Euler Approximation of a Discontinuous Black–Scholes Hedge Integral

Summary

The document asks about simulating an integral driven by a Black–Scholes asset process, with the integrand switching according to whether the asset price is below a strike. The author describes this as a naive hedge for a European put and notes that such a hedge does not work in practice. The central questions are whether the integrand's discontinuity rules out a Milstein-type scheme and whether Euler discretization can approximate the stochastic integral.

This is a numerical-method question rather than a worked analysis: it gives no convergence result, derivation, simulation, or comparison of schemes. It highlights that discontinuous, path-dependent trading rules raise approximation questions, but the document alone does not establish error behavior or hedge performance. Results would depend on the precise discretization, assumptions, and convergence criterion, none of which are developed in the post.

Key ideas

• The target is a stochastic integral whose integrand depends on whether the asset price is below the strike.
• The post frames the integral as a simplified put-hedging rule.
• It asks whether discontinuity affects the availability of a Milstein-type approximation.
• It also asks whether Euler discretization approximates the integral, without supplying a result.

Tags

# Regularity requirement for convergence of Euler scheme for stochastic integral?


# Regularity requirement for convergence of Euler scheme for stochastic integral?












Let $S_t$ be follow Black Scholes, then I am interesting in simulating the process

$\int ^t _0 e^{-rt}1_{\{S_t\leq K\}}dS_t$

which is like a naive hedge of a European put, which does not work in practice.

- Am I correct to say no Milstein Type scheme exists due to discontinuous derivative

- Does the Euler scheme produce a process which approximate this stochastic integral?

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