Lookback Option Bias Correction and Weak Convergence Rates

Summary

The document asks how a continuity correction for a discretely sampled diffusion’s minimum improves the weak convergence rate of a lookback option payoff. The approximation adjusts each simulated grid value by a term proportional to the local diffusion coefficient and the square root of the time step, using a constant of about 0.58. The correction is attributed to earlier work by Broadie, Glasserman, and Kou, with an extension by Gobet, and is discussed in the context of multilevel Monte Carlo.

The cited result is that the correction reduces the minimum’s discretization error from order square-root-of-step-size to a smaller-order term. The question is how this leads to first-order weak convergence overall, rather than merely a slight improvement. The document states Giles’s conclusion but does not supply the explanation or proof. Its scope is numerical approximation of a lookback payoff under a discretized diffusion; it does not establish that the stated rate applies without the assumptions in the cited analyses.

Key ideas

• Lookback payoffs depend on the path minimum, which is missed between discrete simulation points.
• A continuity correction shifts grid values using the local diffusion coefficient and the square root of the time step.
• The cited result improves the minimum’s discretization error from order square-root-of-step-size to smaller order.
• The document asks how that improvement yields first-order weak convergence but does not answer the question.
• The stated convergence rates depend on the assumptions of the underlying analyses.

Tags

# Weak convergence of Lookback payoff with correction term


# Weak convergence of Lookback payoff with correction term












In this article on the Multilevel Monte Carlo method on page 8, http://people.maths.ox.ac.uk/gilesm/files/mcqmc06.pdf, Giles uses a correction term to improve the weak convergence rate of the lookback option payoff $$P = e^{-rT}(S_T - \min_{0\leq t\leq T} S_t)$$ when using the approximation $$\hat{S}_{\min} = \min_{0\leq n \leq N}(\hat{S}_n-\beta^{*}b(nh,\hat{S}_n)\sqrt{h})$$ where $\hat{S}_n$ are the grid values of a discretized diffusion process $S_t$ defined via $dS_t = a(t,S_t)dt+b(t,S_t)$ using the step size $h$ and with $\beta^{*}\approx0.58$. This correction term was introduced by Broadie, Glasserman and Kou, and extended by Gobet in http://hal.archives-ouvertes.fr/docs/00/39/64/22/PDF/BoundaryCorrectionGobetMenozziSPAJune09.pdf.

As proven in Gobet's article and noted in the article by Giles, the correction term improves the discretization error to $o(\sqrt{h})$ from the original order $O(\sqrt{h})$. Giles subsequently concludes that this restores overall $O(h)$ weak convergence from the original weak convergence order $O(\sqrt{h})$. I do not understand how a discretization error of $o(\sqrt{h})$ is sufficient to achieve the overall convergence order. How can removing the leading error term in the discretization improve the result to such a degree?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.