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MA9070 Simulation and Machine Learning
创建日期：2024-08-06 15:36:29
标签： MA9070
Simulation and Machine Learning

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MA9070 Simulation and Machine Learning


1 Methods for Asian options (50% of project credit)

1.1 Overview

An Asian option is an option where the payoff is not determined by the underlying price at

maturity, but by the average underlying price over some preset time interval. Asian options

were originated in Asian markets to prevent option traders from attempting to manipulate the

price of the underlying on the exercise date.

There are a variety of Asian options. We will consider one with the following payoff:

max(

1

N

N

∑

n=1

Sn?K,0),

where Sn are daily closing prices of the underlying and K is the fixed strike price. The option

corresponding to this payoff function is called a Fixed Strike Asian Call Option with Discrete

Arithmetic Average.

For the underlying process we will use the geometric Brownian motion

dSt = rStdt+σ(St , t)StdWt , (1)

allowing for the possibility that the volatility can depend on the current time t and current

value of underlying asset St . We refer to this as the local volatility model.

The aim of Part 1 of the project is to price Asian options by Monte-Carlo simulations,

employing different variance reduction techniques.

1.2 Particulars

Unless otherwise specified, use the following parameters:

The strike price is K = 110.

The interest rate is r = 0.05.

The local volatility is given by the function

σ(S, t) = σ0(1+σ1 cos(2πt))(1+σ2 exp(?S/50)). (2)

where σ0 = 0.2, σ1 = 0.3 and σ2 = 0.5. Time t is in years.

Assume there are 260 (working) days in a year.

Fix the number of sample paths to be N_paths = 1000.

2

1.3 Computational Tasks

Programme the local volatility Eq. (2) in a Python function and then write separate

functions to price an Asain option:

– without variance reduction (naive method);

– antithetic variance reduction.

– control variates (see below).

Use Euler time stepping with a time step of one day. Each function should return

option price and variance.

After you have fully tested your code, compare the different methods you have imple-

mented. For this, fix the time to maturity (expiry) to 3 years, i.e., T = 3. Then price

the option for three values of the spot price S0 = S(t = 0):

S0 < K, S0 = K, S0 > K

You are free to choose sensible values of S0 to give a good assessment of how the

methods are performing under different situations. For each method you have imple-

mented, evaluate the option at three values of S0. From the variances you can obtain

95% confidence intervals for each case.

Write code to plot option price as a function of spot price over the range S0 = 10

to S0 = 180. You only need to plot the option price using the method that gives the

smallest variance.

Using a method of your choice, programme a function to compute the delta for the

Asian option. You only need to implement one method, but ideally it should be a

method with small variance. Write code to plot the delta over the same range of spot

prices as the previous item.

1.4 Report contents

See general discussion of report contents in Sec. 3 and 4. The report should follow the

structure of the computational tasks with the aim to produce a report that leads the reader

clearly through the tasks undertaken. The report should summarize the overall picture of how

the various reduction methods perform and the dependence of option prices and deltas on S0.

A few specific things to consider for this part of the project are:

Your Python code should be commented so that it is clear how you have implemented

each variance reduction technique.

In addition, to make the report understandable independently of the Python code, you

include markdown cells that briefly state which variance reduction methods you have

3

implemented. You do not need to give an analysis of the variance reduction. These can

be short explanations of a few sentences.

Report the results of your runs for different methods and different S0 in a clear

and understandable form. Discuss the benefits and/or disadvantages of the different

methods. Taking into account additional cost of variance reduction computations,

determine which method is the most efficient for this problem.

The plots of option price and corresponding delta should be clear. You can be creative

here and plot prices and deltas for a few values of the time to maturity T to show

the evolution with time to maturity. You could also contrast Asian option prices with

the European counterparts. You should summarize and discuss your plots, possibly

including from a financial perspective.

1.5 Control variates

There are three possible control variates one can consider:

1. ZT ,

2. e?rT max(ZT ?K,0),

3. max

((

∏Nn=0Zn

) 1

N+1 ?K,0

)

,

where Zt is governed by geometry Brownian motion

dZt = rZtdt+σ Zt dWt , (3)

where r and σ are constant.

The volatility in our model varies, but not too much, so one can expect that the discounted

payoff for the Asian option computed along a geometric Brownian path in the local volatility

model will be highly correlated with a corresponding constant-volatility geometric Brownian

path. In practice, one simulates (3) alongside the simulation of (1). From these simulations,

the different control variates depending on Zt are available. A simple choice for σ is σ(S0,0),

(why?). Other choices are possible and might be better.

The first control variate is just the value of ZT at the final time, and hence has a known

expectation (mean), just as was used for European options. The second is the discounted

payoff for a European call option, and hence the expectation is given by the Black Scholes

formula.

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