Gram Charlier Method

Gram-Charlier Introduction

The purpose of the Gram-Charlier Model is to incorporate both the skewness and kurtosis of the underlying share into the option valuation

Developed by Backus, Foresi and Wu (2004)

The reason for doing this is that is we look at historic share prices we can see that the level of skew and kurtosis they exhibit is not consistent with that predicted by the lognormal model

This spreadsheet shows this feature with FTSE100 data

Key idea is to use the third and fourth moments of the cumulant generating function to include skew and kurtosis and then invert the cumulant generating functions to produce a probability density function which better approximates actual share returns and then integrate this up as in the Black Scholes model to produce a closed form valuation method for options

The shortcoming of this model is that it does not allow for variable volatility

The Gram Charlier Model

Setup of Maths

Let $S_t$ be the price of the underlying security at time $t$.

The one-period logarithmic return $Y_t$ is given by:

$$Y_t = ln S_t - ln S_{t-1}$$

The return over T periods is

$$Y^{T}_{t+1}=ln S_{t+T} - ln S_t = \displaystyle \sum_{k=1}^T Y_{t+k}$$

Let $K (\theta )$ be the cumulant generating function of $Y_{t+1}$. Then

$$K (\theta ) = \displaystyle \sum _{i=1}^{\infty} \kappa_i \frac{\theta ^i}{i !}$$

where $\kappa_i$ is the $i^{th}$-order cumulant of $Y_{t+1}$

Suppose that the one-period returns are independent and identically distributed. Then the cumulants of the sum $Y^T_{t+1}$ are $T\times\kappa_i$, for $i=1,2,...,$ In particular, the mean and variance of $Y^T_{t+1}$ are respectively $\mu_T=T \mu$, $\sigma^2_T=T \sigma^2$

It turns out that $\kappa_3$ and $\kappa_4$ etc give us the higher moments albeit indirectly:

Skew: $\gamma_1=E \left[ \left( \frac{X- \mu}{\sigma}\right)^3\right] = \frac{\kappa_3}{\kappa_2^{1.5}}$

Kurtosis: $\gamma_2=E \left[ \left( \frac{X- \mu}{\sigma}\right)^4\right] = \frac{\kappa_4}{\kappa_2^{2}}$

The aside above shows us that doubling the time period doubles all of the cumulants so it is easy to see that:

$\gamma^T_1=\frac{\gamma_1}{\sqrt{T}}$, $\gamma^T_2=\frac{\gamma_2}{T}$

Let $Z_T=\frac{Y^T_{t+1}-\mu_T}{\sigma_T}$, be the standardized T-period return of the underlying security. (This just keeps things a little more simple)

Using the above results and the first four terms on the Gram-Charlier expansion, Backus, Foresi and Wu (2004) gives the following probability density function for $Z_T$:

$$f(Z_T)=\phi(Z_T) \left(1- \frac{\gamma^T_1}{3!} \phi^{(3)} (Z_T) + \frac{\gamma^T_2}{4!} \phi^{(4)} (Z_T) \right)$$

where $\phi(x)$ is the probability density function of the standard normal distribution and $\phi^{(i)}(x)$ is the $i^{th}$ order derivative of $\phi(x)$

By direct differentiation,

$$\phi^{(3)}(x) = (3x-x^3) \phi(x)$$

$$\phi^{(4)}(x) = (x^4-6x^2+3) \phi(x)$$

The time-t price of a European call option maturing at time T is:

$$C(t)=e^{-rT}E \left[ max(S_{t+T} - K, 0)|F_t \right]$$

$$ =e^{-rT} \displaystyle\int_{ln(K/S_t)}^{\infty} (S_t e^x - K)g(x) dx$$

where $r$ is the risk free rate of interest, $K$ is the strike price of the call and $g(x)$ is the probability density function of $Y_{t+1}^T$ under a risk neutral probability measure

We now simply substitute in for $g(x)$ the probability density function we have developed above:

$$C(t)=e^{-rT}\displaystyle\int_{Z^*}^{\infty} (S_t e^{r_T + \sigma_T Z_T} - K)\phi(Z_T) \left(1- \frac{\gamma^T_1}{3!} \phi^{(3)} (Z_T) + \frac{\gamma^T_2}{4!} \phi^{(4)} (Z_T) \right) dZ_T$$

where $r_T=rT$ and $Z^*=\frac{ln(K/S_t)-r_T}{\sigma_T}$

'All' that remains is to crunch through the great big integration

backus, Foresti and Wu obtained the following approximation to the call price:

$$C(t) \approx S_t \Theta(d_1) - K e^{-rT} \Theta(d_2) + S_t \phi(d_1) \sigma_T \left(\frac{\gamma_1^T}{3!} (2 \sigma_T-d_1) - \frac{\gamma^T_2}{4!}(1-d^2_1+3d_1 \sigma_T - 3 \sigma^2_T)\right)$$

where

$$d_1 = \frac{ln(S_t/K) + (r+\sigma^2/2)T}{\sigma_T}$$

$$d_2 = d_1 - \sigma_T$$

As you can see this reduces to the Black-Scholes formula when $\gamma^T_1=\gamma^T_2=0$