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r /> c) Discuss the results and any deficiencies that this valuation method may have as well as how such shortcomings may be resolved.

[20 marks]

Task 3
a) Collect spot prices for an energy commodity for which you can get corresponding forward or futures price series for a few months into future (e.g. 1st to 12th month).
b) Assuming spot prices are mean reverting, estimate parameters of a mean reverting model, and use the estimated parameters of the model to construct the forward curve (e.g. 1st to 12th month) for the chosen commodity.
c) Estimate price volatility of each forward/futures contract (1 to 12 month ahead) and plot the volatility term structure. Then estimate the implied mean reversion rate by fitting exponential function to the term structure of estimated forward volatilities, and compare the implied mean reversion rate with the one estimated in question b.
d) Use the constructed forward curve from (a) to price a 12month fixed for floating swap, and compare the price of the swap based on your model with the price of swap based on actual/market futures or forward prices. Comment on your results by discussing any factor(s) that may cause a difference between swap prices based on your estimated forward curve and the one based on market prices.
e) Finally, use your mean reversion model for spot price to price a 6 month fixed for floating swaption with 12 month to maturity (assume market price of risk is zero).
[30 marks]

Task 4
a) Collect daily spot prices for natural gas or electricity in a particular market for the past 5 years.
b) Estimate parameters of a GBM process for the spot price series.
c) Estimate the parameters of a Mean Reversion Jump-diffusion model for the spot prices. Use the recursive filtering approach with a three standard deviation band to estimate the jump coefficients for the model.
d) Today’s date is 30 June 2013. Run 10000 simulations of the GBM and MRJD models estimated above to find the premium of 6 average price (Asian) call and put options on the spot price of the underlying commodity expiring in 1, 2, 3, 4, 5 and 6 months (i.e. 21, 42, 63, 84, 105, and 126 days). The averaging period for the settlement of the options is based on the last 21 trading days of each contract. Consider the following strike prices for call and put options: At-the-money, 5% above and below and 10% above and below the spot price level on 30 May, respectively. In your simulations you may assume that the market price of risk is equal to zero.
e) Finally, calculate the implied volatilities that correspond to the estimated option premia for the MRJD model and plot the resulting volatility surface across strikes and maturities. You can calculate the implied volatilities by feeding the option premium estimated using Monte Carlo simulation into the Black-Scholes option pricing model. Comment and discuss your results.

[30 marks]

If you are interested you can perform the following as an exercise but will not be marked!
a) Calculate the Delta and Gamma of the options valued under the MRJD model in Task 4 d.
b) Estimate the 1% 1-day VaR for each option, using Delta Approximation, Delta-Gamma Normal Approximation, and Delta-Gamma Approximation.
c) What would be the 1% 1-day VaR of portfolio of all the options?

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