Lesson 7.3: Risk-neutral pricing, change of measure and no-arbitrage
Learn why derivatives are priced by replication, what risk-neutral probabilities are, how Girsanov's theorem works and why the real-world return drops out.
By the end of this lesson you will be able to define arbitrage, price an option in a one-step binomial model by replication, compute the risk-neutral probability and explain that it is not a forecast, state the two fundamental theorems of asset pricing, describe how Girsanov's theorem changes the drift of Brownian motion by the market price of risk, and price an option by averaging payoffs under the risk-neutral measure. You will show that a call on a share at 100 that can go to 110 or 90 over a period with 5 per cent interest is worth 7.19 and is replicated by half a share and a loan of 42.81, that a risk-neutral probability of up of 0.756 gives this price whatever the real-world probability of up is, that averaging the payoff under the real-world measure instead gives a wrong 8.43 for a six-month call while reweighting the same paths gives 7.15 against the exact 7.16, and that N(d₂) in the Black–Scholes formula, 0.556, is the risk-neutral probability of finishing in the money. You should have read Lessons 6.1, 6.2, 7.1 and 7.2. The mathematics is expectations, exponentials and the exponential martingale of Lesson 7.1. This course is education about how markets and research work, not investment advice.
Lesson 7.2 ended on a surprise: in the Black–Scholes equation the expected return of the share drops out. Two people with different beliefs about where the share is going, one expecting 5 per cent a year and the other 50 per cent, must agree on the price of the option. This lesson explains why, and turns that surprise into a general and very practical method. The method, risk-neutral pricing, is how almost every derivative in the world is valued, whether it is a plain option, a complex structured product or a credit derivative. The idea is simple, and so it is easy to misread, so we take it step by step.
No-arbitrage and the law of one price
An arbitrage is a trading strategy that costs nothing now, can never lose money, and has some chance of making money. A free lunch. Markets with active traders tend to remove arbitrage quickly, since anyone who sees one trades it, and prices move until it is gone. We have already used the idea: put-call parity in Lesson 6.1 holds because if it failed, a riskless profit could be locked in.
Assuming no arbitrage leads to the law of one price: two assets that give the same payoffs in every possible circumstance must have the same price today. This one principle is enough to price an option, if the option's payoff can be copied by a portfolio of other assets. The price of the option is then the cost of the copying portfolio, the replicating portfolio.
Pricing by replication: the one-period binomial model
The simplest possible market has one share and one bank account, and one period. The share is worth S today and, at the end of the period, either uS (up) or dS (down), where u > 1 > d. Money in the bank grows by a factor e^(rΔt), written R below. For no arbitrage we need d < R < u: otherwise one asset would always beat the other.
A call option with strike K pays Vu = max(uS − K, 0) if the share goes up and Vd = max(dS − K, 0) if it goes down. Build a portfolio of Δ shares and an amount B in the bank that gives the same payoffs:
in the up state: Δ × uS + B × R = Vu
in the down state: Δ × dS + B × R = Vd
Two equations, two unknowns. Subtract to find the hedge ratio Δ = (Vu − Vd) ÷ (S (u − d)), and then B from either equation. The option's price is the cost of the portfolio, Δ S + B.
Worked example 1. The share is at S = 100 and goes to 110 or 90 (u = 1.1, d = 0.9). The interest rate is 5 per cent, so R = e^0.05 = 1.0513. A call has a strike of 100: Vu = 10 and Vd = 0.
Δ = (10 − 0) ÷ (100 × 0.2) = 0.5 of a share.
In the down state, 0.5 × 90 + B × 1.0513 = 0, so B = −45 ÷ 1.0513 = −42.81: a loan.
The price is 0.5 × 100 − 42.81 = 7.19.
Half a share financed by a loan of 42.81 pays 10 in the up state (55 − 45) and 0 in the down state (45 − 45), exactly what the option pays. So the call is worth 7.19, whatever anyone believes about the probability of the share going up. If it traded for more, you would sell it and buy the replicating portfolio for 7.19 and keep the difference; if for less, the reverse.
