FIN 452 & 537, Spring 2023 Advanced Derivative Securities Homework #4
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FIN 452 & 537, Spring 2023
Advanced Derivative Securities
Homework #4
Due Tuesday, February 14, 2023
1. Use the put-call parity to derive the Greeks of a put option, and write a VBA function that computes the value of the put and its Greeks. Create an Excel worksheet in which the user inputs S, K, r, G, q, and T, and compute the put value and its Greeks using the VBA function you write.
2. The delta of a digital call option that pays $1 when S(T) > K is e −rT n (d2)
GS√T .
Write a VBA function that computes the delta of a digital call option. Create an Excel worksheet in which the user inputs K, r, G, q, and T, compute the delta of the digital call option for stock prices S = .01K, .02K, … , 1.99K, 2K (i.e., S = iK/100 for i = 1, … , 200), and plot the delta against the stock price.
3. The file “DataForProblem4-3.txt” contains price data for call options on the S&P 500 index. Import this data file into an Excel worksheet. The first column lists various exercise prices, the second column gives the bid price, and the third column shows the ask price.
The prices were obtained on January 22, 2003 when the options, expiring in February 2003, have 30 days to maturity, i.e., T = 30/365. At the time the quotes were downloaded, the S&P 500 was at 884.25. According to the CBOE, the dividend yield on the S&P 500 was 1.76%. Use 1.25% for the risk-free interest rate.
1) Compute and plot the implied volatility against the exercise price using the ask price as the market price for the option.
2) Compute and plot the implied volatility against the exercise price using the bid price as the market price for the option. Does it work for all exercise prices? If not, what is wrong? Does this indicate there is an arbitrage opportunity?
4. Suppose an investor invests in a portfolio with price S and constant dividend yield q . Assume the investor is charged a constant expense ratio a (which acts as a negative
dividend) and at date T receives either his portfolio value or his initial investment, whichever is higher. This is similar to a popular type of variable annuity. Letting D denote the number of dollars invested in the contract, the contract pays
s(0)
at date T. We can arrange the payoff function as
max (。, 
) = 。+ max (0, 
-。) = 。+ e-aT。max (0, 
- eaT) .
Thus, the contract payoff is equivalent to the amount invested plus a certain number of call options written on the gross holding period return eqTs(T)/s(0). Note that Z(t) = eqts(t)/s(0) is the date-t value of the portfolio that starts with 1/s(0) units of the asset (i.e., with a $1 investment) and reinvests dividends. Thus, the call options are call options on a non-dividend-paying portfolio with the same volatility as s and an initial price of $1. This implies that the date-0 value of the contract to the investor is e-rT。plus
Exp( –alpha*T)*D*Black_Scholes_Call(1,Exp(alpha*T),r,sigma,0,T) .
1) Create a VBA function to compute the fair expense ratio, that is, find“ such that the date-0 value of the contract is equal to 。.
Hint: Modify the Black_Scholes_Call_Implied_Vol function. You can use“ = 0 as the lower bound. Because the value of the contract is decreasing as“ increases, you can find an upper bound by iterating until the value of the contract is less than 。.
2) Create an Excel worksheet in which you input r and o. Compute the fair expense ratio for different maturities T. How does the fair expense ratio vary with T? Why?
Save your VBA codes and results (including plots and narrative answers) for all four problems in an Excel Macro-Enabled Workbook with the file extension “ .xlsm.” Also, name the worksheets “Problem 1,” “Problem 2,” “Problem 3,” and “Problem 4,” respectively. Submit your Excel file electronically on Canvas with the rest of your work [i.e., derivation of the Greeks of a put option (the first part of Problem 1)].
2023-02-16