Question about using Ito's lemma in Gamma PnL

Question

While deriving the delta hedge error if we hedge with implied vol, and the true vol is different, we say that the PnL of the call option is:

$$dC=C_tdt+C_SdS+0.5C_{ss}<QV>dt - (1)$$

Where $<QV>$ is the 'realised quadratic variation' of the stock price, and not the incorrect implied vol. While I understand this from the mathematical perspective (change in a function depends on the actual change/dynamics of the independent variable), and I also understand that this call price must 'drift' at less than the risk free rate (therefore create an arbitrage with the correct call price). However, I don't see how I 'realize' this PnL.

Consider the case where I have bought a call in a market where there is no option liquidity. 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. How do I know what is the correct value to mark my call value tomorrow? Is there a market mechanism that will force the value of my call to be given by the above equation? Does this mean I will have to remark the volatility in my model everyday, to be consistent with the PnL?

Edit: I'm trying to ask the same question in a different way. Let $<QV>$ be the actual quadratic variation and $<MV>$ be the implied quadratic variation of the stock price. Then:

$$dC(t,S_t;MV)=C_tdt+C_SdS+0.5C_{ss}<MV>dt$$ where derivatives are taken at the implied vol.

$$dC(t,S_t;QV)=C_tdt+C_SdS+0.5C_{ss}<QV>dt$$ where derivatives are taken at the true vol.

However, in equation 1, the derivatives are at the implied vol, whereas quadratic variation is at the true vol. I am not sure what function $C$ is in equation (1). It's certainly not the functions in LHS of (2) and (3). Can somebody explain what call price function is involved in equation (1)?

Answer 1

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: $𝑑𝐶_{mkt}=\frac{\partial C_{BS}(S,t|\hat\sigma)}{\partial t}dt+\frac{\partial C_{BS}(S,t|\hat\sigma)}{\partial S}dS+\frac{1}{2}\frac{\partial^2 C_{BS}(S,t|\hat\sigma)}{\partial S^2}dS^2$

• If implied vol has changed: $𝑑𝐶_{mkt}=\frac{\partial C_{BS}(S,t|\hat\sigma)}{\partial t}dt+\frac{\partial C_{BS}(S,t|\hat\sigma)}{\partial S}dS+\frac{1}{2}\frac{\partial^2 C_{BS}(S,t|\hat\sigma)}{\partial S^2}dS^2+\frac{\partial^2 C_{BS}(S,t|\hat\sigma)}{\partial \sigma\partial S}dSd\sigma+\frac{1}{2}\frac{\partial^2 C_{BS}(S,t|\hat\sigma)}{\partial \sigma^2}(d\sigma)^2+...$

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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 $PnL = 𝐶_𝑡𝑑𝑡+𝐶_𝑆𝑑𝑆+0.5𝐶_{𝑠𝑠}dS^2$. 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 $$PnL=𝐶_𝑡𝑑𝑡+𝐶_𝑆𝑑𝑆+0.5𝐶_{𝑠𝑠}dS^2+C_{\sigma}d\sigma+C_{\sigma S}d\sigma dS+0.5C_{\sigma \sigma}(d\sigma)^2$$

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

• $PnL_{marktomkt} = 𝐶_𝑡𝑑𝑡+𝐶_𝑆(20)+0.5𝐶_{𝑠𝑠}20^2+C_{\sigma}0.02+C_{\sigma S}(20)(0.02)+0.5C_{\sigma\sigma}0.02^2 =PnL_{marktomodel}+unexplained\ PnL$

• 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 = $\frac{1}{2}\Gamma_{BS}E[dS^2]=\frac{1}{2}(\Gamma_{BS}S^2)\hat\sigma^2dt$. 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 $C(S_0,t_0|\hat\sigma_0)=MktPrice(S_0,t_0)$
• If recalibrate, then $MktPrice(S_1,t_1)=C(S_1,t_1|\hat\sigma_1)$ $$C(S_1,t_1|\hat\sigma_1)-C(S_0,t_0|\hat\sigma_0)=𝐶_𝑡𝑑𝑡+𝐶_𝑆𝑑𝑆+0.5𝐶_{𝑠𝑠}dS^2+C_{\sigma}d\sigma+C_{\sigma S}d\sigma dS+0.5C_{\sigma \sigma}(d\sigma)^2+...$$
• Therefore $$MktPrice(S_1,t_1)-MktPrice(S_0,t_0)=𝐶_𝑡𝑑𝑡+𝐶_𝑆𝑑𝑆+0.5𝐶_{𝑠𝑠}dS^2+C_{\sigma}d\sigma+C_{\sigma S}d\sigma dS+0.5C_{\sigma \sigma}(d\sigma)^2+...$$
• Without recalibration, then $MktPrice(S_1,t_1)\neq C(S_1,t_1|\hat\sigma_0)$ $$C(S_1,t_1|\hat\sigma_0)-C(S_0,t_0|\hat\sigma_0)=𝐶_𝑡𝑑𝑡+𝐶_𝑆𝑑𝑆+0.5𝐶_{𝑠𝑠}dS^2+C_{\sigma}0+C_{\sigma S}0 dS+0.5C_{\sigma \sigma}(0)^2+...=𝐶_𝑡𝑑𝑡+𝐶_𝑆𝑑𝑆+0.5𝐶_{𝑠𝑠}dS^2$$