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

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.
13. [15 points] Use the results of the previous two questions to derive an explicit formula
for C0 in terms of the given parameters for this problem. You do not have to substitute
expressions for a and b in your answer: once you have defined a and b you can use them as
known parameters.
Next, we want to check this result by calculating a Monte Carlo estimate of the price C0 for the
case where T = 10 (years), r = −0.27%, (per year, note the minus sign!), σ = 21% (annually)
and S0 = K = 715 (euro). These values roughly correspond to the values for the Dutch AEX
index in the second week of April 2021.