(a) List the assumptions of the Black-Scholes model. Consider a derivative whose payoff at expiry time...

 

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(a) List the assumptions of the Black-Scholes model. Consider a derivative whose payoff at expiry time T' is given by Cr(ST) = max (S²-K, 0), where K > 0 is the strike price and ST is the price of the underlying share at expiry. This type of derivative is known as a power option, since the payoff depends on S, to the power of some fixed number (2 in this case). Derivatives of this type have to be cash-settled. (b) Carefully sketch the graph of Cr as a function of St. (c) We showed in the lectures that the Black-Scholes price Vo at time 0 of a vanilla European option with payoff function Vr(ST) is given by √27 Vr (So exp[(r = o ²/2)T + o√T z]) e-z²/²dz, assuming that the underlying share pays no dividends during the lifetime of the option. The various symbols here have their usual meanings, and you may assume that So, and T all are strictly positive. By using this result, or otherwise, prove that the Black-Scholes price Co at time 0 of the power option described above is given by Co= Sexp [(r + o²)T] (h+) - KeT(h_), where Hint: For full marks in parts (d), (e) and (f), your final answers should each be written in the most compact form possible. h+ log(So/VK) + (r+ 30²/2)T OVT h = h, -20√T, and Þ(r) is the cumulative distribution function of the standard normal distribution. E (d) By (partially) differentiating the expression for Co with respect to So, find the formula for the delta A of this power option. (e) By differentiating once again, find the formula for the gamma I. (f) Finally, find the formula for the theta , defined here as E aco ƏT E [ E [ (a) List the assumptions of the Black-Scholes model. Consider a derivative whose payoff at expiry time T' is given by Cr(ST) = max (S²-K, 0), where K > 0 is the strike price and ST is the price of the underlying share at expiry. This type of derivative is known as a power option, since the payoff depends on S, to the power of some fixed number (2 in this case). Derivatives of this type have to be cash-settled. (b) Carefully sketch the graph of Cr as a function of St. (c) We showed in the lectures that the Black-Scholes price Vo at time 0 of a vanilla European option with payoff function Vr(ST) is given by √27 Vr (So exp[(r = o ²/2)T + o√T z]) e-z²/²dz, assuming that the underlying share pays no dividends during the lifetime of the option. The various symbols here have their usual meanings, and you may assume that So, and T all are strictly positive. By using this result, or otherwise, prove that the Black-Scholes price Co at time 0 of the power option described above is given by Co= Sexp [(r + o²)T] (h+) - KeT(h_), where Hint: For full marks in parts (d), (e) and (f), your final answers should each be written in the most compact form possible. h+ log(So/VK) + (r+ 30²/2)T OVT h = h, -20√T, and Þ(r) is the cumulative distribution function of the standard normal distribution. E (d) By (partially) differentiating the expression for Co with respect to So, find the formula for the delta A of this power option. (e) By differentiating once again, find the formula for the gamma I. (f) Finally, find the formula for the theta , defined here as E aco ƏT E [ E [

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