Sign Changes in Displaced Lognormal Local Volatility Models

Summary

The document asks why an Euler simulation of a displaced lognormal process can produce very large negative prices when the factor multiplying Brownian noise becomes negative. It observes that changing the sign of a diffusion coefficient does not change the law of an ideal stochastic differential equation, since Brownian motion and its negative have the same distribution.

The discussion presents a numerical modeling issue rather than a complete resolution: a finite-step Euler approximation can behave badly when the state-dependent coefficient changes sign, even if the continuous-time diffusion is distributionally equivalent under a sign change. The post gives no derivation, correction method, or simulation evidence. It is therefore useful as a prompt to distinguish continuous-time model properties from discretization behavior, but readers need further analysis to choose a stable scheme or determine whether the intended model preserves positive prices.

Key ideas

• The document studies Euler simulation of a displaced lognormal diffusion with time-varying coefficients.
• A negative diffusion coefficient does not by itself change the law of the continuous-time process.
• A discrete Euler step can produce problematic negative values when the diffusion factor changes sign.
• The post raises the issue but does not provide a proven correction or numerical comparison.

Tags

# Problem of negative local volatility:


# Problem of negative local volatility:












Consider the displaced log-normal process: $$dS(t) = \lambda(t)(a(t)+b(t)S(t))dW(t), S(0) = S_0>0, $$ where $W(t)$ is a one-dimensional Brownian motion.

We suppose that $(\forall t \ge 0) : \lambda(t)\ge0$ and that there is no restrictions on $t\to a(t)$ and $t\to b(t)$.

This is a local volatility model used to describe the dynamic of the price of an underlying $S$.

I implement an Euler scheme to approximate $S(T)$ at a given horizon $T$.

I realize something wrong with my implementation: In fact, if there exists $t'$ such that $a(t')+b(t')S(t')<0$, the values of $\{S(t'')\}_{t'\le t''\le T}$ are negative and very large and the calculations using $S(T)$ are false. But it works fine if we have $(\forall t\ge 0): a(t')+b(t')S(t')\ge0$.

For me, If I define $\sigma(t,S(t)) = \lambda(t)(a(t)+b(t)S(t))$ so that we can write $dS(t) = \sigma(t,S(t))dW(t)$, the sign of $\sigma(t, S(t))$ doesn't matter because $W(t)$ and $-W(t)$ have the same probability law.

Have you an idea about the source of the problem? And how can we correct it?

Thank you in advance for any help you can provide!

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