FE Problem Set #18
June 29, 2009
Note:The following questions were part of the Mid Term Examination of Risk Latte's Certificate in Financial Engineering (CFE) Course in Hong Kong.The answers are available only to CFE students..
The volatility of a particular stock index is extremely high and the prices of European style call options are prohibitively high. A structurer in a bank therefore decides to structure a slightly exotic product such that the payoff will be that of a standard European style call option whose notional amount declines as the call goes in the money (Amortizing option). Price this payoff using both the Monte Carlo simulation and closed form Black-Scholes model.
A Log contract is an extremely important building block in volatility derivatives. The payoff of a Log contract is given by:
In the above formula, is the spot (asset) price at maturity and is the strike price. In a closed form, with interest rates at r, dividend yield at q and volatility equal to , the value of this contract is given by:
Answer the following based on the above:
- Is the log contract an option? Explain your answer.
- A non-dividend paying asset is trading at 100. The risk free rate is 8% and the volatility is 35%. What is the value of a 3 month Log contract struck at 90?
- What are the delta and the gamma of this contract? Plot the Delta and Gamma of this contract as a function of the asset price.
If is a uniform random number between 0 and 1 then using a model of Geometric Brownian motion (GBM), the asset price path in a Monte Carlo simulator would be given by: where, is the inverse cumulative Normal distribution function. Therefore, an in the money path, where the asset price at maturity is greater than the strike price, would be given by:
This equation puts a bound on the uniform random numbers, is such a way that only certain random numbers will generate "in the money" calls and puts. What are those bounds?
If the interest rate is r and if time t then the discount factor can be expressed by the following continuous time formula:
- If the interest rate is assumed to be constant (no term structure of rates) across all time buckets between 0 (today) and T (maturity) then for the time bucket find a closed form solution for the above discount factor in terms of r and T .
- Using that closed form solution find out what would be the present value of all payments of $1 per year for the next 10 years if the interest rate remains constant at 5%.
In the world of exotic financial options, the payoff of an Asymmetric Power option is given by:
And the payoff of a Symmetric Power option is given by:
It is easier to find closed form solution for an Asymmetric Power option and thereby value it. However, estimating closed form solution for Symmetric Power option is difficult. For N = 2 in the above Power option payoffs, a trader has the value of an Asymmetric Power option. , how would he intuitively or otherwise value a Symmetric Power option,
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