Subject Code : EF482
Assignment Task :

1. Derive the price of share digitals in the Black-Scholes model by using the risk-free asset as the numeraire.
 

2. Consider delta and gamma hedging a short call option, using the underlying and a put with the same strike and maturity as the call. Calculate the position in the underlying and the put that you should take, using the analysis in Sect. 3.6. Will, you ever need to adjust this hedge? Related your result to put-call parity.
 

3. The ?le CBOEQuotes.txt (available at www.kerryback.net) contains price data for call options on the S&P 500 index. The options expired in February 2003, and the prices were obtained on January 22, 2003. The ?rst column lists various exercise prices. The second column gives the bid price and the third column the asking price. Using this data to compute and plot the implied volatility against the exercise price. The market price can be approximated by ask, bid or average of bid-ask prices.1 The option has 30 days to maturity (so T = 30/365). At the time the quotes were downloaded, the S&P 500 was at 884.25 and the dividend yield was 1.76%. Use 1.25% for the risk-free rate. 
• Identify an underlying with many options written on it (e.g., S&P 500 stock index). Using these options (download from Bloomberg or any other data provider) repeat the exercise above. How and where to ?nd out the dividend yields and risk-free rate? You can either use a 1-month Treasury or LIBOR as the risk-free rate (these can be download from Bloomberg).2 Identify a couple of dates that the underlying had large positive or negative returns and plot the implied volatilities and compare them with the ones in dates with small returns.
 

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 α (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 annuity. Letting D denotes the number of dollars invested in the contract, the contract pays 

date t

Thus, the contract payo? 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 reinvest dividends. Thus, the call options on a non-dividend paying portfolio with the same volatility as S and initial price of $1. This implies that the date-0 value of the contract to the investor is

logrithm

 

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  • Posted on : October 07th, 2018

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