Economic Modelling · Black-Scholes derivative-pricing model

Risk-Neutral Valuation and the Black-Scholes PDE Explained

Updated 11 October 2026 · Fact-checked

Risk-neutral valuation prices a derivative as the discounted expected payoff, with the share growing at the risk-free rate instead of its real-world drift. It works because delta hedging removes risk and no-arbitrage forces the hedged portfolio to earn the risk-free rate. That condition gives the Black-Scholes PDE.

Understand Risk-Neutral Valuation and the Black-Scholes PDE

Start with the problem. A derivative's payoff depends on a share price S. The share price is random, so you might think the option price must depend on investors' attitude to risk and on the share's expected return μ. The Black-Scholes argument shows it does not.

The idea is to build a portfolio of one option and −Δ shares. Choose Δ so that the random part of the portfolio cancels over a very short time. The portfolio is then riskless over that instant. If it were riskless and earned more than the risk-free rate, you could borrow at the risk-free rate and make a sure profit. That is arbitrage. So the portfolio must earn exactly the risk-free rate r.

This condition, applied with Ito's lemma, gives the Black-Scholes PDE. The drift μ cancels out. Only r, the volatility σ and the share price remain. Since the PDE contains no risk preferences, you may solve it in any world that has the same PDE. The simplest is a world where investors are risk neutral.

That world is the risk-neutral measure Q. Under Q, every traded asset has expected growth at rate r. The share follows dS = rS dt + σS dW̃, where W̃ is a Q-Brownian motion. Under the real-world measure P, the share follows dS = μS dt + σS dW. The volatility σ is the same under both. Only the drift changes.

So the price at time t of a derivative paying X at time T is V(t) = e^(−r(T−t)) E_Q[X | F_t]. Use P for projecting real outcomes, such as risk capital and expected profit. Use Q for pricing. Risk-neutral does not mean investors are risk neutral. It is a pricing device that gives the no-arbitrage price.

Key rules to remember

Share price under P: dS = μS dt + σS dW; Geometric Brownian motion in the real world. μ is the real-world drift.
Share price under Q: dS = rS dt + σS dW̃; Replace μ by r. Volatility σ is unchanged. Assumes a constant risk-free rate and no dividends.
Delta-hedged portfolio: Π = V − ΔS, with Δ = ∂V/∂S; Holding −Δ shares per option removes the dW term over a short interval.
Black-Scholes PDE: ∂V/∂t + ½σ²S² ∂²V/∂S² + rS ∂V/∂S − rV = 0; Holds for any derivative on a non-dividend share under the model assumptions. The payoff comes from the boundary condition at T.
Risk-neutral pricing: V(t) = e^(−r(T−t)) E_Q[V(T) | S(t)]; Expectation is under Q, discounting at the risk-free rate.
Terminal condition for a call: V(S, T) = max(S − K, 0); For a put use max(K − S, 0).
Black-Scholes call price: C = S N(d1) − K e^(−rτ) N(d2); τ = T − t. N is the standard normal distribution function.
d1 and d2: d1 = [ln(S/K) + (r + ½σ²)τ] ÷ (σ√τ); d2 = d1 − σ√τ; For a non-dividend share. N(d2) is the Q-probability the call finishes in the money.
Black-Scholes put price: P = K e^(−rτ) N(−d2) − S N(−d1); Consistent with put-call parity: C − P = S − K e^(−rτ).
Market price of risk: λ = (μ − r) ÷ σ; Links P and Q: dW̃ = dW + λ dt.

How to solve Risk-Neutral Valuation and the Black-Scholes PDE questions

Use this method for derivation questions, pricing questions and questions asking you to explain the measures.

1State the assumptions: share follows geometric Brownian motion, constant r and σ, no dividends, no transaction costs, continuous trading, no arbitrage, short selling allowed.
2Write the share dynamics dS = μS dt + σS dW and apply Ito's lemma to V(S, t) to get dV.
3Form the portfolio Π = V − ΔS and choose Δ = ∂V/∂S so the dW terms cancel.
4Argue by no-arbitrage that the riskless portfolio earns r: dΠ = rΠ dt. Equate and simplify to reach the PDE. Note that μ has dropped out.
5For a pricing question, set the payoff as the boundary condition, or switch to Q and compute V = e^(−rτ) E_Q[payoff].
6Under Q, S(T) is lognormal with mean parameter ln S + (r − ½σ²)τ and variance σ²τ. Compute d1 and d2 and evaluate N(·) from tables.
7Check the answer: a call price should be non-negative, at least S − K e^(−rτ), and at most S.
8State the conclusion in words and say which measure you used.

Quickest way: Price directly under Q

When to use it: Use when you are given S, K, r, σ and time and need a price, not a derivation.

Compute τ and the discount factor e^(−rτ).
Compute d1 and d2 using the formulas.
Read N(d1) and N(d2) from the tables and apply C = S N(d1) − K e^(−rτ) N(d2).
Get a put from put-call parity: P = C − S + K e^(−rτ).
Do a sense check against the lower bound S − K e^(−rτ).

