The 2nd post in a series of 5 (first post here) in which the Black-Scholes-Merton Model for pricing European put and call options is derived.

1. Brownian Motion as a Symmetric Random Walk Limit

This follows Stephen Blyth's Excellent Book An Introduction to Quantitative Finance closely, with embellishment using python and some additional notes. This is probably:

• not very helpful if you pragmatically want to price an option
• overwhelming if you don't like math
• may miss some of the contexts you'd want if you have a lot of mathematical maturity

In the last post, we derived Brownian motion by considering a symmetric random walk (one with an equal likelihood of each change) and taking the limit as the time difference between steps goes to 0. Now that we know what Brownian motion is, we can use that to intuit stochastic differential equations.

For \(\Delta t > 0\), we can write a difference equation such as

\(S_{t + \Delta t} - S_t = \mu \Delta t + \sigma (W_{t + \Delta t} - W_t) = \mu \Delta t + \sigma \sqrt{\Delta t} W\), \(W \sim \mathcal{N}(0, 1)\)

That is, we say that the difference between a stock at time \(t + \Delta t\) and time \(t\) is the weighted sum of two functions \(\mu\) and \(\sigma\).

If we fix \(\mu\) and/or \(\sigma\) we can view the effect that it has on the outcome process:

Intuitively, we see that \(\mu\) dictates our slope. It adds (or removes) a certain amount at each timestep and is only dictated by the length of the timestep.

\(\sigma\) dictates the strength of the Brownian motion effect on the difference. Higher \(\sigma\) can lead us to some pretty rocky territory which can be intuitively seen when we look at the middle row of the plot which isolates the effect of \(\sigma\) by fixing \(\mu\) at 0. We can also see that consistent steady returns from \(\mu\) can 'tame' the variability introduced by \(\sigma\) if we look at the upper or lower right plots.

We write the limits as \(\Delta t \to 0\):

\(dS_t = \mu dt + \sigma d W_t\)

or more generally

\(dS_t = \mu(S_t, t)dt + \sigma(S_t, t) d W_t\)

This is a stochastic differential equation.

The meaning of \(dW_t\) is non-trivial, although the discrete analog \(\sqrt{\Delta t}W\) where \(W \sim \mathcal{N}(0, 1)\) is helpful for conceptual understanding.

A reasonable choice of functions \(\mu(S_t, t), \sigma(S_t, t)\) to model the behavior of a stock price is to:

• assume that the drift \(\mu\) of a stock is proportional to its price, \(\mu(S_t, t) = \mu S_t\)
  • with no random term, the ODE would be \(dS_t = \mu S_tdt \implies S_t = S_0e^{\mu t}\)
• For \(\sigma(S_t, t)\), we might assume that the uncertainty about the stock's percentage return is constant
  • i.e. \(p(S_t) = 100 \to \not \in (99, 101) \equiv p(S_t) = 50 \to \not \in (49.5, 50.5)\)
  • Then over small \(\Delta t\), \(\mathrm{Var}(\frac{\Delta S_t}{S_t}) \approx \sigma^2 \Delta t \implies \mathrm{Var}(\Delta S_t) \approx \sigma^2 S_t^2 \Delta t\)

Therefore, one possible process for the evolution of the stock is

\(dS_t = \mu S_t dt + \sigma S_t d W_t\)

which is the definition that the stock follows Geometric Brownian Motion. [1]

The plot above looks like the jump processes we saw earlier without any jumps.

Geometric Brownian motion is used widely in finance. It has a bunch of characteristics that make it a natural fit for modeling stocks:

• it looks like a stock
• it quacks like a stock
• it waddles like a stock

It is not a duck though, because:

• It does not account for volatility changing over time
  • If we assume that volatility \(\propto t, S_t\), then we have a local volatility model
  • If we assume that the volatility is it's own random variable driven by a different equation (usually another Brownian motion), then we have a stochastic volatility model [2]
• It does not account for jumps caused by unpredictable events or news
  • Attempts to model these discontinuities are given by using jump processes as briefly alluded to above

The next post in this series will take a look at Ito's lemma. We will need it for some of the steps that will follow in our quest to derive the Black-Scholes formula.

Note [1] Most plots look pretty cool in 3D. Here is a Brownian surface:

Note [2] Stochastic volatility models are important. They resolve a shortcoming of the Black-Scholes model that we are deriving, namely the assumption that the underlying volatility is constant over the life of the derivative and unaffected by changes in the price of the underlying security. By modeling this change in volatility as its own stochastic process, the model gets more accurate.