In Chapters 10–17, you priced options by building a delta-hedging portfolio, cancelling the random dW terms, and solving the resulting Black-Scholes partial differential equation. It worked — but it required significant mathematical machinery. Chapter 18 offers a far more elegant path to the exact same answer: instead of solving a calculus problem, compute a probability-weighted average.
Build a delta-hedging portfolio. Cancel the dW terms. Impose no-arbitrage. Solve the Black-Scholes PDE — a second-order partial differential equation in two variables. Verify the solution satisfies boundary conditions.
Result: c = xN(d₁) − Ke⁻ʳᵀN(d₂)
Change the probability measure using Girsanov's Theorem. In the new measure, everything is a Martingale. Price = discounted expected payoff. Compute the expectation of a log-normal random variable.
Same result: c = xN(d₁) − Ke⁻ʳᵀN(d₂)
Both methods arrive at the same formula. The risk-neutral method is preferred not because it is more "real" — the risk-neutral world is explicitly a mathematical fiction — but because it converts a calculus problem (solving a PDE) into a probability problem (computing an expectation). Expectations are often far easier to compute, especially for complex derivatives.
In the real world, investors demand more return for taking more risk. A volatile stock must offer a higher expected return than a government bond — otherwise no rational investor would hold it. This extra return is the Market Price of Risk. It reflects the compensation investors require for bearing uncertainty.
Risk-Neutral Pricing asks: what if nobody cared about risk? If investors were completely indifferent to volatility, they would be satisfied with the risk-free rate r on every asset — no matter how wildly it swings. In this hypothetical world, pricing becomes dramatically simpler.
In the real world, the stock drifts at α (above the risk-free rate r). In the risk-neutral world, the drift is replaced by r — the market price of risk is removed from the equation.
The market price of risk θ = (α − r)/σ is the number of "units of volatility" per unit of excess return. It quantifies how much compensation the market demands for bearing each unit of uncertainty. Girsanov's Theorem uses exactly this number to shift from ℙ to ℙ̃.
How do we mathematically force a risky stock (drifting at α) to drift at the risk-free rate r instead? Girsanov's Theorem provides the answer: we change the probability measure. We do not alter the price paths themselves — we reweight them. The same collection of possible paths exists; we simply shift probability mass from high-drift paths to low-drift paths until the average exactly equals r.
Girsanov's Theorem does not alter the paths — it reweights them. The tinted glasses change what is "average," not what is "possible."
The key insight: a hedge portfolio that eliminates risk does not care which way the price goes. It works in the real world and in the risk-neutral world equally well — because the volatility σ is identical in both. Girsanov allows us to shift the drift without affecting the hedge, which means the option price computed in the risk-neutral world is automatically the correct price in the real world too.
Once you are in the risk-neutral world, pricing any derivative — from a simple call to the most exotic structured product — reduces to the same three steps. Every asset, in the risk-neutral world, is a Martingale when discounted at the risk-free rate. Pricing is just computing an expected value.
Applying this formula to a European call option — where V(T) = max(S(T)−K, 0) and S(T) is log-normal under ℙ̃ — produces exactly the Black-Scholes formula c = xN(d₁) − Ke⁻ʳᵀN(d₂). The PDE derivation in Chapter 13 and the expectation computation here are two routes to the identical answer. The expectation route is typically faster for new derivatives.
Understanding the difference between the two probability measures is essential. They describe the same price paths — but with different purposes and different drift assumptions.
| Feature | Real World ℙ | Risk-Neutral World ℙ̃ |
|---|---|---|
| Stock drift | α — includes risk premium above r | r — only the risk-free rate |
| Volatility σ | Identical — σ | Identical — σ (unchanged by Girsanov) |
| Expectation used for | Forecasting — "where will the stock be?" | Pricing — "what is the option worth today?" |
| Discounted price Martingale? | No — e⁻ʳᵗS(t) has positive drift | Yes — e⁻ʳᵗS(t) is a Martingale under ℙ̃ |
| Brownian Motion | W(t) | W̃(t) = W(t) + θt where θ = (α−r)/σ |
| Is it "real"? | Yes — describes actual market dynamics | No — a mathematical fiction for pricing |
| Used for | Predicting returns, portfolio construction | Pricing and hedging derivatives |
The most important entry in this table: we do not use the risk-neutral world because we believe it is real. We use it because it is mathematically convenient. It converts an initial-capital-required problem (how much do I need today to perfectly replicate this payoff?) into an expectation computation. The answer is the same in both worlds — only the method differs.
Here is the most counter-intuitive consequence of the entire theory. An investor who believes NIFTY will return 25% per year and an investor who believes it will return 5% per year will compute the exact same price for every option on NIFTY. Their disagreement about α is entirely irrelevant to the option price.
Alpha is the first thing Girsanov removes. Once you enter the risk-neutral world, α has been replaced by r — permanently and irreversibly.
This result is not a flaw — it is a feature. The option price reflects only the cost of hedging (which depends on σ and r), not the investor's forecast of direction (which depends on α). Two parties who disagree wildly about where a stock is going can still trade an option at a mutually agreed price — because the option price is determined by the replication cost, not by the drift belief.
Simulate many stock paths under the real-world measure ℙ (using drift α) and under the risk-neutral measure ℙ̃ (using drift r). Compute the average discounted payoff in both worlds. No matter how you set α, the risk-neutral price always matches the BSM formula — and the real-world price does not.
Blue = risk-neutral payoff distribution (drift = r) | Red = real-world payoff distribution (drift = α) | Gold line = BSM fair price
Chapter 18 — the four ideas of risk-neutral pricing:
The key takeaway: We do not use the risk-neutral world because we think it is real. We use it because it is the mathematically cleanest way to compute the initial capital required to hedge a derivative perfectly. It transforms a hard PDE into a simple expectation — and the answer is always correct in the real world too.