Exercise 2.2 - Monte-Carlo Valuation (Multi-Asset)

Dynamics for two stocks, S1 and S2, are given below.

S1 = S2 = 50

Rf = 3%

T = 1 Year

σ 1 = 20%

σ2 = 30%

p = 0.25   Correlation between S1 and S2

No dividends

Monthly time steps

Calculate the following using 100 (or more) Monte-Carlo simulations.

a) Value of a call option on the individual stocks with K = 50?

b) Value a basket option (Sum of S1 and S2) with one share of each stock and a strike of 100?

c) Value of the basket if you change the correlation to 0% and 50%?

d) Value the basket option if the portfolio is rebalanced each month to the original 50%/50% weights of S1 and S2?

Exercise 2.3 - Dynamic Delta Hedging

For the following values,

S = 100,    μRW= 8%, σRW= 15%

K = 100

Rf = 3%

σIV  25%    (Implied volatility)

T = 1 Year

No dividends

Calculate the  delta  hedge  for  a  sold  position  of  100,000  call options like in the book using 100 (or more) trials. Stock paths are generated with μRW  and σRW .

a) Calculate the average value and standard deviation of the trials using Weekly (52) steps?

b) Calculate the average value and standard deviation of the trials using Daily (252) steps?

c) Compare to the Black-Scholes value?

d) Redo the analysis with σRW  = 20%?

e) Redo the analysis with μRW    = 4%?

Exercise 2.4 Least Squares Monte-Carlo

For the Least Squares Monte Carlo example in Hull section 27.8, redo the exercise using a 100 or more scenarios. Generate your own risk-neutral random stock prices with rf  = 3% and σ =  20%. Strike price is 110 and initial stock price is 100.

Exercise 2.5 - Monte-Carlo Valuation (Time-varying Rates and volatility)

For the following values,

S = 100;    K=100;  T = 5 Year

Rf = rates from problem 1.1

Volatility Spot Term Structure

σ(t) = 0.15 + 0.01*t

Dividends

$1 paid at the end of each year

Use monthly time steps

Calculate the following using 100 (or more) Monte-Carlo simulation scenarios:

d) Calculate the following

•  European Call option on the price return

•  European Call option assuming the dividends are reinvested (Total return)