Applying Itô's Lemma to Derivative Pricing

1. Why Ordinary Calculus Fails: The Need for a New Chain Rule

In standard calculus, if you have a function $f(x)$ and $x$ changes by a small amount $dx$, the change in the function, $df$, is approximated by the first term of a Taylor series: $df\approx f^{\prime }(x)dx$. The second-order term, involving $(dx{)}^2$, is ignored because it's vanishingly small.

However, as we learned, Brownian motion is not a "smooth" process. Its changes are so erratic that the squared increment, $(d{B}_t{)}^2$, is not zero; it's equal to $dt$. This single, non-intuitive property means the standard chain rule breaks down.

The following video provides an excellent intuitive contrast between ordinary and stochastic calculus and explains why this second-order term suddenly becomes important.

The key insight is that the "quadratic variation" of Brownian motion is non-zero. The sum of the squares of its tiny jumps doesn't disappear—it accumulates at a deterministic rate equal to the passage of time. This forces us to keep the second derivative term from the Taylor expansion, leading to Itô's Lemma.

2. The Formula: Itô's Lemma

Let's formalize this. Suppose an underlying asset price, $S_t$, follows a general Itô process (GBM is a specific example of this):
$d{S}_t=a(S_t,t)dt+b(S_t,t)d{B}_t$
where $a$ is the drift term and $b$ is the diffusion (volatility) term.

Now, consider a derivative whose price, $V$, is a function of both the underlying price and time, $V(S_t,t)$. Itô's Lemma tells us that the change in the derivative's price, $dV$, is given by:
$dV=\left(\frac{\partial V}{\partial t}+a\frac{\partial V}{\partial S}+\frac{1}{2}{b}^2\frac{{\partial }^{2}V}{\partial S^2}\right)dt+b\frac{\partial V}{\partial S}d{B}_t$

Let's dissect this important formula:

• $\frac{\partial V}{\partial t}$: This is the change in the option's value purely due to the passage of time. In the world of options, you'll come to know this as Theta.
• $a\frac{\partial V}{\partial S}$: This is the change in $V$ due to the deterministic drift of $S$. The term $\frac{\partial V}{\partial S}$ is the option's sensitivity to the underlying's price, known as Delta.
• $b\frac{\partial V}{\partial S}d{B}_t$: This is the new stochastic component of the derivative's price change. It's driven by the same random shock $d{B}_t$ that drives the underlying, scaled by the volatility function $b$ and the option's Delta.
• $\frac{1}{2}{b}^2\frac{{\partial }^{2}V}{\partial S^2}dt$: This is the Itô term or convexity correction. It's the new, non-intuitive term that arises from $(d{B}_t{)}^2=dt$. The term $\frac{{\partial }^{2}V}{\partial S^2}$ is the option's Gamma. This term accounts for the effect of the underlying's volatility on the derivative's value.

3. First Application: Solving the GBM Puzzle

Before we tackle the main learning outcome, let's use Itô's Lemma to solve the mystery from our last lesson. We start with the SDE for GBM:
$d{S}_t=\mu S_{t}dt+\sigma S_{t}d{B}_t$
So, our drift and diffusion functions are $a=\mu S_t$ and $b=\sigma S_t$.

We want to find the process followed by $V=\ln (S_t)$. First, we find the partial derivatives of $V$:

• $\frac{\partial V}{\partial t}=0$ (since $\ln (S_t)$ has no explicit dependence on $t$)
• $\frac{\partial V}{\partial S}=\frac{1}{S_t}$
• $\frac{{\partial }^{2}V}{\partial S^2}=-\frac{1}{S_t^2}$

Now, we plug these into the Itô's Lemma formula:
$dV=\left(0+(\mu S_t)\left(\frac{1}{S_t}\right)+\frac{1}{2}(\sigma S_t{)}^2\left(-\frac{1}{S_t^2}\right)\right)dt+(\sigma S_t)\left(\frac{1}{S_t}\right)d{B}_t$
Simplifying the terms:
$d(\ln S_t)=\left(\mu -\frac{1}{2}{\sigma }^2\right)dt+\sigma d{B}_t$
This elegant result shows that the logarithm of the stock price follows a simple Brownian motion with a constant drift $\left(\mu -\frac{1}{2}{\sigma }^2\right)$ and constant volatility $\sigma$.

To get the final stock price, we can integrate this simple SDE from $t$ to $T$:
$\ln (S_T)-\ln (S_t)={\int }_t^T\left(\mu -\frac{1}{2}{\sigma }^2\right)d\tau +{\int }_t^T\sigma d{B}_{\tau }$
$\ln (S_T)=\ln (S_t)+\left(\mu -\frac{1}{2}{\sigma }^2\right)(T-t)+\sigma (B_T-B_t)$
Exponentiating both sides gives us the exact analytical solution from the previous lesson:
$S_T=S_t\exp \left(\left(\mu -\frac{1}{2}{\sigma }^2\right)(T-t)+\sigma (B_T-B_t)\right)$
The drift correction term $-\frac{1}{2}{\sigma }^2$ is not an arbitrary adjustment; it is a direct mathematical consequence of applying Itô's Lemma to the logarithm function due to its convexity.

