Rough Volatility Model — A Potential Solution for Three Quantitative Finance Puzzles
From 1970 to now, we have seen the development of volatility modeling, from constant volatility Black Scholes model to stochastic volatility model and local volatility model. Before we ask the question “what is the next-generation model,” we should ask ourselves: what are new good properties that the model should have, and what puzzles are not well handled by the current model so that we want to solve them by the new model. In this blog, we list three puzzles troubling both academia and industry in the quantitative finance area, also, the Rough Volatility Model, a potential solution that might solve all of them altogether.
Puzzle 1: the Link between Volatility Model and Market Microstructure
The traditional stochastic volatility model is a reduced form model in the risk-neutral measure. There is no linkage between the underlying’s market microstructure and the volatility model. This disadvantage restricts the volatility model’s ability to absorb useful information from underlying high-frequency data. Intuitively, more volatility data are always helpful because changing measures from the historical measure to the pricing measure does not influence the diffusion terms. Assume we calibrate the Heston model from option prices, and now we are given stock tick data; how can we use the high-frequency stock data to improve our calibration of the Heston model? We need a joint market microstructure model to answer the question, bridging stock underlying and option pricing. Rough Vol model is the long-term limitation of the Hawkes processes. There are four assumptions of Hawkes, which look reasonable:
- a. Orders are mainly driven by other orders, meaning that most of the orders have no real economic motivations but are sent by algorithms in reaction to other orders;
- b. Leverage effect: sell orders have more impact on the order book (liquidity asymmetry);
- c. Arbitrage-free: high-frequency statistical arbitrage is almost impossible; and
- d. Traders usually split a large order into small parts.
The relation between the volatility and market microstructure makes it possible for the derivative market to absorb information in the underlying market and improve calibration performance.
Puzzle 2: Sample Frequency and Prediction Horizon
Finance is so amazing partly because of its fractality: in different sample frequencies, such as high frequency, intraday, daily, monthly, it behaves differently. The players in distinct frequencies are different: with varying horizons of investment and risk-return trade-offs. Practically we can ask how we can use high-frequency data to improve our prediction ability in lower frequencies. Assume we are a portfolio manager with monthly turnover; we want to make a prediction in monthly frequency. We have stock tick data, but it does not help a lot. The usual way to resample our data into the lower frequency to make the sample period is the same as our prediction step length. In the resampling step, we would lose most of our information. However, there are some other potential solutions to solve this problem; for example, we can build a neural network encoder and decoder to find the optimal way to resample. Rough volatility gives a different direction: Fractality. For self-similar objects, we can zoom in/out and note they will be approximately the same. The building block of the rough Volatility model is fractional Brownian motion, which has very few parameters to control properties across different scales, in other words, in different sample frequencies. This puzzle is linked with the first puzzle, and both are related to using high-frequency data to calibrate our model and improve our prediction on a longer horizon. In comparison, the first puzzle is from an economic perspective, while the second puzzle is from a data perspective.
Puzzle 3: Exponential Kernel to Power Kernel
There are generally two modeling blocks, power-law decay and exponential decay, to model decaying processes. The two blocks look similar by naked eyes, but the main difference is the asymptotic effect: exponential decay gets faster than power decay when we go to infinity. For example, in the probability distribution function, power decay has a fat tail while exponential decay does not. In the autocorrelation spectrum, exponential has short-term memory, while power decay has long-term memory. Generally, an exponential kernel is easier to model, but a power kernel is more common in a real market. For example, the atm skew term structure follows the power-law t^-0.44. A rough volatility model can be regarded as a combination of randomness with a power kernel, which gives us a flexible building block to model long-term memory.
In conclusion, there are three motivations behind the rough volatility model. However, the rough volatility model is not perfect. One problem is the calculation speed: because it has long-term memory, the non-markovian property results in any numerical calculation would depend on Monte Carlo, which is slow and hard to converge.