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# This question is on option pricing using Black-Scholes The Black-Scholes pricing problem for European binary call Op tion is given by the partial differeutial equat...

## Question

###### This question is on option pricing using Black-Scholes The Black-Scholes pricing problem for European binary call Op tion is given by the partial differeutial equation and final condition aV 02V OV 5 > E tr5' CtV =0. V(ST) = 952 ds 5 < E V (S,t) is the option value, S is the underlying asset price, is time. T is expiry; is the constant risk-free interest rate, is the constant volatility and E is the strike price. By introducing the following change of variables log { = 29"(T _ t)

This question is on option pricing using Black-Scholes The Black-Scholes pricing problem for European binary call Op tion is given by the partial differeutial equation and final condition aV 02V OV 5 > E tr5' CtV =0. V(ST) = 952 ds 5 < E V (S,t) is the option value, S is the underlying asset price, is time. T is expiry; is the constant risk-free interest rate, is the constant volatility and E is the strike price. By introducing the following change of variables log { = 29"(T _ t), ~(r.t) =V(S.t) . show that the problem can be reduced to 0" ~ O~ {1 I > 0 aw (1.0) = 0r2 Or I <0 where a = 2r/o"_ Show that the term ay can be eliminated by further transformation of variable (1,0) =e-au (1.t) Finally show that the drift term will disappear by putting =I+(a = 1)t' . T =t resulting in the diffusion equation and final condition 0 u 7> 0 u (:. 0) 0-2 3<0 The solution of the above diffusion equation is given in terms of the fundamental solution by (y-=)?[4t ~> u(at) = f(u)e dy: u(:.0) = f(:) = 2V 4u Use this to calculate the option value and show that it is given by V(S.t) r(T-')N(d2)-  #### Similar Solved Questions

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