Variation

We apply the above theorems to the European Put. We assume the same set up as Boness and later Black Scholes, i.e. a lognormal process on the stock price S. In this case, we can calculate the variation of the payoff as the strike price K. The Boness Formula, of which the Black Scholes formula is a special case, can be reinterpreted as an integral over u where u varies from 0 to 1. Once this reinterpretation is done, we can prove that the total variation is finite and equal to the strike price.

Let the stock price process be

$$S = e^{\mu t + \sigma \sqrt{t} z}$$ (2.15)

where z is a standard mean zero, variance 1 normal. We can write

$$z = F^{-1}(u)$$ (2.16)

where F is the cumulative standard normal distribution function. So u varies from 0 to 1. If we write

$$u = F(z)$$ (2.17)

it is easier to interpret the relation of z and u. When $z$ is $-\infty$, $F(z)=0$, so u=0. When z is plus infinity, F is 1, so u is 1.

The payoff function for a European put is

$$P = max[0,K-S ]$$ (2.18)

This is a monotonic decreasing function of S, i.e. for larger S the value of the put gets smaller or stays the same. The exponential function, standard normal cumulative and its inverse are all monotonic increasing functions, so their compositions are as well. So P is a monotonic decreasing function of u. We can also see check this with a few points. If u is zero, z is minus infinity, so S is zero. In that case, P is K. If u is 1, z is plus infinity, S is infinity, so P is 0. It is clear that as u increases from 0, z increases from $-\infty$. Therefore S increases, therefore K-S decreases, and so P decreases from K to 0. After reaching 0, P stays at 0 as u increases further.

By the theorems in Carothers in Chapter 13, the Put Payoff Function therefore has a variation equal to K, since its maximum value is K and its minimum value 0. When the range of integration covers S=0 to S=K, the variation is K.

Theorem 6 (Variation of Put)   The Variation of a European put's payoff for the lognormal model is its strike price.

Discounting this strike price at constant interest rates can only decrease it, so K is still a bound for the variation of such discounted values and therefore for the European put option price.