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
denotes the number of shares of underlying stock. Suppose that
(t) = 5(0) + lo dc(s, S(s)) - 1. A(s) dS(s)
which means (by the equivalence between (5.5) and (5.6))
= - AdS(t).
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
Here r is the risk-free interest rate for the money market.
Show that if we choose
and let c(t, S) satisfy
8c - rc = 0,
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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