5.3 Risk-Neutrality and Martingales
53
dictum about the no-arbitrage law: “There is no such thing as a free lunch, just a
quick exchange of snacks.”
The notion of a fair process leads to the concept of a martingale, which describes a
stochastic process in which the outcome is un-biassed in the sense that the expectation
value of tomorrows share value, knowing the share values on all previous days, is
the value of today. In other words, the share value can go up or down with equal
probability. For a stochastic variable x this can be written as
E(x n+1 |x n , . . . x 0 ) = x n ,
(5.19)
where the left hand side denotes the expectation value of x n+1 conditioned on previous
values x j with j ≤ n. Thus the best guess of tomorrows value is equal to todays value.
If the x were to denote share prices and the n labels the days, we compare value at
different days and in that case we need to take discounting into account, but the
discussion above should have made clear that we need to discount with the risk-free
rate r f such that the martingale condition now reads
E(e
−r f T S T |S 0 ) = S 0
(5.20)
with the interpretation that tomorrows share value S T , discounted to today, conditioned on today’s value S 0 , is S 0 . In a fair game we cannot foresee the next roll of the
dice so-to-speak. Evaluating expectation values of any function g(S) is then done by
weighted averaging with r f such that we obtain
E(g(S)) = =g(S) =
g(S)) r f (S, t)d S ,
(5.21)
which constitutes a method to evaluate expectation values in a risk-neutral way using
the weighting function r f , where r f d S is the martingale measure.
5.4 Dynamic Hedging
We now take a closer look at the mechanics of dynamic or −hedging, namely the
continuous adjustment of the ratio of shares to options in a risk-free portfolio. First
consider the left plot in Fig. 5.1, which shows the price of a European call option as a
function of the share price for different times until maturity. At maturity, the price of
the option is obviously the payoff function, shown as the solid black line. At earlier
times the price of the option is higher than the payoff function, which reflects the
expectation that there is a finite probability that the option will end up in the money,
which means it will create a profit for the option holder, because the share price S at
maturity exceeds the strike price K . If the share price is below the strike price—out
of the money—the option price is low, but not zero, because there is still a chance
that an upward motion of the share price can make the option profitable at maturity.
53
dictum about the no-arbitrage law: “There is no such thing as a free lunch, just a
quick exchange of snacks.”
The notion of a fair process leads to the concept of a martingale, which describes a
stochastic process in which the outcome is un-biassed in the sense that the expectation
value of tomorrows share value, knowing the share values on all previous days, is
the value of today. In other words, the share value can go up or down with equal
probability. For a stochastic variable x this can be written as
E(x n+1 |x n , . . . x 0 ) = x n ,
(5.19)
where the left hand side denotes the expectation value of x n+1 conditioned on previous
values x j with j ≤ n. Thus the best guess of tomorrows value is equal to todays value.
If the x were to denote share prices and the n labels the days, we compare value at
different days and in that case we need to take discounting into account, but the
discussion above should have made clear that we need to discount with the risk-free
rate r f such that the martingale condition now reads
E(e
−r f T S T |S 0 ) = S 0
(5.20)
with the interpretation that tomorrows share value S T , discounted to today, conditioned on today’s value S 0 , is S 0 . In a fair game we cannot foresee the next roll of the
dice so-to-speak. Evaluating expectation values of any function g(S) is then done by
weighted averaging with r f such that we obtain
E(g(S)) = =g(S) =
g(S)) r f (S, t)d S ,
(5.21)
which constitutes a method to evaluate expectation values in a risk-neutral way using
the weighting function r f , where r f d S is the martingale measure.
5.4 Dynamic Hedging
We now take a closer look at the mechanics of dynamic or −hedging, namely the
continuous adjustment of the ratio of shares to options in a risk-free portfolio. First
consider the left plot in Fig. 5.1, which shows the price of a European call option as a
function of the share price for different times until maturity. At maturity, the price of
the option is obviously the payoff function, shown as the solid black line. At earlier
times the price of the option is higher than the payoff function, which reflects the
expectation that there is a finite probability that the option will end up in the money,
which means it will create a profit for the option holder, because the share price S at
maturity exceeds the strike price K . If the share price is below the strike price—out
of the money—the option price is low, but not zero, because there is still a chance
that an upward motion of the share price can make the option profitable at maturity.