ECMT3150: Assignment 2 (Semester 2, 2020)
Lecturer: Simon Kwok
Due: 6pm, 20 November 2020
1. Suppose we are in a single-period market. The interest rate is zero. There is
a risky stock with a current price of S0 = 1. At the end of the period, it may
rise to $2 or drop to $0.5 with equal probability (this de…nes the statistical
probability measure P). Also available in the market is a European call with
strike price $1 expiring at the end of the period, and a money account (so
borrowing/lending is possible at zero interest rate).
(a) What is the average payo¤ of the stock under P?
(b) What is the average payo¤ of the call under P?
(c) Find the distribution of the stock price at time 1 under the risk-neutral
probability measure Q [Hint: fStg is a martingale process under Q].
(d) What is the no-arbitrage price of the call at time 0? [Hint: use (c) and
the …rst fundamental theorem of asset pricing]
(e) How can you construct a portfolio (using the risky stock and the money
account) that replicates the payo¤ of the call option at time 1?
(f) What is the no-arbitrage price of your portfolio in (e) at time 0?
(g) Suppose now there is a European put option available with identical strike
price and expiry date as the European call. Using (d) and the put-call
parity, compute the no-arbitrage price of the put at time 0.
2. Let dX(t) = (bX(t) + c)dt + 2
p
X(t)dW (s). Suppose X(t) > 0 for all t 0.
Find the law of motion of Y (t) =
p
X(t). Express the stochastic di¤erential
equation in terms of Y (t) instead of X(t).
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3. Mimi is an intern working in Goldman Sachs. One day, his boss asked him to
form the following portfolio:
long 1 European put with strike = $30
long 1 European put with strike = $40
short 2 European puts with strike = $35
All the calls in the portfolio are European options expiring in 8 months written
on the same non-dividend paying stock. The underlying stock is currently
trading at $35 per share. Assume a constant cash (i.e., risk-free) rate of 1%,
and that the stock price follows a gBm with a constant volatility of 10%.
(a) Draw the portfolio’s payo¤ at maturity as a function of the spot price at
maturity.
(b) What is the cost of the porfolio?
(c) Compute the “delta”of the portfolio. How could Mimi hedge against the
risk of this portfolio at this moment?
(d) It turns out that only European calls instead of puts are available in the
option market at the moment. To avoid being sacked on the spot, please
help Mimi form another portfolio consisting of only European calls but
yielding identical payo¤ as the above portfolio at maturity. [Hint: Use
the put-call parity.]
4. (Ex: 7.1 of Tsay (2010, 3rd ed)) Consider the daily returns of GE stock from
January 2, 1998, to December 31, 2008. The data can be obtained from CRSP
or the …le d-ge9808.txt. Convert the simple returns into log returns. Suppose
that you hold a long position on the stock valued at $1 million. Use the tail
probability 0.01. Compute the value at risk of your position for 1-day horizon
and 15-day horizon using the following methods:
(a) The RiskMetrics method.
(b) A Gaussian ARMA–GARCH model.
(c) An ARMA–GARCH model with a Student-t distribution. You should
also estimate the degrees of freedom.
(d) The traditional extreme value theory with subperiod length n = 21.
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