15.1 Valuing Options with ARCH-Models

Consider an economy in discrete time in which interest and proceeds are paid out at the end of every constant, equally long time interval. Let $ S_t, t=0,1,2,\ldots$ be the price of the stock at time $ t$ and $ Y_t = (S_t-S_{t-1})/S_{t-1}$ the corresponding one period return without dividends. Assume that a price for risk exists in the form of a risk premium which is added to the risk free interest rate $ r$ to obtain the expected return of the next period. It seems reasonable to model the risk premium dependent on the conditional variance. As a basis we assume an ARCH-M-Model (see Section 12.2.3) with a risk premium, which is a linear function of the conditional standard deviation:

$\displaystyle Y_t$ $\displaystyle =$ $\displaystyle r + \lambda \sigma_t + \varepsilon_t$ (15.1)
$\displaystyle {\cal L} ( \varepsilon_t \mid {\cal F}_{t-1})$ $\displaystyle =$ N$\displaystyle (0, \sigma_t^2)$ (15.2)
$\displaystyle \sigma_t^2$ $\displaystyle =$ $\displaystyle \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2.$ (15.3)

In (14.3) $ \omega$, $ \alpha$ and $ \beta$ are constant parameters that satisfy the stationarity and non-negativity conditions. The constant parameter $ \lambda$ can be understood as the price of one unit of risk. $ {\cal F}_{t}$ indicates, as usual, the set of information available up to and including time $ t$. In order to simplify the notation our discuss will be limited to the GARCH(1,1) case.

The above model is estimated under the empirical measure $ P$. In order to deal with a valuation under no arbitrage, similar to Black-Scholes in continuous time (see Section 6.1), assumptions on the valuation of risk must be met. Many studies have researched option pricing with stochastic volatility under the assumption that the volatility has a systematic risk of zero, that is, the risk premium for volatility is zero. Duan (1995) has identified an equivalent martingale measure $ Q$ for $ P$ under the assumption that the conditional distribution of the returns are normal and in addition it holds that

$\displaystyle {\mathop{\text{\rm Var}}}^P(Y_t\mid {\cal F}_{t-1}) = {\mathop{\text{\rm Var}}}^Q(Y_t\mid {\cal F}_{t-1})$ (15.4)

$ P$ a.s.. He shows that under this assumption a representative agent with, for example, constant relative risk aversion and a normally distributed relative change of aggregate consumption maximizes his expected utility. The assumption (14.4) contains a constant risk premium for the volatility that directly enters its mean.

In order to obtain a martingale under the new measure a new error term, $ \eta_t$, needs to be introduced that captures the effect of the time varying risk premium. When we define $ \eta_t =
\varepsilon_t + \lambda \sigma_t$, (14.4) leads to the following model under the new measure $ Q$:

$\displaystyle Y_t$ $\displaystyle =$ $\displaystyle r + \eta_t$ (15.5)
$\displaystyle {\cal L} ( \eta_t \mid {\cal F}_{t-1})$ $\displaystyle =$ N$\displaystyle (0, \sigma_t^2)$ (15.6)
$\displaystyle \sigma_t^2$ $\displaystyle =$ $\displaystyle \omega + \alpha(\eta_{t-1} - \lambda \sigma_{t-1})^2 + \beta
\sigma_{t-1}^2.$ (15.7)

In the case of a GARCH(1,1) model according to Theorem 12.10 the variance of the stationary distribution under the empirical measure $ P$ is $ {\mathop{\text{\rm Var}}}^P(\varepsilon_t) =
\omega/(1-\alpha-\beta)$. For the Duan measure $ Q$ the variance of the stationary distribution increases to $ {\mathop{\text{\rm Var}}}^Q(\eta_t) =
\omega/\{1-\alpha(1+\lambda^2)-\beta\}$, because the volatility process under the new measure is determined by the innovations from an asymmetric and not a symmetric Chi squared distribution. Later on we will see that changes in the unconditional variance depend in a critical way on the specification of the news impact curve.

