FE Problem Set #09
Team Latte Mar 20, 2006
Problem #1
A trainee trader who had an academic background in Mathematics was hastily given a crash course in derivatives pricing using BlackScholes formula by the bank's HR department before starting work on the trading floor. He was assigned to fixed income trading team and on his first day of work he was required to price a European call option on the one year US Treasury Bill. The spot was trading at 98.00, the strike was 100, the risk free rate was 5% and the volatility of the rate was 10%. Plugging these numbers on to the BlackScholes formula he got the value of a one year call option on the US Treasury bill as 5.71.
Upon seeing this result, his boss was furious and went into a fit. The result was completely wrong. Intuitively (with as little mathematics as possible) explain what went wrong with the trader's calculation?
(Hint : pay close attention to the value of the option upon expiration. What is the payoff at the end of one year on a US Treasury bill where the price is capped at 100?)
Problem #2
The short term interest rate differential (spread) between the U.S. and Japan is currently 4%. If this spread has a volatility of 5% then what is the probability that this spread will hit 5% in the next one year? Assume that the spread follows a driftless arithmetic Brownian motion.
(Hint: This is a very simple First Exit Time problem. Also, remember that by reflection principle if a point y is hit by a Brownian path X( t ) at time T( y ) < t , then it is equally likely that X( t ) with either be above or below y .)
Problem #3
The optimal weight for an investment in private equity is given by:
Where, is the risk aversion factor, is the market's beta (public equity beta) and and are the volatilities of private equity and public equity (market) portfolios and is the return on the market (public equity) portfolio. In the above formula is positive and is the excess return for the private equity portfolio and is the expected excess return on the public equity (market) portfolio. It is to be noted that the multiplier must be positive.
 Explain the behaviour of this allocation to private equity as a function of market's beta.
 What happens to the allocation to private equity if ?
 What would be the optimum allocation to private equity portfolio when the expected excess return for the private equity portfolio is exactly equal to the expected excess return for the public equity portfolio?
Problem #4
You are a quantitative analyst with the equity derivatives trading desk in a small local bank. A trader is trading options on a highly illiquid and volatility stock index. The call and put option prices across various maturities and strikes are not easily available due to lack of trading. However, short term (1month and 3month) ATM options are relatively more liquid. The trader wants to capitalize on the big volatility mispricings (across smile and term structure) by buying and selling of vol.
You are required to model the smile and the time dependence of the implied volatility and create a surface. The idea is capture the smile and the time dependence as accurately as possible in a model and mark to model the trades, more as a risk management rather than a trading tool. You have three math models to choose from:
Where, is the implied volatility function (that generates the surface), is the strike price, is the spot, T is the time to maturity and and are constants.
Which of the above models will you use? Explain your answer with examples (choose your own data)
Problem #5
You are one of the product controllers of a medium sized European bank and one of your major responsibilities is to evaluate the accurate price of options in a certainilliquid equity market where your bank's exposure is big. A company called LiquidQuotes collects options prices (bid and ask quotes) from the product control functions of several participation banks on a daily basis and after excluding the outliers (extreme values on either side) creates an average price from the remaining quotes. LiquidQuotes then provides to each of the participating banks the information about the average price and how many standard deviations away is a bank's particular quote from the consensus value.
One day you find that you are expelled from the contributing group and LiquidQuotes informs you that you are no longer part of this process. Upon enquiry you realize that over the last three months your bank's (traders') price on these options have been consistently been more than 3 standard deviation away from the market's average (consensus) estimate. This means something is wrong with your pricing.
Your trader and the quant insist that your bank's pricing model is far better than anyone else's and it is state of the art. You also realize that the trader has been playing this market very aggressively and the exposure is extremely large. You are alarmed and worried.
You approach the Chief Risk Officer of the bank to discuss the situation. How will you discuss this with him? What would be your suggestions to resolve this situation?
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