📂Stochastic Differential Equations

Derivation of Black-Scholes Model

Model ¹

At time point $t$, let’s say the price of $S_{t}$ units of the underlying asset $1$, and assume that $S_{t}$ undergoes Geometric Brownian Motion. That is, for Standard Brownian Motion $W_{t}$, drift $\mu \in \mathbb{R}$, and diffusion $\sigma^{2} > 0$, $S_{t}$ is the solution to the following Stochastic Differential Equation. $$ d S_{t} = S_{t} \left( \mu dt + \sigma d W_{t} \right) $$ When a risk-free rate $r \in \mathbb{R}$ is given, the price $F = F \left( t, S_{t} \right)$ of $1$ units of the derivative at time $t$ follows the following [Partial Differential Equation](../../categories/Partial Differential Equations). $$ r F = {{ \partial F } \over { \partial t }} + r S_{t} {{ \partial F } \over { \partial S_{t} }} + {{ 1 } \over { 2 }} \sigma^{2} S_{t}^{2} {{ \partial^{2} F } \over { \partial S_{t}^{2} }} $$

Variables

• $F \left( t, S_{t} \right)$: Derivatives refer to financial instruments such as futures and options.
• $S_{t}$: Underlying Assets are the commodities traded in derivatives, such as currency, bonds, and stocks.

Parameters

• $r \in \mathbb{R}$: Represents the interest rate of Risk-free Assets. A typical example of a risk-free asset is a deposit.
• $\sigma^{2} > 0$: Represents the market’s Volatility.

Explanation

Contrary to common misconceptions about derivatives, futures and options were created as means of hedging against uncertain futures. It was a way to reduce risk by paying a certain premium, even if it cost a bit. The problem was the lack of an appropriate method to price them, and traders traded derivatives based on experience. The Black-Scholes model is an equation that made it possible to mathematically explain the price of such derivatives.

Commonly, the contributors to the Black-Scholes model (1973) are cited as Fischer Black, Myron Scholes, and Robert K. Merton, who introduced the ‘hedge-based derivation’ in this post. Unfortunately, Black passed away in 1995, and Scholes and Merton were awarded the Nobel Prize in Economics in 1997. After the discovery of the Black-Scholes-Merton equation, the options market developed dazzlingly, and a new sub-discipline called financial engineering emerged in academia.

The Comedy of Black

According to Wikipedia², Black frequently changed majors during his PhD studies and had trouble settling down in one field. He switched from physics to mathematics, to computer science, and to artificial intelligence, but eventually made a significant contribution in economics.

Although no reliable reference was found, it is said that Black, during his physics major, realized he couldn’t survive among the surrounding geniuses and became a pioneer in economics/finance, where there were no pioneers actively using mathematics, thus making a name for himself in a wasteland without science monsters.

The Tragedy of Scholes

According to Namuwiki³, Scholes caused a sensation at the 1997 Nobel Economics Prize press conference by saying he would invest the prize money in stocks. The hedge fund Scholes was managing went bankrupt in 1998 due to overconfidence and excessive leverage, following Russia’s default. After the crisis, Scholes managed to return profits to investors and continued as a fund manager until retiring just before the subprime mortgage crisis erupted.

Assumptions

Before delving into the derivation, let’s check some assumptions.

Factors such as commissions, taxes, and dividends are not considered

Think of it as not considering resistance, temperature, or atmospheric pressure in a physics model, which are not the focus of the study. Additionally, it’s assumed that the trend $\mu$ and $\sigma$ are simply constants.

Derivatives depend on the underlying assets and timing

If the price of a derivative is independent of the underlying asset, there’s no need to use the terms ‘derivative’ and ‘underlying’. It’s reasonable for the price of a derivative to change as the price of the underlying asset changes. If it doesn’t change over time (if it’s constant), then there’s no point in pondering the price of a derivative. Thus, it’s assumed that $F$ is a function of at least two factors $t$ and $S_{t}$, even if we can’t specify the exact form. $$ F = F \left( t, S_{t} \right) $$

The underlying asset undergoes Geometric Brownian Motion

The primary application of Geometric Brownian Motion GBM is to explain the price fluctuations of underlying assets like stock prices. It assumes that the change in asset prices is proportional to the asset’s price, and that the price cannot become negative unless the asset is delisted, among other favorable assumptions. $$ d S_{t} = S_{t} \left( \mu dt + \sigma d W_{t} \right) $$

Let’s assume the price $p_{t}$ of some stock follows GBM. The return, defined as taking the log of dividing the closing price on day $t$ by the closing price on day $t-1$, $$ r_{t} = \nabla \log p_{t} = \log {{ p_{t} } \over { p_{t-1} }} $$ aligns with our intuition that the return should be positive if the price increases and negative if it decreases, regardless of the stock’s size. As explained in the “Log-normal Distribution” section, this return follows a normal distribution, focusing on the essence of growth and decay rather than simple fluctuations.

