A bank account growing at the riskless rate r turns
1 today into e^{rt} at time
t. To compare a risky price S_t against
that benchmark we strip out the guaranteed growth — we discount — giving the
discounted price
\tilde{S}_t = e^{-rt} S_t.
Under the real-world measure \mathbb{P} a stock typically drifts
upward faster than the bank (that excess is its risk premium), so
\tilde{S}_t drifts up too — it is not fair. The central trick of
arbitrage pricing is to change measure: find a new probability measure
\mathbb{Q}, equivalent to
\mathbb{P}, under which the discounted price is a fair game.
An equivalent martingale measure (EMM), also called a
risk-neutral measure, is a measure
\mathbb{Q} \sim \mathbb{P} under which the discounted price is a
martingale:
\mathbb{E}_{\mathbb{Q}}\big[\tilde{S}_t \mid \mathcal{F}_s\big] = \tilde{S}_s \qquad \text{for all } s \le t.
The word equivalent is doing real work: by the
Radon–Nikodym
relation, \mathbb{Q} and \mathbb{P} must
agree on what is possible — we are allowed to reweight the odds, never to conjure or forbid an
outcome.
Pricing by discounted expectation, derived line by line
Here is the payoff for all that machinery. Consider a claim that pays
H_T at maturity T (a call option pays
(S_T - K)^+, say). Let V_t be its
arbitrage-free price at time t. We will show
V_0 = \mathbb{E}_{\mathbb{Q}}\big[e^{-rT} H_T\big]
— today's price is just the discounted expected payoff under
\mathbb{Q}.
Step 1 — discount the value process. Define the discounted price of the
claim itself, \tilde{V}_t = e^{-rt} V_t. The no-arbitrage principle
says any tradeable asset must be priced consistently with the stock, so — exactly like
\tilde{S}_t — the discounted claim price is a
\mathbb{Q}-martingale:
\mathbb{E}_{\mathbb{Q}}\big[\tilde{V}_t \mid \mathcal{F}_s\big] = \tilde{V}_s.
Step 2 — apply the martingale property from 0 to
T. Take s = 0 and
t = T. Conditioning on
\mathcal{F}_0 (which knows nothing but constants) is just an
ordinary expectation:
\tilde{V}_0 = \mathbb{E}_{\mathbb{Q}}\big[\tilde{V}_T \mid \mathcal{F}_0\big] = \mathbb{E}_{\mathbb{Q}}\big[\tilde{V}_T\big].
Step 3 — substitute the discount factors. At time
0 there is nothing to discount, so
\tilde{V}_0 = e^{0}V_0 = V_0. At maturity the claim is worth its
payoff, V_T = H_T, so
\tilde{V}_T = e^{-rT} H_T. Plugging both in,
V_0 = \mathbb{E}_{\mathbb{Q}}\big[e^{-rT} H_T\big] = e^{-rT}\,\mathbb{E}_{\mathbb{Q}}[H_T],
the last equality pulling out the constant discount factor. Price a claim by taking
its expected payoff under \mathbb{Q} and discounting. The
whole subject of derivative pricing is the project of computing this one expectation for ever
more elaborate H_T.
For a market with riskless rate r:
-
(First FTAP) the market admits no arbitrage
iff there exists an equivalent martingale measure
\mathbb{Q} \sim \mathbb{P} under which every discounted price
\tilde{S}_t = e^{-rt} S_t is a martingale.
-
(Second FTAP) the market is complete (every claim can be
replicated by trading) iff that EMM is unique.
-
When it exists, the arbitrage-free price of a claim paying
H_T is its discounted
\mathbb{Q}-expectation,
V_0 = \mathbb{E}_{\mathbb{Q}}[e^{-rT} H_T].
Discounting by e^{-rt} is a choice of numéraire
— the yardstick we measure every price in. Here the yardstick is the bank account
B_t = e^{rt}, and \tilde{S}_t = S_t / B_t
is the stock priced in units of the bank account. A martingale measure is one in
which prices-in-units-of-the-numéraire are fair; change the numéraire (to the stock itself,
or to a bond) and you change which measure does the job — a flexibility that pays off in the
forward-measure tricks of interest-rate modelling.
The name risk-neutral comes from a striking fact. Rearranging the
martingale condition \mathbb{E}_{\mathbb{Q}}[\tilde{S}_t] = \tilde{S}_0
back into undiscounted terms gives
\mathbb{E}_{\mathbb{Q}}[S_t] = e^{rt} S_0,
so under \mathbb{Q} the stock is expected to grow at exactly
the riskless rate r — the same as the bank. Every asset earns
r on average; the risk premium has been reweighted away. It is not
that investors are truly indifferent to risk, but that the pricing measure behaves
as if they were. \mathbb{Q} is a computational device, not a
forecast — real-world frequencies still live under
\mathbb{P}.