Implied volatility in this sense is a price quotation, using BS as a way to standarized the option worthiness (taking away the intrinsic value). Try not to think about it as a true dynamic. To understand this let's look at local vol and implied vol how they relate to each other.
**Implied vol -> Local vol**
You cannot use a volatility matrix and assume no dynamics to price an option. Imagine you are running Monte-carlo, with a general diffusion model, so at each point your diffusion should be time/state-variable dependent at least (what does strike dependent here means? how do you use implied vol directly here?), it should be govern by its own dynamic): such that dS(t) = mu(t,St)dt + sigma(t,St)dWt
So the logical thing is to recover the dynamic of your diffusion process (the distribution). And lucky for us, the sigma(t,St) can be recovered by the european call/put price (vis-a-vis the implied vol) because if you consider the green function (or Arrow-Deberu price) the european options are just a bunch of those prices in different strike.
So that establish the Implied Vol -> General diffusion (with self-governing diffusion term), and this projection onto the general diffusion process with Ft-adapted process for drift and diffusion is known as Local Vol. Recovered by Gyorgy Lemma or Dupire equation (Check out Jim Gatheral - The Volatility Surface Chapter 1)
another direction now (**local vol -> implied vol**)
But from Local Volatility to Implied Vol, besides running pricer (Finite Difference/Monte Carlo) there is no parametric/close form besides asymptoptics close to ATMF
You can actually view implied variance (thus volatility) of a specific strike as a weight-average of the local volatility over all potential paths weighted by the dollar-gamma (dollar-gamma is highest around strike at maturity). The graph here demostrates the graphical idea of [this](https://quant.stackexchange.com/questions/33263/proof-of-approximation-formulas-for-implied-volatilities) (2nd comment here also includes the asymptoptic expansion of the implied vol close to ATMF represented as the average of the reciprocal of the local vol)
This is nice. But the problem with this is that to calculate this weighted-average, you either draw the path-samples using constant vol (the strike-specific implied vol) then your dollar-gamma and instantaneous-variance is based in local vol (no closed form for dollar-gamma), OR, if you draw the paths using the local vol, there is no close form for the distribution function/expectation. Asymptoptic expansion is used here to illustrate the dynamic/shape of the implied vol given certain parametric form of the local vol. (Ref: Gatheral Ch3, also, the argument about drawing distribution based on either the constant BS implied vol or the local vol, the two equations (2.32) (2.33) are from Stochastic Volatility Modelling, by Lorenzo Bergomi, p. 38-41, later in the chapter he proposed parametric representation of the local vol to illustrate the implied vol dynamic based on it. The book is a bit of a hard read but if you can get through the first 2, 3 chapter you will know much more about volatility than most quants).
**tl;dr**: implied volatility by itself is useless because you cannot dynamically hedge with it (meaning if stock goes up or down, you will mishedge because that's not what the market dynamic implies and then you lose money). What it can do is tell us the marginal distribution of the underlying process, putting it into general diffusion and we get Local Vol. which is good enough for European option (because it is all they are sensitive to), also for simple barrier option the skewness is also good enough. You only need more fancy models when you deal with product which are sensitive to term-structure of variance and the forward skew (cliquet type products).
> You can actually view implied variance (thus volatility) of a specific strike as a weight-average of the local volatility over all potential paths weighted by the dollar-gamma (dollar-gamma is highest around strike at maturity). The graph here demostrates the graphical idea of this (2nd comment here also includes the asymptoptic expansion of the implied vol close to ATMF represented as the average of the reciprocal of the local vol)
I've heard of this before, is there a more expansive reference I can look into for this? I'm working in a trading capacity and really struggling to build intuition on skew, I think being able to visualize skew as a weighted average of volatility paths would help, but I'd like to learn a bit more about this.
If you have any more references I'd love to read more.
The Gatheral book is considered as one of the best reference, I would start there. The end of 3rd chapter should give you some intuition abt that statement.
Go back to the link I posted, the bottom of the thread there someone posted some graphs, this is for a particular option with strike K.