4. Core Application: Deriving the Black-Scholes PDE

Now, we arrive at the heart of today's lesson and a cornerstone of modern finance. We will use Itô's Lemma to model how an option's price changes and, through a brilliant hedging argument, derive the famous Black-Scholes Partial Differential Equation (PDE).

The following reading explains this derivation. Your background in finance, particularly the no-arbitrage principle, will be very helpful here.

Let's walk through the logic from the article step-by-step. This is the application mentioned in the learning outcome.

Step 1: Define the Dynamics of the Option Price
We begin with our derivative, $V(S,t)$, and the underlying asset, $S$, which follows GBM: $dS=\mu Sdt+\sigma Sd{B}_t$.
Applying Itô's Lemma directly to $V(S,t)$ gives us the dynamics of the option price:
$dV=\left(\frac{\partial V}{\partial t}+\mu S\frac{\partial V}{\partial S}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}\right)dt+\sigma S\frac{\partial V}{\partial S}d{B}_t$
This equation tells us precisely how the option price changes over an infinitesimal time step. It has a predictable part (the $dt$ term) and a random part (the $d{B}_t$ term).

Step 2: Construct a Riskless Portfolio
The genius of Black, Scholes, and Merton was to construct a portfolio that eliminates the random component. The portfolio consists of:

• One long option (+V)
• A short position in $\Delta$ units of the underlying asset (-$\Delta S$)

The value of this portfolio is $\Pi =V-\Delta S$.

Step 3: Analyze the Change in the Portfolio's Value
The change in the portfolio's value, $d\Pi$, is simply $d\Pi =dV-\Delta dS$. Let's substitute our expressions for $dV$ (from Step 1) and $dS$ (from GBM):
$d\Pi =\left[\left(\frac{\partial V}{\partial t}+\mu S\frac{\partial V}{\partial S}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}\right)dt+\sigma S\frac{\partial V}{\partial S}d{B}_t\right]-\Delta \left[\mu Sdt+\sigma Sd{B}_t\right]$
Now, group the deterministic ($dt$) and random ($d{B}_t$) terms:
$d\Pi =\left(\frac{\partial V}{\partial t}+\mu S\frac{\partial V}{\partial S}-\Delta \mu S+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}\right)dt+\left(\sigma S\frac{\partial V}{\partial S}-\Delta \sigma S\right)d{B}_t$
Notice that the entire random part of the portfolio's change is contained in the second bracket. We can eliminate this randomness completely by choosing our hedge ratio $\Delta$ cleverly. If we set:
$\Delta =\frac{\partial V}{\partial S}$
the entire $d{B}_t$ term vanishes! This is the concept of Delta Hedging.

Step 4: Apply the No-Arbitrage Principle
With this choice of $\Delta$, the change in our portfolio is now entirely deterministic (risk-free):
$d\Pi =\left(\frac{\partial V}{\partial t}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}\right)dt$
From basic finance principles, a risk-free investment must earn the risk-free interest rate, $r$. Therefore, the change $d\Pi$ must be equal to the portfolio's value $\Pi$ earning interest for a period $dt$:
$d\Pi =r\Pi dt=r(V-\Delta S)dt$
Substituting our choice for $\Delta =\frac{\partial V}{\partial S}$, we get:
$d\Pi =r\left(V-\frac{\partial V}{\partial S}S\right)dt$
Now we have two expressions for the same risk-free change $d\Pi$. Equating them gives:
$\left(\frac{\partial V}{\partial t}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}\right)dt=\left(rV-rS\frac{\partial V}{\partial S}\right)dt$
Finally, we cancel $dt$ from both sides and rearrange all terms to one side to arrive at the celebrated Black-Scholes Partial Differential Equation:
$\frac{\partial V}{\partial t}+rS\frac{\partial V}{\partial S}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}-rV=0$
This is a monumental result. We started with the random behavior of a stock and, using Itô's Lemma and a no-arbitrage argument, derived a deterministic equation that governs the price of any derivative written on that stock.

Conclusion

In this lesson, we demystified and applied Itô's Lemma, one of the most fundamental tools for a quant trader. We saw how it extends the chain rule to the stochastic world and explains the "convexity correction" term in financial models.

Key Takeaways:

• Itô's Lemma is the chain rule for stochastic processes. It's necessary because the squared increment of Brownian motion, $(d{B}_t{)}^2$, is not zero but equals $dt$.
• The lemma introduces a new term involving the second partial derivative ($\frac{{\partial }^{2}V}{\partial S^2}$, or Gamma), which accounts for the impact of volatility.
• We used Itô's Lemma to derive the process for $\ln (S_t)$ and prove why the analytical solution for GBM contains the $\mu -\frac{1}{2}{\sigma }^2$ drift term.
• Most importantly, by applying Itô's Lemma to a derivative's value $V(S,t)$ and forming a delta-hedged portfolio, we eliminated all randomness and used a no-arbitrage argument to derive the Black-Scholes PDE.

Preview of the Next Lesson:
We have successfully derived the Black-Scholes PDE. The next logical step is to solve it. In the next module, we will begin our deep dive into options pricing, starting with intuitive models like binomial trees before tackling the full Black-Scholes-Merton formula, which is the solution to the PDE we just derived. We will outline the conceptual steps for this derivation and finally arrive at the famous pricing formulas for European calls and puts.