FE Problem Set #13
August 8, 2006
A Dollar-Yen trader, X, is given a stop loss of $200,000 in any given month and another Dollar-Yen trader, Y, is given a stop loss of $300,000. Assuming that the volatility of Dollar-Yen is constant and the P & L of both the traders were random, which one of the traders X and Y will have a higher expected return? Explain your answer.
( Hint :
The P & L of almost all traders follow ArcSine Law of Brownian motion . )
You have three one year call option prices of $4, $3.50 and $3 for three corresponding contiguous strikes of 90, 100 and 110 on a particular stock (underlying). Is it possible to calculate the probability density function of the strike from the above data? If so, how would you do it? Explain with methodology. If not, explain why so.
( Hint : The price of a European call option is given by . Differentiating this function twice with respect to K will give the risk neutral probability density function . .)
A swap dealer of a bank has the following book:
|| Fixed Rate
|| Tenor of
|| Receiver or Payer
|| the Swap
|| (Fixed Rate)
|| 7 years
|| $10 million
|| 3 years
|| $ 35 million
|| 1 year
|| $17 million
|| 2 years
|| $53 million
|| 4.5 years
|| $14 million
All coupons are payable semi-annually. The current 5 year US Treasury Note is priced to yield 5.75% (semi-annual bond basis).
- What is the equivalent position (in 5 year US Treasury Note) of the dealer's swaps book?
- How will the dealer hedge the above book? (Remember T Note futures have a face value of $100,000 each).
( Hint: Decompose the above book into bucket cash flows and calculate the DV01, dollar value of a basis point change, of the bucket cash flows. Then divide the DV01 of bucket cash flows by the DV01 of the 5 year T Note, which is constant. This will give you the hedge ratio. Then using the hedge ratio calculate the T Note Baseline equivalent for each bucket cash flows. Sum them up and you will get the equivalent position in T Note cash . )
Arvind Swamy is an astute venture capitalist and works for a successful Silicon Valley VC firm. He is planning to invest $10 million into a start up telecommunications venture and is trying to estimate what should be the percentage share of the company that he should ask for from the founding entrepreneurs. Arvind's team of analysts makes revenue projections for the start up using market and competitor data and finds out that the net income in the year 5 would be around $35 million. Further, there are a few competitors who are in similar technology space and a couple of them are listed on NASDAQ. The average P/E of those stocks is 11. Arvind wants a target rate of return of 30% for taking this risk. Currently there are 1.5 million common (ordinary) shares outstanding in the company which is all held by the founders.
What is the implied pre-money and post-money valuation of the start up?
( Hint: Estimate the per share value of the start up using discounted terminal value, whereby the discount rate is the target rate required by the VC and then multiply the existing number of shares of the company with the per share value to get pre-money value of the company and multiply the expanded number of shares, i.e. including VC's investment, with the per share value to get the post-money value of the company .)
An analyst at the exposure management department of a bank is trying to estimate the 95% MLE (maximum likely exposure) from a vanilla one year at the money call option on a stock. The current price of the stock is 50 and the (implied) volatility of the stock is 14%. The stock pays no dividend and the risk free interest rate is 4%. What is the 95% MLE of this option?
( Hint: MLE is a measure of the credit risk and in this case 95% MLE should be calculated using 95% worst case stock price. The 95% worst case stock price is given by: )
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