I have a portfolio of equities and equity options and I am working on a P&L attribution report using revaluation. I am having trouble calculating the implied volatility part of the P&L. I would like my model to account for Day1's volatility skew and movements in volatility skew from Day1 to Day2. I am looking at a skew curve that plots relative stock prices versus implied volatility. In this way, an option that is 10% out-of-the-money (OTM) one day could be 5% OTM the next day, and thus its point on a volatility skew would move.It would be something like this (here is a chart that illustrates this *):1) From Day1 to Day2, if the skew doesn't move, but the underlying price moves, both the option's relative moneyness and implied volatility will have changed. This implied volatility change results in some P&L change that I would call "P&L Implied Volatility" in my report.2) From Day1 to Day2, if the skew does move but the underlying price remains constant, the implied volatility will again have changed resulting in some P&L change. This time however I would call it "P&L Skew" in my report.3) If from Day1 to Day2, the skew and the underlying price move, I can calculate both "P&L Skew" and "P&L Imp Vol".What I am having trouble with is determining what P&L is attributable to underlying price movements. If I always treat underlying price movements as some function of volatility skew, then I have no way of determining what P&L to assign simply to underlying price movements. On the one hand, I understand that implied volatility movements are tightly related to underlying price movements in a way that splitting up their effects on P&L may not make sense. On the other hand, it doesn't feel right to say that no P&L is attributable to underlying price movements, and that it's all either "P&L Skew" or "P&L Imp Vol".Is there a preferred way to attribute P&L to skew, movements in skew, and movements in the underlier?* In the chart, the skew for Day1 is in green and the skew for Day2 is in blue. The option's implied volatility is 40% on Day1 and 30% on Day2. The option is 20% OTM on Day1 and 15% OTM on Day2. Half of the implied volatility movement is due to the skew movement from Day1 to Day2. The other half of the implied volatility movement is due to the underlying price movement from Day1 to Day2. I don't know how to attribute any P&L to the underlying stock price movement (Called "P&L Delta").P&L Skew = BS(Stock1, IV = 35%, X, Div, IntRate, Exp) - BS(Stock1, IV = 40%, X, Div, IntRate, Exp)P&L Imp Vol = BS(Stock2, IV = 30%, X, Div, IntRate, Exp) - BS(Stock1, IV = 40%, X, Div, IntRate, Exp) - P&L Skew

It seems to me it is just a matter of total derivatives.Say we write for a call price C = c(T,S,K, I(T,m)) where c = BS formula, I = implied vol and m = moneyness = K/S Then (*) dC = c_T dT + c_S dS + c_I dIYou can also substitute dI = I_T dT + I_m dmSo the second term in (*) picks up Delta and the two terms in dI pick up the two effects in your graph beyond Delta.

In your equation, is "c_I dI" the partial derivative of the change in the option price with respect to the change in the implied volatility? If so, isn't that just the vega? And if so, then is your substitution simply a disaggregation of vega into a time component and a moneyness component? I can see how this would work for my purposes if I was taking an exposure-based approach to attributing my P&L. I think it would look something like this:1: Vega = V, Quantity = qty, Contract Multiplier = mult (= 100)2: Vega Exposure = V * qty * mult3: P&L-Vega: = Vega Exposure * (Spot2 - Spot1)Now if I do the same for the change in the option price with respect to the change in the underlying spot price (delta), would I be double-counting the underlying stock price movement? The movement would appear in the delta component calculation and also in the moneyness part of the vega component calculation. I understand mathematically that the fact that I'm using a partial derivative probably is what prevents this double-counting, but conceptually it seems like I would be counting the spot price movement twice (and also the time movement).I'm instead using a revaluation-based approach to attributing P&L. The equivalent process that I am using now for calculating P&L-Vega is:P&L-Vega = (BS(Spot1, Strike, IV2, DivY, IntRate, ExpDate) - BS(Spot1, Strike, IV1, DivY, IntRate, ExpDate)) * qty * mult* where BS is Black-Scholes and the only input that changes is the implied volatility (IV1 -> IV2)

Yes, c_I = Vega and yes, it is a disaggregation of vega into a time component and a moneyness component. Since time and moneyness are dT and dm, I don't understand why you are writing 3.dT = T2 - T1 and dm = moneyness2 - moneyness1. There is no reason to write Spot2-Spot1, except for the C_S=Delta term in (*),which has dS.My suggestion is that you try to implement (*) directly, writing out the differentials as I just did,and see if it doesn't solve your issues or at least see where you get stuck. Unlike your approach, in (*), there is correct accounting for the P&L due to stock price moves: (i) an effect due to Delta exposure and(ii) a *separate* effect due to Vega (and skew) exposure. The latter is the term in (*): c_I x I_m x dm = Vega x (IV skew slope) x (the change in moneyness).

I think we are taking two approaches to calculating P&L effects. I am using a revaluation-based approach that changes the inputs to the option valuation equation and sees what happens. My P&L-Delta calculation is this:1) P&L-Delta = [BS(spot2, etc) - BS(spot1, etc)] * quantity * multiplierYou are explaining an exposures- or sensitivites-based approach that uses greeks. Your P&L-Delta calculation is this.2) P&L-Delta = delta * (spot2 - spot1) * quantity * multiplierI think my approach won't work because it can't account for second-order effects (according to question #5 here).With your approach, I will need to calculate the sensitivity of the implied volatility to both time and moneyness. Is it like this?3) I_T = IV(spot1, option mid, strike, div, intRate, time2, maturity) - IV(spot1, option mid, strike, div, intRate, time1, maturity)4) I_m = IV(spot2, option mid, strike, div, intRate, time1, maturity) - IV(spot1, option mid, strike, div, intRate, time1, maturity)Where: IV is my calculation for the impliedVolatility from the option's market price. Time2 is one day after time1. Spot2 is a 1% moneyness movement above spot1.Also I did have that equation in 3 in my last post wrong. I did mean to type (impVol2 - impVol1).

Your 3) looks good for my I_T dT. It picks up the one-day change in IV due solely to the one-day passage of time, holding spot unchanged.For 4), we want I_m dm, the one-day change in IV due solely to the one-day change in spot (which changes the moneyness), holding time unchanged.Because of that, I would use Spot2 as the actual spot price on day 2, not some 1% moneyness move price.With that change, 4) looks good for my I_m dm. If you implement my formula (*), with 3), 4) as revised, and a similar approach to c_T dT + c_S dS, thenI would expect (*) to account for most of the PnL on most days, with a small unexplained residual. The likelyexceptions (days with a big residual) would be days with a big jump in the stock price and/or Vega, say after an earnings release by a firm. Also, if day 2 is expiration, the approximations are likely poor. Referring to your cited link, whether or not there is another version of what I wrote that makes it an exactaccounting identity (the 'revaluation approach'), with no residual, I don't know. Ahh ... the joys of performance attribution.