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A Gram-Charlier distribution has a density that is a polynomial times a normal density. For option pricing this retains the tractability of the normal distribution while allowing nonzero skewness and excess kurtosis. Properties of the Gram-Charlier distributions are derived, leading to the definition of a process with independent Gram-Charlier increments, as well as formulas for option prices and their sensitivities. A procedure for simulating Gram-Charlier distributions and processes is given. Numerical illustrations show the effect of skewness and kurtosis on option prices.

Gram-Charlier series are expansions of the form

This paper has three main goals: (1) define and study the properties of the family of Gram-Charlier distributions; (2) define a Gram-Charlier process and derive its basic properties; (3) apply those to European options. The formulas we give for European option prices and Greeks apply to Gram-Charlier distributions of any order, and we use four- and six-parameter Gram-Charlier distributions in our examples. Two numerical illustrations show how option prices are affected by the skewness and kurtosis of returns. This paper can a reference for those using Gram-Charlier distributions in option pricing but also in statistics.

Most previous applications to option pricing have assumed that

It has been observed that option prices have nonconstant implied volatilities, meaning that log returns do not have a normal distribution under the risk-neutral measure. There is a wide literature on modelling log returns to fit observed option prices, the main alternatives to Brownian motion being stochastic volatility models (where the parameter

Jurczenko et al. [

The paper by León et al. [

Almost all previous authors have used Gram-Charlier distributed log returns over a single time period. This has an obvious downside, in that it becomes tricky, if not impossible, to preserve consistency between the prices of options with different maturities. Section

The layout of the paper is as follows. In Section

In Section

The normal density function is denoted as

For a fixed

Let

The class of Gram-Charlier distributions just defined includes all distributions with density

Generating functions are convenient when dealing with orthogonal polynomials. One is

Suppose

(a)

(b)

(c) The representation of the

(d) All Gram-Charlier distributions are determined by their moments.

(e) The set of valid

(f) The first six moments of the

(g) The first six cumulants of the

(h) The following hold for any

(i) Suppose

(j) Suppose

(k) If

(l) The law of the square of

Parts (a) and (b) were proved above, and (c) follows directly from (b). To prove (d) it is sufficient to note the existence of

Part (f) is found by expanding the mgf in (b) as a series in

For part (i), it is sufficient to consider the case

Turning to (j), the first equivalence

When

(a) If

(b) The tails of the

The tails of the Gram-Charlier distributions are thicker than those of the normal distribution but are still “thin” because they are in the limit smaller than any exponential function

Here the exact region for the

The

The set of

Almost all previous authors have assumed that

We now show, using an example that can be worked out explicitly, that it is not always best to use normalized data when fitting Gram-Charlier distributions, because choosing another affine transformation of the data may well yield a much better fit.

The “standard” Gram-Charlier expansion for a function

Let us first calculate

Let us now consider a random variable

There is an easy solution: use a different scaling for

Gram-Charlier approximations to the density

Gram-Charlier approximations to the density

Feasible region for the skewness and excess kurtosis of the

The one-dimensional Cameron-Martin formula may be stated as follows: if

Suppose that

It is sufficient to calculate the mgf of

If

Simulation is required for many kinds of options, and it turns out that the Gram-Charlier distributions are very easy to generate, as we now show; there is no need to invert their distribution functions. When estimating some quantity

The simplest way to find the distribution of the sum of two independent Gram-Charlier distributed variables is to multiply their moment generating functions. Suppose

The above raises the question of whether there is a continuous-time process that has Gram-Charlier distributed increments. There is such a process with normal increments (Brownian motion), and it is moreover a Lévy process.

The only Lévy process with Gram-Charlier distributed increments is Brownian motion.

It is sufficient to show that, besides the normal distribution, any Gram-Charlier distribution cannot be infinitely divisible. If

More precisely, this says that if we exclude Brownian motion, no increment of any Lévy process can have a Gram-Charlier distribution. Any Gram-Charlier process with independent increments must be discrete-time.

