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日期：2023-01-30 09:23

Individual Question for Student with Keats ID

21201322

Problem Setup

A trader knows that the price of a stock, St, will evolve according to the stochastic differential equation:

dSt = µSt dt + σ(t)St dWt.

Here µ is a constant, σt is given by the deterministic function

σ(t) = (

0.14 + 0.14 ×

t

0.60 t < 0.60

0.28 otherwise

,

and Wt is a Brownian motion.

A trader manages a portfolio of options, the stock and a risk free bankaccount and rebalances the portfolio at times t ∈ T := {0, δt, 2δt, . . . , Nδt = T}.

They ensure that at each time t ∈ T \ {T} they are holding q1 + q2∆t units of

the stock and q3 units of an option with payoff

max{−ST + 181, 0} + max{−ST + 104, 0}

and maturity T. Here, ∆t is the delta of the option.

They start with an initial bank balance b0 and withdraw the money needed

to make any purchases from this account and place the proceeds of any sales

into the account. The bank account is risk free and grows with continuously

compounded constant interest rate r. At time T they liquidate their stock and

options holdings and place all the proceeds in the bank balance, to obtain bT .

The quantities q1, q2 and q3 are constant real numbers.

You should assume that the price of the option in this model can be computed

using the formula for the equivalent option in the Black–Scholes model but

substituting the value σt,T in place of the volatility in the Black-Scholes formula

where

σt,s := s

1

s − t

Z s

t

σ(u)

2 du.

As well as only being able to follow the strategy described above the trader

must pay transaction costs. These are given by a charge equal |q|θSt for buying

or selling q units of the stock at time t.

In this exercise you should estimate the values q1, q2 and q3 which maximize

the trader’s expected utility assuming that their utility function is given by

u(bT ) = − exp(−λbT ) and where bT is their final bank balance. When performing numerical optimization it is important to choose a sensibly scaled objective

function, for this reason it is recommended that you optimize the equivalent

quantity

φ = −

1

b0λ

log(−E(u(bT )).

1

Your answer should take the form of an essay and should cover the following

points:

1. How St can be approximately simulated using the Euler–Maruyama scheme

for St.

2. How St can be exactly simulated using the alternative scheme described

below. You should use this exact method to generate M simulations of

S

α

t where the index 1 ≤ α ≤ M denotes the scenario.

3. You should generate a plot which shows how accurately the mean of ST

can be estimated using the Euler-Maruyama scheme for different step sizes

δt. You should use M simulations of ST to estimate the mean.

4. You should discuss the financial interpretation of the trading strategies

when all but one of the qi

’s are equal to zero.

5. You should provide all the difference equations needed to compute b

α

T

, the

final bank balance in scenario α.

6. You must give a clear mathematical statement of the optimization problem

you have solved, including details of any approximations you have used.

7. You should discuss the accuracy of any approximations you have used.

8. When θ = 0, how do the optimal quantities depend upon the value of δt?

Interpret your results.

9. When θ = 0, how do the optimal quantities depend upon the value of λ?

Interpret your results.

10. You must START your essay by including a completed version of the table

of results shown in 2 where the quantity φ(q1, q2, q3) denotes the value of

φ obtained for a given set of quantities. The parameter values you should

use are given in 1. The prices quoted for the options will not correspond

to the model the trader is using, you can assume that the trader has some

inside information.

Exact simulation method

Suppose that for a fixed grid size δt, z0 is given and that for t ∈ T zt+δt =

(µ −

1

2

σ

2

t,t+δt)δt + σt,t+δt

√

δtt where the t are independent, identically distributed normal random variables and a is a constant. You should show that

zt is normally distributed with a mean and variance that do not depend upon

δt. You may then assume that if we simulate zt in this way it gives an exact

simulation for the stochastic differential equation

dzt = (µ −

1

2

σ

2

t

) dt + σt dWt.

You should be able to use this fact in order to devise a method to simulate St

exactly.

2

Parameter Value

b0 129.00

S0 140.00

µ 0.14

r 0.04

T 1.40

θ 0.01

N 32.00

M 100000.00

λ 0.01

Option bid price at time 0 41.36

Option ask price at time 0 42.17

Table 1: Parameter values

Keats ID 21201322

φ(0, 0, 0)

φ(1, 0, 0)

φ(0, 1, 0)

φ(0, 0, 1)

Optimal φ

Table 2: Required table of results

3