How do investors' expectations drive asset prices?

where for notational convenience we write Φ^ for . Φ^ is called the
pricing kernel. To simplify notation in the remainder the index (к) is omitted.
Furthermore, it is well known from Girsanov’s theorem that the process
W, which is given by

follows a standard Brownian motion on the space

Thus, technically the transformation is done by multiplying the asset
price under the probability measure P with a suitable P-martingale.⁵ To get
some economic intuition we apply these results in a simplified economy.

4.2 Interpretation in a simplified economy

Consider the following market with no arbitrage opportunities: On the fil-
tered probability space (Ω^z, Q^,, Q^,_t, P^z) with a one-dimensional Brownian mo-
tion W^z defined on it the forward price is given by

dF_t = F_tμ_tdt + F_tΣ_t dW_t^z ,   0 « t « T,             (6)

Fo = F > 0,

with μ, Σ deterministic functions depending on t and F_t and Σ = 0. Thus,
this market is complete and there is a unique martingale measure. With
к := d define Φ_t = exp (— ʃθ κ_udW_u — ɪ ʃθ ∣κ_w∣² du) , then applying Itô’s
formula it is easily seen that Φ_tF_t is a martingale and thus

F = E*' (F_τ ∣Q^l_t) , 0 « t « T,

₁ FJ ∙    ₁ r. _{1 1}

where P^z is defined by

dP'               _

dP = ⅛ .  ⁰ «^t « ^t .

The instantaneous covariance, i.e. the quadratic variation, between this
process Φ and the forward price process F is equal to the instantaneous
drift μ of the forward price process.

⁰For a detailed derivation see for example Karatzas, Shreve [21] and Musiela, Rutkowski
[25].

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