Ok, thanks for your patience Alan. Maybe one last point: Starting from the BS-equation, one uses ITO to get the dynamics of the option [$]C[$] to be priced:[$]dC_t = \frac{\partial{C_t}}{\partial{t}}dt + \frac{\partial{C_t}}{\partial{S}}dS_t + 0.5 \frac{\partial^2{C_t}}{\partial{S^2}}(dS_t)^2 \, (*)[$]which is later used (by using the hedging-argument) to arrive at the valuation equation that has to be satisfied by [$]C[$]. I think my problem lies in inrepreting (*). Isn't this equation (at first) the [$]\mathbb{P}[$]-dynamic of [$]C[$]? On the other hand there is no initial price [$]C[$] to speak of.. I guess I need to think about that a little longer.QuoteIf you can't get to that understanding, then your best bet is to accept that the prescription you have been given(in a world where the model is valid, hedge ratios use Q-parms) is correct, even though you don't get it.Ok, would this also hold for a minimum-variance hedge in say a SV-model? (for BS minimum variance hedge should be identical to delta-hedging).bestobs

QuoteIsn't this equation (at first) the [$]\mathbb{P}[$]-dynamic of [$]C[$]? YesQuoteIf you can't get to that understanding, then your best bet is to accept that the prescription you have been given(in a world where the model is valid, hedge ratios use Q-parms) is correct, even though you don't get it.Ok, would this also hold for a minimum-variance hedge in say a SV-model? (for BS minimum variance hedge should be identical to delta-hedging).Yes, if you are hedging away both sources of uncertainty (stock price and vol.) in the small-time limit.

QuoteQuoteIsn't this equation (at first) the P-dynamic of C?YesOk. So, the [$]\frac{\partial{C}}{\partial{S}}[$] in the [$]\mathbb{P}[$]-dynamics is not the same one appearing in the BS-valuation-formula?QuoteYes, if you are hedging away both sources of uncertainty (stock price and vol.) in the small-time limitI meant a minimum-variance hedge using the underlying only (see Bakshi (1997) equation (21) ). But I guess, this should be similar. If we leave the small-time limit and want to hedge the terminal variance, this argument no longer holds?Thanksobs

QuoteOriginally posted by: observer84QuoteQuoteIsn't this equation (at first) the P-dynamic of C?YesOk. So, the [$]\frac{\partial{C}}{\partial{S}}[$] in the [$]\mathbb{P}[$]-dynamics is not the same one appearing in the BS-valuation-formula?Under BS theory, [$]C[$] is (ultimately) uniquely determined; that means so then is [$]\frac{\partial{C}}{\partial{S}}[$].In your (*) the coefficients of the differentials (the partials) are the same under [$]P[$] or [$]Q[$] dynamics.The only part of (*) that changes with the measure is the Ito expansion of [$]dS[$]. Before you ask, under SV models, once the volatility SDE adjustments are fixed by the market, the same holds:C and all the various partials of C are (ultimately) fixed. The only part (of the generalization of (*))that changes with the measure are the Ito expansions of [$]dS[$] and [$]dV[$].

QuoteBefore you ask, under SV models, once the volatility SDE adjustments are fixed by the market, the same holds: C and all the various partials of C are (ultimately) fixed. The only part (of the generalization of (*)) that changes with the measure are the Ito expansions of dS and dV. Ah ok. Thanks a lot for that explanation. So basically, the partial derivative [$]\frac{\partial{C}}{S}[$] based on [$]\mathbb{P}[$]-parameters in a SV-model does not make a lot of sense to look at actually.So, this argument actually hinges on a market fixing the prices. If I tried to do a minimum variance hedge for a non-tradable asset (like [$]V_t[$] in Heston-world where we have no options):[$]MinVar[dV_t - h dS_t)[$](for whatever reason, a better example for [$]V_t[$]would prob. be weather and we have no weather-derivatives)..one would rather use the dynamics of the physical measure (of course there would have to be some correlation among [$]V[$] and [$]S[$]?)bestobs

QuoteOriginally posted by: observer84QuoteBefore you ask, under SV models, once the volatility SDE adjustments are fixed by the market, the same holds: C and all the various partials of C are (ultimately) fixed. The only part (of the generalization of (*)) that changes with the measure are the Ito expansions of dS and dV. Ah ok. Thanks a lot for that explanation. So basically, the partial derivative [$]\frac{\partial{C}}{S}[$] based on [$]\mathbb{P}[$]-parameters in a SV-model does not make a lot of sense to look at actually.So, this argument actually hinges on a market fixing the prices. If I tried to do a minimum variance hedge for a non-tradable asset (like [$]V_t[$] in Heston-world where we have no options):[$]MinVar[dV_t - h dS_t)[$](for whatever reason, a better example for [$]V_t[$]would prob. be weather and we have no weather-derivatives)..one would rather use the dynamics of the physical measure (of course there would have to be some correlation among [$]V[$] and [$]S[$]?)bestobsIn continuous-time, I don't think it matters what measure you choose for the this last problem.Under any measure, you'll have[$]dS = (\cdots) dt + \sigma_S \, dB[$], [$]dV = (\cdots) dt + \sigma_V \, dW[$], and[$]dS \, dV = \rho \,\sigma_S \, \sigma_V \, dt[$].The MinVar (rate) is [$]\sigma_V^2 (1 - \rho^2)[$] and this is achieved by [$]h = \rho \frac{\sigma_V}{\sigma_S}[$]; all 3 terms on the rhs are invariant under measure changes

@AlanThanks, that makes sense. Statically hedging the terminal variance on the other hand would not work accordingly, I guess?bestobs

latex it -- what exactly is the objective function being minimized?

[$]minVar_t(V_T - h S_T)[$]with [$]h[$] being the hedge-ratio, V the non-tradable asset and S our underyling.

QuoteOriginally posted by: observer84[$]minVar_t(V_T - h S_T)[$]with [$]h[$] being the hedge-ratio, V the non-tradable asset and S our underyling.I think you may have finally found a case that depends on the [$]P[$]-parameters,assuming the everyday interpretation that Var means 'variance-under-P'. Congratulations.Given a [$]P[$]-model evolution (SDE), I would develop the joint pdf of [$](S_T,V_T)[$], compute the variance,and minimize. Most or all steps would be numerical.

As always, thanks a lot for your answer. Now, if we go back to the case of a tradable asset C (which is driven by V_t and S_t):[$]minVar_t(C(S_t,V_t) - hS_t) [$]we have variance and covariance terms:[$]Var_t(C(S_t,V_t) - hS_t) = Var_t(C) + Var(hS)+ 2Cov(C,hS) [$]I wonder under which measure one has to compute these moments from? It seems to be different than hedging continuously..best,obs

Your question is not well-posed, since [$]minVar_t[\text{something known at t}] =0[$] I'll fix it up and give it to back to you:In a continuous-time world in which the Heston '93 model holds,how would you compute (the single static) [$]h[$] in[$]\min_h \text{Var}_0[C(S_t,V_t) - h S_t][$]?Here [$]0 < t < T[$], where [$]T[$] is the option expiration.Discuss the role of the [$]P[$] and [$]Q[$] parameters in your computation, all of which are assumed known. Show your work and no asking for answers at Wilmott forums

My bad, for clarification, what I meant was terminal variance of course:[$]min_h Var_t(C_T - h S_T)[$]I guess, I will have to implement it then Just thought it might be more obvious beforehand.bestobs