20.4 Measure of the Implied Volatility's Risk

The market value $ {P_t}$ of a portfolio consisting of $ w$ different options is dependent on changes of the risk free interest rate $ r_t$, the prices $ S_t$ of the financial underlying, the time to maturity $ \tau$ and the individual implied volatilities $ \sigma_I$. Changes in the portfolios value can be analytically approximated using the following Taylor approximation, where it is assumed that the options are all based on the same underlying.

$\displaystyle \Delta {P_t}$ $\displaystyle =$ $\displaystyle \sum^w_{u=1} \Big \{ \frac{\partial V_{ut}}
{\partial \sigma_I}
\...
...tial V_{ut}}{\partial t}\Delta t
+ \frac{\partial V_{ut}}{\partial r}\Delta r_t$  
    $\displaystyle \qquad + \frac{\partial V_{ut}}{\partial S}\Delta S_t
+ \frac{1}{2} \frac{\partial^2 V_{ut}}{\partial S^2}
(\Delta S_t)^2 \Big \}$  
$\displaystyle \Delta {P_t}$ $\displaystyle =$ $\displaystyle \sum^w_{u=1}
\left \{\frac{\partial V_{ut}}{\partial \sigma_I}
\D...
... \Delta r_t + \Delta_u \Delta S_t
+\frac{1}{2}\Gamma_u (\Delta S_t)^2 \right \}$  

Here $ V_{ut}$ describes the price of the $ u$-th option with a time to maturity $ \tau_u$ at date $ t$, and $ \Theta_u, \rho_u,
\Delta_u, \Gamma_u$ are the characteristic values described in Section 6.3 of the $ u$-th option. In practice option traders often insert ``Vega'' positions directly. In doing so they create portfolios whose profit and loss profile can be determined by the changes in the implied volatilities of the respective options, see Taleb (1997). Portfolios of this kind are called $ (\Delta,
\Gamma)$ and $ \Theta$ neutral. The sensitivity of the option price under consideration to the changes in the volatilities is measured by the variable $ {\cal V}$ (``Vega'' - see (6.29)).

A well known strategy in utilizing the forecasted changes in the maturity structure of implied volatilities consists of buying and selling so called ``Straddles'' with varying maturities. A straddle is constructed by simultaneously buying (``Long Straddle'') or selling (``Short Straddle'') the same number of ATM Call and Put options with the same time to maturity. If a trader expects a relatively strong increase in the implied volatility in the short maturities and a relatively weaker increase in the longer maturities, then he will buy straddles with a short time to maturity and sell longer maturity straddles at a suitable ratio. The resulting option portfolio is ( $ \Delta, \Gamma$) neutral and over a short time frame $ \Theta$ neutral, i.e., it is insensitive with respect to losing value over time. The Taylor series given above can thus be reduced to:

$\displaystyle \Delta {P_t} \approx \sum^{w}_{u=1}
\left \{\frac{\partial V_{ut}}{\partial \sigma_I}
\Delta \sigma^{(t)}_{I,t}\right \}$     (20.6)

The first differences of the implied volatilities can now be given as linear combinations of the principal components. By substituting the volatility indices $ \sigma_{I,t}$, which are temporally next to the actual implied volatility $ \hat{\sigma}_I(\tau^*_u)$, one obtains the following representation given (19.3):
$\displaystyle \Delta {P_t} \approx \sum^{w}_{u=1}\left \{\frac{\partial V_{ut}}
{\partial \sigma_I} \left(\sum^2_{l=1}b_{jl}y_{lt}\right)\right \}$     (20.7)

The number of principal components used in the previous expression can be reduced to the first two without any significant loss of information.

The following ``Maximum Loss'' (ML) concept describes the probability distribution of a short-term change in the portfolios value dependent on changes in the value of the underlying factors. The change in value of a $ (\Delta,
\Gamma)$ neutral option portfolio is substantially determined by the changes in the implied volatilities of the options contained in the portfolio. To determine the ``Maximum Loss'' it is necessary to have an adequate exact representation of the future distribution of the changes to the volatility of the options with varying time to maturity.

The ``Maximum Loss'' is defined as the largest possible loss of a portfolio that can occur over a specific factor space $ A_t$ and over a specific holding period $ \tau$. The factor space $ A_t$ is determined by a closed set with $ {\P(A_t)} = \alpha$. Here $ \alpha$ is set to 99% or 99.9%. The ML definition resembles at first sight the ``Value-at-Risk'' Definition (see Chapter 15). There exists, however, an important difference between the two concepts: In calculating the ``Value-at-Risk'' the distribution of the returns of the given portfolio must be known, whereas the ML is defined directly over the factor space and thus has an additional degree of freedom, see Studer (1995).

In our analysis we have divided the maturity structure of the implied volatilities into two principal components, which explain a considerable portion of the variability of the structure curve. Thus the first two principal components represent the risk factors used in the ML model. The profit and loss profile of each portfolio held is determined by the corresponding changes in the risk factors using a suitable valuation model. In order to obtain this, a valuation of the underlying portfolios must theoretically occur for every point in the factor space. In the practical application the factor space is probed over a sufficiently small grid of discrete data points $ y^{z}_1
(z=1,...,N_1)$, during which the other risk factor is in each case held constant. Due to the orthogonality properties of the principal components, the profit and loss function $ PL()$ is additive with $ PL(y^{z_1}_1, y^{z_2}_2) = PL(y^{z_1}_1)+PL(y^{z_2}_2)$.

Under the assumption of multivariate, normally distributed principal components confidence intervals can be constructed for the ``Maximum Loss'' over the total density


$\displaystyle \varphi_2(y)=\frac{1}{(2\pi)\sqrt{\mathop{\rm {det}}\Lambda_2}}\exp(-\frac{1}{2}y^\top \Lambda^{-1}_2y),$     (20.8)

with $ y=(y_1,y_2)^\top $. Here the matrix $ \Lambda_2$ represents the $ 2 \times 2$ diagonal matrix of the eigenvalues $ \lambda_k$, $ k$ = 1,2. The random variable $ y^\top \Lambda^{-1}_2y = X^2_1 +
X^2_2$ has a Chi-square distribution. The confidence interval for an existing portfolio is then $ A_t = \{y ; y^\top \Lambda^{-1}_2y
\leq c_{\alpha}\}$, $ c_{\alpha}$, where $ c_{\alpha}$ is the $ \alpha$ quantile of a random variable with a Chi-square distribution and 2 degrees of freedom.