and note that, since Σ2/λ plays a role of small parameter in the problem, the noise R can be considered as weak and we can find a formal iterative R-dependent solution of the last equation. In the lowest non-trivial order we have the equation:
which after averaging (using equation 3) over all possible realizations of the fluctuations R give us an equation for the average derivative price V ̄ ≡ ⟨V ⟩R up to and including terms proportional to Σ2/λ:
This is an integro-differential equation (6) which in the limit λ → ∞ or Σ → 0 reduces to the Black-Scholes equation and accounts for local arbitrage opportunities and the corresponding market reaction to them. It is interesting to note that due to properties of the integrand on the right hand side of equation 6, the integration is effectively limited to the interval from time t to the expiration date. It means that any mispricings which happened in the past do not influence the derivative price, as one would expect, and the only relevant contribution comes from future mispricings.