sparse grid method in the libor market model. option valuation and the

Risk-Neutral Pricing Another important implication of no-arbitrage assumption is
reflected in the Arbitrage Theorem:
Theorem 2.1.1. In a market with N assets and n states at time T 1 , there is no arbitrage
if and only if there is a vector ψ, strictly positive in all coordinates, such that I = Dψ
For the proof, an interested reader is referred to [Etherage, 2002].
The vector ψ is known as a state price vector. Let us define |ψ| 1 = ∑ n
i=1 ψ i. If each
side of the relation I = Dψ is multiplied with a scalar:
1
I = Dψ 1
(2.1)
|ψ| 1
|ψ| 1
then the product ψ 1
|ψ| 1
can be interpreted as a probability distribution vector. Given
such a vector, the right-hand side of (2.1) can be interpreted as the expectation operator
with respect to the probability distribution ψ 1
|ψ| 1
. As a result, the following relation
holds for each asset comprising our economy:
I i T 0
= |ψ| 1 E(I i T ), i = 1 . . . N.
Conveniently enough, it appears 4 that the coefficient |ψ| 1 is nothing but a risk-free
discount factor e −rT .
For example, we can look at the expected rate of return of the stock setting the
probabilities of up and down movements to be the p and 1 − p, and given the previous
result:
E(S T1 ) = pS T0 u + (1 − p)S T0 d = S T0 e rT (2.2)
which implies that:
p = erT − d
u − d
As we can see, the expectation does not rely on real-world probabilities of up or
down movements of the stock, just on the magnitude of change in the price, u and d.
The same price will be observed whether P(u) = 0.5 or, for instance, P(u) = 0.1.
Such perspective on price formation correlates with the key notion of Efficient Markets
Theory - Efficient Markets Hypothesis (EMH). The so-called ’weak form’ of EMH
states that the current price of the stock already reflects the investors’ expectations of
possible price movements [Bodie et al., 2004]. Even though probabilistic distribution
of stock price movements is not present in this valuation framework, we could treat p
and 1 − p as pseudo-probabilities of future payoff value. With this result, we can conclude
that given a probability distribution vector, time zero price of any asset defining
the market can be found as the expectation with respect to this probability measure,
discounted at a risk-free rate. The world where the stock-prices grow on average at a
risk-free rate we shall call “risk-neutral” and the normalized state price vector will be
addressed as risk-neutral probability measure Q.
4 This can be justified by pricing a portfolio that replicates a zero-coupon discount bond, paying one unit
of currency at time T 1 . E.g.:
B 0 = B 0 e rT (ψ 0 + · · · + ψ n ) =⇒ |ψ| 1 = (ψ 0 + · · · + ψ n ) = e −rT .
5