Hey fellow quants,

Today, let’s dive into the fascinating world of forecasting asset price behavior, specifically volatility and density, using the power of options data. It's not just about predicting whether a stock will go up or down; it’s about understanding the distribution of possible future prices, and that’s where the real edge lies.

Volatility: More Than Just a Number

Volatility forecasting is crucial for several reasons: risk management, derivatives pricing, hedging strategies, and even portfolio optimization. It's not a directly observable quantity, so we need to define a target, which is often a sum of squared returns over a forecast horizon. This is key because our forecast is about predicting the variability of returns, not the returns themselves. Now, how do we make these forecasts?

There are a few main approaches, and each uses a different type of data:

• Historical Prices: We can use models like ARCH (autoregressive conditional heteroscedastic) or HAR (heterogeneous autoregressive) models, using past returns to predict future volatility. Daily returns are calculated as rt = ln(pt) - ln(pt-1). Intraday, we can also calculate the realized variance RVt = ∑j=1N rjt2, where rjt is the return for sub-period j in day t. This gives us a more granular view of volatility.

• Option Prices: This is where it gets particularly interesting. Options prices reflect market expectations about future volatility. We can extract this information using:

  • Specific pricing models like the Black-Scholes model to compute implied volatilities (IV).

  • Model-free measures like the VIX index. The VIX is calculated from a range of option prices without making assumptions about price dynamics. It provides a risk-neutral expectation of the integrated variance over the time to expiry. The formula is:

    ∫0T EQ(Vt)dt = 2∫0F (1/X2)PT(X)dX + 2∫F∞ (1/X2)CT(X)dX,

    where PT(X) and CT(X) are the prices of put and call options with strike price X, and F is the forward price.

It’s crucial to understand that option prices are risk-neutral. This means that they reflect expectations under a probability measure where all assets have the same expected return (the risk-free rate). However, real-world expectations may differ due to the variance risk premium. For example, because of people’s willingness to pay a premium to hedge against volatility increases, option-implied volatility tends to be higher than subsequently realized volatility.

When comparing volatility forecasts, we need a loss function to evaluate the accuracy of our predictions. Typically, we use the mean squared error (MSE). Also, parameters must be estimated using only information available at the time the forecasts are made.

Integrating Option Prices into ARCH Models

You might be thinking, "How can we blend these approaches?" We can integrate implied volatility into ARCH models. A basic ARCH model for conditional variance ht is: ht = ω + α(rt-1 - μt-1)2 + βht-1, where rt is the return at time t, μt the conditional mean of rt and St-1 is an indicator function of negative returns. We can enhance it by adding implied volatility, vt-1:

• ht = ω + α(rt-1 - μt-1)2 + βht-1 + δvt-1

This model (S1) assumes that squared returns and squared implied volatilities have the same lag structure. But, we can add flexibility:

• ht = ω + α(rt-1 - μt-1)2 + βht-1 + δvt-1* Where the lags are applied only to the components of the returns. In the second model (S2), we would expect β to be larger than *β if the options market contains forward-looking information not included in past returns.

Empirical evidence shows that option prices do contain incremental information for forecasting volatility that is not found in historical returns.

Density Forecasting: Beyond Point Predictions

Volatility is important, but sometimes we need the full picture – the density forecast. This means estimating the probability of different price levels at a future time. We can do this in a few ways:

• Simulating ARCH models: Generate many possible future price paths using a calibrated ARCH model. This is a Monte Carlo approach, and can give you the full distribution of potential prices.

• Estimating risk-neutral densities from option prices: This approach is often done using mixtures of lognormal distributions, since single lognormal distributions are contradicted by market prices.

The risk-neutral density fQ(x) can be found from the second derivative of call option prices using the formula: fQ(x) = erT∂2C(X)/∂X2. A lognormal mixture is a combination of two or more lognormal distributions, defined by: fQ(x) = p ψ(x; F1, σ1) + (1 - p) ψ(x; F2, σ2), where ψ denotes the lognormal density.

Then, we can transform the risk-neutral density to a real-world density by using a utility function, such as one with constant relative risk aversion (CRRA). The real-world density fP(x) is proportional to fQ(x) divided by marginal utility : fP(x) ∝ fQ(x) / u'(x).

Real-world densities and risk aversion

The risk aversion parameter γ in the CRRA utility function ( u(x) = x(1-γ) / (1-γ) ) changes the risk-neutral distribution so that :

• the expected value of future price TS is higher.

• the skewness of the distribution of TS is closer to zero. In a mixture of lognormals this can be achieved by transforming the means and probabilities of the two lognormals .

In short, risk aversion shifts the distribution to the right, increasing the expected return and reducing the negative skew.

Important Caveats

Remember, that there are a few challenges when forecasting using options data:

• Option liquidity: Some options are not actively traded, which makes it difficult to obtain accurate price information.

• Data quality: Option prices may not be perfectly contemporaneous with asset prices, and bid-ask spreads can add noise .

• Extrapolation: Any method for density forecasting needs extrapolation outside the range of traded prices. Ideally, the probability of the asset price falling outside of this range will be small .

Out-of-Sample Evidence

Out-of-sample testing is critical for assessing forecast accuracy. Generally, implied volatility (VIX) provides more accurate forecasts of realized volatility than historical volatility or intraday volatility measures. Also, forecasts tend to be more accurate when the target is the sum of squared intraday returns and improve with longer horizons. Finally, combining implied volatility with historical information can improve accuracy.

Final Thoughts

Forecasting volatility and density is a complex and evolving field. The approaches described above offer a robust toolkit, but the best methods for forecasting volatility and densities depend on many choices: the forecast horizon, the data used, and the loss function. Keep exploring, stay critical, and never stop learning.

Until next time, happy quanting!