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which shows that the return of a delta-hedged portfolio (first two term)
is exactly balanced by the return on a bank deposit (last two terms).
The equation describes the price of stock options (call, put and indeed
any derivative security where the price
depends only on the
spot price and time) in a market that is parametrized by the volatility
and the spot rate
.
An important finding is that the price of an option does not explicitly
depend on how rapidly the underlying asset grows: the drift
does not explicitly appear in the Black-Scholes equation. The reason
is that the option price has been calculated relative to the underlying,
so that the drift is here already accounted for in an implicit manner.
In mathematics (3.4#eq.4) is known as a
partial differential equation
(since it has partial derivatives up to first order in time
and second order in the asset price
) of backward parabolic type
(backward because both these derivatives have the same sign,
parabolic since this combination of coefficients yields exactly one
characteristic).
In physics, it bears a strong resemblance with the heat equation, except
that here it is the price of an option that is diffusing with respect to
the underlying rather than heat diffusing in space.
To solve the Black-Scholes equation backward in time, a final
condition needs to be imposed at expiry
reproducing the
terminal payoff
discussed in
chapter 2.
Boundary conditions have to be imposed on the price boundaries
and are obtained from additional no-arbitrage arguments.
A vanilla put option, for example, is increasingly unlikely to be
exercised when the price of the underlying gets very large, so that
the value of a put
when
.
On the lower boundary, (3.3.1#eq.1) shows that the log-normal price
increments of the underlying vanish
when
, so that the problem becomes deterministic: a put option
that is certain to be exercised for a price
at the expiry can be
discounted back to the present value
.
Similar arguments will be used to derive the conditions for a call option
in section 4.4.
SYLLABUS Previous: 3.3.3 Evaluate an expectancy
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