Solved by a verified expert :7. Suppose the index option smile is symmetric, but you expect it to
steepen on both sides. What option strategy would you adopt?
8. (Requires Writing Code) Using the following parameters, price call
options for a range of seven strike prices with the Merton jump model.
S = 100
K = {70, 80, 90, 100, 110, 120, 130} T = 0.5
years
rf = 3% sigma = 0.30 mu = -0.05 gamma = 0.50 lambda = 0.5
Now with the seven option prices (one for
each strike price), nd out what the implied volatility is in the Black-Scholes
model. You will need to write program code to nd the implied volatility.
Once you have the seven corresponding implied
volatilities, plot them against the strike prices. What shape does your options
smile have?
10. (Requires Writing Code) Write a program to simulate monthly
returns for two years
from a process where returns r are drawn from
a normal distribution with mean 10% and standard deviationt, which follows the risk-neutral
process:
t+1 = tex; x N(0; 1)
The initial stock price is $100 and the initial0 = 0:15. Each month the stock
price grows as follows:
St+1 = Stert; rt N(0:10;
t)
(a) Price call options for strikes: 90,100,110 with = 0:1. Assume the
interest rate is zero.
(b) Now set = 0 and reprice the options for these strikes. Compare
your results with those in (a) and comment.
11. (Requires Writing Code) Write a program to implement the
Derman-Kani model for n periods. The inputs are the current stock price and a
volatility surface. Your output will be the Derman-Kani tree of stock prices.
12. For a negatively skewed stock return process, what GARCH model
would you use? Why?
Sundaram
& Das: Derivatives – Problems and Solutions . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 230
13. (a) What happens empirically to the option smile with increasing
maturity? (b) How is the smirk typically di erent from the smile? (c) Which
markets are characterized by smiles, and which ones display smirks? (d) What is
the volatility surface?
14. (Requires Writing Code) Does put-call parity hold in the extended
Black-Scholes models? Explain.
