Pstat 170 Fall 2022 – Assignment #5
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Pstat 170 Fall 2022 – Assignment #5
Problem 1
Consider a stock paying continuous dividends of δ = 1%. Assume r = 0.06,σ = 0.32 and today’s stock price of S0 = 33.
a) You sell 100 Calls with strike K = 34 and expiration in 55 days (assume 365 days in a year). You also construct a Delta hedge to manage your risk. If tomorrow (1 day later) stock price rises to $34.50, find your net profit/loss from your hedged portfolio. Hint: don’t forget about dividends!
b) Repeat this problem for the case of selling 100 Puts with strike K = 34 and all other parameters staying the same. Compare the answers you get in (a) and (b).
Problem 2
Consider a Black-Scholes model with r = 6%, σ = 0.25, S(0) = 50 and δ = 0. Suppose you sold ten (10) 90-day Puts with strike K = 50 (at price P1 (0) determined by the Black-Scholes formula). You wish to hedge your risks and in particular are worried about the stock moving up and down.
1. Using the underlying stock as well as 60-day Puts with strike K = 50 (priced at P2 (0) again using the B-S formula), construct a Delta-Gamma neutral portfolio.
2. Suppose that you also invest in bonds such that the initial value of your portfolio is exactly zero. Thus your overall portfolio is x shares of stock, short 10 contracts of 90-day Puts, y contracts of the 60-day Puts and z zero-coupon bonds. You should write out explicitly the numeric values of x,y,z above.
V (0) = xS(0) − 10C1 (0) + yC2 (0) + z,
Produce a table listing the value of your portfolio tomorrow V (
) in the cases that the stock goes to S(
) = 48, 50, 52.
Problem 3
Consider a Black-Scholes model with σ = 0.3,S(0) = 50,r = 0.05,δ = 0.03. Suppose you use a bear spread, selling a European Put option with K = 40 and buying a Put with K = 45. Both options expire at T = 1.
1. Find the aggregate Delta, Gamma, Vega, Theta and Rho of the option portfolio.
2. Recall that the Greeks are partial derivatives of the portfolio value with respect to the given parameter. Instead of using calculus to find the derivative, we may use a finite-difference approximation. For example, to approximate Vega we can use Taylor expansion:
Price(σ0 + ϵ) − Price(σ0 )
ϵ
for a small perturbation ϵ . This means you compute the portfolio value plugging in σ0 for volatility, then plugging-in σ0 + ϵ for volatility and look at the difference.
Use the above method to approximate the Vega of the above bull-spread portfolio. Use ϵ = 0.002. Compare to the true Vega in part 1.
3. Assuming the stock price has real growth rate of α = 0.1 (meaning E[St] = S0 eαt ) and using the log-normal model, find the real-world probability that (a) the Bear spread will pay zero; (b) the Bear spread will pay exactly $5; (c) the Bear spread will pay between 0 and 5 dollars.
Problem 4
Consider a Gap Call (see Section 14.5 in text) that pays
f(ST ) = 
Assume S0 = 100, σ = 0.15, r = 0.06, δ = 0.05 and T = 1. We use a Black-Scholes model, so that ST is log-normal.
. Compute the risk-neutral probability Q(ST > 103)
. Compute the expected stock price (under Q) conditional on ST > 103, i.e EQ [ST |ST >
103] (cf. formula (18.28) in textbook).
. Using a Monte Carlo simulation with 200 simulations of ST , find the approximate
price of this Gap Call today. Compare to the exact answer given by the formula (14.15) in textbook.
Problem 5
We consider implementing Delta hedging using discrete-time rebalancing. Namely, let us take options with T = 4/52 or 4 weeks until maturity, r = 0.04, δ = 0 and σ = 0.25. We will use weekly rebalancing. The initial share price is S(0) = 100, the contract is a long Put and the strike is K = 100. Throughout we will use the Black-Scholes Price/Delta.
For each scenario of realized prices below, compute the total Delta-hedging P&L, which is the future value at T of the rebalancing costs at t = 1, 2, 3 weeks plus, the final net profit (Put payoff minus the value of the hedging portfolio). Below tk := k/52 refers to the end of the k-th week; t4 = T is the contract maturity at 4 weeks.
1. S(t1 ) = 100,S(t2 ) = 100,S(t3 ) = 100,S(t4 ) = 100
2. S(t1 ) = 100,S(t2 ) = 99,S(t3 ) = 98,S(t4 ) = 97
3. S(t1 ) = 100,S(t2 ) = 99,S(t3 ) = 100,S(t4 ) = 99
4. S(t1 ) = 100,S(t2 ) = 96,S(t3 ) = 97,S(t4 ) = 99
Summarize in words which scenarios generate better hedging P&L.
2022-12-06