Stochastic Calculus, 2023-2024 Assignment 2: Monte Carlo pricing of Asian options

Stochastic Calculus, 2023-2024

 

Assignment 2: Monte Carlo pricing of Asian options

 

All computations should be done in Matlab, Python or R. Please submit, via Canvas:

 

(a) a report (as a PDF file), which contains your results and explains in detail how you have

 

obtained them;

 

(b) all Matlab / Python / R files you wrote to accomplish this (a Jupyter notebook is possible,

 

but even then a PDF version of the notebook should also be submitted); the code should

 

contain clarifying comments as much as possible.

 

This is a group assignment; you may work in teams of two or three and submit a common report.

 

Please put the names of the group members in the files that are submitted, as in Name1Name2.pdf

 

(or Name1Name2Name3.pdf), and Name1Name2.m (or Name1Name2Name3.m).

 

It is important that you do your own programming; copies of other students’ programs (or of

 

programs found on the internet) are not acceptable. 100 points can be earned in total; points for

 

each subquestion are indicated.

 

We wish to obtain the no-arbitrage price of certain financial derivatives that are based on a

 

stock market index St . The derivatives have a payoff which depends on an average over values

 

of the stock price process {St}0≤t≤T at different times. We start by considering a put option on

 

the geometric average over the stock prices at the initial time, the final time, and halfway the

 

lifetime of the option:

 

CT = (K − HT ) +, HT =

 

3

 

q

 

S0S1

 

2 T ST .

 

(1)

 

Time is measured in years, so T is a positive integer in this expression, representing the number

 

of years after which the payoff is received. The strike K is a given strictly positive number and

 

we assume that the stock price process follows the Black-Scholes model without dividends so

 

dSt = rStdt + σStdWt Q ,

 

dBt = rBtdt,

 

for given parameters S0 > 0, B0 > 0, µ > r and σ > 0, with WQ a standard Brownian Motion

 

under the risk-neutral measure Q. An explicit formula for C0, the value of this put option at

 

time zero, can be derived.

 

1. [15 points] Show that

 

C0 = e −rT E Q[(K − S0e aT +b √ T Z) +]

 

for suitably chosen values a and b if Z is a stochastic variable which has a standard normal

 

distribution under Q. Do this by first rewriting the payoff HT in terms of the two stochastic

 

variables W

 

Q

 

1

 

2

 

T = W

 

Q

 

1

 

2

 

T − W0 Q and WT Q − W

 

Q

 

1

 

2

 

T .

 

2. [10 points] Show that for a standard normal random variable X we have that

 

E[e mX1X≤d] = e

 

1

 

2m2

 

Φ(d − m),

 

with Φ the cumulative distribution function for X, using an explicit integration over the

 

standard normal density function

Stochastic Calculus, 2023-2024 Assignment 2: Monte Carlo pricing of Asian options

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