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Does someone know if there is a strong solution for this SDE : $$\frac{dS_t}{S_t}=\sigma(S_t)dW_t$$ where $$\sigma(S)=\begin{cases} 1\;\;\;S>1\\2\;\;\;S\leq 1 \end{cases} $$ $S_0=1$ and $W_t$ is a brownian motion ?

It's the first try for a model in which a discontinuity in the volatility appears when a stock is cheaper than a fixed level.

This kind of local volatility appears when you construct the Dupire's local volatility surface from an implicit volatility surface which isn't two times differentiable.

I tried to construct $S_t$ from $W_t$ by refining $W_t$ with the bridge technique but it doesn't seem to converge.

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Am I smelling a push by some group/individual with certain undisclosed motivations to propose a new BS model? This sounds awfully identical to some earlier question where an idea was pushed without full disclosure. (quant.stackexchange.com/questions/8666/…). The earlier wiki-sandbox proposal looks similar in its idea to what you show, yet no data is shown that demonstrate that the suggested approach improves results over the standard BS model (en.wikipedia.org/wiki/User%3aStockequation2/sandbox) –  Matt Wolf Aug 7 '13 at 8:15
    
I aggree that this question seems related to the one you mentioned. Neithertheless, I'm not related to the other one. This question is linked to the Dupire's local volatility model when the implicit volatility surface is not two times differentiable. –  Sylvestre Aug 7 '13 at 8:39
    
I still do not get why you would have two volatilities? –  Matt Wolf Aug 7 '13 at 12:44
    
If you try to compute the local volatility surface from a implicit volatility surface which isn't two times differentiable, you will get discontinuity in the local volatility. What I would like to do is to understand the behaviour of the stock path when it goes through this gap. The simple sde that I propose here is a toy model to investigate this. –  Sylvestre Aug 7 '13 at 12:49
    
one is independent of the other. The implied volatility surface expressed expectations of future return variations of the underlying. There are different interpolation techniques used in practice between two surface points but you still end up with one implied volatility measure. On the other hand the future stock price path is modeled through a separate process that can be driven by any number of Brownian Motions and a volatility measure. Whether a stock price path has discontinuities/jumps/whatever is expressed through the stock price model not the option model. –  Matt Wolf Aug 7 '13 at 12:58

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