carlo (MC) simulations can be employed as a simple flexible method, which

computes an estimate of an option price by running a large number of simulations and

then computing the average of the discounted pay-offs. The simulations generate a

series of random stock price paths which are then used to create the option price. The

random stock price paths are generated through the GBM. The standard deviation and

standard error of the processes are critical in identifying the accuracy of the model. Control Variates

MC simulations are computationally inefficient when used alone. Control variate

techniques can be used to improve the accuracy of option pricing and efficiency of the

MC process. Two such techniques are:

Antithetic Reduction

This method constructs an asset, which is exactly negatively correlated with an

underlying asset. A portfolio containing two negatively correlated assets will have a

much lower variance than a portfolio containing one asset. This is represented in the

MatLab script by filling alternate rows of the stockPrices array with a one randomstock motion having a positive term and the next with a negative term. This is more

accurate than two separate simulations as the variance reduction removes the

probability spike associated with one option i.e. when one pays off, the other doesnt

and vice versa. It is also more computationally efficient as one simulation is run

instead of two.

Delta variate

This control technique involves calculating the options delta at each step in the

process and using this to populate the delta variate array with the rebalanced hedge

amount. The options value is then derived from subtracting the sum of the delta

variate (hedge amounts) from the normally calculated payoff i.e. max(S t-K, 0). This

reduces the standard deviation of the payoff and therefore increases the accuracy of

the option price.

The dynamic rebalancing portfolio strategy creates a hedged payoff with a standard

deviation much smaller than an unhedged position. Line 27 of our code introduces

erddt which allows us to compute E[St] as the asset evolves at the risk free rate. Beta

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is set at minus one, which is used in line 38 of the code to subtract the sum of the

variates noted from the extent the option is in the money. The payoff of the hedged

portfolio is computed after the end of the time step loop as are the mean and standard

error. Since the mean of the control variate is zero, the mean gives us an estimate of

the option price.

Power options

Power options are a class of exotic option whos pay-off at maturity is related to

the power of the stock price (n=1,2,3...). Normally n would not exceed 3. For a power

option on a stock with a strike price X and a time to maturity of T, the pay-off on a

call option is Max((St ^n- X),0) and put option is Max((X - St ^n ),0). Normally a

power call option would be issued only if it was very deep out of the money, which is

why our at the money example below is priced so high i.e. potential huge pay-off.

Results

Having performed MC simulations (code attached) on a plain MC model, MC model

with antithetic reduction and MC model with delta control variate we noted the

following results for a varying amount of steps:

Monte Carlo (No controls)

Simulations Steps Option Price Std Dev Std Error

10 100 2.47E+10 5.247+010 1.659+E010

100 100 1.03+E10 3.542+E10 3.542+E09

1000 100 1.225+E010 9.111+E010 2.88+E009

Monte Carlo + Delta VariateSimulations Steps Option Price Std Dev Std Error

10 100 9.46+E03 3.64+E03 1.15E+03

100 100 1.08+E04 4.79+E03 478

1000 100 1.06+E04 4.47+E03 141.3

Monte Carlo + Antithetic Reduction

Simulations Steps Option Price Std Dev Std Error10 100 11,080.00 6,205.00 1,962.00

100 100 9,855.00 4,977.00 498

1000 100 9,957.00 5,033.00 159

Monte Carlo+Antithetic Reduction+Delta VariateSimulations Steps Option Price Std Dev Std Error

10 100 NaN NaN NaN

100 100 NaN NaN NaN

1000 100 NaN NaN NaN

It is clear from these results that for an increase in simulations, a corresponding

decrease in standard error is noted. We have standardized the steps in each Monte

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Carlo to 100 for each simulation to highlight the impact of the simulations. As

expected, the introduction of the control variates has substantially decreased the

option price, standard deviation and standard error for reasons outlined above.

When combining both the antithetic and delta control variate, answers received were

NaN for all simulations. We consider this a coding error as our understanding of the

control variates tells us the hybrid answers should fall between the higher delta variate

and lower antithetic variate.

Based on the simulations ran we noted only one instance when the simulation took

longer than zero seconds (MC with delta variate took .000016 seconds). In the real

world, when prices are needed instantly, there may be a trade-off between accuracy

and time taken to run model i.e. the more simulations the more accuracy but time

taken to complete lengthens also. The use of fewer simulations to arrive at a less

accurate price as a guideline may result. This is the case when 100,000+ simulations

and a variety of control variates are in use.