The risk-neutral probability
There is a way to write the answer as an expectation. Algebra on the formula for the price gives
price = e^(−rΔt) × [ p × Vu + (1 − p) × Vd ], where p = (e^(rΔt) − d) ÷ (u − d)
The number p is the risk-neutral probability of an up move. In the example, p = (1.0513 − 0.9) ÷ 0.2 = 0.7564, and the price is e^(−0.05) × (0.7564 × 10 + 0.2436 × 0) = 0.9512 × 7.564 = 7.19, the same number as before. Three points deserve emphasis.
1. p is not a forecast. It is a number constructed so that the share itself, under p, has an expected return equal to the risk-free rate: p × 110 + (1 − p) × 90 = 83.20 + 21.92 = 105.13, which equals 100 × e^0.05. A world in which this holds is a world in which investors do not demand a reward for risk, hence the name risk-neutral.
2. The real-world probability does not enter. The number p depends only on u, d and r.
3. Using the real-world probability gives the wrong price. Suppose the true chance of the share going up is 60 per cent. The discounted expected payoff is e^(−0.05) × 0.6 × 10 = 5.71, which is below the arbitrage-free price of 7.19. Even at 70 per cent it is 6.66. The only probability that gives 7.19 is p = 0.756.
Real-world probability of up · Discounted expected payoff · Arbitrage-free price40% · 3.80 · 7.19
50% · 4.76 · 7.19
60% · 5.71 · 7.19
70% · 6.66 · 7.19
75.6% (equal to p) · 7.19 · 7.19
An option priced at 5.71 because someone believes in a 60 per cent chance would be an arbitrage: buy it for 5.71, sell the replicating portfolio for 7.19, and keep 1.48 for sure. The market price is pinned by replication, not by opinion. The reason an option on a share that is more likely to rise is not more expensive than the formula says is that its expected payoff is not what you pay for. What you pay is the cost of the hedge.
Try your own tree below, and note that changing the real-world probability does not move the price.
The fundamental theorems of asset pricing
The binomial example is a special case of a general theory, due to Michael Harrison, David Kreps and Stanley Pliska in the late 1970s and early 1980s, and extended by Freddy Delbaen and Walter Schachermayer.
First fundamental theorem: a market has no arbitrage if and only if there exists at least one probability measure Q, equivalent to the real-world measure P, under which the discounted price of every traded asset is a martingale. "Equivalent" means the two measures agree on which events are possible and which are impossible, though they may disagree on how probable the possible ones are.
Second fundamental theorem: the market is complete, meaning that every payoff can be replicated, if and only if Q is unique.
The consequence is the pricing rule of the whole subject. The price today of a derivative with payoff V at time T is
price = e^(−rT) × E^Q[ V ]
the discounted expected payoff under Q, the risk-neutral measure. In a complete market this gives the unique arbitrage-free price. The binomial model and the Black–Scholes model with one share are both complete. Models with random volatility or jumps are incomplete: there are many measures Q, each consistent with no arbitrage, and the market's option prices select which one applies. This is what calibration does (Lesson 8.4).
Changing the measure: Girsanov's theorem
In continuous time, the share follows dS = μ S dt + σ S dB under the real-world measure P, with expected return μ. We want a new measure Q under which the discounted share is a martingale, so that its drift equals the risk-free rate r. The tool is Girsanov's theorem: you can change the drift of a Brownian motion by changing the probability measure, if you reweight the paths properly.
Define the market price of risk θ = (μ − r) ÷ σ. It is the excess return per unit of volatility, the Sharpe ratio of the share. Define a new process W^Qt = Bt + θ t, which is Bt with a drift added. Girsanov's theorem says there is a measure Q under which W^Q is a standard Brownian motion, with density
dQ/dP = exp( −θ BT − ½ θ² T )
This is exactly the exponential martingale of Lesson 7.1 with σ replaced by −θ, and its expected value under P is 1, as a density should be. Substitute B = W^Q − θ t into the share's equation:
dS = μ S dt + σ S (dW^Q − θ dt) = (μ − σθ) S dt + σ S dW^Q = r S dt + σ S dW^Q
Under Q, the drift of every share is the risk-free rate, whatever its real-world expected return, and the volatility is unchanged. The change of measure removes μ and leaves σ, which is the mathematical reason the Black–Scholes equation of Lesson 7.2 does not contain μ. Volatility is a property of the path itself (it is the quadratic variation of Lesson 7.1) and cannot be changed by reweighting, but the drift can.
