# 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?

• A local volatility model would actually be $dS(t)/S(t) = \sigma(t,S(t)) dW(t)$ not $dS(t) = \sigma(t,S(t)) dW(t)$. Nothing guarantees prices from going negative in the latter "arithmetic" version. – Quantuple Nov 29 '17 at 9:16
• @Quantuple: Yes. Let me write the correspondent Euler scheme: $$S(t_{i+1})-S(t_i)=\lambda(t_i)\big(a(t_i)+b(t_i)S(t_i)\big)\big(W(t_{i+1})-W(t_i)\big)$$ – Zoro-X Nov 30 '17 at 19:30