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Why Zero Is Not an Absorbing Boundary for an Additive-Noise Process

Article Quant Q&A · Author: Valentin
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Summary

The document questions a boundary condition for a travel-time option whose state includes a mean-reverting process with additive Brownian noise and its time integral. In the familiar geometric Brownian motion example, zero is absorbing: once the underlying reaches zero, it remains there, fixing the future average and making the discounted payoff a valid value at that state.

That reasoning does not transfer to the stated process. Its Brownian increment continues to move the state even when the current value is zero, so the future integral and terminal payoff are not determined by the current state being zero. The proposed payoff expression therefore cannot be justified by the absorbing-state argument alone. The document raises the issue but supplies no resolution from the paper; whether the expression is appropriate would require examining the model’s domain, PDE formulation and any additional boundary or terminal conditions.

Key ideas

  • In geometric Brownian motion, zero is absorbing and can determine the future path.
  • The process in the question has additive Brownian noise and can move away from zero.
  • At zero, the future integrated state and option payoff are not fixed by the current state alone.
  • The usual absorbing-state derivation does not justify the stated boundary value.
  • The source poses the issue without resolving the paper’s PDE conditions.

Tags

OptionsDerivatives pricingStatistics
Full text
# Option pricing boundary condition


# Option pricing boundary condition












I am currently working on this paper "https://arxiv.org/abs/2305.02523" about travel time options and I am stuck at Theorem 14 page 20. The proof is similar to Theorem 7.5.1, "Stochastic Calculus for finance II, continuous time model" from Shreve. Just a different underlying process. I do not understand the last boundary condition. Here the underlying process is given by \begin{align*} dX_t=(-a_1X_t+\gamma_t)dt+dW_t\\ \end{align*} \begin{align*} Y_t=\int_{0}^{t}X_udu\ \end{align*} The boundary condtion is given by \begin{align*} v(t,0,y)=e^{-r(T-t)}\max(\frac{y}{T}-K,0). \end{align*} Derivation from Shreve ($dX_t=rX_tdt+X_tdW_t$):

> If $X_t=0$ and $Y_t=y$ for some variable $t$, then $X_u=0, \ \forall u \in[t,T]$ and so $Y_u$ is constant on $[t,T]$ and therefore $Y_T=y$ and the value of the Asian call option at time t is $e^{-r(T-t)}\max(\frac{y}{T}-K,0)$.

I understand his derivation (see Pricing PDE of Asian option by Shreve) Problem: The author of my paper gives no explanation to this boundary condition. But I don't think I can apply Shreve's explanation to my process, since it has no $X_t$ before the $dW_t$ term and can therefore take non-zero values again once it was zero?!

Thank you very much for your help!

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