Worked example 2. A share has an expected return of 10 per cent and volatility of 20 per cent, and the interest rate is 6 per cent. The market price of risk is θ = (0.10 − 0.06) ÷ 0.20 = 0.20. A share with an expected return of 25 per cent and the same volatility has θ = 0.95. Under Q both drift at 6 per cent, and the options on both are worth the same: option prices depend on σ and r only.
Pricing by averaging under Q
With the drift of the share under Q equal to r, pricing a call is an expectation. Under Q, ST = S₀ exp((r − σ²/2)T + σ W^QT), and
Call = e^(−rT) × E^Q[ max(ST − K, 0) ]
Evaluating this expectation with the normal distribution gives the Black–Scholes formula. The two terms have a clean meaning. Under Q, the probability that the call finishes in the money is N(d₂). The first term, S N(d₁), is the discounted expected value of the share received when the call finishes in the money, which is an expectation under a third measure, the one that uses the share itself as the unit of account (a change of numeraire), under which the probability of finishing in the money is N(d₁). Changing the numeraire, to the share or to a bond that matures at T, is a standard way of simplifying prices in interest-rate and currency products.
We checked all of this by simulation for the six-month call of Lesson 6.2: share at 100, strike 100, expected return 10 per cent, interest 6 per cent, volatility 20 per cent.
Averaging the payoff under the real-world measure. Simulate 400,000 paths with the drift of 10 per cent, discount the average payoff at 6 per cent, and you get 8.43, wrong.
Reweighting the same paths by dQ/dP. Multiply each path's payoff by exp(−θ BT − ½ θ² T) with θ = 0.20, whose average is 1.0000, and the discounted average becomes 7.147, within 0.01 of the exact Black–Scholes price of 7.156.
Probability of finishing in the money under Q. Simulating paths with the risk-neutral drift of 6 per cent, 55.62 per cent finish above the strike, matching N(d₂) = 0.5562.
In practice there is no need to simulate under P and reweight: you simulate directly under Q, with drift r. The reweighting is shown to make the logic visible: Q and P differ by a density, and pricing is done with the one that makes discounted prices martingales.
What "risk-neutral" does and does not mean
The name causes confusion, so be precise.
It does not mean that investors are risk-neutral. Real investors demand a premium for risk. The risk-neutral measure is a mathematical device: a reweighting of real-world probabilities in which the risk premium has been absorbed, so that expected payoffs can be discounted at the risk-free rate.
Q probabilities are not forecasts. They put more weight on bad states than real-world probabilities do, because bad states are the ones investors pay to insure against. The risk-neutral probability of a large fall in an index, read from put prices, is typically higher than the frequency with which large falls occur. This is the skew and the variance risk premium of Lesson 6.3, seen from the other side.
P is for risk, Q is for pricing. To estimate the chance of a loss, a value at risk or the expected profit of a strategy, you need real-world probabilities, which means estimating μ from data, with all its noise (Lesson 1.3). To price a derivative, you use Q, where μ plays no part.
· Real-world measure P · Risk-neutral measure QDrift of a share · μ, the expected return · r, the risk-free rate
Volatility · σ · σ, unchanged
Estimated from · Historical data and judgement · Market prices of traded options and bonds
Used for · Risk, forecasting, strategy evaluation · Pricing and hedging derivatives
Probabilities · Frequencies of events · Prices of insurance, with risk premia built in
## Practical consequences
Monte Carlo pricing. Simulate paths with drift r, average the discounted payoff, and add variance-reduction techniques. Complex products, such as path-dependent options, are priced this way (Lesson 8.1).
Model-free relations. Any relation that follows from replication, such as put-call parity, holds in every model.
Calibration. Because Q is defined by market prices, models are chosen so that their Q-prices match the traded options. A model that prices the options of today correctly can still predict the hedge ratios badly.
Hedging and pricing are linked. The price is the cost of the hedge, so a trader who cannot hedge cannot rely on the price. Illiquid or jump-prone assets break the argument, because the hedge is imperfect.
Risk premia and arbitrage-free models. Interest-rate and credit models are built by specifying the dynamics directly under Q and fitting them to today's curve.
