52
5 Black-Scholes Differential Equation
5.3 Risk-Neutrality and Martingales
In this section we illustrate how the two approaches, from Sects. 4.6 and 5.2 are
related. Consider two stochastic processes, one for the share price S that was already
used in Sect. 4.6 and a second one for a stochastic process X for which the growth
rate ρ is replaced by r f
d S = ρ Sdt + σ SdW (t)
d X = r f Xdt + σ XdW (t)
(5.18)
where dW is a Wiener-process. We assume that both processes have the same initial
conditions S 0 = X 0 and the same volatility but have different growth rates ρ and r f .
Therefore, the processes are clearly different. If we use the first process with S we
arrive at a price for the call option in (4.40) and if we use the second process with X
we would arrive at (5.17), which was initially derived by creating a risk-free hedged
portfolio. Note that the growth rate ρ depends on the expectation of the option writer
or some market analyst and is therefore somewhat arbitrary.
We can then compare call options c S priced according to process S to options
c X based on process X. To be specific, we consider a call option with strike price
K above the present share value S. This makes the option c S more expensive than
c X . Imagine an option writer selling an option with price c S and at the same time
purchases a less expensive option c X , based on process X which is hedged with the
underlying asset and thus generates a risk-free profit. In this case the package based
on c X produces risk-free profit on top of the difference between the option prices c X
and c S which is already in the pocket of the option writer and therefore also risk free.
The total profit of the option writer is risk-free and higher than the risk-free rate. But
this contradicts the basic assumption that systematic arbitrage or a risk-free profit
above the risk-free rate is not possible. We conclude that writing options based on
expected growth rates will give an unfair advantage to the option writer. In practice
nobody would purchase option c S .
We find that, at least for simple options, we can use the hypothetical process X
with the same volatility as the underlying share but with risk-free growth rate r f
to calculate the fair, no-arbitrage, price of the option by integrating the expectation
values of the payoff function using the log-normal distribution function (4.32) but
with ˆ
ρ = ρ − σ
2
/2 replaced by r f − σ
2
/2. This modified distribution function we
denote by r f (S, t). Calculating expectation values using this modified distribution
function is called risk-neutral valuation and allows us to calculate option prices fairly.
Central in the discussion of efficient markets is that arbitrage—unfair advantage—
to make a risk-free profit above the risk-free rate, is prohibited. Yet, there are arbitrageurs who exploit minute imbalances to make just such a profit. This is similar
to the physical law of conservation of energy and yet, there are vacuum fluctuations
where energy conservation is violated, provided it happens rapid enough, limited
by Heisenberg’s uncertainty principle. In the same way arbitrage is prohibited in an
efficient market, but can be briefly violated. In that sense, to paraphrase Wilmott’s [3]
5 Black-Scholes Differential Equation
5.3 Risk-Neutrality and Martingales
In this section we illustrate how the two approaches, from Sects. 4.6 and 5.2 are
related. Consider two stochastic processes, one for the share price S that was already
used in Sect. 4.6 and a second one for a stochastic process X for which the growth
rate ρ is replaced by r f
d S = ρ Sdt + σ SdW (t)
d X = r f Xdt + σ XdW (t)
(5.18)
where dW is a Wiener-process. We assume that both processes have the same initial
conditions S 0 = X 0 and the same volatility but have different growth rates ρ and r f .
Therefore, the processes are clearly different. If we use the first process with S we
arrive at a price for the call option in (4.40) and if we use the second process with X
we would arrive at (5.17), which was initially derived by creating a risk-free hedged
portfolio. Note that the growth rate ρ depends on the expectation of the option writer
or some market analyst and is therefore somewhat arbitrary.
We can then compare call options c S priced according to process S to options
c X based on process X. To be specific, we consider a call option with strike price
K above the present share value S. This makes the option c S more expensive than
c X . Imagine an option writer selling an option with price c S and at the same time
purchases a less expensive option c X , based on process X which is hedged with the
underlying asset and thus generates a risk-free profit. In this case the package based
on c X produces risk-free profit on top of the difference between the option prices c X
and c S which is already in the pocket of the option writer and therefore also risk free.
The total profit of the option writer is risk-free and higher than the risk-free rate. But
this contradicts the basic assumption that systematic arbitrage or a risk-free profit
above the risk-free rate is not possible. We conclude that writing options based on
expected growth rates will give an unfair advantage to the option writer. In practice
nobody would purchase option c S .
We find that, at least for simple options, we can use the hypothetical process X
with the same volatility as the underlying share but with risk-free growth rate r f
to calculate the fair, no-arbitrage, price of the option by integrating the expectation
values of the payoff function using the log-normal distribution function (4.32) but
with ˆ
ρ = ρ − σ
2
/2 replaced by r f − σ
2
/2. This modified distribution function we
denote by r f (S, t). Calculating expectation values using this modified distribution
function is called risk-neutral valuation and allows us to calculate option prices fairly.
Central in the discussion of efficient markets is that arbitrage—unfair advantage—
to make a risk-free profit above the risk-free rate, is prohibited. Yet, there are arbitrageurs who exploit minute imbalances to make just such a profit. This is similar
to the physical law of conservation of energy and yet, there are vacuum fluctuations
where energy conservation is violated, provided it happens rapid enough, limited
by Heisenberg’s uncertainty principle. In the same way arbitrage is prohibited in an
efficient market, but can be briefly violated. In that sense, to paraphrase Wilmott’s [3]