Question about using Ito's lemma in Gamma PnL

Hope this answers your qs, Denote $C_{model}(S,t)=e^{-rT}E_{{model}}[(S_T-K)^+]$

• We model the spot dynamics $S$ with different models, e.g.

• In BS, $$frac{dS}{S}=rdt+sigma dW$$

• $dC_{BS}(S,t)=frac{partial C_{BS}}{partial t}dt+frac{partial C_{BS}}{partial S}dS+frac{1}{2}frac{partial^2 C_{BS}}{partial S^2}dS^2$

• $dC_{BS}(S,t)=frac{partial C_{BS}}{partial t}dt+frac{partial C_{BS}}{partial S}dS+frac{1}{2}frac{partial^2 C_{BS}}{partial S^2}sigma^2S^2dt$

• Note that $dC_{BS}(S,t)$ is only the PnL of option that exists in BS world

Can somebody explain what call price function is involved in equation (1)?

• In equation (1), can you clarify dS is the real world's $dS$ or model $dS$?

• If you mean $dS$ is black scholes world's $dS$ with $frac{dS}{S}=rdt+sigma dW$, then $$dC_{BS}(S,t)=frac{partial C_{BS}}{partial t}dt+frac{partial C_{BS}}{partial S}dS+frac{1}{2}frac{partial^2 C_{BS}}{partial S^2}dS^2$$

• If you mean $dS$ is real world $dS$ with unknown dynamics, I think your equation (1) LHS's $C=C_{mkt}$ and RHS's $C=C_{BS}$, basically you want to explain option P&L observed in real mkt with black scholes greeks

• equation (1) is only valid when implied vol has not changed

• If implied vol has not changed: $??_{mkt}=frac{partial C_{BS}(S,t|hatsigma)}{partial t}dt+frac{partial C_{BS}(S,t|hatsigma)}{partial S}dS+frac{1}{2}frac{partial^2 C_{BS}(S,t|hatsigma)}{partial S^2}dS^2$

• If implied vol has changed: $??_{mkt}=frac{partial C_{BS}(S,t|hatsigma)}{partial t}dt+frac{partial C_{BS}(S,t|hatsigma)}{partial S}dS+frac{1}{2}frac{partial^2 C_{BS}(S,t|hatsigma)}{partial S^2}dS^2+frac{partial^2 C_{BS}(S,t|hatsigma)}{partial sigmapartial S}dSdsigma+frac{1}{2}frac{partial^2 C_{BS}(S,t|hatsigma)}{partial sigma^2}(dsigma)^2+...$

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1. You can 'realize' this PnL $dC$ by selling the option tomorrow

2. If there is no liquidity tomorrow, that means your call does not have a market quote to calculate its new implied vol. Of course you can use the yesterday's implied vol can calculate the delta, gamma and theta P&L and estimate the theo value of the call today, but implied vols are rarely constant in the real world, so it will only be an estimate

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3. PnL

• If you mark to model without re-calibration of parameters, your $PnL = ?_???+?_???+0.5?_{??}dS^2$. Note that this PnL will not equal $dC$ if model parameters changed tomorrow

• Let say your model takes in $sigma$ as a parameter. If you re-calibrate $sigma$, PnL up to second order reads $$PnL=?_???+?_???+0.5?_{??}dS^2+C_{sigma}dsigma+C_{sigma S}dsigma dS+0.5C_{sigma sigma}(dsigma)^2$$

• e.g. spot increase by $20 and implied vol increased by 2%, and you insist to mark to model without re-calibration, your $PnL_{marktomodel} = ?_???+?_?(20)+0.5?_{??}20^2$

• $PnL_{marktomkt} = ?_???+?_?(20)+0.5?_{??}20^2+C_{sigma}0.02+C_{sigma S}(20)(0.02)+0.5C_{sigmasigma}0.02^2 =PnL_{marktomodel}+unexplained PnL$

• The fact that you refused to adjust your parameters despite market implied parameter values has gone up means your model with yesterday's parameters can no longer price your option same as current market quotes

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4. "I come back tomorrow, I mark to model, and therefore my PnL should be given by the difference in model price today and tomorrow, which is just the above equation but with implied vol used as the quadratic variation":

• I think $Gamma PnL=frac{1}{2}Gamma dS^2$, e.g. if spot today is 100 and spot tomorrow is 120, $Gamma PnL=frac{1}{2}Gamma 20^2$

• Expected Gamma PnL in BS = $frac{1}{2}Gamma_{BS}E[dS^2]=frac{1}{2}(Gamma_{BS}S^2)hatsigma^2dt$. Your expected gamma P&L is related to implied vol, but you actual gamma P&L is simply $frac{1}{2}Gamma dS^2$

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5. Is there a market mechanism that will force the value of my call to be given by the above equation?

• There is only one market price, I think you are referring to PnL attribution
• PnL is expanded into different partial derivatives acc to Ito's lemma as you mentioned
• As long as you recalibrate parameters, your partial derivatives will sum up to market's $dC$ (terms in order 3 or above will not make a material difference for most models)
• Denote $C=Model(S,t|sigma)$, and $C(S_0,t_0|hatsigma_0)=MktPrice(S_0,t_0)$
• If recalibrate, then $MktPrice(S_1,t_1)=C(S_1,t_1|hatsigma_1)$ $$C(S_1,t_1|hatsigma_1)-C(S_0,t_0|hatsigma_0)=?_???+?_???+0.5?_{??}dS^2+C_{sigma}dsigma+C_{sigma S}dsigma dS+0.5C_{sigma sigma}(dsigma)^2+...$$
• Therefore $$MktPrice(S_1,t_1)-MktPrice(S_0,t_0)=?_???+?_???+0.5?_{??}dS^2+C_{sigma}dsigma+C_{sigma S}dsigma dS+0.5C_{sigma sigma}(dsigma)^2+...$$
• Without recalibration, then $MktPrice(S_1,t_1)neq C(S_1,t_1|hatsigma_0)$ $$C(S_1,t_1|hatsigma_0)-C(S_0,t_0|hatsigma_0)=?_???+?_???+0.5?_{??}dS^2+C_{sigma}0+C_{sigma S}0 dS+0.5C_{sigma sigma}(0)^2+...=?_???+?_???+0.5?_{??}dS^2$$