The code below prices the one-step call, the naive alternative, the Monte Carlo reweighting and the probabilities of finishing in the money.
python
import numpy as np
from math import exp, log, sqrt
from scipy.stats import norm
1. one-step binomial model: replication and the risk-neutral probability
S, u, d, r, K = 100.0, 1.1, 0.9, 0.05, 100.0
R = exp(r)
pstar = (R - d) / (u - d)
Vu, Vd = max(u S - K, 0), max(d S - K, 0)
delta = (Vu - Vd) / (S (u - d))
bond = (Vu - delta u S) / R
print("p", round(pstar, 4), "| delta", delta, "| loan", round(bond, 3), "| price by replication", round(delta S + bond, 4),
"| price by expectation", round(exp(-r) (pstar Vu + (1 - pstar) Vd), 4))
print("the share under p grows at the risk-free rate:", round((pstar u S + (1 - pstar) d S) / S, 4), "=", round(R, 4))
for p in (0.4, 0.5, 0.6, 0.7, pstar):
print(f" real-world probability {p:.3f}: discounted expected payoff {exp(-r) (p Vu + (1 - p) Vd):.3f}")
2. market price of risk
mu, rf, sigma, T, K2, S0 = 0.10, 0.06, 0.20, 0.5, 100.0, 100.0
theta = (mu - rf) / sigma
print("market price of risk", round(theta, 3))
3. Monte Carlo: real-world average, reweighting by dQ/dP, and the exact formula
rng = np.random.defaultrng(11)
N = 400000
z = rng.standardnormal(N)
STP = S0 np.exp((mu - 0.5 sigma 2) T + sigma sqrt(T) z) # paths under the real-world measure
weight = np.exp(-theta sqrt(T) z - 0.5 theta 2 T) # dQ/dP, since BT = sqrt(T) z
payoff = np.maximum(STP - K2, 0)
d1 = (log(S0 / K2) + (rf + 0.5 sigma 2) T) / (sigma sqrt(T))
d2 = d1 - sigma sqrt(T)
exact = S0 norm.cdf(d1) - K2 exp(-rf T) norm.cdf(d2)
print("discounted real-world average", round(exp(-rf T) payoff.mean(), 3), "| reweighted", round(exp(-rf T) (weight payoff).mean(), 3),
"| exact", round(exact, 3), "| average weight", round(weight.mean(), 4))
4. probability of finishing in the money under Q
zq = rng.standardnormal(N)
STQ = S0 np.exp((rf - 0.5 sigma 2) T + sigma sqrt(T) zq) # paths drifting at the risk-free rate
print("N(d2) =", round(norm.cdf(d2), 4), "| simulated Q(ST > K) =", round((STQ > K2).mean(), 4), "| N(d1) =", round(norm.cdf(d1), 4))
Common mistakes
Pricing derivatives with real-world expected payoffs. It gives the wrong answer, 8.43 instead of 7.16 in our example, and an exploitable one.
Reading risk-neutral probabilities as forecasts. They embed risk premia and are generally more pessimistic than real-world frequencies.
Thinking risk-neutral pricing assumes investors are indifferent to risk. It is a device that follows from no-arbitrage and replication.
Using Q to estimate risks or P to price. Use real-world probabilities for risk and strategy evaluation, and Q for pricing.
Applying the theory to an asset that cannot be hedged. The price is the cost of a hedge, and with jumps, illiquidity or constraints the hedge is imperfect, so the price is a range.
Forgetting that Q is not unique in incomplete markets. Calibration to traded prices selects it.
Mixing up N(d₁) and N(d₂). N(d₂) is the probability under Q, and N(d₁) is the probability under the share numeraire.
Believing that arbitrage never appears. It appears briefly, in data errors and in stress, and the theory describes what prices are pulled towards.
Key takeaways
An arbitrage costs nothing, cannot lose and might gain. No-arbitrage implies the law of one price, so a derivative is worth what its replicating portfolio costs.
In a one-step binomial model with u = 1.1, d = 0.9 and 5 per cent interest, a call at 100 is replicated by 0.5 of a share and a loan of 42.81, so it is worth 7.19. The same price comes from the risk-neutral probability p = (e^(rΔt) − d) ÷ (u − d) = 0.756.
The real-world probability does not enter. Using 60 per cent in place of p gives 5.71, an arbitrage-exploitable price. The price is the cost of the hedge, not the expected payoff.
