Conjuncted: Mispricings Happened in the Past do not Influence the Derivative Price: Black-Scholes (BS) Analysis and Arbitrage-Free Financial Economics. Note Quote.

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It can be shown that the probability (up to a normalization constant) of the trajectory R(·,·) has the form:

P[R(.,.)] ∼ exp[-1/2∑0 dt dt’ dS dS’ R(t, S) K-1(t, S|t’, S’) R(t’, S’)] —– (1)

where the kernel of the operator K is defined as:

K(t, S|t’, S’) = θ (T – t) θ (T – t’)∫0 dτ ds f(τ) θ (t – τ) θ (t’ – τ) e-λ(t + t’ – 2τ) x P (t, S|τ, s)P (t′, S′|τ, s) —– (2)

It is easy to see that the kernel is of order 1/λ and vanishes as λ → ∞. Equation 2, in particular, results in the equality for the correlation function:

⟨R(t, S) R(t′, S′)⟩ = Σ2 · K(t, S|t′, S′) —– (3)

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