Deriving the Black-Scholes-Merton PDE

Introduction

In our last lesson, we established the foundational assumptions of the Black-Scholes-Merton (BSM) world—a theoretical marketplace with continuous trading, no transaction costs, and asset prices following a geometric Brownian motion. We are now ready to put those assumptions to work. Our goal in this lesson is to derive the single most important equation in modern finance: the Black-Scholes partial differential equation (PDE).

This isn't just a mathematical exercise. The derivation reveals the core economic insight behind option pricing. The key idea, known as delta hedging, is that by continuously trading the underlying asset against an option, we can construct a portfolio that is completely immune to the random fluctuations of the market. Since this portfolio is risk-free, its return must equal the risk-free interest rate to prevent arbitrage. This powerful constraint is what gives us the PDE.

We will first build an intuitive understanding of how delta hedging works in practice and then walk through the mathematical steps that formalize this intuition into the final equation.

The Intuition of Delta Hedging

Imagine you have sold a call option. Your position loses value if the stock price goes up. To hedge this risk, you could buy some shares of the same stock. If the stock price rises, your loss on the short call will be offset by the gain on your long stock position. The crucial question is: how many shares should you buy?

The answer is given by the option's delta ( $Δ$ ), which measures the sensitivity of the option's price to a change in the stock price. By holding $Δ$ shares of the stock against your option, you can create a delta-neutral portfolio whose value is momentarily insensitive to small movements in the underlying stock price.

This diagram illustrates the principle of delta hedging. A short call position (top left) has a negative delta, meaning it loses value as the stock price rises. By combining it with a long position in the underlying shares (top right), we can create a new portfolio (bottom) whose delta is zero at a specific stock price (here, at the strike of 100). The payoff diagram becomes tent-shaped, and the delta curve (orange) is flat at that point, indicating local insensitivity to price changes.

Of course, an option's delta is not constant; it changes as the stock price and time to maturity change. Therefore, to maintain the hedge, a trader must continuously adjust their stock holdings. This process is called dynamic delta hedging. In the idealized BSM world of continuous trading, this rebalancing can be done perfectly.

To see how this works in a more practical, numerical setting, let's watch an illustration.

An illustration of Black Scholes’ Delta Hedging

The video "An illustration of Black Scholes’ Delta Hedging" by quantpie provides an excellent step-by-step example of replicating a short call option's payoff by dynamically trading the underlying stock.

Watch the following segments to build your intuition: Initial setup (from the start): Understand the basic scenario of selling an option and using the premium. Rebalancing (the process): Focus on the process of how the number of shares held is adjusted at each step as the stock price and the option's delta change. Notice how this involves either borrowing money to buy more shares or investing cash from selling shares. Improving the Hedge (from discrete to continuous): This is the most crucial part. Observe how increasing the frequency of rebalancing makes the final value of the hedging portfolio converge to the actual payoff of the option. This provides the motivation for the continuous-time model.

This video beautifully illustrates that as rebalancing becomes more frequent, the replication of the option payoff becomes more exact. The BSM derivation is essentially the limit of this process as the time between rebalancing goes to zero.

The Mathematical Derivation of the BSM PDE

Now, let's formalize this logic using the stochastic calculus tools you're familiar with. We will follow a clear, six-step process.

We start with the assumptions from the previous lesson:

The stock price $S_{t}$ follows geometric Brownian motion (GBM): $d S_{t} = μ S_{t} d t + σ S_{t} d W_{t}$ .
The price of a derivative, $V (S, t)$ , is a function of the stock price and time.
The market is frictionless and has a constant risk-free rate $r$ .
There are no arbitrage opportunities.

Our goal is to create a risk-free portfolio, $Π$ , by holding one unit of the derivative (long) and shorting $Δ$ units of the stock.

The value of this portfolio is:

Π = V - Δ S

Step 1: Find the Dynamics of the Derivative Price $d V$

Since $V$ is a function of the stochastic process $S_{t}$ and time $t$ , we must use Itô's lemma to find its differential, $d V$ . Your background in probability theory makes you well-acquainted with this tool. Applying Itô's lemma gives:

d V = \frac{\partial V}{\partial t} d t + \frac{\partial V}{\partial S} d S + \frac{1}{2} \frac{\partial ^{2} V}{\partial S ^{2}} (d S)^{2}

We know from the properties of Itô calculus that $(d S)^{2} = (μ S d t + σ S d W_{t})^{2} = σ^{2} S^{2} d t$ , as all terms with $(d t)^{2}$ or $d t d W_{t}$ go to zero.

Substituting $d S$ and $(d S)^{2}$ into the equation for $d V$ , we get:

d V = (\frac{\partial V}{\partial t} + μ S \frac{\partial V}{\partial S} + \frac{1}{2} σ^{2} S^{2} \frac{\partial ^{2} V}{\partial S ^{2}}) d t + σ S \frac{\partial V}{\partial S} d W_{t}

This equation shows that the change in the option's value has a deterministic part (the $d t$ term) and a random part (the $d W_{t}$ term).