The fundamental theorems: no arbitrage if and only if an equivalent measure Q exists under which discounted prices are martingales, and completeness if and only if Q is unique. The price is e^(−rT) E^Q[payoff].
By Girsanov's theorem, W^Q = B + θt with θ = (μ − r) ÷ σ is Brownian under Q, which has density exp(−θBT − ½θ²T). Under Q every share drifts at r and keeps its volatility, so μ drops out.
In simulation, the real-world average gives 8.43, reweighting by dQ/dP gives 7.147 and the formula 7.156. N(d₂) = 0.5562 is the risk-neutral probability of finishing in the money.
Q is a pricing tool, not a forecast. Use P for risk and strategy evaluation and Q for pricing, and in incomplete markets calibrate Q to traded prices.
Check yourself
1. A share at 50 may go to 60 or 45 in one period, and the interest rate is 4 per cent per period, continuously compounded. What is the risk-neutral probability of an up move?
u = 1.2 and d = 0.9. R = e^0.04 = 1.0408. p = (1.0408 − 0.9) ÷ (1.2 − 0.9) = 0.1408 ÷ 0.3 = 0.469.
2. For that tree, what is the value of a call with a strike of 50?
Vu = 10, Vd = 0. Price = e^(−0.04) × 0.469 × 10 = 0.9608 × 4.69 = 4.51.
3. Why must the price of the option in the example not change if a forecaster says the share is 80 per cent likely to go up?
The option can be replicated by a share and a loan, whose cost does not depend on the chance of either outcome. A price different from the cost of replication would allow an arbitrage.
4. A share has an expected return of 15 per cent, volatility of 25 per cent and the interest rate is 5 per cent. What is the market price of risk, and what is the share's drift under Q?
θ = (0.15 − 0.05) ÷ 0.25 = 0.40. The drift under Q is 5 per cent.
5. An analyst says the option market's put prices imply a 10 per cent chance of a 20 per cent fall in a year, when history says it happens 4 per cent of the time. What explains the gap?
The put prices give risk-neutral probabilities, which put more weight on bad states than real-world frequencies because investors pay to insure against them. The gap is the risk premium, not a forecast that the fall is more likely than history says.
Try this
Set aside forty-five minutes. You need Python and the code in this lesson.
1. Run the code. Check the replication, the risk-neutral probability and the three prices in the Monte Carlo test.
2. Change the real-world expected return from 10 to 25 per cent. Do the reweighted price and the exact price change? What happens to the real-world average?
3. Extend the one-step model to two steps by hand for the call of the lesson. Compute the price by working back from the final payoffs with p, and compare with the formula of Lesson 7.4.
4. Use the one-step simulator to find the up factor at which the risk-neutral probability reaches 1. What does that mean for arbitrage?
5. Compute the risk-neutral probability of a fall of more than 20 per cent for a share with 25 per cent volatility over one year, using N(−d₂) for a put with a strike at 80, and compare with a plausible real-world frequency.
6. Simulate under Q directly, with drift r, and price an at-the-money put. Check put-call parity against the call price.
7. Write a short note: why two investors who disagree about a share's future return must agree on the price of its option, and when that argument fails.
A good result is a replication table showing the same price by two methods, a Monte Carlo comparison of the real-world, reweighted and exact prices, and a clear explanation of why the expected return drops out. The last lesson of this module builds the bridge between the one-step model and the continuous one: binomial trees with many steps, and how they converge to Black–Scholes.
Read more
J. Michael Harrison and David Kreps's 1979 paper "Martingales and Arbitrage in Multiperiod Securities Markets" in the Journal of Economic Theory and Harrison and Stanley Pliska's 1981 paper "Martingales and Stochastic Integrals in the Theory of Continuous Trading" in Stochastic Processes and their Applications set out the theory. Freddy Delbaen and Walter Schachermayer's The Mathematics of Arbitrage is the rigorous account. Steven Shreve's Stochastic Calculus for Finance I (binomial models) and II (continuous time) develop risk-neutral pricing step by step, and John Hull's Options, Futures and Other Derivatives gives an applied treatment with the numeraire changes. Tomas Björk's Arbitrage Theory in Continuous Time is a clear text on the measure change.
Tags: Quant Trader Risk-neutral pricing No-arbitrage Girsanov theorem Change of measure