Lesson 5.1: The Solution (The Black-Scholes-Merton Formula)

That PDE is the problem we need to solve. It's not the answer. A client doesn't want a PDE; they want a price in dollars and cents.

The full mathematical derivation that "solves" this PDE is one of the most complex in finance (it involves transforming it into the "Heat Equation" from physics, solving that, and then transforming it back).

We will not do that here. That is a "pure math" exercise.

Our goal as quants is to take the famous solution and understand what it means. We are going to deconstruct it, piece by piece, so you understand the financial intuition. This is the "Holy Grail" formula of finance.

Let's look at the second half of the formula first. This is the "what you expect to pay" part.

This is the present value of the strike price $K$ you *might* have to pay. Let's break it down into its three pieces.

• $K$: This is the Strike Price (e.g., $100). It's the fixed amount of cash you must pay at expiration (time $T$) to get the stock.
• $e^{-r(T-t)}$: This is the Present Value Discount Factor. This is "Finance 101" from Module 0. You don't pay $K$ today; you pay it in the future (at time $T$). This term calculates the "present value" of that future payment. It answers: "How much money would I have to put in a risk-free bank account today at rate $r$ to have exactly $K$ dollars at time $T$?" It's always less than $K$, because a dollar today is worth more than a dollar tomorrow.
• $N(d_2)$: This is the "Probability Calculator." This is the most brilliant part. You don't always have to pay $K$. You only pay $K$ if the option is "in-the-money" (if $S_T > K$). $N(d_2)$ is the risk-neutral probability that your option will expire in-the-money. $N(\cdot)$ is the "Cumulative Normal Distribution" function. It just tells you the probability of an outcome happening. $N(d_2)$ is a magic number (calculated by $d_2$) that gives us $\text{Prob}(S_T > K)$.

Conclusion for Part 1:

The "Expected Cost" term, $K e^{-r(T-t)} N(d_2)$, is just:

($100 you must pay) × (Discounted to today's value) × (Probability you'll actually have to pay it)

This is a beautiful, intuitive concept.

Now let's look at the first half of the formula. This is the "what you expect to get" part.

This part is more subtle.

• $S_t$: This is the Stock Price today.
• $N(d_1)$: This is another "Probability Calculator."

It's tempting to think this term is just `(Expected Future Stock Price) * (Probability of Getting It)`. It's almost that, but it's much more clever.

The term $S_t N(d_1)$ is the Present Value of the stock you might receive at expiration.

The "Aha!" Moment (Connecting to Module 4):

Remember our "magic" hedge ratio $\Delta$? We proved that to be risk-free, we had to hold $\Delta$ shares of the stock, where $\Delta = \frac{\partial V}{\partial S}$. It turns out that when you solve the Black-Scholes PDE, you find this amazing identity:

The "Delta" of our call option is exactly $N(d_1)$!

Therefore, the "Expected Benefit" term, $S_t N(d_1)$, can be rewritten as:

This is the value of the stock side of our delta-hedged portfolio!

Conclusion for Part 2:

The "Expected Benefit" term is simply the current stock price ($S_t$) multiplied by the option's sensitivity ($N(d_1)$ or $\Delta$). It's the "share" of the stock that the option "acts like" it owns right now.

Let's put our two, newly-translated pieces back together:

The entire Black-Scholes formula is just the value of a leveraged bet on the stock. It's the value of the "Delta-equivalent" shares you "own" ($S_t N(d_1)$), minus the present value of the "loan" you'll need to pay to get them ($K e^{-r(T-t)} N(d_2)$).

The "Magic Calculators" $d_1$ and $d_2$

So what are $d_1$ and $d_2$? They are just the "engines" that calculate the probabilities $N(d_1)$ and $N(d_2)$. You don't need to memorize them, but you must understand what goes into them.

This formula looks like a nightmare, but it's just a "signal-to-noise" ratio.

• The Numerator (The "Signal"):
  • $\ln(S_t/K)$: This is the "value" part. It asks, "Is the option in-the-money now?"
  • $(r + \frac{1}{2}\sigma^2)(T-t)$: This is the "growth" part. It's the total expected drift of the stock in the risk-neutral world (including our Itô Correction term $\frac{1}{2}\sigma^2$!).
• The Denominator (The "Noise"):
  • $\sigma\sqrt{T-t}$: This is the total volatility of the stock from now until expiration. (Notice our "weird scaling" rule $\sqrt{t}$ right there!)

$d_1$ and $d_2$ are just numbers that package up all 5 of our key inputs ($S, K, t, r, \sigma$) and feed them to the $N(\cdot)$ probability calculator.