 # 6. We now present the derivation of Black-Scholes-Merton equation i...

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6. We now present the derivation of Black-Scholes-Merton equation in Lishang Jiang Section 5.2. Let c(t,x) denote the value of an option at time t if the stock price at that time is S(t) = x. We want to derive an equation for c(t,x) and hence obtain its formula. Like in (2.2), construct a portfolio D=C-AS, (7.47) where denotes the number of shares of underlying stock. Suppose that (t) = 5(0) + lo dc(s, S(s)) - 1. A(s) dS(s) (7.48) which means (by the equivalence between (5.5) and (5.6)) = - AdS(t). (7.49) As in (2.3), we can manage to choose A SO that § is not random. By the arbitrage-free principle, we expect that the resulting § behaves like a bank deposit, i.e., (same as (4.1)) = (7.50) Here r is the risk-free interest rate for the money market. Show that if we choose (7.51) and let c(t, S) satisfy 8c - rc = 0, (7.52) then we have (7.50). [Hint: Use Itô formula to calcualte dc and then stick the result into the right hand side of (7.49). If we want to enforce (7.50), we should set the right hand side of (7.49) equal to rddt which is rc(t,S(t))dt - r^(t)S(t)dt.]

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