Energy Derivatives-Final exam

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Energy Derivatives Final Exam Name: Jaimin Pandya Student ID:2292119 Q.1 Ans: a) F 82.39 -0.019353 ln(f/K) K 84 0.006035 Tau*(0.5sigma^2) Tau 0.20273973 -0.013318 r 0.05 0.109865 sigma*sqrt(tau) sigma 0.244 -0.121218 d1 -0.231083 d2 0.451759 n(d1) 0.408625 N(d2) 2.866717 call price 0.548241 N(-d1) 0.591375 N(-d2) 4.460479 put price 4.460479 0.447203 call delta -0.553826 put delta 0.042349275 Gamma 14.5432357 Vega The Price of the 84 Strike calls is $2.86 . The revenue dealer realizes on sales is $5,720,000 .

b) The delta of this call is 0.447203 , the Gamma of this call is 0.042 and the Vega of this call is 14.54 . c) The delta of the entire 2 million bbl. position is -894405.9648 . (Call delta *2000000 negative sign indicates short position). The position gamma is -84698.54. d) A negative Vega of -29M means a 1% volatility rise could sink position value by -$29M! This high negative Vega exposes the firm to potentially devastating losses if volatility surges. e)

Q-2 Ans 0 Options Tree 0 0 0 0 0 0 0 0 0.00529144 0 0 0.02705066 0.01021523 0 0 0.07229487 0.04731285 0.01972067 0 0.14143632 0.11447134 0.08186153 0.0380711 0 0.20597664 0.17709632 0.13974005 0.07349691 0.29144533 0.2659438 0.23445167 0.23445167 0.39834631 0.38377094 0.38377094 0.52229587 0.52229587 0.52229587 0.65080678 0.65080678 0.77002758 0.77002758 0.88062987 0.98323668 Delta Tree 0 0 0 0 -0.0226237 0 -0.1003167 -0.0470787 0 -0.2249598 -0.1843676 -0.0979684 -0.3708095 -0.3599963 -0.3329125 -0.2038674 -0.5291467 -0.5504002 -0.5871718 -0.7130793 -0.7865012 -1 -0.8905931 -1 -1 -1 -1 -1 The price of American put by the binomial model is $0.141. The delta of Put is -0.370. Buy 0.370 to replicate the short put.

Q.3 Ans c 3.73 F 82.31 K 82 r 5% tau 0.20273973 (F-k)*e-rT 0.30687341 Theritical p 3.42312659 Actual p 3.35 Arb Profit 0.07312659 Buy Put Sell Call Sell Futures

Q.4 Ans. Lambda in the Black model measures an option’s price sensitivity to volatility changes. It’s always positive, meaning option prices increase as volatility rises. It is influenced by time to expiration, moneyness, and underlying asset volatility. It’s used to assess risk, develop strategies, and hedge positions. The more convexity, the more valuable is volatility. The option’s price is more sensitive to changes in volatility when the strike price is close to the futures price. This is reflected in the d1 and d2 terms in the equations. When F is close to X, the d1 and d2 values will be close to 0, making the N(d1) and N(d2) terms (which represent probabilities) more sensitive to changes in σ. On the other hand, if the strike price is far from the futures price, the option’s price will be less sensitive to changes in volatility. This is because the N(d1) and N(d2) terms will be close to 1 or 0, making them less sensitive to changes in σ.

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