### Long Term GARCH Forecast with Options

#### Definition

Consider an asset’s log-return series ${r}_{t}=\mu +{\epsilon }_{t}$, where $\mu$ is the expected return and ${\epsilon }_{t}$ is a zero-mean white noise. The total log-return between date $t$ and date $t+k$ is then naturally defined as:

$r t,t+k = ∑ i=1 k rt+k$

The standard definition of the $k$-day ahead $\mathrm{VaR}$ of a position in this asset is the 1% or 5% quantile of the return distribution for ${r}_{t,t+k}$. V-Lab’s long run risk measures use simulation based methods to calculate the $\mathrm{VaR}$ at horizons of $k=30$ and $k=365$.

#### Estimation

One way to calculate $\mathrm{VaR}$ is to simulate future realizations of the return process and use the resulting simulations to calculate $\mathrm{VaR}$. For both of the models prescribed in the models section of the documentation, a volatility model is fit to historical data on each day. The resulting model is then simulated ahead 10,000 times, for a horizon of 1-year in advance. All simulations are bootstrapped - that is historical shocks to the return process are drawn at random to simulate each path. $\mathrm{VaR}$ is then calculated using both the 1% and 5% quantile of the 10,000 simulated return paths. Finally, logarithmic returns are converted back to arithmetic returns.

Since a pure statistical model such as $\mathrm{GJR-GARCH}$ may not “know” current macroeconomic conditions, V-Lab uses a variation of the $\mathrm{GJR-GARCH}$ model to incorporate information contained in options prices. Options are in intuitive source of information for forecasting, since they are forward-looking financial instruments. The spirit of this $\mathrm{GJR-GARCH}$ with options model is very similar to the $\mathrm{SPLINE-GARCH}$ model of Engle and Rangel (2008). Namely, a low-frequency component of volatility is calibrated so that the future expected volatility from the model matches the term structure of option implied volatilities in the market. Formally, this means the return process is modeled as:

The deterministic ${q}_{t}$’s in the above formulation are set to match the term structure of implied volatilities from options markets as follows. Denote the conditional variance of the total logarithmic return from date $t$ to date $t+k$ by ${V}_{t,t+k}$. According to our model above, this is easily calculated as:

$Vt,t+k = 𝔼t ∑ j=1 k r t+j 2 = ∑ j=1 k 𝔼t g t+j 𝔼t g t+j = 1+θj-1 g t+1 - 1$

Now, denote the option implied volatility on date $t$ for horizon $j$ by ${\sigma }_{t,t+j}$. Our goal is to calibrate the series of ${q}_{t}$’s so that ${V}_{t,t+j}={\sigma }_{t,t+j}^{2}$ for all $j=1,...,k$. In order to do this, notice our model implies:

$Vt,t+j-Vt,t+j-1 = qt+j𝔼t g t+j = qt+j 1+θj-1 g t+1 - 1$

So we can construct the ${q}_{t}$’s as follows:

$qt+j = σt,t+j2 - σt,t+j-12 1+θj-1 g t+1 - 1 , j>1 qt+1 = q$

where $q$ is the unconditional variance of the historical return series. Since there are not options that expire each day, the series of ${\sigma }_{t+j}$’s are constructed by cubic spline interpolation of available option implied volatility data.

To estimate the parameters of the $\mathrm{GJR-GARCH}$ component of the return series, ${q}_{t}$ is assumed to be a constant $q$ in-sample. However, for simulation V-lab uses the ${q}_{t}$ series constructed from option implied volatilities. Bootstrapped simulation based $\mathrm{VaR}$ is then calculated analogously to the baseline $\mathrm{GJR-GARCH}$ model.

#### Bibliography

Engle, Robert F., The Risk that Risk Will Change. Journal Of Investment Management (JOIM), Fourth Quarter 2009.

Engle, R. F. and J. G. Rangel, 2008. The Spline-GARCH Model for Low-Frequency Volatility and Its Global Macroeconomic Causes. Review of Financial Studies 21(3): 1187-1222.

Glosten, L. R., R. Jagannathan, and D. E. Runkle, 1993. On The Relation between The Expected Value and The Volatility of Nominal Excess Return on stocks. Journal of Finance 48: 1779-1801.

Zakoian, J. M., 1994. Threshold Heteroscedastic Models. Journal of Economic Dynamics and Control 18: 931-955.