Risk-free assets grow according to Malthusian growth

The Malthusian growth model is the simplest model describing the growth of a population without any limitations or interventions and can be used as an assumption to explain the proliferation of risk-free assets in economics/finance. The risk-free rate is assumed to be a constant $r$, and the financial income is proportional to the size of asset $N_{t}$, so it can be expressed by the following [Ordinary Differential Equation](../../categories/Ordinary Differential Equations). $$ {{ d V_{t} } \over { d t }} = r V_{t} $$

Arbitrage-free pricing: There’s no value difference between portfolios

• A mathematical explanation of the portfolio will be further detailed in this proof.

The assumption of arbitrage-free pricing means that all portfolios we consider are balanced at the same value. For instance, if the value of portfolio $A$ is higher than $B$, a rational market participant would increase the proportion of the more valuable $A$ to make a profit, so there’s no reason to consider $B$. Therefore, it’s assumed that the portfolios we consider are already in a state where no further profits can be made through such arbitrage.

Frictionless market: There are no restrictions on division and short selling

Anyone who has traded stocks knows that there are minimum bidding units, so you can’t trade exactly the amount you want, and there are restrictions on short selling in the Korean stock market, where borrowed short selling (borrowing stocks) is the principle. Being able to divide trading units as desired and short sell without any restrictions can be considered as having no friction opposing actions.

Derivation

Part 1. Portfolio Composition

Let’s assume we can only hold three types of assets:

• Underlying Asset: Let’s say we hold $s$ units.
• Derivative: Let’s say we hold $f$ units.
• Risk-free Asset: An asset that is neither an underlying asset nor a derivative, which can be considered as cash.

If we denote the value of all assets we hold at time $t$ as $V_{t}$, and if $S_{t}$ was the price of $1$ units of the underlying asset and $F \left( t , S_{t} \right)$ was the price of $1$ units of the derivative, it can be represented as follows. $$ V_{t} = f F \left( t, S_{t} \right) + s S_{t} $$ Composing a portfolio means adjusting the amounts of $f$ and $s$, in other words, strategizing on how to invest. Assuming that the trading volume generated by such portfolio composition significantly affects the market is irrational, so the prices of the underlying asset and derivative are independent of the choices of $f$ and $s$. In other words, the mathematical discussions that follow do not change regardless of how $f$ and $s$ are determined.

It’s important to note that $V_{t}$ is not the sum of the total assets. It’s easier to understand if you think of it as looking only at the stock balance, not the cash account. Let’s think about what portfolios could be:

• Savings $V_{t} = 0$: Clear the stock account and put everything into savings to receive interest. It might seem trivial to a scholar who knows nothing but mathematics, but it’s a legitimate strategy for dealing with market crashes or recessions.
• Individual Investor $V_{t} = 5 S_{t}$: Individuals should not touch derivatives. Most individual investors in countries where short selling is banned have this type of portfolio. For example, if $S_{t} = 81,200$ is the stock price of Samsung Electronics, this portfolio is my friend ‘Kim Soo-hyung’s account holding $5$ shares of Samsung Electronics.
• Hedge $\displaystyle V_{t} = 1 \cdot F- {{ \partial F } \over { \partial S_{t} }} \cdot S_{t}$: Consider buying a call option $1$ and short selling the underlying asset ${{ \partial F } \over { \partial S_{t} }}$. If the price of the underlying asset increases significantly on the expiration date of the option, the call option will bring in a large profit, and if the price of the underlying asset falls, profit will have been made from the short sale early on.

The options mentioned in the explanation of hedging and to be covered later are European Options, which are usually known as ‘options that can only be exercised on the expiration date’. American Options can be exercised at any time before maturity, but that’s not our concern here, so don’t be intimidated by the terms European and American.