The distribution of the exponential of a Gram-Charlier distributed variable will naturally be called “log Gram-Charlier”, as we do for the lognormal: if

The log Gram-Charlier distribution is not determined by its moments. More precisely, there is a noncountable number of other distributions that have the same moments as any particular log Gram-Charlier distribution.

There is a well-known way to construct a family of distributions that have the same moments as the lognormal (Feller [

The formulas below hold for any vector

As previous authors have done, we consider a market with a risky security

The market model may have one or more periods, but since we consider the pricing of ordinary European puts and calls only the distribution of the log return for the whole period

The risky security has price

Suppose that a risky security pays dividends at a constant rate

Absence of arbitrage implies that

The price of a European put can be found from the put-call parity identity

The option price formulas are of the form “Black-Scholes plus correction term.” Observe, however, that the values of

The previous literature includes formulas for sensitivities of Gram-Charlier option prices, but only in the case of the four-parameter

Let

The following lemma is obtained by elementary calculations.

Lemma 10.

From

Finally, turn to the sensitivities with respect to

Hardy [

A single premium

In [

The no-arbitrage price of the EIA ratchet premium option described above is

We now show the effect of skewness and kurtosis on the ratchet premium option values; this is done using the four-parameter distribution

Our application involves an annual ratchet, and so the option

We use the parameters

Ratchet premium option prices as a function of skewness and excess kurtosis, with

The effect of varying

The parameters fitted by maximum likelihood for the

Table

Comparison of ratchet option values computed with the Black-Scholes formula and Gram-Charlier distributions. Rows A to F are based on the four-parameter

| | | | | | | | B-E | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A | −0.0042 | 0.1685 | 0 | 0 | 0 | 0 | 0 | 0 | 0.419 | 0 | 109.26 | 0 | 100.00 | 0 |

B | −0.0035 | 0.1685 | 0 | 0 | −0.1150 | 0.0360 | −0.6898 | 0.8634 | 0.443 | 5.5 | 107.60 | −1.5 | 98.92 | −1.1 |

C | −0.0034 | 0.1685 | 0 | 0 | −0.1749 | 0.1021 | −1.0493 | 2.4508 | 0.478 | 13.9 | 105.42 | −3.5 | 97.50 | −2.5 |

D | −0.0051 | 0.1685 | 0 | 0 | 0.1749 | 0.1021 | 1.0493 | 2.4508 | 0.446 | 6.3 | 107.39 | −1.7 | 98.78 | −1.2 |

E | −0.0043 | 0.1685 | 0 | 0 | 0 | 0.1667 | 0 | 4.0000 | 0.493 | 17.5 | 104.59 | −4.3 | 96.96 | −3.0 |

| ||||||||||||||

| | | | | | | | B-E | | | | | | |

| ||||||||||||||

F | 0.0027 | 0.1595 | 0 | 0 | 0 | .0 | 0 | 0 | 0.441 | 0 | 107.69 | 0 | 100.00 | 0 |

G | 0.0451 | 0.1595 | −.3054 | .09542 | −0.12384 | 0.06120 | −0.5437 | 0.5091 | 0.438 | −0.7 | 107.90 | 0.2 | 100.14 | 0.14 |

Let us compare this with the approach in [

Consider a European lookback call option with fixed strike, with payoff

Figure

Lookback option prices as a function of skewness and excess kurtosis.

This paper tries to provide a framework for applying Gram-Charlier series to option pricing. We have shown that option prices may be significantly affected by varying skewness or kurtosis away from their Gaussian values. The practical consequence is that using the Black-Scholes formula (

Convergence of Gram-Charlier and other expansions in option pricing and other applied problems is an area for further research. The heuristic derivation of the Gram-Charlier/Edgeworth series of a density

The Gram-Charlier type B series are based on the Poisson distribution, rather than the normal. There are several other expansions that can be used. For instance, series of Laguerre polynomials may be used for densities on the positive half-line (a convergent Laguerre series has some numerical success in pricing Asian options; see [

The authors declare that they have no competing interests.

The authors thank the Committee for Knowledge Extension and Research (CKER) of the Society of Actuaries for partially funding this project.