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Hi,I would like to know if it's possible to use singular perturbation method to calculate an implied volatility approximation of a mixture model. I remind that a mixture model is a discrete stochastic volatility model. Brigo & Mercurio show that an asset who follows a mixture model satisfies the following equation:dS(t)/ S(t) =A(S(t),t)dWtWe cannot separate A(S(t),t) into a function of S(t) and a function of t:A(S(t),t) <> f(S(t))g(t) so I cannot apply straightforwardly the result of Pat's article "Equivalent Black Volatility".Pat wrote on the top of the page 3 in this article:“Singular perturbation techniques can be used to solve many related pricing problems. For example these techniques can be used to solve intrinsically time-dependent modelsdF =A(S(t),t)dWtto obtain accurate equivalent vol formulas for calls and puts.”I'm looking for some more details.kac.

Kac,That's a very interesting question. I look forward to Pat's answer, on this and on other related problems.Is Pat's paper available on the forum? It must have been there, but I cannot locate it off hand. If not, could you please post it?

You can found it in the SABR topic. This is one of the first messages.kac

Nothing new ?kac

You can use a technique known as "differential effective media" theory (known to physicists as transparency theory) ... basically, you write A(t,S(t)) = Aavg + [A(t,S(t))-Aavg] and then at the end of the problem, one picks the Aavg so that the contributions to the answer (price?) from[A(t,S(t))-Aavg] just happen to be zero, order by order. This essentially yields the "effective" average value of A for the medium.This is pretty close to what is done in the dynamic SABR analysis of the last(?) appendix in the SABR paper ... this analyssi should be simpler (since there is no stochastic vol equation), but I never drove that solution all the way to a nuts and bolts answer.

Pat, Can you give us a reference to transparency theory? I haven't heard these words in an averaging problem before. Do you have Whitham averaging in mind by any chance? What would stochastic volatility correspond to in fluid mechanics?

For the historical record, I would like to mention that I wrote down a singular perturbation approach to volatility skew in 1986 using functional path integral techniques, and described the work in a SIAM talk in 1993. The perturbation expansion around a nominal constant volatility is singular, involving a “velocity-dependent” potential, and requiring the Schwinger formalism for the discretization specification. There is a “mass renormalization” involving counterterms, and the final results are finite as dt goes to zero. Some work was done with an explicit ansatz for the volatility function vol(x) with underlying variable x. More details are in my book, Ch. 42, App. C.--------

QuoteOriginally posted by: kacHi,I would like to know if it's possible to use singular perturbation method to calculate an implied volatility approximation of a mixture model. I remind that a mixture model is a discrete stochastic volatility model. Brigo & Mercurio show that an asset who follows a mixture model satisfies the following equation:dS(t)/ S(t) =A(S(t),t)dWtWe cannot separate A(S(t),t) into a function of S(t) and a function of t:A(S(t),t) <> f(S(t))g(t) so I cannot apply straightforwardly the result of Pat's article "Equivalent Black Volatility".Pat wrote on the top of the page 3 in this article:“Singular perturbation techniques can be used to solve many related pricing problems. For example these techniques can be used to solve intrinsically time-dependent modelsdF =A(S(t),t)dWtto obtain accurate equivalent vol formulas for calls and puts.”I'm looking for some more details.kac.Well, first of all you do not need any singular pertrubation techniques for Brigo-Mercurio approach. Their local volatility is designed specifically to return the value of the option that is equal to the weighted average of two Black-Scholes prices. It is a trivial exercise to extract in implied volatility from that pricesecond, for time-dependent local (and stochastic) volatility models, I have some methods that work quite well. My paper is coming out in the May issue of Risk -Vladimir

Hi Pitebarg,which paper's coming out in RISK?"A Stochastic volatility forward LIBOR model ..."?If so, have u tried pricing multi-callables under this framework?

QuoteOriginally posted by: MFordeHi Pitebarg,which paper's coming out in RISK?"A Stochastic volatility forward LIBOR model ..."?If so, have u tried pricing multi-callables under this framework?a flavor of it, yes (without actually any Libor market model stuff)answer to you second question is yes. See the paper you cite for test results

Just saw this from the latest issue of "Mathematical Finance". May be helpful.THE BLACK-SCHOLES EQUATION REVISITED: ASYMPTOTIC EXPANSIONS AND SINGULAR PERTURBATIONSMartin Widdicks, Peter W. Duck, Ari D. Andricopoulos, David P. Newton

The "transparency" techniques would reduce to Whitham's averaging techniques in the appropriate circumstances