The mathematics of option pricing can appear intimidating especially if you don't know much about stochastic calculus and have only a rudimentary knowledge of random processes. The good news is that the basic ideas are rather easy to grasp with only a solid background in basic probability theory. What follows is an attempt to explain option pricing without using stochastic calculus or the Black-Scholes equation.

An option has an uncertain payoff value. This is why they are hard to price. The only way to price them is to come up with an expected payoff value which you then discount using the risk free interest rate. If V is the value of the option then the equation is:

\[\tag{1}V = e^{-rT}E[f(S(T))]\]

\(r\) is the risk free interest rate, \(T\) is the the length of time to expiration, \(f(S(T))\) is a function of the price of the underlying at expiration, represented by \(S(T)\), and \(E[ ]\) means find the expected value.

There are some assumptions behind this formula. The first is that payoff or exercise of the option only occurs at expiration. This is true for European style options. There are also American style options that can be exercised any time up to expiration. For some of these options, expected payoff values at times other than expiration have to be taken into account. This makes them harder to price. To keep things simple I will stick to European style options.

The other assumption is that the payoff value is a function only of \(S(T)\), the price of the underlying at expiration. There are some options for which this is not true. The value of barrier options for example, depend on whether \(S(t)\) did or did not breach certain levels (barriers) before expiration. These are called path dependent options and their payoff value can in general depend on the entire price history \(S(t)\). What follows applies only to options that are not path dependent.

So the problem of pricing an option comes down to determining the expected value of its payoff function, \(f(S(T))\). To calculate a mathematical expectation you need a probability distribution for \(S(T)\). The only way to objectively arrive at one is to come up with a model for how \(S(t)\) behaves during the life of the option.

The model will depend on what type of asset \(S(t)\) represents. Stocks will behave differently than bonds for example. The following discussion will be limited to stocks.

If \(S(t)\) represents the price of a stock as a function of time then one widely accepted model is that it is a lognormal random process. Symbolically you have:

\[\tag{2}S(t) = S(0)e^{W(t)}\]

where \(W(t)\) represents a series of normal random variables, one for each value of \(t\), called a Wiener process,

\[\tag{3}W(t)=N(\mu t, \sigma^2t)\]

This is a normal random variable with mean equal to \(\mu t\) and variance equal to \(\sigma^2t\). It has a probability density function (pdf) given by:

\[\tag{4}p(W(t)) = \frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-(W(t)-\mu t)^2}{2\sigma^2}}\]

From eq. 2 you can see that this is also the pdf for

\[\tag{5}\ln\frac{S(t)}{S(0)}\]

The values of \(\mu\) and \(\sigma\) can be estimated from historical data or you can guess what their values might be over the life of the option. There is always the question of whether or not these values remain constant or even if this model has any validity. My goal here is just to show how to value the option given this model.

The value of \(\mu\) can also be found from a simple arbitrage argument. If you calculate the expectation value of \(S(t)\), using the above pdf, you get:

\[\tag{6}E[S(t)] = S(0)e^{(\mu+\frac{\sigma^2}{2})t}\]

This increases in time just like an interest rate investment. The principle of no arbitrage says that the rate of increase should equal the risk free interest rate. This means \(\mu\) must equal

\[\tag{7}\mu = r - \frac{\sigma^2}{2}\]

So the point is that given the risk free interest rate, \(r\), the volatility, \(\sigma\), and the pdf you can, in principle, calculate the expectation of \(f(S(T))\). This is not to say that it will be possible to get an analytical expression for the expectation of every possible function of \(S(T)\). One case where it is possible is for a simple call option with strike price \(X\), where

\[\tag{8}f(S(T)) = \mathrm{max}(S(T)-X,0)\]

The expectation is then given by the following integral:

\[\tag{9}E[f(S(T))] = \int_{\frac{X}{S(0)}}^{\infty}(S(T)-X)p\left(\ln\frac{S(T)}{S(0)}\right)\frac{dS(T)}{S(T)}\]

Putting everything together and with some manipulation you get the following expression for the value of the option.

\[\tag{10}V = \frac{e^{-rT}}{\sqrt{2\pi\sigma^2T}}\int_{\ln(X)}^\infty (e^w-X) e^{\frac{-(w-\ln(S(0)) - (r-\frac{1}{2}\sigma^2)T)^2}{2\sigma^2T}}dw\]

This is the integral form of the Black-Scholes equation for a simple call option. It is possible to solve the integral in this case to get the well known formula for the value of a European call option on a stock with no dividend.

\[\tag{11}V = S(0)\Phi(d_1)-Xe^{-rT}\Phi(d_2)\]

where the function \(\Phi()\) is the cumulative distribution function for a normal random variable and the variables \(d_1\) and \(d_2\) are:

\[\tag{12}d_1 = \frac{\ln(\frac{S(0)}{X})+(r+\frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\]

\[\tag{13}d_2 = d_1 - \sigma\sqrt{T}\]

You can get similar solutions for european style put options and some barrier options. For options that can not be solved analytically there are three commonly used solution methods.

There is the binomial lattice method which converts the problem into a form where \(S(t)\) can take on only a set of discrete values at discrete time steps. The probabilities are then given by a binomial distribution as opposed to the normal distribution in the continuous case. In the limit as the number of values and time steps increases to infinity, the discrete problem converges to the continuous one.

Another alternative is to numerically solve the Black-Scholes partial differential equation using finite difference approximations of the partial derivatives. You end up with a large number of equations that have to be solved for the option value. It is in some ways similar to the binomial lattice method.

Finally there is Monte Carlo simulation. Here the time evolution of \(S(t)\) is simulated using a stochastic differential equation (sde) to move \(S(t)\) forward in time by small increments. The particular sde used will depend on the model. The value of \(f(S(T))\) is averaged over a large number of simulation runs to get an estimate for the expectation value. For european style options this is probably the most flexible method since you can use an arbitrarily complex sde for the time evolution of \(S(t)\). To get good results, a large number of simulation runs need to be done. This can make the method slow.

So these are some of the basic ideas behind option pricing. Presented with no stochastic calculus and no Black-Scholes equation (although we did end up with an integral form of the B-S eq.). Much of the details are left out, only one example for european style call options was given, but you should get the basic idea. To dig deeper the following two books are helpful.

Investment Science, David G. Luenberger.

Derivatives: the theory and practice of financial engineering, Paul Wilmott.

© 2010-2012 Stefan Hollos and Richard Hollos

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