- Implied Variance
- Value-at-Risk
- Key concepts
- Models
- Volatility-based
- Quantile Regression
- Historical Simulation

- Model evaluation

- Implied volatility is very different from ARCH and Realized measures
- Market based: Level of volatility is calculated from options prices
- Forward looking: Options depend on future price path
- “Classic” implied relies on the Black-Scholes pricing formula
- “Model free” implied volatility exploits a relationship between the second derivative of the call price with respect to the strike and the risk neutral measure
- VIX is a Chicago Board Options Exchange (CBOE) index based on a model free measure
- Allows volatility to be directly traded

- Prices follow a geometric Brownian Motion

- Constant drift and volatility
- Price of a call is

- Invert to produce a formula for the volatility given the call price $C(T,K)$

In [2]:

```
bsiv()
```

- VIX is continuously computed by the CBOE
- Uses both out-of-the-monry calls and puts

- $Q(K_{i})$ is the mid-quote for a strike of $K_{i}$, $K_{0}$ is the first strike below the forward index level
- VIX appears to have information about future realized volatility that is not in other backward looking measures (GARCH/RV)
- Computes area under curves defined by OOM options

In [4]:

```
plot_20()
```

In [6]:

```
plot_60()
```

- Defining VaR
- Changed in the future return distribution and VaR
- Volatility-based VaR models
- CaViaR
- Weighted Historical Simulation
- VaR model evaluation

The $\alpha$ percentage Value-at-Risk (%VaR) of a portfolio is defined as the largest return such that the probability that the return on the portfolio over some period of time is less than -%VaR is $\alpha$

$$ \mathrm{Pr}(r<-\text{%}\textrm{VaR})=\alpha$$where $r$ is the percentage return on the portfolio.

- I will use VaR interchangeably with %VaR
- Usually interested in
*conditional*VaR

- VaR has an inverse relationship to the mean of a portfolio
- The mean always enters additively

- Higher mean reduces VaR

In [9]:

```
plot_mean_shift()
```