Phase 9 - Lesson 9.6
Risk-Neutral Pricing: Foundations
Why a derivative's price is a discounted expectation under a special probability measure.
Builds on: 9.4 built the measure change; 9.5 gave the PDE view.
Leads to: Phase 14–15 extend this to full derivatives pricing and volatility.
Leads to: Phase 14–15 extend this to full derivatives pricing and volatility.
Learning Objectives
Click a status chip to cycle: Not started → In progress → Studied → Practiced → Needs review → Mastered.
- Distinguish the physical measure \(\Prob\) from the risk-neutral measure \(\mathbb{Q}\) and state what each is for.
- State the risk-neutral pricing formula as a discounted \(\mathbb{Q}\)-expectation.
- Explain the role of a replicating/self-financing portfolio and no-arbitrage.
- Connect risk-neutral pricing to Girsanov (9.4) and Feynman–Kac (9.5).
- Estimate a call price by Monte Carlo under \(\mathbb{Q}\) and compare to Black–Scholes.
Key Vocabulary
- Physical measure \(\Prob\)
- The real-world probability estimated from data; used for risk, forecasting, and P&L.
- Risk-neutral measure \(\mathbb{Q}\)
- An equivalent measure under which discounted tradable prices are martingales; used only for pricing.
- Numeraire
- The asset prices are expressed in; the money-market account \(B_t=e^{rt}\) is the standard choice.
- Self-financing portfolio
- A trading strategy whose value changes only through asset returns, with no external cash in/out.
- No-arbitrage
- No self-financing strategy turns zero initial wealth into a sure profit; equivalent to existence of \(\mathbb{Q}\).
- Replication
- Matching a derivative's payoff with a dynamic portfolio of the underlying and cash.
Intuition & Motivation
Intuition
Investors demand extra return \(\mu-r\) for bearing risk, so real-world averages are hard to price with - you would need every investor's risk appetite. The trick: find an equivalent measure \(\mathbb{Q}\) in which every tradable asset is expected to grow at just the risk-free rate \(r\). Under \(\mathbb{Q}\) nobody is compensated for risk (hence ‘risk-neutral’), pricing collapses to simple discounted averaging, and - crucially - the answer is still correct in the real world because \(\mathbb{Q}\) and \(\Prob\) agree on what is possible.
Two measures, two jobs
| Question | Measure | Drift of \(S\) |
|---|---|---|
| What return should I expect / how risky is this? | Physical \(\Prob\) | \(\mu\) |
| What is this derivative worth today? | Risk-neutral \(\mathbb{Q}\) | \(r\) |
They are equivalent (same null sets) but assign different probabilities. Girsanov (9.4) is the explicit bridge: the market price of risk \(\theta=(\mu-r)/\sigma\) is exactly the drift adjustment.
Theorem - Risk-neutral pricing formula
In a complete, arbitrage-free market with money-market account \(B_t=e^{rt}\), the time-\(t\) price of a derivative paying \(V_T\) at \(T\) is
\[V_t=\E_{\mathbb{Q}}\!\big[e^{-r(T-t)}\,V_T\mid \mathcal{F}_t\big],\quad\text{equivalently }\ \frac{V_t}{B_t}=\E_{\mathbb{Q}}\!\Big[\frac{V_T}{B_T}\Big|\mathcal{F}_t\Big].\] (9.8)
Discounted prices are \(\mathbb{Q}\)-martingales: today's discounted value is the expected discounted future value.
Worked Example - Why the formula holds (replication + no arbitrage)
1
Build a self-financing portfolio of the stock and cash that replicates the payoff \(V_T\) (possible in a complete market).
2
Its discounted value \(V_t/B_t\) has, under \(\mathbb{Q}\), zero drift (the stock's discounted price \(e^{-rt}S_t\) is a \(\mathbb{Q}\)-martingale from 9.4), so it is a martingale.
3
A martingale satisfies \(V_t/B_t=\E_{\mathbb{Q}}[V_T/B_T\mid\mathcal F_t]\), which rearranges to (9.8).
4
No-arbitrage forces the derivative's market price to equal the replicating portfolio's value - else a riskless profit exists. Hence (9.8) is the price.
Three views of the same price
- Probabilistic (this lesson): \(V_0=e^{-rT}\E_{\mathbb{Q}}[V_T]\).
- Girsanov (9.4): \(\mathbb{Q}\) is obtained by shifting the drift \(\mu\to r\) with density \(Z\).
- Feynman–Kac (9.5): the same \(V_0\) solves the pricing PDE \(v_t+rSv_S+\tfrac12\sigma^2S^2v_{SS}-rv=0\) - the Black–Scholes equation.