The restriction to a quadratic or symmetric news impact curve is not always optimal, as many empirical studies of stock returns have indicated. Within the framework of the above mentioned model these assumptions can lead to a non-linear news impact function $ g(\cdot)$. The following model is a semi-parametric analogue to the GARCH model. Under the empirical measure $ P$ we obtain

$\displaystyle Y_t$ $\displaystyle =$ $\displaystyle r + \lambda \sigma_t + \varepsilon_t$  
$\displaystyle {\cal L}_P (\varepsilon_t \mid {\cal F}_{t-1})$ $\displaystyle =$ N$\displaystyle (0, \sigma_t^2)$  
$\displaystyle \sigma_t^2$ $\displaystyle =$ $\displaystyle g(\varepsilon_{t-1}) + \beta \sigma_{t-1}^2.$  

Under the Duan martingale measure $ Q$ the model changes to
$\displaystyle Y_t$ $\displaystyle =$ $\displaystyle r + \eta_t$  
$\displaystyle {\cal L}_Q (\eta_t \mid {\cal F}_{t-1})$ $\displaystyle =$ N$\displaystyle (0, \sigma_t^2)$  
$\displaystyle \sigma_t^2$ $\displaystyle =$ $\displaystyle g(\eta_{t-1} - \lambda \sigma_{t-1}) + \beta
\sigma_{t-1}^2.$  

One notices that as soon as an estimator of $ g(\cdot)$ under $ P$ is known it can immediately be substituted under the measure $ Q$.

In this general specification the estimation without additional information on $ g(\cdot)$ is a difficult matter, since iterative estimation procedures would be necessary in order to estimate the parameters $ \lambda, \beta$ and the non-parametric function $ g$ at the same time. Therefore we will consider a specific, flexible parametric model: the Threshold GARCH Model, see Section 12.2. With this model the news impact function can be written as:

$\displaystyle g(x) = \omega + \alpha_1 x^2 \boldsymbol{1}(x<0) + \alpha_2 x^2 \boldsymbol{1}(x\ge 0)$

To motivate this model consider fitting a very simple non-parametric model $ Y_t = \sigma (Y_{t-1})\xi _t$ to the returns of a German stock index, the DAX, where $ \xi_t$ is independent and identically distributed with mean 0 and variance $ 1$. The estimator of the news impact curve $ \sigma^2(\cdot)$ is given in Figure 14.2. To get an idea of the underlying distribution of the returns a non-parametric estimator of the return distribution has been added in Figure 14.1 over a smoothed normal distribution. Obviously $ g(\cdot)$ is not symmetric around zero. The TGARCH model captures this phenomenon when $ \alpha_1 > \alpha_2$. Other parametric models can describe these properties as well but the TGARCH model in the case of stock returns has proven to be extremely flexible and technically manageable as claimed, for example, in Rabemananjara and Zakoian (1993).

Fig. 14.1: Kernel estimation of the density of DAX returns (solid line) against a kernel estimation of a normal distribution (dotted line) with the same mean and variance. A bandwidth of 0.03 is used and a quadratic kernel function $ K(u)=15/16(1-u^2)^2\boldsymbol {1}(\vert u\vert<1)$. The tails have been eliminated from the figure.

Fig. 14.2: Local linear estimation of the news impact curve for the DAX. The model is $ Y_t = \sigma (Y_{t-1})\xi _t$. Shown is the estimator of the function $ \sigma ^2(y)$ with a bandwidth of 0.03. The tails have been eliminated from the figure.

Remember that the innovations are normally distributed. Thus it follows for the TGARCH model that the unconditional variance, similar to Theorem 12.10, under the measure $ P$ is $ {\mathop{\text{\rm Var}}}^P(\varepsilon_t) = \omega/(1-\bar{\alpha}-\beta)$, where $ \bar{\alpha} = (\alpha_1 + \alpha_2)/2$. The following theorem gives the unconditional variance for $ \eta_t =
\varepsilon_t + \lambda \sigma_t$ under $ Q$.

