Consider a portfolio containing t units of the risky asset and M t dollars of the

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Consider a portfolio containing Δt units of the risky asset and Mt dollars of the riskless asset in the form of a money market account. The portfolio is dynamically adjusted so as to replicate an option. Let St and V (St,t) denote the price process of the underlying asset and the option value, respectively. Let r denote the riskless interest rate and Πt denote the value of the self-financing replicating portfolio. When the self-financing trading strategy is adopted, we obtain

t = At St + Mt and dt dt = At dSt + r M, dt,where r is the riskless interest rate. Explain why the differential term Stt does not appear in dΠt . The asset price dynamics is assumed to follow the Geometric Brownian process:

d St St =  dt + o dZt.In order that the option value and the value of the replicating portfolio match at all times, show that the number of units of asset held is given by

Ar =

How should we proceed in order to obtain the Black–Scholes equation for V ?

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