Options and Black-Scholes formula (Part 2)

Kirill Bogomolov

6 min readMar 11, 2023

Part 2

In part 1, we discussed the difficulties of the resulting equation:

We have no idea about the drift parameter $\mu(t)$. Intuitively, this is the “risk” parameter for a specific stock, coin, or token (but formally, the risk parameter is defined in another way, there will be a formula below). And this parameter can be time-dependent.
The current equation is quite difficult to solve. We can’t just integrate that because the right part has non-constant coefficients $\mu(t)S$ and $\sigma S$.

Let’s solve problems one by one

Let’s step aside a little and remember how the value of a risk-free asset is generally determined.

A risk-free asset is, for example, a bond. That is, such instruments that are guaranteed to bring income in the future with a rate of r, called the risk-free interest rate. For example, such instruments are used to protect themselves from inflation.

For simplicity, we assume that the risk-free interest rate is constant.

Then the differential equation for the cost of a risk-free asset is given by the equation

The bond equation doesn’t have a stochastic part, and drift r is usually known to anyone (for example, r=0.1 means that the annual risk-free rate equals to 10%)

Let’s suppose that our \mu parameter is not an arbitrary function, but a risk-free rate. In other words, if our equation would be defined as follows:

then the equation would have 1 less unknown parameter, and we could move to problem 2.

In short, it would be cool to get rid of the unknown drift. But is there any way to do it?

It turns out yes!

Recalling the mathematical analysis (and omit some conditions of the theorems), there is a theorem that allows switching between measures (Radon-Nikodym theorem), and the ‘transition coefficient’ between measures is called the Radon-Nikodym derivative and is defined as follows:

𝜈, 𝜆 — Lebesgue measures

In our case, the Radon-Nikodym derivative is not a ‘real’ function, but a stochastic process.

Girsanov’s Theorem and risk-neutral measure

Given the equation

we want to get

At first glance, it seems that this is impossible. In fact, it turns out that we just need to switch to another probability measure — “risk-neutral”.

Firstly, I am going to remind what’s a martingale

Martingale — is a stochastic process for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values.

We have the equation in the measure P:

But S_t is not a martingale in the measure P, because there’s a drift:

Our objective:

Find a measure Q such that the parameter 𝜇(t) does not appear at all.

Recall that equation for the bond is given by:

Let’s consider the discounted price of the underlying asset under the measure P:

Such a process is not a martingale in measure P. But if we make such a process a martingale, then we will see that in measure Q we get rid of the unknown drift 𝜇(t).

Let’s write a differential equation for that process. Using Ito’s lemma for that process, we get:

Substituting the equations for dS_t and dB_t:

Simplifying this expression and canceling the terms above dt, we obtain the final formula for the process S_t / B_t

And now the most important thing

If we make a substitution:

then process

is a martingale

Let’s now see how the process S_t will look like under such a change of measure.

Now let’s express dW_t in terms of dW~_t and substitute it into the equation for dS_t:

Finally, we get

You may ask:

Why is dW~_t a Wiener process under the measure Q?

Girsanov’s theorem gives an answer to this.

Let the process {𝛼_t} is adapted to the natural filtration generated by Wiener process {W_t} (under the measure P)

Let’s define the ‘stochastic exponential’:

If Z_t is a positive martingale and

then Z_t is the Radon-Nikodym derivative for the new measure Q,
and the process

is a Wiener process under the measure Q

In other words,

We have made the following change of measure:

That is, in our case

It means the process Z_t is given as follows:

It is easy to check that the process Z_t is indeed a positive martingale and has Ep[Z_t] = 1:

consider Ep[Z_t] as the product of the exponents
note that one expectation of exponent is a mean of a log-normal random variable, and then you will see that expectation indeed equals to 1

Thus, the process

is indeed a Wiener process under measure Q.

Thus, we have moved to such a probability measure, where the unknown parameter 𝜇(t) becomes equal to the interest rate 𝑟. Moreover, if we consider the discounted (by risk-free rate) price 𝑆(t) under such a measure, then such a process will be a martingale. That’s why such a measure is also called a martingale measure.

P.S. it’s essential that the price must be discounted.