QuantPrep | AI-Powered Learning for Quantitative Finance

Build a portfolio $\Pi = -V + \Delta S_t$ .
Apply Itô's Lemma to $d\Pi$ .
Choose $\Delta$ to "kill" the random $dW_t$ term.
Show that the subjective drift $\mu$ "magically" canceled out.
Argue that this risk-free portfolio must earn the risk-free rate $r$ .
This left us with a complex Partial Differential Equation (PDE) to solve.

This method is brilliant, but it's very difficult. What if the option is complex? What if we have multiple random factors? This "delta-hedging" method becomes a nightmare.

This lesson is about a different way to get the same answer. It's a "shortcut" that is more powerful, more intuitive, and the foundation of all modern quant finance. This is Risk-Neutral Valuation.

Part 1: The "Real World" vs. The "Risk-Neutral World"

This is a "thought experiment." We need to understand the two "worlds" we can live in.

World 1: The "Real World" (Physical Measure $\mathbb{P}$ )

This is the world we live in, the world that news anchors report on.

Investors are "Risk-Averse": We don't like risk. If a safe government bond pays 5%, we would *not* buy a risky stock unless we *expected* it to make more (e.g., 8% or 10%).
The Drift $\mu$ : The stock's expected return $\mu$ is subjective (my guess is different from yours) and greater than $r$ ( $\mu > r$ ).
The SDE (Our GBM Model):
$dS_t = \mu S_t dt + \sigma S_t dW_t^{\mathbb{P}}$
(We use $W_t^{\mathbb{P}}$ to show this is a "real-world" random walk).

The Problem: We can't use this SDE to price an option, because we can't agree on what $\mu$ is.

World 2: The "Magic" Risk-Neutral World (Measure $\mathbb{Q}$ )

Now, let's *imagine* a parallel universe. In this "magic" world, all investors are "risk-neutral."

Investors are "Risk-Neutral": They *do not care* about risk. A 5% return from a risky stock is just as good as a 5% return from a safe bond.
The Drift $r$ : In this world, investors don't need to be "bribed" with a higher return to take on risk. Therefore, **every single asset** (stocks, bonds, etc.) must have an expected return exactly equal to the risk-free rate, $r$ .
The SDE (The "Magic" SDE):
$dS_t = r S_t dt + \sigma S_t dW_t^{\mathbb{Q}}$
(Note: The volatility $\sigma$ is the *same*! But the drift is now $r$ , and the random path $dW_t^{\mathbb{Q}}$ is a different "magic world" path).

Part 2: The "Aha!" Moment (Why Are We Allowed to Do This?)

This seems like a silly "thought experiment." Why are we allowed to just "jump" into a "magic world" where $\mu=r$ ?

Because our PDE derivation in Module 4 *proved* it's allowed!

Remember the two "magic tricks" from Lessons 4.2 and 4.3?

We "killed" the random $dW_t$ term by choosing $\Delta = \frac{\partial V}{\partial S}$ .
When we did that, the $\mu$ term **canceled out completely**.

Our final Black-Scholes-Merton PDE...

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

...**does not contain $\mu$ anywhere.**

The price of an option is independent of the stock's expected return $\mu$ .

This means that we will get the **exact same price** for the option whether we do our calculation in the "real world" (using $\mu=10\%$ ) or in the "magic world" (using $\mu=5\%$ ).

So, if we're *guaranteed* to get the same answer, why not just *pretend* we live in the "magic" risk-neutral world from the start? The math becomes *infinitely* easier. We just set $\mu = r$ and solve the problem.

Part 3: The New Pricing Formula (The "Shortcut")

This "magic world" gives us a powerful new way to price derivatives, known as the Fundamental Theorem of Asset Pricing.

It states that in a "no-free-lunch" (no-arbitrage) market, the fair price of any derivative ( $V$ ) today is simply:

"The expected payoff of the derivative at expiration, calculated in the 'risk-neutral world' (where $\mu=r$ ), and then discounted back to today using the risk-free rate $r$ ."

The Risk-Neutral Valuation Formula ("The Shortcut")

V(t) = \mathbb{E}^{\mathbb{Q}} \left[ e^{-r(T-t)} \cdot \text{Payoff}(S_T) \right]

Let's translate this:

$V(t)$ : The option's fair price today.
$\mathbb{E}^{\mathbb{Q}}[\dots]$ : The "Expected Value" (the average payoff) calculated in our **"magic" $\mathbb{Q}$ world**.
$e^{-r(T-t)}$ : The "Present Value" discount factor (from Lesson 0.1).
$\text{Payoff}(S_T)$ : The option's payoff at expiration (time $T$ ).

Part 4: A Concrete Example (Call Option)

Let's see how this "shortcut" gets us the Black-Scholes formula.

Problem: Price a Call Option, where $\text{Payoff} = \max(S_T - K, 0)$ .

Old Way (PDE Method): We must solve this complex PDE from Module 4:
$\frac{\partial V}{\partial t} + r S_t \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} - rV = 0$
This is a "pure math" problem that is very difficult.
New Way (Risk-Neutral Method): We just solve this statistics problem:
$C = e^{-r(T-t)} \cdot \mathbb{E}^{\mathbb{Q}}[ \max(S_T - K, 0) ]$
To solve this, we just need to know the statistics of $S_T$ in our "magic world." In Lesson 3.4, we'll (optionally) show that the *solution* to our "magic" SDE ( $dS_t = r S_t dt + \dots$ ) is:
$S_T = S_t \exp\left( (r - \frac{1}{2}\sigma^2)(T-t) + \sigma W_T^{\mathbb{Q}} \right)$
This is a log-normal distribution.

When you (or rather, a PhD-level statistician) calculate the "expected value" of a log-normal distribution, the answer that pops out is *exactly* the Black-Scholes formula:

\mathbb{E}^{\mathbb{Q}}[ \max(S_T - K, 0) ] = S_t e^{r(T-t)} N(d_1) - K N(d_2)

Now, plug this into our "Shortcut" formula and multiply by $e^{-r(T-t)}$ :

C = e^{-r(T-t)} \left[ S_t e^{r(T-t)} N(d_1) - K N(d_2) \right]

C = S_t N(d_1) - K e^{-r(T-t)} N(d_2)

We get the *exact same answer* as the PDE method, but with statistics instead of a complex PDE.

Lesson 6.1: The "Other Way" - Risk-Neutral Valuation