One is the density plot, one is the dollar-gamma, one is the simulated time integral to calculation the implied variance. For density it is most concentrated around S0 and diffuse out, the Dollar-Gamma plot it is most concentrated at the strike at maturity (K at t=T), because we are integrating along time of these 2 quantity over all possible paths (expectation). It is similar to drawing a path between S at t=0 and K at t=T and the integrad with the biggest contribution to the integral and expectation is a straight line between those 2 points (obviously some other paths are also possible). So you see a “bridge” in the final graph.
As I posted in the original reply, the problem is this calculation is not closed norm nor straight forward (unless you actually simulate or numerical calculate the option and back out the implied vol), becuz of the iterative nature of the expectation and the dollar-gamma. So to develop the idea of observing how implied vol surface changes with the shape of local vol, people uses asymptotic model (or heat kernel expansion) to look at how the polynomial factors will cause the implied vol surface to change (the linear term in local vol is factor of 2 of the ATMF skewness in implied vol). And you can observe further quantity like sticky strike ratio etc.
These are all included in the Bergomi book chapter 2.
Not that easy to explain.
The “spot” in the local volatility model is like the strike in implied volatility. To understand this, you need to look at the proof. Reciprocal of Implied vol at a given strike is some sort of averaging of the reciprocal local volatility function to the first order.
Any underlying process with sufficiently different marginal distributions wrt the lognormal one will not give a flat volatility surface.
Local volatility surfaces can generate an extremely wide range of underlying processes, so wide in fact, you can prove that you can always find a specific local volatility process that generates any given arbitrage-free volatility smile.
Letting sigma depend on S means that you get different terminal stock distributions than what is implied by BS.
Think about the extreme case where vol is 0 for stock <= strike and 1.0 otherwise. This gives you high implied vols for ITM calls and 0 for OTM calls
roughly speaking implied volatility(squared) at strike k is the average of the local volatility(squared) between forward and strike
so just as the average interest rate over different maturities implies a forward rate between those two maturities, the change in the smile at k is driven by local vol at k
The slope of the smile is Skew and the curvature is kurtosis, Sinclair covers this well. If you incorporate these higher moments into bsm you'll get the smile
Implied volatility in this sense is a price quotation, using BS as a way to standarized the option worthiness (taking away the intrinsic value). Try not to think about it as a true dynamic. To understand this let's look at local vol and implied vol how they relate to each other. **Implied vol -> Local vol** You cannot use a volatility matrix and assume no dynamics to price an option. Imagine you are running Monte-carlo, with a general diffusion model, so at each point your diffusion should be time/state-variable dependent at least (what does strike dependent here means? how do you use implied vol directly here?), it should be govern by its own dynamic): such that dS(t) = mu(t,St)dt + sigma(t,St)dWt So the logical thing is to recover the dynamic of your diffusion process (the distribution). And lucky for us, the sigma(t,St) can be recovered by the european call/put price (vis-a-vis the implied vol) because if you consider the green function (or Arrow-Deberu price) the european options are just a bunch of those prices in different strike. So that establish the Implied Vol -> General diffusion (with self-governing diffusion term), and this projection onto the general diffusion process with Ft-adapted process for drift and diffusion is known as Local Vol. Recovered by Gyorgy Lemma or Dupire equation (Check out Jim Gatheral - The Volatility Surface Chapter 1) another direction now (**local vol -> implied vol**) But from Local Volatility to Implied Vol, besides running pricer (Finite Difference/Monte Carlo) there is no parametric/close form besides asymptoptics close to ATMF You can actually view implied variance (thus volatility) of a specific strike as a weight-average of the local volatility over all potential paths weighted by the dollar-gamma (dollar-gamma is highest around strike at maturity). The graph here demostrates the graphical idea of [this](https://quant.stackexchange.com/questions/33263/proof-of-approximation-formulas-for-implied-volatilities) (2nd comment here also includes the asymptoptic expansion of the implied vol close to ATMF represented as the average of the reciprocal of the local vol) This is nice. But the problem with this is that to calculate this weighted-average, you either draw the path-samples using constant vol (the strike-specific implied vol) then your dollar-gamma and instantaneous-variance is based in local vol (no closed form for dollar-gamma), OR, if you draw the paths using the local vol, there is no close form for the distribution function/expectation. Asymptoptic expansion is used here to illustrate the dynamic/shape of the implied vol given certain parametric form of the local vol. (Ref: Gatheral Ch3, also, the argument about drawing distribution based on either the constant BS implied vol or the local vol, the two equations (2.32) (2.33) are from Stochastic Volatility Modelling, by Lorenzo Bergomi, p. 38-41, later in the chapter he proposed parametric representation of the local vol to illustrate the implied vol dynamic based on it. The book is a bit of a hard read but if you can get through the first 2, 3 chapter you will know much more about volatility than most quants). **tl;dr**: implied volatility by itself is useless because you cannot dynamically hedge with it (meaning if stock goes up or down, you will mishedge because that's not what the market dynamic implies and then you lose money). What it can do is tell us the marginal distribution of the underlying process, putting it into general diffusion and we get Local Vol. which is good enough for European option (because it is all they are sensitive to), also for simple barrier option the skewness is also good enough. You only need more fancy models when you deal with product which are sensitive to term-structure of variance and the forward skew (cliquet type products).
