FE Problem Set #02
Team Latte Nov 03, 2005
Problem #1
The price of a call option on an underlying is given by where S is the spot (underlying)and t is the time variable. A very small shift of the magnitude is made to the spot price and the corresponding call option price becomes . Then another identical but opposite shift of the magnitude is made to the spot price and the corresponding call option price becomes . Using only the above information compute the gamma of the call option in terms of the above call prices. (Hint: you can use Taylor Series expansion and ignore higher order terms).
Problem #2
A portfolio of options is approximated by a Taylor Series expansion as:
where, is the change in the option portfolio value, is the change in the underlying spot price, is the delta of the portfolio and is the gamma of the portfolio. What is the value of the portfolio change under a worst case scenario? What will happen to the worst case scenario if the gamma of the portfolio is small or negative. Discuss.
Problem #3
You are the head of FX options sales and a customerfor a trade idea. He is bullish on JPY in the short term and expects it to get strong. However, given the high volatility of JPY (an implied volatility of 14%) and higher expectation of volatility built in option prices he feels that vanilla JPY Call / USD Put are expensive. Also, he does not want to pay too much up front premium but rather stagger the premium payment in stages over the life of the option. Assume that the Dollaryen spot is trading at 115.00, and the implied volatility term structure is: 3 month implied volatility is 14.5%, 6 month implied volatility is 14%, and one year implied volatility is 12%. Further assume that JPY risk free is 2% and USD risk free of 4%.
Assuming no skew across strike prices, what kind of an option will you suggest to him and why. Discuss in details with different pricing.
Problem #4
A hedge fund manager is invested in three asset classes, S1, S2 and S3. His funds under management is US$100 million and he is fully invested in S1, S2 and S3 with respective percentages as 30%, 25% and 45%. The respective annualized volatilities of S1, S2 and S3 are 25%, 27% and 30%. The correlation between the asset classes is given by:
(a) What is the 95% 1 month VaR of the fund manager's portfolio?
(b) Which asset class is contributing the greatest risk to the manager's portfolio.
(c) If 5 yr US T Bonds yield 4.7% and the fund manager's net profit (after expenses) from the portfolio was $3.85 million in the performance year then how did his return compare with that of an investor whose entire money was locked in 5 year US Treasuries.
Problem #5
The volatility of a portfolio is given by . The portfolio is invested in two assets, Stocks and Bonds with the proportion invested in stock equal to . The stocks have a volatility of and bonds have a volatility of and the correlation between the two is
If all parameters are known except for the amount invested in the stocks then estimate the value of as a function of all other parameters.
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