Which one is your volatility — Constant, Local or Stochastic?
Analyzing the path of EURUSD derived from the market vol surface by using the Heston Model
There are three main volatility models in the finance: constant volatility, local volatility and stochastic volatility models.
Before the stock market crash of 1987, the Black-Scholes (B-S) model which was built on geometric Brownian motion (GBM) with constant volatility and drift was the dominant model. In this model, stock price is the only source of randomness and it can be hedged with the underlying stock with a return distribution as log-normal. In the B-S model, the stock price S is described by the following stochastic differential equation (SDE), where W is a standard Brownian motion:
The above Stochastic Differential Equation (SDE) has the following analytical solution (under Itô’s interpretation):
After the crash of 1987, the word ‘skew’, which used to mean happiness or when something good has happened, meant something completely different: ‘skew’ indicates that very small and very large returns are realized more often than the normal distribution would imply. So, traders started to price especially lower strike options with higher volatilities (check out the implied vol curve pre- and post- 1987). Supply and demand for different options also played an important role in shaping the volatility curve.
Local Volatility (LV) model, which is actually a simple extension of B-S model, addressed the need to incorporate the skewness into the pricing model and hence implies a non-normal distribution. In LV models, the volatility of the underlying, called as ‘instantaneous volatility,’ is modeled as a deterministic function of time t, and the time-t value of the underlying asset (S):
This is a very intuitive formula just by thinking of the fact that volatility can be viewed as an asset itself. For example, VIX index, a real-time market index that represents the market’s expectation of 30-day forward-looking volatility derived from the price inputs of the S&P 500 index options, can be traded through derivatives. Check out the non-constant nature of the VIX index:
The implied volatility surface can be transformed to LV surface, which is known as calibration of LV model of Dupire. Dupire (1994) shows that there exists a model that can match option prices quoted on the market and that there is an explicit formula for how to construct this model from observed options’ prices under the assumption that they can be interpolated across continuous strikes and maturities. (At the time Dupire actually didn’t think of local volatility as a realistic model for the evolution of actual volatility.)
Since the asset price is driven by a single Brownian motion, every payoff is a result of a unique self-financing replicating portfolio consisting of cash and the underlying asset, and that is to say that its price is uniquely defined as the initial value of the replicating portfolio (market completeness). The advantage of being complete means that derivatives written on S can be hedged perfectly. While BS and LV models are complete, stochastic volatility models are not. On the other hand, while market completeness is very convenient from a theoretical point of view, it is not necessarily a realistic property of financial markets, as we all can deal with derivatives, right?
Anyways. If we compare implied volatility surface with local volatility surface, we can see that the fit is pretty good. Actually, the LV models, unlike B-S model, are very good and sometimes even perfect at fitting arbitrage-free surface of implied volatilities, i.e. ‘smile’, via Dupire’s formula.
However, LV model has a major defect: The model only uses today prices and makes no assumptions about their behavior over time! Even though it fits the value of the smile as of today, the model gives an almost constant smile for long maturities, leading to a flattening of the forward smile (i.e. no smile in the future)! How absurd! Therefore, it leads to unreasonable skew dynamics and underestimates the volatility of volatility. This is very unrealistic especially for exotic options which depend on the forward smile such as cliquet options and other forward starting options.
Moreover, LV models are not good bad at providing stability and they may need to be re-calibrated frequently, such as hourly !!
Nonetheless, with some adjustments made to local volatility prices, especially adding some jumps to the model to capture the complex dynamics of stock price processes, these models can be used to price several barrier options.
As an upgrade/alternative to LV model, a stochastic volatility (SV) model can produce a more realistic forward volatility surface, where the smile is almost self similar, compared to LV models which flatten out the forward volatility curve and vanish the smile. LV models assume that skew in the future will be what we see the forward skew today in local vol, whereas SV models assume the skew in the future is similar to the skew today. That’s why SV model is likely to overprice options, whereas the LV and BS models tends to underestimate options.
In SV models, the asset price and its volatility are both assumed to be random processes and can change over time, hence it gives more realistic dynamics of the volatility smile. In an SV model, promoted in the late 1980s by Hull and White, the volatility of the underlying is modeled as a deterministic function of some auxiliary process, which is usually modeled as a diffusion.
The most well-known and important stochastic volatility model is the Heston model. It is a two-factor model and assumes separate dynamics for both the stock price and instantaneous volatility.
As a super simple example, the below code shows a simple application of Heston model to EURUSD to find the currency path in 1M, 2M, and 3M’s time.
If you compare the results with the Bloomberg forward quotes below, you can see how our results match those quotes given below perfectly.
Also now we have the path of the currency in each simulation and we can price our exotic options based on these paths (below graphs).
In the above example, Heston model parameters were determined by calibrating to the market observed implied volatility smile for European options (surface given in the below graph). Ok, I admit I actually didn’t calculate them by using tedious formulas but got the recalibrated parameters directly from Bloomberg to save some time:)
SV models can produce rich spot-vol dynamics, such as large negative short term forward skews or large forward smile curvatures. However, they also have some drawbacks. They are harder to calibrate than local vol models, which is why I took them directly form Bloomnberg, and they may sometimes not exhibit enough smile for options with short maturities. And for products which depend only on terminal distributions, the fit of the volatility surface may be poor. Hedging ratios in SV models also depend strongly on the parameters and so they are sensitive with respect to changes in parameters. A simpler model can even do better since its parameters can be calibrated more efficiently and robust while hedging errors can get “averaged out”.
There are other models to overcome these shortcomings. One of them is local-stochastic volatility (LSV) models, which try to get the positive sides of local and stochastic volatility models: They model the volatility as a function of time t and the underlying asset S with additional stochastic process. There are also Path-dependent volatility (PDV) models which tries to combine benefits from both LV and SV by trying to fit the market smile and produce a wide variety of joint spot-vol dynamics at the same time.
Another alternative is using ARCH/GARCH models. Even though ARCH/GARCH represent discrete-time models for stock returns, whereas SV models are continuous-time models, discrete-time models for stock returns attract quite a lot of attention in econometrics literature and offer robust results. ARCH/GARCH can capture tail risks, volatility clustering and dependence without correlation.
All in all, it can be concluded that there is no “right” model but “suitable” model. Your choice of a vol model depends on what you are gonna use it for. For example, if you want to price exotic options, go with the SV models. For less complicated and exotic products, local vol models can perform better.
