of the PDE
start with a more general argument. Let
f(S,t) denote the value of any derivative security (e.g., a call
option) at time t when the stock price
is S. Note that, by assumption,
the derivative price does not depend on the path of S, but only on the current value of S.
recall that for the stock price
consider the riskless hedge portfolio in Chapter 2, topic 2.4.
share of stock.
number of derivative securities, k, is
value of this portfolio, V, is given
the change in the value is
the df equation and the dS equation into this portfolio and rearranging, we get
important fact about this equation is that dV
is independent of the stochastic term z.
This means that all the risk of the derivative can be hedged by holding
of the derivative.
dV is instantaneously deterministic, the portfolio return must equal
the instantaneous risk-free return. In
for dV and V, we get
is called the Black-Scholes (partial) differential equation.
The solution to this equation, subject to the appropriate boundary
condition for f, determines the value
of the derivative security. For a
European call option, the boundary condition is
solution of the differential equation for the call option is given by the Black-Scholes
can also apply Ito’s Lemma to derive the implied distribution of stock prices
from a geometric Brownian motion model of stock prices.
The next topic Distribution of Stock
Prices provides a technical overview of this issue.
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Financial Trading System