Consider the Black-Scholes equation for the option price V(S, t) if theunderlying does not pay out dividends.(i) Show that V(S,t) = AS, where S is the underlying price, t is time,and A is a constant, is a solution to the Black-Scholes equation.(ii) For the solution V(S,t) = AS, define the value A (Delta) used forhedging.

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Answered: Consider the Black-Scholes equation for…

Essentials Of Investments

11th Edition

ISBN:9781260013924

Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.

Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.

Chapter1: Investments: Background And Issues

Section: Chapter Questions

Problem 1PS

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The Black-Scholes equation for the value of an option V(S,t), where S is the value of the underlying asset and t is time, can be transformed into the diffusion equation for a new variable u(x,7) using S = Ee™, t = T − 27/0², k = 2r/o² and - (k-1) x (k+1)² + T V=Eu(x,T) exp 2 4 where r is the risk-free interest rate, σ is the volatility, E is the strike price, and T is the expiry date. The diffusion equation has the solution u(x,T): = 1 TT or L 2√Tπ 1 uo (s) exp 4T : (x - s)²) ds. Determine, showing all steps, the function uo(s) for a European asset- or-nothing call with the pay-off A(S) = SH(S – E). -

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