Options, futures, and swaps in derivative forms are priceless risk management tools in finance. Their application, however, is widely well known to be extremely reliant on volatility, i.e., magnitude of variation in asset prices. Standard models, including the widely hailed Black-Scholes model, usually demand volatility to be constant or change in accordance with a straightforward linear trend.
The reality, especially in market meltdown or surprise technological breakthrough, is that volatility is not stable at all; it fluctuates between rather different regimes (e.g., extended peace, extended high apprehension, or one spike). This renders the marvelous task of equitable derivatives pricing and hedging off market risk yet more challenging.
The Flaw in the Classic Model
The Black-Scholes-Merton (BSM) Model of the 1970s was an advancement to also give the closed-form solution of European-style option prices. Its greatest simplification, though, is also its greatest weakness in contemporary markets:
- Constant Volatility Assumption: BSM further assumes volatility ($\sigma$) of the underlying asset to be constant for the life of the option.
- Volatility Smile: Real-world practicalities cause market option prices to illustrate that volatility is not level. More costly out-of- or in-the-money options (options with strike price far from asset value currently) than near-the-money options. When plotted versus the strike price, it generates a “smile” or “skew,” in opposition to BSM assumption.
This paradox states that volatility is thought to change by traders, and therefore modelers must step up to improved models.
Better Models: Stochastic Volatility
The solution to better models is to make volatility a stochastic (randomly changing) process rather than a parameter. This gives rise to models known as Stochastic Volatility (SV) models:
1. Heston Model
Heston model is the most widely used SV model. It describes two correlated stochastic processes:
- The asset price is a geometric Brownian motion (like in BSM).
- The volatility squared (variance) is an asset price-independent mean-reverting stochastic process (the CIR process).
The primary advantage is to allow for correlation of volatility and asset price—a property known as the leverage effect. Prices decline when volatility increases (e.g., during crashes), a real market move and volatility skew behavior.
2. Rough Volatility (rV) Models
Subsequent more coarse volatility models assume the log of realized volatility to be a “fractional Brownian motion,” i.e., its path is much more rough than standard models would suggest. That’s an empirical characteristic: today’s volatility is extremely highly correlated with tomorrow’s, and the correlation weakens very slowly. rV models performed better in explaining short-term behavior of the implied volatility surface.
Regimes of Volatility and Jump Processes
Even in extremely volatile markets, regime transitions are not everywhere continuous; they are occasionally jumps (abrupt, enormous, unforeseen movements, frequently caused by geopolitical crises or flash crashes).
Jump diffusion models such as the Merton Jump-Diffusion model cause asset prices to jump randomly in enormous, discontinuous jumps at random times, more accurately modeling crisis dynamics.
More sophisticated models combine SV with regime-switching dynamics:
- Regime-Switching Models: Regime-switching models assume that the market is switching between a finite set of “states” or regimes (e.g., “High Volatility,” “Low Volatility,” “Crisis”). Price dynamics, e.g., the volatility parameter, vary according to regime, and regime switching is modeled in terms of probabilities.
- Pricing Implications: These models have direct application to derivatives pricing, especially complex exotic options, whose price depends heavily on the likelihood that the market is in stress regime throughout the lifetime of the option.
Hedging Volatility Uncertainty
Successful hedging—the delicate art of building a portfolio which hedges off risk—is very difficult with stochastic volatility.
BSM assumes Delta Hedging, in which the volatility is constant and, hence, perfect hedging by the underlying alone is achievable. Perfect hedging by the underlying alone is not feasible when volatility is stochastic.
- Greeks and SV: In SV models, an item to bear in mind is the Vanna (derivative of Delta with respect to volatility) and Volga (option value convexity with respect to volatility).
- Vanna-Volga Hedging: Sophisticated traders use a Vanna-Volga hedge, taking advantage of liquid options (i.e., for calibrating Heston models) in hedge portfolios to counter non-linear stochastic volatility risks. It is hedge delta (sensitivity to price) and hedging exposure to volatility itself.
Using these sophisticated quantitative models, banks are able to price complex derivatives more accurately, reduce risk exposure during crisis situations, and stabilize global markets.