We will derive the Black-Scholes equation from the last example, a portfolio that hedges a derivative with a spot short sale $$ V_{t} = 1 \cdot F \left( t, S_{t} \right) - {{ \partial F } \over { \partial S_{t} }} \cdot S_{t} $$ Since it’s perfectly hedged, this portfolio is a risk-free asset, and the increment over time $t$ is $$ d V_{t} = d F - {{ \partial F } \over { \partial S_{t} }} d S_{t} $$ Here, since $S_{t}$ is assumed to follow Geometric Brownian Motion, substituting $d S_{t}$ for $\displaystyle S_{t} \left( \mu dt + \sigma d W_{t} \right)$ yields $$ d V_{t} = d F - {{ \partial F } \over { \partial S_{t} }} S_{t} \left( \mu dt + \sigma d W_{t} \right) $$ Considering the assumption of arbitrage-free pricing, the increment of this portfolio should be the same as that of a risk-free asset portfolio. If we assume there’s a price difference between the portfolios, profit could be made by liquidating one portfolio and investing in the other. Since we assumed risk-free assets grow according to Malthusian growth, the risk-free rate $r$ can be expressed by the following [Ordinary Differential Equation](../../categories/Ordinary Differential Equations). $$ {{ d V_{t} } \over { d t }} = r V_{t} $$ Organizing these, we get $$ \begin{align*} d V_{t} =& d F - {{ \partial F } \over { \partial S_{t} }} S_{t} \left( \mu dt + \sigma d W_{t} \right) \\ d V_{t} =& r V_{t} dt \end{align*} $$ thus, $$ \begin{equation} r V_{t} dt = d F - {{ \partial F } \over { \partial S_{t} }} S_{t} \left( \mu dt + \sigma d W_{t} \right) \label{1} \end{equation} $$ is obtained. Now, let’s use Itô calculus to find $dF$.

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Part 2. Itô Calculus

Itô’s Lemma: Given an Itô Process $\left\{ X_{t} \right\}_{t \ge 0}$, $$ d X_{t} = u dt + v d W_{t} $$ for function $V \left( t, X_{t} \right) = V \in C^{2} \left( [0,\infty) \times \mathbb{R} \right)$, if we set $Y_{t} := V \left( t, X_{t} \right)$, then $\left\{ Y_{t} \right\}$ is also an Itô Process, and the following holds. $$ \begin{align*} d Y_{t} =& V_{t} dt + V_{x} d X_{t} + {{ 1 } \over { 2 }} V_{xx} \left( d X_{t} \right)^{2} \\ =& \left( V_{t} + V_{x} u + {{ 1 } \over { 2 }} V_{xx} v^{2} \right) dt + V_{x} v d W_{t} \end{align*} $$

In Geometric Brownian Motion, distributing $S_{t}$ according to the distributive law gives $$ d S_{t} = \mu S_{t} dt + \sigma S_{t} d W_{t} $$ and, from Itô’s Lemma, since $u = \mu S_{t}$ and $v = \sigma S_{t}$, we obtain $$ d F = \left( {{ \partial F } \over { \partial t }} + {{ \partial F } \over { \partial S_{t} }} \mu S_{t} + {{ 1 } \over { 2 }} {{ \partial^{2} F } \over { \partial S_{t}^{2} }} \sigma^{2} S_{t}^{2} \right) dt + {{ \partial F } \over { \partial S_{t} }} \sigma S_{t} d W_{t} $$ Substituting this into $\eqref{1}$ for $d F$ gives $$ \begin{align*} r V_{t} dt =& d F - {{ \partial F } \over { \partial S_{t} }} S_{t} \mu dt - {{ \partial F } \over { \partial S_{t} }} \sigma S_{t} d W_{t} \\ =& \left( {{ \partial F } \over { \partial t }} + {\color{Red}{{ \partial F } \over { \partial S_{t} }} \mu S_{t}} + {{ 1 } \over { 2 }} {{ \partial^{2} F } \over { \partial S_{t}^{2} }} \sigma^{2} S_{t}^{2} \right) dt + {\color{Blue}{{ \partial F } \over { \partial S_{t} }} \sigma S_{t} d W_{t}} \\ & - {\color{Red}{{ \partial F } \over { \partial S_{t} }} S_{t} \mu dt} - {\color{Blue}{{ \partial F } \over { \partial S_{t} }} \sigma S_{t} d W_{t}} \\ =& {{ \partial F } \over { \partial t }} dt + {{ 1 } \over { 2 }} {{ \partial^{2} F } \over { \partial S_{t}^{2} }} \sigma^{2} S_{t}^{2} dt \end{align*} $$ Since the portfolio’s value $V_{t}$ was defined as $\displaystyle V_{t} = F- {{ \partial F } \over { \partial S_{t} }} S_{t}$, substituting this yields $$ r \left( F- {{ \partial F } \over { \partial S_{t} }} S_{t} \right) dt = {{ \partial F } \over { \partial t }} dt + {{ 1 } \over { 2 }} {{ \partial^{2} F } \over { \partial S_{t}^{2} }} \sigma^{2} S_{t}^{2} dt $$ Rearranging the equation for $rF$, we obtain the desired equation. $$ r F = {{ \partial F } \over { \partial t }} + r S_{t} {{ \partial F } \over { \partial S_{t} }} + {{ 1 } \over { 2 }} \sigma^{2} S_{t}^{2} {{ \partial^{2} F } \over { \partial S_{t}^{2} }} $$

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