Theorem 15.1   The unconditional variance of the TGARCH(1,1) model under the equivalent martingale measure $ Q$ from Duan is

$\displaystyle {\mathop{\text{\rm Var}}}^Q(\eta_t) = \frac{\omega}{1-\psi(\lambda)(\alpha_1-\alpha_2) - \alpha_2(1+\lambda^2) - \beta}$ (15.8)

where

$\displaystyle \psi(u) = u \varphi(u) + (1+u^2)\Phi(u)$

and $ \varphi(u),
\Phi(u)$ are the density and the distribution function of the standard normal distribution.

Proof:
Let $ Z_t = \eta_t/\sigma_t - \lambda$. Under measure $ Q$ it holds that $ {\cal L}(Z_t \mid {\cal F}_{t-1}) =$   N$ (-\lambda,1)$. The conditional variance $ \sigma_t^2$ can be written as

$\displaystyle \sigma_t^2 = \omega + \alpha_1 \sigma_{t-1}^2 Z_{t-1}^2
\boldsymb...
...\sigma_{t-1}^2 Z_{t-1}^2 \boldsymbol{1}(Z_{t-1} \ge 0)
+ \beta \sigma_{t-1}^2.
$

By calculating the expected value it can be shown that for the integral over the negative values it follows that:

$\displaystyle {\mathop{\text{\rm\sf E}}}^Q[Z_{t}^2 \boldsymbol{1}(Z_t<0)\mid {\cal F}_{t-1}]$ $\displaystyle =$ $\displaystyle \frac{1}{\sqrt{2\pi}} \int_{-\infty}^0 z^2 e^{-\frac{1}{2}
(z+\lambda)^2} dz$  
  $\displaystyle =$ $\displaystyle \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\lambda (u-\lambda)^2 e^{-\frac{1}{2}
u^2} du$  
  $\displaystyle =$ $\displaystyle \frac{\lambda}{\sqrt{2\pi}}e^{-\frac{1}{2}\lambda^2} +
(1+\lambda^2)\Phi(\lambda)$  
  $\displaystyle \stackrel{\mathrm{def}}{=}$ $\displaystyle \psi(\lambda).$ (15.9)

Because of

$\displaystyle {\mathop{\text{\rm\sf E}}}^Q[Z_{t}^2\mid {\cal F}_{t-1}] = \frac{...
...
\int_{-\infty}^{\infty} z^2 e^{-\frac{1}{2}
(z+\lambda)^2} dz = 1 + \lambda^2
$

it follows for the positive values that

$\displaystyle {\mathop{\text{\rm\sf E}}}^Q[Z_{t}^2 \boldsymbol{1}(Z_t \ge 0)\mid {\cal F}_{t-1}] = 1 + \lambda^2 - \psi(\lambda).$ (15.10)

Thus we obtain

$\displaystyle {\mathop{\text{\rm\sf E}}}^Q[\sigma_t^2] = \omega + \alpha_1 \psi...
...sf E}}}^Q[\sigma_{t-1}^2] + \beta {\mathop{\text{\rm\sf E}}}^Q[\sigma_{t-1}^2].$ (15.11)

Since the unconditional variance is independent of $ t$, the theorem follows. $ {\Box}$

The function $ \psi$ is positive and $ \psi(\lambda)> 1/2$ for the realistic case $ \lambda>0$. We can make the following statement about the changes in the unconditional variance: for $ \alpha_1 =
\alpha_2$ in (14.8), one obtains the GARCH(1,1) results. For $ \alpha_1 > \alpha_2$ (the case of the leverage effect) the increase in the unconditional variance is even stronger than the symmetric GARCH case. For $ \alpha_1 < \alpha_2$, the unconditional variance is smaller as in the leverage case, and we can distinguish between two cases: when the inequality