For a European call under \(\mathbb{Q}\), \(S_T=S_0e^{(r-\sigma^2/2)T+\sigma\tilde W_T}\) is lognormal, and (9.8) evaluates in closed form to the Black–Scholes price \(C=S_0N(d_1)-Ke^{-rT}N(d_2)\).
Interactive: Monte-Carlo the call under \(\mathbb{Q}\)
Common Mistakes to Avoid
- Discounting real-world expected payoffs \(\E_\Prob\) at \(r\). You must use \(\E_{\mathbb{Q}}\); mixing \(\Prob\) payoffs with \(r\) discounting double-counts or ignores the risk premium.
- Believing \(\mathbb{Q}\) is the ‘true’ probability. It is a pricing device; frequencies of outcomes are governed by \(\Prob\).
- Using \(\mu\) anywhere in an option price. Under \(\mathbb{Q}\) the drift is \(r\); \(\mu\) cancels out.
- Forgetting to discount, or discounting the wrong horizon \((T-t)\).
Quant Practitioner Tips
- Slogan: price = discounted expected payoff, but expectation under \(\mathbb{Q}\), not \(\Prob\).
- Completeness (replication) is what makes \(\mathbb{Q}\) unique; in incomplete markets many \(\mathbb{Q}\) exist and prices form a range.
- Monte Carlo error shrinks like \(1/\sqrt{N}\); to halve it, quadruple the paths (or use variance reduction).
Knowledge Check
Q1 Medium
The risk-neutral price of a claim paying \(V_T\) at \(T\) is:
Q2 Easy
Under the risk-neutral measure \(\mathbb{Q}\), a non-dividend stock has drift:
Q3 Medium
What guarantees the existence of a risk-neutral measure \(\mathbb{Q}\)?
Practical Exercise
A digital option pays $1 if \(S_T\gt K\), else 0, with \(S_0=100,K=100,r=0.03,\sigma=0.2,T=1\). (a) Write its price as a \(\mathbb{Q}\)-expectation. (b) Show it equals \(e^{-rT}N(d_2)\) and compute it. (Recall \(d_2=0.05\) from the Black–Scholes numbers in this phase.)
▶ Show full solution
(a) \(V_0=e^{-rT}\E_{\mathbb{Q}}[\ind_{S_T\gt K}]=e^{-rT}\,\mathbb{Q}(S_T\gt K)\).
(b) Under \(\mathbb{Q}\), \(\log S_T\sim\Normal(\log S_0+(r-\tfrac12\sigma^2)T,\ \sigma^2 T)\). Then \(\mathbb{Q}(S_T\gt K)=N(d_2)\) with \(d_2=\tfrac{\log(S_0/K)+(r-\sigma^2/2)T}{\sigma\sqrt T}=0.05\).
So \(V_0=e^{-0.03}N(0.05)\approx0.9704\times0.5199\approx0.5046\), about $0.50.
After the reveal, answer for yourself: Why is a digital's price essentially \(N(d_2)\) discounted - the same \(d_2\) appearing in the vanilla call?
Lesson Summary
Risk-neutral pricing values a derivative as \(V_t=\E_{\mathbb{Q}}[e^{-r(T-t)}V_T\mid\mathcal F_t]\): discount at \(r\) and average under the measure \(\mathbb{Q}\) in which discounted tradables are martingales. \(\mathbb{Q}\) (from Girsanov) replaces the physical drift \(\mu\) with \(r\); no-arbitrage and replication justify it; Feynman–Kac turns it into the Black–Scholes PDE. The physical measure \(\Prob\) stays for risk, not pricing.
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
▶ Show retrieval prompts & answers
Q: State the risk-neutral pricing formula and name the measure.
A: \(V_t=\E_{\mathbb{Q}}[e^{-r(T-t)}V_T\mid\mathcal F_t]\), where \(\mathbb{Q}\) is the risk-neutral (equivalent martingale) measure under which discounted asset prices are martingales.
A: \(V_t=\E_{\mathbb{Q}}[e^{-r(T-t)}V_T\mid\mathcal F_t]\), where \(\mathbb{Q}\) is the risk-neutral (equivalent martingale) measure under which discounted asset prices are martingales.
Q: Why does the stock's real-world drift \(\mu\) not appear in an option price?
A: Because pricing uses \(\mathbb{Q}\), where Girsanov replaced \(\mu\) by \(r\). Replication makes the price independent of the risk premium, so \(\mu\) cancels.
A: Because pricing uses \(\mathbb{Q}\), where Girsanov replaced \(\mu\) by \(r\). Replication makes the price independent of the risk premium, so \(\mu\) cancels.
Completion Checklist
- I can explain the core ideas in my own words
- I worked the derivations/examples by hand
- I completed the interactive workbench(es)
- I passed the knowledge check
Confidence / mastery rating
Personal notes