7

Why is pricing with local volatility models are problem with exotics, mainly due to "the volatility surface is the market's current view of volatility and this will change in the future meaning the exotic options will no longer be consistent with market prices" (from Quant Job Interview Questions and Answers)

What does it mean by the vol surface is the current view of vol (I didnt think vol models were predictive of the future anyway) and why is this better if you use stochastic volatility models instead?

Trajan
  • 2,512
  • 9
  • 34
  • 66
  • Summaries of the conclusions are best, to get a headline idea. People can dig deeper into individual issues if they want to – Trajan Dec 12 '18 at 17:00

3 Answers3

17

1. What does it mean by the vol surface is the current view of vol?

The local volatility model is calibrated to vanillas prices (and equivalently their implied volatilities), which reflect the market's view of the volatility, in order to use it to use it to price other options that one will hedge with the vanillas.

Where a Black-Scholes model (no smile) will not be able to match the options implied volatilities at all strikes (smile). Local volatility models will. Given a continuous surface of call options prices, that is twice-differentiable in strike and once in time, Dupire's formula gives the unique risk-neutral diffusion (no jumps) process that is compatible with european option prices:

  • $dS_t = (r − q)S_t dt + σ(t, S_t)S_tdW_t$
  • with: $\sigma(t, S)^2 = 2 \frac{\frac{\partial C}{\partial T} + qC +(r-q)K\frac{\partial C}{\partial K}}{K^2\frac{\partial^2C}{\partial K^2}} |_{K = S, T = t}$
  • where $r$ is the interest rate, $q$ the div yield, $C$ the function giving the call price, $K$ the strike and $T$ the expiry.

For more info, see Dupire's and Derman and Kani's seminal papers:

2. Why is this better if you use stochastic volatility models instead?

The local volatility models will be able to match the value of the smile as of today, but because the smile flattens for long maturities, the model gives an almost constant smile for these maturities, leading to a flattening of the forward smile (i.e. smile in the future), which is unrealistic.

This is not desirable when the exotic option you are concerned with depends on the forward smile (e.g. ratchet option). In this case, one needs a model which will give realistic smile dynamics.

Stochastic volatility models give more realistic dynamics of the volatility smile. However, they come with their issues/challenges.

For example, they may be harder to calibrate than local vol models. Furthermore, they may sometimes not exhibit enough smile for options with short maturities. To overcome this second issue, stochastic volatility models are either:

  • combined with jumps in the underlying.
  • combined with local volatility (local-stochastic vol models).
byouness
  • 2,210
  • 2
  • 13
  • 28
  • 1
    The leads to a flattening of the forward smile is not obvious why this is true at all – Trajan May 07 '18 at 18:20
  • Also it is not clear to me what desirable smile dynamics is – Trajan May 07 '18 at 18:22
  • 1
    The forward vol/smile are given by the local vol function $\sigma(S,t)$. This function is deterministic and is calibrated on vanilla options prices. Vanilla options implied vol is flat for long maturities (see here for an explanation of this empirical fact: https://quant.stackexchange.com/questions/16670/why-does-the-volatility-smile-flatten-as-maturities-increase), which gives a flat local vol function for times in the future. – byouness May 07 '18 at 19:07
  • Desirable smile dynamics are characteristics that the implied volatility actually exhibits. e.g. at a given time, the vol is decreasing as a function of the option maturity, however for a given option maturity, although the level and orientation of the smile curve might change over time, its rough shape stays very much the same, etc.

    Everything of course depends on what you want to use the model for. It has to be accurate for the characteristics that your product is sensitive to.