$\displaystyle \alpha_1 < \alpha_2 \frac{2\psi(\lambda)-1-2\lambda^2}{2\psi(\lambda)-1}$ (15.12)

is fulfilled then the unconditional variance under $ Q$ is actually smaller than under $ P$. If (14.12) is not fulfilled, then we obtain as above $ {\mathop{\text{\rm Var}}}^P(\varepsilon_t) \le {\mathop{\text{\rm Var}}}^Q(\eta_t)$. Indeed the quotient on the right hand side of (14.12) takes on negative values for realistic values of a unit of the risk premium, (for example for small positive values), so that in most empirical studies (14.12) can not be fulfilled.

Naturally the stationary variance has an effect on an option's price: the larger (smaller) the variance is, the higher (lower) is the option price. This holds in particular for options with longer time to maturity where the long-run average of the volatility is the most important determinant of the option's price. Therefore, an option can be undervalued when a GARCH model is used and at the same time a leverage effect is present.

A second feature of the Duan approach is that under $ Q$ and with positive risk premia the current innovation is negatively correlated with the next period's conditional variance of the GARCH risk premium, whereas under $ P$ the correlation is zero. More precisely, we obtain $ {\mathop{\text{\rm Cov}}}^Q(\eta_t/\sigma_t,
\sigma_{t+1}^2) = -2 \lambda \alpha {\mathop{\text{\rm Var}}}^Q(\eta_t)$ with the GARCH parameter $ \alpha$. It is obvious that small forecasts of the volatility under $ Q$ (that influences the option's price) depend not only on the past squared innovations, but also on their sign. In particular a negative (positive) past innovation for $ \lambda>0$ leads to the fact that the volatility increases (falls) and with it, the option price. The following theorem claims that the covariance is dependent on the asymmetry of the news impact function when a TGARCH instead of a GARCH model is used.

Theorem 15.2   For the TGARCH(1,1) model the covariance between the innovation in $ t$ and the conditional variance in $ t+1$ under the equivalent martingale measure $ Q$ from Duan is given by

Cov$\displaystyle ^Q(\frac{\eta_t}{\sigma_t}, \sigma_{t+1}^2) = -2 {\mathop{\text{\...
...a \alpha_2 + \{ \varphi(\lambda) + \lambda \Phi(\lambda)\}(\alpha_1-\alpha_2)],$ (15.13)

where $ {\mathop{\text{\rm Var}}}^Q(\eta_t)$ follows from the previous theorem.

Proof:
First the conditional covariance is deterimined:

$\displaystyle {\mathop{\text{\rm Cov}}}^Q_{t-1}(\frac{\eta_t}{\sigma_t},\sigma_{t+1}^2)$ $\displaystyle =$ $\displaystyle {\mathop{\text{\rm\sf E}}}^Q_{t-1}\left[\frac{\eta_t}{\sigma_t}\s...
...
\omega {\mathop{\text{\rm\sf E}}}^Q_{t-1}\left[\frac{\eta_t}{\sigma_t} \right]$  
  $\displaystyle +$ $\displaystyle \alpha_1
{\mathop{\text{\rm\sf E}}}^Q_{t-1}\left[\frac{\eta_t}{\sigma_t}(\eta_t-\lambda\sigma_t)^2
\boldsymbol{1}(\eta_t-\lambda\sigma_t<0)\right]$  
  $\displaystyle +$ $\displaystyle \alpha_2
{\mathop{\text{\rm\sf E}}}^Q_{t-1}\left[\frac{\eta_t}{\s...
...}(\eta_t-\lambda\sigma_t)^2
\boldsymbol{1}(\eta_t-\lambda\sigma_t \ge 0)\right]$  
  $\displaystyle +$ $\displaystyle \beta \sigma_t {\mathop{\text{\rm\sf E}}}^Q_{t-1}\left[ \eta_t \right] ,$ (15.14)

