Risk-Neutral Pricing and Martingale Measures

Risk-neutral pricing values a replicable payoff as its discounted expectation under a probability measure in which tradable assets, after discounting by a numeraire, are martingales. “Risk neutral” does not describe investors’ beliefs or forecast returns. It is an accounting measure selected by no-arbitrage, and it is the reason the expected stock drift in Black-Scholes pricing is the financing rate rather than an equity-risk-premium estimate.

No-arbitrage in measure language

Take a money-market account B_t satisfying:

dB_t = r_t B_t dt
B_t = exp(integral_0^t r_u du)

The fundamental theorem of asset pricing, in its useful desk form, says:

Market property	Measure statement
No arbitrage	At least one equivalent local martingale measure Q exists
Complete market	That measure is unique
Incomplete market	Several valid Q measures may price non-hedgeable risk

“Equivalent” means Q assigns zero probability to exactly the same impossible events as the physical measure P. Under Q, every traded asset S with dividends D has a discounted gains process that is a martingale:

S_t / B_t + integral_0^t (1 / B_u) dD_u
  = E_Q[ S_T / B_T + integral_0^T (1 / B_u) dD_u | F_t ]

The relation prevents a free expected excess return after financing. It does not claim realized returns will equal rates.

Girsanov in one factor

Under P, let an equity without dividends follow:

dS/S = mu dt + sigma dW_P

Define the market price of risk lambda = (mu - r) / sigma and a new Brownian motion:

dW_Q = dW_P + lambda dt

Then:

dS/S = r dt + sigma dW_Q

Girsanov’s theorem makes this change legitimate under regularity conditions. Economically, the drift adjustment removes the compensation investors require for bearing Brownian risk. The volatility is unchanged because an equivalent measure cannot change instantaneous uncertainty, only how paths are weighted.

Pricing by conditional expectation

For a payoff H_T that can be replicated:

V_t = B_t * E_Q[ H_T / B_T | F_t ]

With deterministic rates:

V_t = exp(-r (T-t)) * E_Q[H_T | F_t]

This is equivalent to solving a pricing PDE or constructing a hedge. Each representation gives a different production tool: PDE grids handle early exercise, trees give transparent regression tests, and Monte Carlo handles high-dimensional path dependence.

import numpy as np

def european_call_mc(s0, strike, rate, vol, maturity, n=1_000_000, seed=19):
    rng = np.random.default_rng(seed)
    z = rng.standard_normal(n)
    st = s0 * np.exp((rate - 0.5 * vol**2) * maturity
                     + vol * np.sqrt(maturity) * z)
    payoff = np.maximum(st - strike, 0.0)
    pv = np.exp(-rate * maturity) * payoff.mean()
    stderr = np.exp(-rate * maturity) * payoff.std(ddof=1) / np.sqrt(n)
    return pv, stderr

The standard error is part of the result. A point estimate without its Monte Carlo uncertainty is not a validation.

Numeraires and forward measures

The money-market account is not mandatory. For any strictly positive tradable numeraire Nt, there is a measure Q^N under which St/N_t is a martingale:

V_t = N_t * E_QN[H_T / N_T | F_t]

This choice often simplifies dynamics:

Numeraire	Natural measure	Typical application
Money market B	risk-neutral	generic discounting
Zero-coupon bond P(t,T)	T-forward	caplets, bond options
Stock with reinvested dividends	share measure	digital/asset-or-nothing options

For a payoff at T, pricing under the T-forward measure avoids carrying stochastic discount factors in the expectation. In rate models, this is a computational and calibration advantage, not notation.

Completeness is a model choice

In a one-stock, one-Brownian Black-Scholes market, delta trading spans the only source of risk and makes Q unique. Add independent stochastic volatility, default, liquidity risk, or jumps without enough liquid hedge instruments, and the market is incomplete. The model must then choose a risk premium for unspanned risk or calibrate directly to liquid option prices.

That distinction explains why Heston parameters are often quoted under Q, while historical volatility studies use P. Mixing the two—estimating physical mean reversion and inserting it unmodified into an option pricer—is a frequent source of unexplained P&L.

Operational checks

Before trusting a risk-neutral engine, verify:

1. Discounted simulated tradables have stable means.
2. Put-call parity holds within numerical tolerance:

C - P = S_0 exp(-qT) - K exp(-rT)

1. A change of numeraire produces the same price.
2. Calibration instruments reprice using the same curves and conventions as the market.

The last item catches more production errors than elegant theory: day count, collateral discounting, dividend timing, and settlement conventions can dominate a small model improvement.

Key takeaways

• Risk-neutral probabilities are pricing weights, not forecasts of physical returns.
• Discounted traded gains are martingales under an equivalent martingale measure.
• Replication, PDEs, and discounted expectations are equivalent pricing views.
• Changing numeraire changes the convenient measure, never the arbitrage-free price.
• In incomplete markets, the choice of risk premium is an explicit modeling assumption.