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3.4 Hedging an option with the underlying (BlackScholes)
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Take a lognormal distribution of the price increments
in (3.3.1#eq.1)
modeling the price evolution of a share
with a volatility
in
an efficient market where all the riskfree portfolios earn the same
deterministic spot rate
.
For simplicity, neglect the payment of dividends
and assume
that the underlying trades continuously without transaction costs
(extensions are considered in exercises 3.03 & 3.04).
Now construct a portfolio combining an option (or indeed any kind of
stock market derivative) with a value
and an unspecified,
but constant number
of the underlying share. The initial
value of this portfolio and its incremental change per timestep are

(3.4#eq.1) 
Using Itô's lemma (3.3.2#eq.2) to substitute for the stochastic
value of the option price increment
and choosing
for a lognormal walk with (3.3.1#eq.1), this yields

(3.4#eq.2) 
The random term in
can now be eliminated by continuously hedging
the portfolio so as to maintain the number of shares equal to
.
With this choice, the portfolio becomes deterministic (riskfree) and,
using the standard noarbitrage arguments,
must earn the same amount as the riskfree return
payed when
an amount of cash
is invested at a spot rate
during a small
time
. Equating both earnings

(3.4#eq.3) 
leads directly to the celebrated BlackScholes equation [3]

(3.4#eq.4) 
which shows that the return of a deltahedged 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 BlackScholes 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 BlackScholes 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 noarbitrage 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 lognormal 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
Up: 3 FORECASTING WITH UNCERTAINTY
Next: 3.5 Hedging a bond