Implied Volatilty and Value-at-Risk Modeling

Kevin Sheppard

Today

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

Implied Variance

• 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

Black-Scholes Implied Volatility

• Prices follow a geometric Brownian Motion

$$\text{d}{S}_{t}=\mu S_{t}\text{d}t+\sigma S_{t}\text{d}W_{t} $$

• Constant drift and volatility
• Price of a call is

$$C(T,K)=S\Phi(d_{1})+Ke^{-rT}\Phi(d_{2})$$$$d_{1}=\frac{\ln\left(S/K\right)+\left(r+\sigma^{2}/2\right)T}{\sigma\sqrt{T}},\,\,d_{2}=\frac{\ln\left(S/K\right)+\left(r-\sigma^{2}/2\right)T}{\sigma\sqrt{T}} $$

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

$$\sigma_{t}^{Implied}=g\left(C_{t}(T,K),S_{t},K,T,r\right)$$

Black-Scholes Smile

bsiv()

Model-free Implied Variance

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

$$\sigma^{2}=\frac{2}{T}e^{rT}\sum_{i=1}^{N}\frac{\Delta K_{i}}{K_{i}}\frac{Q(K_{i})}{K_i}-\frac{1}{T}\left(\frac{F_{0}}{K_{0}}-1\right)^{2} $$

• $Q(K_{i})$ 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

20% Annualized Volatility

plot_20()

60% Annualized Volatility

plot_60()

Value-at-Risk

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

Percentage Value-at-Risk

Defined

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

$$ \mathrm{Pr}(r_{t+1}<-\text{%}\textrm{VaR}_{t+1}|\mathcal{F}_{t})=\alpha$$

The Effect of Changes in Mean

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

$$ \text{VAR} = -\mu + \text{other terms} $$

• Higher mean reduces VaR

Mean Shifts

plot_mean_shift()