# Problem setup

A trader believes that the volatility of a particular stock over the time period from time 0 to T will follow a stochastic process. They represent their views by writing down a model for the evolution of the stock price and its volatility in terms of a correlated Brownian motion ((WP1)t,(WP2)t) with covariance matrix

1 ρ ρ

1. (1)

The trader believes the stock will evolve according to the stochastic differ ential equation:

dSt = µSt dt + σtSt d(WP1)t)

Here σt which the trader assumes satisfies the stochastic differential equation t = a(m(t) − σt)dt + v(t)d(WP2)t

In this formula m(t) and v(t) are two deterministic functions of time which rep resent the trader’s beliefs about likely changes to the volatility. These functions

are

m(t) =

0.19 t < 0.40

0.24 otherwise, v(t) = 0.20

This model we have just described is a P-measure model representing the trader’s beliefs about the actual evolution of stock prices.

When pricing derivatives the trader uses the Q-measure model dSt = rSt dt + σtSt d(WQ1)t

where r is the constant interest rate of a risk-free bank account and t = a(m(t) − σt)dt + v(t)d(WQ2)t

The covariance matrix of ((WQ1)t,(WQ2)t) in the Q-measure is also given by (1). This is called a Q-measure model because it does not represent the trader’s beliefs, instead it is used to determine the price of derivatives with maturity T using the formula:

e−rT EQ(payoff)

Here EQ denotes the expectation in the Q-measure model.

The numerical values of the parameters in these models are shown in Table 1. Throughout this question the units of price are dollars and the units of time are years.

At time 0 the trader has a portfolio consisting of b = 31.29 in a risk-free bank account together with a number of European options on the stock price ST all with maturity T. The quantities they hold of different options are described in Table 2. The initial bank balance is intentionally negative.

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 Parameter Value a  µ  r  ρ  S0  σ0  T 1.00  0.08  0.03  -0.20  120.00  0.19  0.80

Table 1: Numerical values of parameters

 Quantity Option type Strike 2  2  1 Put  Put  Call 120  125  130

Table 2: The initial portfolio excluding the quantity b in the risk-free bank account

Perform the tasks below. The percentage of your mark associated with each task is shown in square brackets.

1. Simulate the volatility σt from time 0 to T in the Q-measure using the Euler scheme and plot a fan diagram of the result which should show the fifth, fiftieth and ninety-fifth percentiles together with a sample path. [20%]
2. Use the Monte Carlo pricing method to estimate the value of the trader’s portfolio at time 0 (which should include the money held in the risk-free account). Do this by using the Euler method with 100 steps to simulate the log of the stock price and you should use 105scenarios to compute the price. Compute a 95% confidence interval for your answer. [20%]
3. Use the same technique to estimate the total delta of the portfolio. [10%]
4. Apply the antithetic sampling method to the computation of the price and discuss whether it improves the accuracy of your Monte Carlo method, [10%]
5. Consider the process

dSt = rSt dt + σ0St d(WQ1)t

Notice that the coefficient σ0 is constant. As a result it is possible to compute the expectation of terms such as (ST − K)+ in the Q-measure

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 ID Price by simple Monte Carlo Width of 95% confidence interval Delta by simple Monte Carlo Optimal value of q

Table 3: Required table of results

analytically. Use this observation to show how you can improve your estimate for the price of the portfolio using the control variate method. Discuss the effectiveness of this technique. [10%]

1. Suppose that trader decides to change their portfolio at time 0 by pur chasing q units of stock. They then hold this portfolio till time T when they receive or pay any option payoffs and sell their stock holding, leaving them with a final wealth wT . Find the value of q that maximises

EP(u(wT ))

where

u(x) = 1 exp

wT |b|

There are no restrictions on whether q is positive or negative and the trader is allowed to have a negative quantity in the risk-free account. Discuss how the answer depends upon the value of µ, making sure you describe the financial relevance of your observations. [10%]

The remaining 20% of your mark will be determined by the overall quality of your assignment. This will include credit for any interesting mathematical, computational or financial observations and interpretations you may make. For example you might want to comment on how you have validated your answers.

Be sure to include sufficient mathematical detail that the reader can under stand what you have done without reading this list of tasks or your code. You may assume that the reader is familiar with the models and notation defined in the Problem Setup section above.

You must finish your essay by completing a table of results as show in Table 3. Make sure to include your ID which has been completed for you.

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