Step 2: Find the Dynamics of the Portfolio $d Π$

The change in the value of our hedged portfolio $Π = V - Δ S$ is given by:

d Π = d V - Δ d S

Now, we substitute the expressions for $d V$ (from Step 1) and $d S$ (from the GBM definition) into this equation. This looks complicated at first, but the next step will simplify it dramatically.

d Π = [(\frac{\partial V}{\partial t} + μ S \frac{\partial V}{\partial S} + \frac{1}{2} σ^{2} S^{2} \frac{\partial ^{2} V}{\partial S ^{2}}) d t + σ S \frac{\partial V}{\partial S} d W_{t}] - Δ [μ S d t + σ S d W_{t}]

Step 3: Eliminate the Risk (The Delta Hedge)

To make the portfolio risk-free, we must eliminate the source of randomness, which is the $d W_{t}$ term. We can achieve this by collecting all the $d W_{t}$ terms and setting their combined coefficient to zero.

d Π = (...) d t + (σ S \frac{\partial V}{\partial S} - Δ σ S) d W_{t}

To make the coefficient of $d W_{t}$ zero, we must choose $Δ$ such that:

σ S \frac{\partial V}{\partial S} - Δ σ S = 0

This gives us the magic hedge ratio:

Δ = \frac{\partial V}{\partial S}

The number of shares we need to short is exactly the option's delta!

Step 4: The Risk-Free Portfolio's Dynamics

By setting $Δ = \frac{\partial V}{\partial S}$ , the $d W_{t}$ term vanishes. Let's see what happens to the $d t$ term.

d Π = (\frac{\partial V}{\partial t} + μ S \frac{\partial V}{\partial S} + \frac{1}{2} σ^{2} S^{2} \frac{\partial ^{2} V}{\partial S ^{2}} - Δ μ S) d t

Substituting $Δ = \frac{\partial V}{\partial S}$ , the $μ S \frac{\partial V}{\partial S}$ and $- Δ μ S$ terms cancel each other out. This is a remarkable result: not only is the risk gone, but the stock's expected return $μ$ has also disappeared from the equation!

We are left with a simple, deterministic expression for the change in our portfolio's value:

d Π = (\frac{\partial V}{\partial t} + \frac{1}{2} σ^{2} S^{2} \frac{\partial ^{2} V}{\partial S ^{2}}) d t

Step 5: Invoke the No-Arbitrage Principle

Our portfolio $Π$ is now instantaneously risk-free. The no-arbitrage assumption dictates that any risk-free investment must earn exactly the risk-free interest rate, $r$ . Therefore, the change in the portfolio's value must be:

d Π = r Π d t

Substituting the definition $Π = V - Δ S = V - \frac{\partial V}{\partial S} S$ , we get:

d Π = r (V - \frac{\partial V}{\partial S} S) d t

Step 6: Arrive at the Black-Scholes PDE

We now have two expressions for $d Π$ (from Step 4 and Step 5). Since they must be equal, we can set them equal to each other:

(\frac{\partial V}{\partial t} + \frac{1}{2} σ^{2} S^{2} \frac{\partial ^{2} V}{\partial S ^{2}}) d t = r (V - \frac{\partial V}{\partial S} S) d t

Dividing by $d t$ and rearranging gives the celebrated Black-Scholes-Merton Partial Differential Equation:

\frac{\partial V}{\partial t} + r S \frac{\partial V}{\partial S} + \frac{1}{2} σ^{2} S^{2} \frac{\partial ^{2} V}{\partial S ^{2}} - r V = 0

For a complete walkthrough of this derivation, please review the following text. It mirrors the steps we just took and provides a solid reference.

Black-Scholes PDE: Derivation & Delta Hedging Explained

This article from mbrenndoerfer.com provides an excellent, detailed derivation of the Black-Scholes-Merton PDE.

Please read the following sections to solidify your understanding of the steps we just covered: Dynamics of the Derivative Price: This covers Step 1, the application of Itô's lemma. Constructing the Hedged Portfolio: This covers Steps 2, 3, and 4, showing how to create the riskless portfolio. Pay close attention to how choosing \Delta eliminates both the random term and the drift term \mu. Applying the No-Arbitrage Principle: This covers Step 5, equating the portfolio's return to the risk-free rate. The Black-Scholes-Merton PDE: This section presents the final equation from Step 6.

The video below also provides a narrated summary of the entire derivation, which can help reinforce the sequence of operations.

Black Scholes PDE Derivation using Delta Hedging

The video "Black Scholes PDE Derivation using Delta Hedging" by quantpie visually walks through the mathematical steps.

Watch from the beginning to the derivation to see the concepts we've discussed put together in a continuous flow.

Conclusion

In this lesson, we bridged the gap between the intuitive idea of hedging and the formal mathematics of stochastic calculus. By combining an option with a dynamically adjusted position in the underlying stock, we constructed a portfolio that is free from market randomness. The no-arbitrage principle then forces this riskless portfolio to grow at the risk-free rate, which leads directly to the Black-Scholes-Merton PDE.

Key Takeaways:

Delta Hedging: The core logic is to create a delta-neutral portfolio ( $Π = V - Δ S$ ) whose value is insensitive to small changes in the stock price. The correct hedge ratio is the option's delta, $Δ = \frac{\partial V}{\partial S}$ .
Continuous Rebalancing: In the BSM model, this hedging is done continuously, which perfectly eliminates all risk.
No-Arbitrage: A perfectly hedged, risk-free portfolio must earn the risk-free rate, $r$ . This is the economic principle that drives the pricing relationship.
The BSM PDE: The resulting equation, $\frac{\partial V}{\partial t} + r S \frac{\partial V}{\partial S} + \frac{1}{2} σ^{2} S^{2} \frac{\partial ^{2} V}{\partial S ^{2}} = r V$ , is a universal law that the price of any derivative $V$ must obey in the BSM world, regardless of investor risk preferences or the asset's expected return $μ$ .

We have derived the equation that governs the price of an option. In our next lesson, we will examine the famous solution to this PDE: the Black-Scholes formula. We will then learn how to apply this formula to calculate the prices of European calls and puts, a fundamental skill for your CM2 exam.

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