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 FE Problem Set #16 Team LatteDecember 29, 2007 Note: Solutions to most of these problems are available only to our online Trainees . Problem #1* Assuming the risk free rate to be zero and using put-call parity show that the equation: is equivalent to the equation: Where, and are call and put option prices respectively with strike and maturity T. (Note: This problem has application for deriving model free volatility and variance from a set of call and put option prices. The above expressing can be tested on an Excel spreadsheet very easily.) Problem #2 Structure a hybrid TARN (target redemption note) linked to Nikkei225 stock index with a 15 year maturity and a TARN level of 15%. Coupons are paid annually and the first two coupons (on year 1 and year 2) are fixed at 5%. Show the payoff formula for years 3 to 14 and the final year, i.e. year 15; Estimate the IRR (internal rate of return) of the note for various redemption scenarios (early redemption in year 3, 4, 5 and so on) Analyze the binary risk (the TARN level acts as a digital barrier) in the note; Analyze the risks embedded in the call spreads (TARN can be decomposed into lookback call spreads) embedded in the note; Problem #3 You are structuring an equity linked note tied to a major equity index. The volatility in the market is very high and you are convinced that empirically the realized volatility is inversely proportional to the index level. One of your main concern is the estimation of the participation rate for the note. You want to make the participation rate dynamic such that the investor gets smaller exposure to the index in a falling market and enhanced exposure in a rising market. Suggest some solution to this problem. Problem #4 The following short answer questions pertain to math applications and coding in Excel/VBA. If you are writing a function in VBA and if you want all arrays in the function to begin indexing from one and not zero (as is the default) what declaration should you put on top of the module in the VBA editor? In a numerical integration routine why is Simpson's rule a better approximation than the Trapezoidal rule? In what way is the Simpson's three-eights rule a refinement over the Simpson's rule? A function takes three parameters as input: volatility, interest rates and a third parameter declaration of the type of the option, such as a call option or a put option. How would declare the function in a module? Analyze the risks embedded in the call spreads (TARN can be decomposed into lookback call spreads) embedded in the note; How would you find out the square root of the complex number,Z   where and Problem #5 The following short answer questions pertain to option pricing and probability distributions: What is a free boundary problem and what is the optimal stopping time for an American option?** What is the probability density function for a Poisson distribution which has only one parameter , with ? What is the probability density function for Weibull distribution which has three parameters, the location parameter , the scale parameter and the shape parameter ? Is there any constraint on these parameters? In which area of quantitative finance is a Weibull distribution used? What happens if we change the sign of the volatility (from positive to negative) in a Black-Scholes option pricing model? Any comments and queries can be sent through our web-based form. More on FE Problem >> back to top More from FE Problems More from Articles Searching for the Most Beautiful Equation in Finance Where does a Black Swan Come from? Napoleon on Wall Street: Advent of the Stochastic Volatility Models The Remarkable Power of the Monte Carlo Method Mean Reversion and the Half-Life of Interest Rates 