     – byouness May 07 '18 at 19:21
  • Sorry I still dont see the connection between why if the current surface is flat for long maturities why that gives flat local vol in the future? – Trajan May 09 '18 at 17:50
  • This is because the initial smile you calibrate the model on has decreasing smile: $\frac{\partial \Sigma}{\partial K} \rightarrow 0$ when $T\rightarrow \infty$. The local vol is a function of this smile: $\sigma(t,S)= f\left(\frac{\partial \Sigma}{\partial K}| K = S, T = t \right)$. So, the local vol will get flatter the more we move into the future. This leads to the forward implied vol, which is a function of the local vol $\sigma(t, S)$, that is also flatter as a result. – byouness May 09 '18 at 20:16
  • Heres the main problem I have, how does SV incorporate forward vol (and LV doesnt) – Trajan May 11 '18 at 19:04
  • 1
    It doesn't. The difference is in SV models the vol is random driven by its own Brownian while in LV models it's not, the randomness of vol comes only from the stock, via a deterministic function. SV models give realistic scenarios for the evolution of the vol surface as a result, while LV models don't. – byouness May 11 '18 at 21:13
  • right that wasnt the answer i was expecting – Trajan May 11 '18 at 21:15
  • I hope things are a bit more clear now :) – byouness May 11 '18 at 21:18
  • no sorry im really confused – Trajan May 11 '18 at 21:18
  • As far as I can see a vol surface at time 0 flattens to maturity – Trajan May 11 '18 at 21:19
  • and I dont see whats wrong with LV models having decling forward smile? – Trajan May 11 '18 at 21:20
  • What's wrong with LV models having a declining smile is that it will misprice some products that depend on the forward smile, e.g. forward options, ratchet options, etc. – byouness May 12 '18 at 22:21
  • the notion of "desired" future implied vol dynamics has been established by what is observed empirically. The best way to have a clue of what "desired" should be per asset class is to do a back test with historical data. I.e. Take sample of dates $t_1<t_2<T$ and compare the spot smile for $t_1-T$ vs the "future" smile $t_2-T$. – noob-mathematician Dec 13 '20 at 14:58
  • @byouness you said that the IV smile flattens with maturity ("but because the smile flattens for long maturities") and also the LV smile flattens as well ("leading to a flattening of the forward smile (i.e. smile in the future), which is unrealistic"), so I dont understand what is the issue? The LV model cant match the flattening exactly? – Skittles Dec 24 '23 at 08:55
  • 1
    @Skittles, so if you look at IV surface now you will see typically that the LT smile is flatter than the ST, this leads to smile flattening with LV models. For example, the smile for 1Y expiry today is fine but the smile for 1Y expiry in 10Y (implied by the LV model) will be much flatter. In practice, however, if you look today at the 1Y smile, then wait until 10Y have passed and look at the 1Y smile again, it won't be flatter. That is what the history suggests. – byouness Dec 24 '23 at 18:54
  • @byouness "Implied volatility for far-dated options is relatively flat compared to near-dated, as both skew and kurtosis decay with maturity." (Trading Volatility, C. Bennett), so we can say IV flattens with maturity, while the SV matches that, the LV flattens too much. – Skittles Dec 25 '23 at 09:35
  • @Skittles, indeed, the implied vol $\Sigma(t, \tau)$ decays with the time to maturity $\tau$, and this leads to the local vol flattening with time $t$ (in a Monte Carlo simulation or grid for PDE resolution for example). The SV doesn't have this shortfall. – byouness Dec 26 '23 at 10:14
8

The following paper is helpful for understanding the point you raise:

Hagan et al.: Managing Smile Risk, January 2002, Wilmott 1:84-108

The main point is given in the paper:

[...] the dynamics of the market smile predicted by local vol models is opposite of observed market behavior: when the price of the underlying decreases, local vol models predict that the smile shifts to higher prices; when the price increases, these models predict that the smile shifts to lower prices. Due to this contradiction between model and market, delta and vega hedges derived from the model can be unstable and may perform worse than naive Black-Scholes’ hedges.

You can find the details on page 5ff.

The following questions (and answers therein) may also be helpful:

vonjd
  • 27,437
  • 11
  • 102
  • 165
5

The following source contains detailed answers to your questions in a research paper from ETH Zürich. van der Weijst, Roel (2017). "Numerical Solutions for the Stochastic Local Volatility Model" http://resolver.tudelft.nl/uuid:029cbbc3-d4d4-4582-8be2-e0979e9f6bc3

Alistar
  • 51
  • 1
  • 1