Common mistakes in Risk-Neutral Valuation and the Black-Scholes PDE

Leaving μ in the PDE or in the risk-neutral price.
Students carry the real-world drift from the share model into the pricing step.
Fix: After hedging, μ cancels. Under Q the drift is r. Never use μ to price a derivative.
Saying risk-neutral valuation assumes investors are risk neutral.
The name suggests it.
Fix: Say that the no-arbitrage price is the same as it would be in a risk-neutral world. It is a pricing technique, not a claim about behaviour.
Changing σ when moving from P to Q.
Students think the whole share model changes.
Fix: Only the drift changes. Volatility is unchanged, because the change of measure only shifts the Brownian motion by a drift.
Forgetting that Δ is held constant over the short interval when forming dΠ.
Students differentiate ΔS as if Δ were changing.
Fix: State that Δ is fixed over dt, so dΠ = dV − Δ dS.
Discounting at μ or forgetting to discount at all.
Students compute the expected payoff and stop.
Fix: Always multiply E_Q[payoff] by e^(−rτ).
Mixing up d1 and d2, or using σ² instead of σ√τ in the denominator.
Formula memorised without structure.
Fix: Remember d2 = d1 − σ√τ and that the denominator is σ√τ. Check that N(d2) ≤ N(d1) for a call.

Worked examples

Example 1

Derive the Black-Scholes PDE for a derivative V(S, t) on a non-dividend share with dS = μS dt + σS dW and constant risk-free rate r.

Show the solution

By Ito's lemma, dV = (∂V/∂t + μS ∂V/∂S + ½σ²S² ∂²V/∂S²) dt + σS ∂V/∂S dW.
Form Π = V − ΔS with Δ held fixed over dt. Then dΠ = dV − Δ dS.
Substitute dS = μS dt + σS dW. The dW coefficient is σS(∂V/∂S − Δ).
Choose Δ = ∂V/∂S. The dW term vanishes, so the portfolio is riskless over dt.
The dt terms give dΠ = (∂V/∂t + ½σ²S² ∂²V/∂S²) dt, since the μS ∂V/∂S terms cancel.
No arbitrage requires dΠ = rΠ dt = r(V − S ∂V/∂S) dt.
Equate: ∂V/∂t + ½σ²S² ∂²V/∂S² = rV − rS ∂V/∂S.
Rearrange to ∂V/∂t + ½σ²S² ∂²V/∂S² + rS ∂V/∂S − rV = 0.

Answer: ∂V/∂t + ½σ²S² ∂²V/∂S² + rS ∂V/∂S − rV = 0. The drift μ does not appear, so the price does not depend on the share's real-world expected return.

Example 2

A non-dividend share has price ₹100. The risk-free force of interest is 5% pa and σ = 20% pa. Using the risk-neutral approach, find d1 and d2 for a 1-year European call with strike ₹100, and then the price. Use N(0.35) = 0.6368 and N(0.15) = 0.5596. Take e^(−0.05) = 0.95123.

Show the solution

τ = 1, S = K = 100, so ln(S/K) = 0.
d1 = [0 + (0.05 + ½ × 0.04) × 1] ÷ (0.2 × 1) = 0.07 ÷ 0.2 = 0.35.
d2 = 0.35 − 0.2 = 0.15.
Call price C = 100 × N(0.35) − 100 × 0.95123 × N(0.15).
First term: 100 × 0.6368 = 63.68.
Second term: 95.123 × 0.5596 = 53.23 (95.123 × 0.5596 = 53.2304).
C = 63.68 − 53.23 = 10.45.
Check: lower bound S − K e^(−rτ) = 100 − 95.123 = 4.877. The price 10.45 is above this and below 100.

Answer: d1 = 0.35, d2 = 0.15 and the call price is about ₹10.45.

Exam tips

In derivation questions, marks go to each logical step: Ito's lemma, the choice of Δ, the no-arbitrage argument, and the final PDE. Show all four.
Write down the assumptions before you start. Examiners often ask for them separately, and they also help you answer limitation questions.
When asked to contrast P and Q, say which parameters change (drift) and which do not (volatility). Say P is used for real-world projection and Q for pricing.
For numerical questions, show d1, d2, the N values and the discount factor. Method marks are given even if your table reading is slightly off.
In computer-based papers, you can price by simulating S(T) under Q with drift r − ½σ² for the log price and discounting the average payoff at r. Do not use μ.

Practice questions from Black-Scholes derivative-pricing model

Risk-Neutral Valuation and the Black-Scholes PDE in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Risk-Neutral Valuation and the Black-Scholes PDE: frequently asked questions

Why does the Black-Scholes PDE not contain the share's expected return?

Delta hedging removes the random term, and with it the need to compensate investors for bearing that risk. The riskless portfolio must earn r, so μ cancels. The option price depends only on S, K, r, σ and time.

What is the difference between the real-world and risk-neutral measure?

Under the real-world measure P, the share grows at its true drift μ. Under the risk-neutral measure Q, it grows at the risk-free rate r. Volatility is the same. You use Q to price derivatives and P to project real outcomes.

Does risk-neutral valuation mean the option's real expected return equals r?

No. The expectation under Q is a pricing tool. Under P, an option typically has a different expected return from r, because it is a leveraged position in the share. Q probabilities are not real-world probabilities.

Can I use risk-neutral pricing for any derivative?

You can when the model is arbitrage-free and the derivative can be replicated, as in Black-Scholes with continuous hedging. Real markets have costs, jumps and stochastic volatility, so the price is a model price rather than an exact one.