# The Black-Scholes Partial Differential Equation

## Table of Contents

## Attention Conservation Notice

The 4th post in a series of 5 in which the Black-Scholes-Merton Model for pricing European put and call options is derived.

- Brownian Motion as a Symmetric Random Walk Limit
- Stochastic Differential Equations & Geometric Brownian Motion
- Itô's Lemma

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

## Recap

In the last post we learned a cool new way of taking the derivative of a random variable and time (Itô's Lemma) and learned that if we apply it to geometric Brownian motion then we can can that the distribution of \(\log{S_T|S_t}\) is log-normal. As a result, we can predict where the next time step in a stock modeled by geometric Brownian motion is likely to fall.

Let's apply our shiny new tool to do some option pricing.

## Call \(\to\) Call'

Consider the price of a call (any derivative works) \(C_{K}(t, T)\)
which is a function of the stock price and a strike price \(K\). We haven't really discussed
calls in this series yet (outside of the *attention conservation
notice*), so here is a brief definition if you aren't familiar:

A

call[1] option is a financial contract that gives the person owning the option the right (but not the obligation) to buy the underlying security (stock, bond, rare holographic charizard in mint condition) at a specified price (thestrike price) within some time period. The owner of the call option makes a profit when the underlying asset increases in price over the strike price.

Back to our call \(C_{K}(t, T)\):

If we attempt to differentiate the function of our call price with Ito's lemma, we get to split it into a time term, a term about the random variable, and a term reflecting the quadratic variation of the underlying Brownian motion and get the following stochastic differential equation:

\(\partial C_K(t, T) = (\frac{\partial C_K(t, T)}{\partial t} + \frac{\partial
C_K(t, T)}{\partial S_T} \mu S_t + \frac{1}{2}\frac{\partial^2 C_K(t,
T)}{\partial S_t^2} \sigma^2 S_t^2)dt + \frac{\partial C_K(t,
T)}{\partial S_t}\sigma S_t dW_t\) **[2]**

## Constructing a simple portfolio

Suppose we create a portfolio \(\Pi\) that consists of long **[3]** one
option and **[3]** short \(\frac{\partial C_K(t, T)}{\partial S_t}\) stock.

Then at time \(t\), the price of our portfolio is

\(\Pi_t = C_K(t, T) - \frac{\partial C_K(t, T)}{\partial S_t}S_t\)

## Portfolio \(\to\) Portfolio'

using the expressions for \(\partial C_K(t, T)\) and \(\partial S_T\) above:

\(d \Pi_t = (\frac{\partial}{\partial t}C_k(t, T) + \frac{1}{2}\frac{\partial^2 C_K(t, T)}{\partial S_t^2}\sigma^2 S_t^2) dt\)

This portfolio instantaneously at time t has no exposure to the term
\(dW_t\). Therefore, it is instantaneously a replicating portfolio for
the money market account **[4]** and must grow at rate \(r\) (the risk-free interest
rate). Then

\(d \Pi_t = r \Pi_t dt \implies \frac{\partial}{\partial t}C_k(t, T) + \frac{1}{2}\frac{\partial^2 C_K(t, T)}{\partial S_t^2}\sigma^2 S_t^2 = r (C_K(t, T) - \frac{\partial C_K(t, T)}{\partial S_t}S_t)\)

Rearranging:

\(\frac{\partial}{\partial t}C_k(t, T) + \frac{1}{2}\frac{\partial^2 C_K(t, T)}{\partial S_t^2}\sigma^2 S_t^2 + \frac{\partial C_K(t, T)}{\partial S_t}S_t - rC_K(t, T) = 0\)

This is the **Black-Scholes partial differential equation** for a European
derivative contract. **[5]**

## Black-Scholes Partial Differential Equation Usage

The solution to our equation above, under the boundary conditions \(C_K(T, T) = (S_t - K)^+\) (the value of our call at expiration time \(T\) is the profit of the underlying security minus the strike price), is the tried and true Black-Scholes model. We will go this extra step in the next (and final) post in this series.

## Financial Interpretation of the PDE

The key insight behind our PDE is that if we disregard transaction costs, we can perfectly hedge our call option at a given time \(C_K(t, T)\) by buying and selling the underlying asset (the \(\frac{\partial C_K(t, T)}{\partial S_t}\) short stock in our portfolio \(\Pi\)) in just the right way to completely eliminate risk. If this combination exists, then there is an exact way to price the option in order to have our "riskless" portfolio.

We can also interpret the initial 2-sided form:

\(\frac{\partial}{\partial t}C_k(t, T) + \frac{1}{2}\frac{\partial^2 C_K(t, T)}{\partial S_t^2}\sigma^2 S_t^2 = r C_K(t, T) - r\frac{\partial C_K(t, T)}{\partial S_t}S_t\)

The left-hand side consists of a time decay term
\(\frac{\partial}{\partial t}C_k(t, T)\) and is called *theta* (\(\theta\))
and a term denoting the convexity of the derivative value with respect
to the underlying value \(\frac{1}{2}\frac{\partial^2 C_K(t,
T)}{\partial S_t^2}\sigma^2 S_t^2\) which is called *gamma* \(\gamma\).

The right-hand side is the riskless return from a long position in the derivative and a short position consisting of \(\frac{\partial C_K(t, T)}{\partial S_t}\) shares of the underlying security \(S_t\).

The key insight is that the right-hand side is riskless, so we can represent a riskless portfolio as a sum of \(\theta\) and \(\gamma\).

For an option, \(\theta\) is typically negative since as time runs out on the option it usually loses value. \(\gamma\) is typically positive, so it reflects the gains in holding the option. The equation states that over any infinitesimal time interval \(\theta\) and \(\gamma\) offset each other such that the result is a return at the riskless rate.

## Next time

We wrap up this series by deriving the Black-Scholes formula

## Extra Notes

**Note [1]**

More information on call options:

https://www.investopedia.com/terms/c/calloption.asp

It is also worth reading about put options as well:

https://www.investopedia.com/terms/p/putoption.asp

**Note [2]**

This is using the most well-known form of Ito's lemma:

\((\frac{df}{dt} + \frac{df}{dS_t} \mu(S_t, t) + \frac{1}{2}\frac{d^2f}{dS_t^2} \sigma^2(S_t, t))dt + \frac{df}{dS_t} \sigma(S_t, t) dW_t\)

**Note [3]**

Long means the investor owns the option here, and short means that the investor owes those stocks to someone, but they haven't bought them yet. Here is more detail: https://www.investopedia.com/ask/answers/100314/whats-difference-between-long-and-short-position-market.asp

**Note [4]**

In this case, the money market account means that we are replicating
an account that provides interest at a rate \(r\). This is deduced
because we are assuming that there is no arbitrage opportunity to be
had. We can also call \(r\) the *risk-free* interest rate.

**Note [5]**

A European derivative contract is a derivative that limits execution
to a given date. This is in contrast to other types of options like an
American option which let you execute on *any* day leading up to
expiration, or Bermudan which let you execute on *specific* days (e.g.
once per month) leading up to the expiration.

For example, suppose you have an option on 100 shares of $STOCK with a $100 strike price and an expiration one month from the day you purchase the option. The cost of the option is $10 per option, for a total cost of $1000 (10 * 100).

When the option expires, it happens to be trading at $150 ( ͝סּ ͜ʖ͡סּ). Then the owner of the European call option has the right to purchase the underlying stock at $100 - exercising their option - and they make a profit of $40 per share ($150 - $100 - $10 / share for a total gain of $40 * 100 = $4000).