Lesson 3.3: Understanding the Full Formula (Translating Math to Finance)

Let's start with the "easy" part, the random term (the "diffusion bin"):

This tells us the total random jiggle of our option, $f$. It's made of two ingredients:

1. $b$ (The "Jiggle Source"): This is the diffusion coefficient (the volatility) of the underlying stock ($X_t$). This is the "engine" of all the randomness.
2. $\frac{\partial f}{\partial x}$ (The "Jiggle Amplifier"): This is the partial derivative of $f$ with respect to $X_t$. In finance, this has a famous name: Delta ($\Delta$).

Physical Meaning (Delta, $\Delta$)

Delta is the "speed" of the option. It's the "Risk-O-Meter" we talked about in Lesson 1.4.

• It answers: "If the stock price ($X_t$) moves by $1, how many dollars does my option price ($f$) move?"
• If $\Delta = 0.5$, a $1 move in the stock causes a $0.50 move in the option.

So, the "New Diffusion" of our option, $(b \cdot \Delta) dW_t$, is just:

(Stock's Volatility) × (Option's Sensitivity to Stock) × (The Random Event)

This makes perfect physical sense. The option's "jiggliness" is just the stock's "jiggliness," amplified or reduced by its own sensitivity, Delta.

Now for the "messy" part, the "drift bin." This is the predictable, non-random change in our option price per second:

This is not one concept. It is the sum of three separate, distinct physical effects that are all happening at the same time. Let's break them down one by one.

Drift Term 1: $\frac{\partial f}{\partial t}$

• What it is: The partial derivative of $f$ with respect to time $t$.
• Financial Name: Theta ($\Theta$)
• Physical Meaning (The "Melting Ice Cube"): This is the "Time Decay" of the option. An option is a decaying asset; it has an expiration date. This term measures how much value the option loses from one second to the next, even if the stock price stays perfectly still.
• For a simple "call" option, this term is negative. $\Theta = -0.05$ means your option "melts" and loses 5 cents every single day.

Drift Term 2: $a\frac{\partial f}{\partial x}$

• What it is: The drift of the stock ($a$) multiplied by the option's sensitivity to the stock ($\frac{\partial f}{\partial x}$, or Delta).
• Financial Name: This is the "Delta-Drift".
• Physical Meaning (The "Moving Walkway"): This is the "normal chain rule" part of the change. It's the predictable gain or loss your option gets just by "going along for the ride" on the stock's own predictable trend ($a$, or $\mu S_t$).
• If the stock has an 8% expected return ($a$), and your option has a Delta of 0.5, this term represents your "share" of that expected return.

Drift Term 3: $\frac{1}{2}b^2\frac{\partial^2 f}{\partial x^2}$

• What it is: This is our "Itô Correction Term" from Lesson 3.1. It's $\frac{1}{2} \times (\text{diffusion})^2 \times (\text{curvature})$.
• Financial Name: The "curvature" $\frac{\partial^2 f}{\partial x^2}$ is called Gamma ($\Gamma$). So this term is $\frac{1}{2}b^2\Gamma$.
• Physical Meaning (The "Skateboard in a Half-Pipe"): This is the most profound part. It's a predictable drift ($dt$) that you earn or pay because of randomness ($b^2$)!
  • Gamma ($\Gamma$) measures the "curvature" or "convexity" of your option's payoff.
  • $b^2$ measures the "jiggliness" (squared volatility) of the stock.
  • This term says that "jiggling" ($b^2$) on a "curved" path ($\Gamma$) creates a net, predictable drift. For a typical call option (which is "convex," or shaped like a smile, $\Gamma > 0$), this term is positive.
  • This is a "volatility profit." It's a small, non-random amount of money the option earns every second as a reward for being exposed to the "jiggles."

Let's put our "translated" formula back together, this time replacing the math with the "Greek" names for the derivatives ($\Delta = \frac{\partial f}{\partial x}$, $\Gamma = \frac{\partial^2 f}{\partial x^2}$, $\Theta = \frac{\partial f}{\partial t}$):

This formula is no longer a "wall of math." It's a complete story that says:

"The total change in an option's price ($dV$) is the sum of two things:

1. A Predictable Drift ($dt$): This drift is a combination of three effects: its Time Decay ($\Theta$), its "share" of the Stock's Drift ($a \cdot \Delta$), and its "Volatility Profit" ($\frac{1}{2}b^2\Gamma$).
2. A Random Jiggle ($dW_t$): This jiggle is just the Stock's Jiggle ($b$), amplified by the Option's Sensitivity ($\Delta$)."