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Consider the case where at t=0, I calibrate my model to the market, but at t=1 my model is no longer able to recover the price in the market, so it needs recalibration. Say I have delta hedged my position. Consider my portfolio PnL in the following 2 situations:

  1. I re calibrate my model, and therefore get some PnL due to a change in the portfolio value, which is instantaneous.

  2. I choose not to re calibrate it, therefore I get a gamma PnL due to an incorrect delta hedge, which is is not instantaneous but realizes in the next time interval.

Is the PnL in (1) related to the PnL in (2)? How should I choose whether to recalibrate or just accept the gamma PnL?

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    $\begingroup$ If in the second scenario at $t=1$ you mark to market (not model) you also get a PnL right? What I'm getting at, if you mark to market you would also get an instantaneous PnL change. $\endgroup$
    – Bob Jansen ♦
    Jul 17, 2020 at 14:21
  • $\begingroup$ Yes, you would. I think I have gotten something wrong here in situation 2. As far as I thought, the trader can either recalibrate and get a PnL, or he can choose not to recalibrate (and thus not MtM either). Is this incorrect? $\endgroup$
    – Arshdeep
    Jul 17, 2020 at 14:24
  • $\begingroup$ The trader can choose to not recalibrate and mark to model but this is a bad idea: when the position is eventually closed, market prices have to be paid, not those of the model. It's better to update the parameter and hedge correctly. $\endgroup$
    – Bob Jansen ♦
    Jul 17, 2020 at 14:37
  • $\begingroup$ I suppose then my question is: how is this PnL due to marking to market related to the hedging error, if at all? Obviously one way to quantify it is just the sensitivity*change in parameter value, but is there a relationship with hedging error? $\endgroup$
    – Arshdeep
    Jul 17, 2020 at 14:41

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To continue the discussion in the comments but in order to not put answer there:

Section 2.6 from these notes by Mark Davis mentioned in this question describes hedging error in the Black-Scholes world.

There is no direct relation between marking to market and hedging error. If you continuously hedge and have a perfect model which is correctly calibrated, there will be no hedging error even if the market prices are temporarily out of whack.

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  • $\begingroup$ I understand the mathematics behind the hedging error; my question is to relate this with the 'remarking' PnL. For example, I know that the vega is just the (risk neutral) expected Gamma PnL. Vega is also the PnL I get when I remark vol; so this tells me that the cost of recalibration (vega) is related to the Expected PnL leak if you do not remark. $\endgroup$
    – Arshdeep
    Jul 17, 2020 at 17:13
  • $\begingroup$ Vega P&L comes from implied vol moves, it is not expected risk neutral gamma PnL if you talking about black scholes. In black scholes, theta PnL paid is indeed expected risk neutral gamma P&L received under zero rates and dividends. $\endgroup$
    – ryc
    Jul 19, 2020 at 11:43
  • $\begingroup$ Check out quant.stackexchange.com/q/39619/848 $\endgroup$
    – Bob Jansen ♦
    Jul 19, 2020 at 12:04

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