256 Frequently Asked Questions In Quantitative Finance
of a zero-coupon bond maturing at timeTare bought:
dGt=
αGt
σS
dS+
G−ασGStS
e−r(T−t)
d(e−r(T−t)).
Such a strategy is self financing because the values of
the stock and bond positions add up toG. Because
of the existence of such a self-financing strategy and
because at timet=Twe have thatGTis the call payoff
we must have thatGtis the value of the call before
expiration. The role of the self-financing strategy is to
ensure that there are no arbitrage opportunities.
Thus the price of a call option is
e−r(T−t)EQt[max(ST−K,0)].
The interpretation is simply that the option value is the
present value of the expected payoff under a risk-neutral
random walk.
For other options simply put the payoff function inside
the expectation.
This derivation is most useful for showing the link
between option values and expectations, as it is the
theoretical foundation for valuation by Monte Carlo
simulation.
Now that we have a representation of the option value
in terms of an expectation we can formally calculate this
quantity and hence the Black–Scholes formulæ. UnderQ
the logarithm of the stock price at expiration is normally
distributed with meanm=ln(St)+
(
r−^12 σ^2
)
(T−t)
and variancev^2 =σ^2 (T−t). Therefore the call option
value is
e−r(T−t)
∫∞
lnK−m
v
(em+vx−K)
e−
1
2 x
2
√
2 π
dx.
