Vega

We have already highlighted the dependence of derivative contracts on the volatility of the underlying asset. The BS methodology makes the assumption that the volatility is constant across time, that is why [so far] we have ignored the effect of volatility changes on the value of a portfolio. If we assume the $\Pi =\Pi \left( S,t,\sigma \right)$, then the Taylor's expansion in the beginning of the lecture would become

$$\begin{multline*} \Delta \Pi =\frac{\partial \Pi }{\partial t}\Delta t+\frac{\pa... ...igma }%% \Delta S\Delta \sigma +o\left( \Delta t\right) \text{.} \end{multline*}$$

We therefore define as the vega of a portfolio the sensitivity of its value with respect to volatility changes. We implicitly argue at this point that the volatility of the volatility is small, therefore ignoring the $%% \frac{\partial ^{2}\Pi }{\partial \sigma ^{2}}\left( \Delta \sigma \right) ^{2}$ term. In addition we assume that the asset price and its volatility are uncorrelated, therefore ignoring the $\frac{\partial ^{2}\Pi }{\partial S\partial \sigma }\Delta S\Delta \sigma$ term. Formally

$$\mathcal{V}_{\Pi }=\frac{\partial \Pi }{\partial \sigma }$$.

The underlying asset's price and a forward contract on this asset do not depend explicitly on the volatility, rendering $\mathcal{V}_{S}=\mathcal{V}%% _{F}=0$. Therefore one has to rely on nonlinear contracts [such as options] to make a portfolio vega neutral. If one wants to achieve a gamma-vega hedge, she will have to use two different derivative securities. This is due to the fact that, generally speaking, for a given derivative the values of gamma and vega will be different.

Say that we hold a portfolio with value $\Pi _{1}$ which has a given delta, $%% \Delta _{\Pi _{1}}$, gamma, $\Gamma _{\Pi _{1}}$ and vega $\mathcal{V}_{\Pi _{1}}$. Our goal is to find the position we have to take in two different derivatives of the underlying asset, in order to achieve vega-gamma neutrality. Say that we use two options with prices $C_{1}$ and $C_{2}$, with known deltas, $\Delta _{C_{1}}$ and $\Delta _{C_{2}}$, gammas, $\Gamma _{C_{1}}$ and $\Gamma _{C_{2}}$, and vegas, $\mathcal{V}_{C_{1}}$ and $%% \mathcal{V}_{C_{2}}$. We also use the underlying asset, which has price $S$ to achieve delta neutrality [of course $\Delta _{S}=1$ and $\Gamma _{S}=%% \mathcal{V}_{S}=0$]. The resulting portfolio will be delta-gamma-vega neutral.

Gamma neutrality

We want to buy $w_{C_{1}}$ and $w_{C_{2}}$ units of the two derivative securities to achieve gamma neutrality. The value of the composite portfolio is

$$\Pi _{2}=\Pi _{1}+w_{C_{1}}C_{1}+w_{C_{2}}C_{2}$$.

The gamma of the portfolio will therefore be

$$\Gamma _{\Pi _{2}}=\Gamma _{\Pi _{1}}+w_{C_{1}}\Gamma _{C_{1}}+w_{C_{2}}\Gamma _{C_{2}}$$,

and we demand $\Gamma _{\Pi _{2}}=0$.

Vega neutrality

We also want the above portfolio $\Pi _{2}$ to be vega neutral. The vega of the portfolio will be

$$\mathcal{V}_{\Pi _{2}}=\mathcal{V}_{\Pi _{1}}+w_{C_{1}}\mathcal{V}%% _{C_{1}}+w_{C_{2}}\mathcal{V}_{C_{2}}$$,

and we demand $\mathcal{V}_{\Pi _{2}}=0$. Solving the above system yields

$$\left\{ \begin{array}{c} w_{C_{1}}\Gamma _{C_{1}}+w_{C_{2}}\... ...a _{C_{2}}%% \mathcal{V}_{C_{1}}} \end{array}\right\} \text{.}$$

Delta neutrality

Now we want to take a position in the underlying asset, in order to achieve delta neutrality. Say that we buy $w_{S}$ shares. The composite portfolio will have value

$$\Pi _{3} = \Pi _{2}+w_{S}S$$

$$= \Pi _{1}+w_{C_{1}}C_{1}+w_{C_{2}}C_{2}+w_{S}S$$

The delta of this portfolio is therefore

$$\Delta _{\Pi _{3}}=\Delta _{\Pi _{1}}+w_{C_{1}}\Delta _{C_{1}}+w_{C_{2}}\Delta _{C_{2}}+w_{S}$$.

To achieve delta neutrality, $\Delta _{\Pi _{3}}=0$, one has to buy

$$w_{S} = -\Delta _{\Pi _{1}}-w_{C_{1}}\Delta _{C_{1}}-w_{C_{2}}\Delta _{C_{2}}$$

$$= -\Delta _{\Pi _{1}}+\frac{\Gamma _{\Pi _{1}}\mathcal{V}_{C_{2}}-\... ...athcal{V}%% _{C_{2}}-\Gamma _{C_{2}}\mathcal{V}_{C_{1}}}\Delta _{C_{2}}\text{.}$$

One can easily verify that $\Delta _{\Pi _{3}}=\Gamma _{\Pi _{3}}=\mathcal{V}%% _{\Pi _{3}}=0$.

For a European call option the BS value of vega is given by

$$\mathcal{V}_{C}=S\sqrt{T}\mathcal{N}^{\prime }\left( d_{1}\right)$$   .

Graphically vega is given in figure 8.5

Figure 8.5: Behavior of a Call option Vega. Part (a) gives the behavior of the vega on a three options with specifications $(K,\tau,r,\sigma)$ equal to $(1,0.2,0,0.5)$ for the solid line, $(1,0.2,0,1)$ for the thin line, and $(1,0.5,0,0.5)$ for the dashed line. Part (b) gives the behavior of the vega for a contract which is in-the-money [solid line], at-the-money [thin line], and out-of-the-money [dashed line].

Calculating vega from a model that considers constant volatility is not correct. On the other hand it can be shown that it offers a good approximation of the vega computed using models that assume the volatility to be stochastic, especially for short maturities. One can just remember the fact that implied BS volatilities of short maturity at-the-money options serve as an estimate of the expectation of the average volatility during the [short] life of the option. In this context vega can be used in practice.