where $ {\mathop{\text{\rm\sf E}}}_t(\cdot)$ and $ {\mathop{\text{\rm Cov}}}_t(\cdot)$ are abbreviations of $ {\mathop{\text{\rm\sf E}}}(\cdot\mid {\cal F}_t)$ and $ {\mathop{\text{\rm Cov}}}(\cdot \mid {\cal F}_t)$ respectively. Due to (14.6) the first and the fourth expectation values on the right side of (14.14) are zero. The second conditional expected value is
    $\displaystyle {\mathop{\text{\rm\sf E}}}^Q_{t-1}\left[\frac{\eta_t}{\sigma_t}(\eta_t-\lambda\sigma_t)^2
\boldsymbol{1}(\eta_t-\lambda\sigma_t<0)\right]$  
$\displaystyle =$   $\displaystyle -2\sigma_t^2
\left[\frac{1}{\sqrt{2\pi}}\exp(-\frac{1}{2}\lambda^2) + \lambda
\Phi(\lambda)\right].$ (15.15)

Since $ {\mathop{\text{\rm\sf E}}}^Q_{t-1}\left[\frac{\eta_t}{\sigma_t}(\eta_t-\lambda\sigma_t)^2
\right] = -2\lambda\sigma_t^2$, we can write for the third conditional expected value in (14.14):
    $\displaystyle {\mathop{\text{\rm\sf E}}}^Q_{t-1}\left[\frac{\eta_t}{\sigma_t}(\eta_t-\lambda\sigma_t)^2
\boldsymbol{1}(\eta_t-\lambda\sigma_t \ge 0)\right]$  
$\displaystyle =$   $\displaystyle -2\sigma_t^2 \left[\lambda -
\frac{1}{\sqrt{2\pi}}\exp(-\frac{1}{2}\lambda^2) - \lambda
\Phi(\lambda)\right].$ (15.16)

Inserting (14.15) and (14.16) into (14.14), it follows that

$\displaystyle {\mathop{\text{\rm Cov}}}^Q_{t-1}(\frac{\eta_t}{\sigma_t},\sigma_...
...alpha_2 + \{ \varphi(\lambda) + \lambda \Phi(\lambda) \} (\alpha_1-\alpha_2) ].$ (15.17)

One notices that $ {\mathop{\text{\rm Cov}}}^Q(\eta_t/\sigma_t,\sigma_{t+1}^2) =
{\mathop{\text{\rm\sf E}}}^Q[{\mathop{\text{\rm Cov}}}^Q_{t-1}(\eta_t/\sigma_t,\sigma_{t+1}^2)]$, thus the claim follows immediately. $ {\Box}$

In the following we assume that a positive risk premium $ \lambda$ exists per unit. Three cases can be identified: for $ \alpha_1 =
\alpha_2$ (in the symmetric case) we obtain $ {\mathop{\text{\rm Cov}}}^Q(\eta_t/\sigma_t, \sigma_{t+1}^2) = -2 \lambda \alpha_2
{\mathop{\text{\rm Var}}}^Q(\eta_t)$, i.e., the GARCH(1,1) result. For $ \alpha_1 < \alpha_2$ (the case of the reverse leverage effect) the covariance increases, and when

$\displaystyle \lambda \alpha_2 + \left[\frac{1}{\sqrt{2\pi}}\exp(-\frac {1}{2}\lambda^2) + \lambda \Phi(\lambda)\right](\alpha_1-\alpha_2) < 0,$ (15.18)

the correlation is positive. In the last case, $ \alpha_1 > \alpha_2$ (the leverage case), the covariance is negative and increases with the total.

This also shows that the return of the volatility to a stationary variance under $ Q$ is different from the symmetric GARCH case. The negative covariance in the leverage case is actually larger. This could indicate that options are over (under) valued when for positive (negative) past innovation a TGARCH process with $ \alpha_1 > \alpha_2$ is used for the price process and then mistakenly a GARCH model ( $ \alpha_1 =
\alpha_2$) is used for the volatility forecast.