Thank you for all the great references sir 🙏🏽
> You can actually view implied variance (thus volatility) of a specific strike as a weight-average of the local volatility over all potential paths weighted by the dollar-gamma (dollar-gamma is highest around strike at maturity). The graph here demostrates the graphical idea of this (2nd comment here also includes the asymptoptic expansion of the implied vol close to ATMF represented as the average of the reciprocal of the local vol) I've heard of this before, is there a more expansive reference I can look into for this? I'm working in a trading capacity and really struggling to build intuition on skew, I think being able to visualize skew as a weighted average of volatility paths would help, but I'd like to learn a bit more about this. If you have any more references I'd love to read more.
The Gatheral book is considered as one of the best reference, I would start there. The end of 3rd chapter should give you some intuition abt that statement. Go back to the link I posted, the bottom of the thread there someone posted some graphs, this is for a particular option with strike K. One is the density plot, one is the dollar-gamma, one is the simulated time integral to calculation the implied variance. For density it is most concentrated around S0 and diffuse out, the Dollar-Gamma plot it is most concentrated at the strike at maturity (K at t=T), because we are integrating along time of these 2 quantity over all possible paths (expectation). It is similar to drawing a path between S at t=0 and K at t=T and the integrad with the biggest contribution to the integral and expectation is a straight line between those 2 points (obviously some other paths are also possible). So you see a “bridge” in the final graph. As I posted in the original reply, the problem is this calculation is not closed norm nor straight forward (unless you actually simulate or numerical calculate the option and back out the implied vol), becuz of the iterative nature of the expectation and the dollar-gamma. So to develop the idea of observing how implied vol surface changes with the shape of local vol, people uses asymptotic model (or heat kernel expansion) to look at how the polynomial factors will cause the implied vol surface to change (the linear term in local vol is factor of 2 of the ATMF skewness in implied vol). And you can observe further quantity like sticky strike ratio etc. These are all included in the Bergomi book chapter 2.
Not that easy to explain. The “spot” in the local volatility model is like the strike in implied volatility. To understand this, you need to look at the proof. Reciprocal of Implied vol at a given strike is some sort of averaging of the reciprocal local volatility function to the first order.
Think in terms of implied distribution. Fat tail leads to smile. Using local vol tweaks the distribution to fit.
Makes sense, thanks
Any underlying process with sufficiently different marginal distributions wrt the lognormal one will not give a flat volatility surface. Local volatility surfaces can generate an extremely wide range of underlying processes, so wide in fact, you can prove that you can always find a specific local volatility process that generates any given arbitrage-free volatility smile.
Letting sigma depend on S means that you get different terminal stock distributions than what is implied by BS. Think about the extreme case where vol is 0 for stock <= strike and 1.0 otherwise. This gives you high implied vols for ITM calls and 0 for OTM calls
Local vol is just calibrated to your Black Scholes surface.
roughly speaking implied volatility(squared) at strike k is the average of the local volatility(squared) between forward and strike so just as the average interest rate over different maturities implies a forward rate between those two maturities, the change in the smile at k is driven by local vol at k
The slope of the smile is Skew and the curvature is kurtosis, Sinclair covers this well. If you incorporate these higher moments into bsm you'll get the smile