FINA6542 Quantitative Risk Management (2022-23) Assignment 2
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FINA6542 Quantitative Risk Management (2022-23)
Assignment 2 (Due date: 21 Jan 2023)
Please submit your assignment to the blackboard learning system.
Note: Answers should be given within 4 decimal places whenever necessary.
If you feel necessary, you should upload the corresponding excel, R, python or whatever software you use to the blackboard learning system.
1. You are given that
(i) the stock price process is
dS(t) = 0.16S(t)dt + 0.32S(t)dZ(t),
where Z(t) is a standard Brownian motion under the risk neutral measure.
(ii) the stock pays dividends continuously at a rate proportional to its price and the
dividend yield is 4%, and
(iii) the market price of a 1-year at-the-money European put on the stock is 1.6195.
i) Find the price of a 1-year European call with strike equal to 15.
ii) Find the delta, Gamma and theta of this option as well. For this question, you may want to develop your own version of Black-Scholes Calculator, which should include the price of European call and put as well as the corresponding Greek Letters of these options. This is going to be useful for the rest of the class.
2. Consider a three-period Binomial model. Suppose S(0) = 5, u = 1.1, d = 1/1.1, r = 0.05 and the stock does not pay dividend. The risk-free interest rate here is annual continuously compounded interest rate.
a) Find the price of a 3-year European call option with payoff (c(3, S(3))) equals to max{S(3) 一 K, 0} and K = 5.1.
b) Find the price of a 3-year European put option with payof (p(3, S(3))) equals to max{K 一 S(3), 0} and K = 5.1.
c) Verify that the put-call parity is correct.
d) Use Monte Carlo simulation with 5000 trails to estimate the price of the European put option. To ensure consistency, please use SEED = 123 for simulation.
e) Find the variance of your estimate in 2d). Explain how you can use Monte Carlo simulation to verify your answer is correct.
f) What if the option in 2b) is an American put option?
g) Repeat the same 2d with the American put option and find the variance of the estimate.
Mid Term 2019-2020
3. Suppose Z1, Z2 and F are independent standard normal random variables. Let X1 = 0.2F +^1 一 0.22 Z1
and
X2 = 0.3F +^1 一 0.32 Z2
a) Find the mean and variance of X1 .
b) Find the mean and variance of X2 .
c) Find the covariance between X1 and X2 .
d) Construct the variance-covariance matrix of X1 and X2 . Find the corresponding Cholesky decomposition of this variance-covariance matrix.
Consider the Black-Scholes framework. Let S1 and S2 be two stock prices. In particular,
S1(t) = 100 exp((0.06 一 0.5 * 0.42)t + 0.4^tX1)
and
S2(t) = 110 exp((0.06 一 0.5 * 0.32)t + 0.3^tX2)
e) Find P (S1(2) > S2(2)).
f) Find E(S2(2)|S1(2) = 115).
g) Finding Covariance between S1(2) and S2(2) is difficult. However, using Monte Carlo simulation to estimate is relatively easy. Explain how you can use Monte Carlo Simulation and Cholesky decomposition to estimate the covariance between S1(2) and S2(2). How would you access the accuracy of this estimate?
Bonus Question: Can you estimate the likelihood of early exercise for an American option?
4. (Question for you to practice. You are not required to do this) Derive ∆, Γ, Θ and ρ of a European put option with long position under the Black-Scholes framework. Please do NOT use the results of a Euopean call and apply put-call parity.
2023-01-09