Risk-Neutral Pricing: Foundations
Why a derivative's price is a discounted expectation under a special probability measure.
Leads to: Phase 14–15 extend this to full derivatives pricing and volatility.
Learning Objectives
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- 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
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.
Discounted prices are \(\mathbb{Q}\)-martingales: today's discounted value is the expected discounted future value.
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 \(\mathbb{Q}\)
- 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 \((T-t)\).
- 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
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.)
(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.
Lesson Summary
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
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: 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
Source References
This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.
- Stochastic Calculus for Finance II (Steven Shreve, 2004) foundational - Ch. 5 - Ch. 5: risk-neutral pricing, the fundamental theorems, and Black–Scholes.
- Methods of Mathematical Finance (Karatzas & Shreve, 1998) foundational - Ch. 1-2 - Ch. 1–2: martingale methods, equivalent martingale measures, and contingent-claim valuation.