Stochastic Volatility : pricing and replication

[sraks]

Hi All,

First of all, I would like say that reading "Concept" is clearing a lot of cobwebs in mind. Thanks Dr. Joshi for this book.

Reading the chapter on stochastic volatility, I understand that there is no unique replication strategy for a vanilla (and by extension exotic) options. There is no also unique risk-neutral measure and a price. Herein lies my confusion. In Heston model, there is a unique price for a European option. How can one reconcile the non-existence of a unique risk-neutral measure and the unique price in the Heston model?

Regards,
SR

[sraks]

I think careful reading of the text provides the answer. The "unique"ness of the Heston model or any other SV model for that matter would be in choosing the drift term. Once a drift term has been chosen, a unique risk-neutral measure is implied. Note that the vol-vol does not affect the risk-neutral measure.
It follows that different SV models can imply a different Risk-Neutral measure. Hence, the comment in the book that if your mean reverting vol model is fitting well to the market, it means that others are also using mean-reverting vol model.

[mj]

That is correct.

[sraks]

Thanks Dr. Joshi.

I think I made a mis-statement in an earlier post by saying that the risk-neutral probabilities are not affected by vol-vol. I don't think that's right. This can be explicitly seen through the tree pricing. In tree pricing, one the four equations determining the risk-neutral probabilities would be
E[dv]=vol_drift*dt
LHS would contain terms affected by vol-vol.

Regards,
SR

[mj]

it all comes down to what you mean by "affected" .

In truth, almost everyone just writes down risk-neutral dynamics and doesn't care about the shift from real-world to risk-neutral in any case.