Asian options are options in which the underlying variable is the average price over a period of time. Because of this fact, Asian options have a lower volatility and hence rendering them cheaper relative to their European counterparts. They are commonly traded on currencies and commodity products which have low trading volumes.
Geometric Averaging Closed Form || Turnbull-Wakeman Approximation || Levy Approximation || Curran Approximation || Binomial Method || Trinomial Method || Finite Differences || Monte Carlo Simulation || Other Methods || Known Names & Variants || References || Pricing Models
Asian options are options in which the underlying variable is the average price over a period of time. Because of this fact, Asian options have a lower volatility and hence rendering them cheaper relative to their European counterparts. They are commonly traded on currencies and commodity products which have low trading volumes. They were originally used in 1987 when Banker's Trust Tokyo office used them for pricing average options on crude oil contracts; and hence the name "Asian" option.
The are broadly segregated into three categories; arithmetic average Asians, geometric average Asians and both these forms can be averaged on a weighted average basis, whereby a given weight is applied to each stock being averaged. This can be useful for attaining an average on a sample with a highly skewed sample population.
To this date, there are no known closed form analytical solutions arithmetic options, due to the a property of these options under which the lognormal assumptions collapse - It is not possible to analytically evaluate the sum of the correlated lognormal random variables. A further breakdown of these options conclude that Asians are either based on the average price of the underlying asset, or alternatively, there are the average strike type.
The payoff of geometric Asian options is given as:
The payoff of arithmetic Asian options is given as:
To elaborate on arithmetic averaging, this is seen as being the sum of the sampled asset prices divided by the number of samples:
and for geometric averaging, the average value is taken as:
Where the nth root of the sample values multipled together is taken.
The payoff functions for Asian options are given as:
For an average price Asian:
and and average strike Asian:
Where is a binary variable which is set to 1 for a call, and -1 for a put.
Asian's can be both European style or American style exercise.
Here we look a some of the models to price standard Asian options under a variety of methods.
Kemna & Vorst (1990) put forward a closed form pricing solution to geometric averaging options by altering the volatility, and cost of carry term. Geometric averaging options can be priced via a closed form analytic solution because of the reason that the geometric average of the underlying prices follows a lognormal distribution as well, whereas with arithmetic average rate options, this condition collapses.
The solutions to the geometric averaging Asian call and puts are given as:
Where N(x) is the cumulative normal distribution function of:
which can be simplified to:
The adjusted volatility and dividend yield are given as:
Where is the observed volatility, r is the risk free rate of interest and D is the dividend yield.
As there are no closed form solutions to arithmetic averages due to the inappropriate use of the lognormal assumption under this form of averaging, a number of approximations have emerged in literature. The approximation suggested by Turnbull and Wakeman (TW) (1991) makes use of the fact that the distribution under arithmetic averaging is approximately lognormal, and they put forward the first and second moments of the average in order to price the option.
The analytical approximations for a call and a put under TW are given as:
Where is the time remaining until maturity. For averaging options which have already begun their averaging period, then is simply T (the original time to maturity), if the averaging period has not yet begun, then is .
The adjusted volatility and dividend yield are given as:
To generalise the equations, we assume that the averaging period has not yet begun and give the first and second moments as:
If the averaging period has already begun, we must adjust the strike price accordingly as:
Where we reiterate T as the original time to maturity, as the remaining time to maturity, X as the original strike price and is the average asset price. Haug (1998) notes that if r = D, the formula will not generate a solution.
3) Arithmetic Rate Approximation (Levy)
Levy puts forward another analytical approximation which is suggested to give more accurate results than the TW approximation. We look at the differences later.
The approximation to a call is given as:
and through put-call parity, we get the price of a put as:
Where the variables are the same as defined under the TW approximation.
For the following inputs, we compared the price of an Asian call under the TW approximation to that of Levy's approximation:
Asset Price: 100, Average Price: 95
D: 5%, r: 10%, V: 15%, T: 0, T1: 1, T2: 0.5
We can see that the absolute difference between the 2 approximations are very small, and that the two values can be said to be similar.
Furthermore, transposing the 2 call values as a function of the strike price illustrates the similarity between the two methods.
Curran (1992) produces an approximation of an arithmetic Asian option based on a geometric conditioning approach. This model looks towards our knowledge of the geometric distribution as well as of the underlying asset price at a particular point in time. By taking the natural logarithm of both of these at each point in time, we can then condition the underlying asset price on the geometric distribution and integrate accordingly.
Like any other options, Asian options can be priced using lattice/tree methods. The additional consideration which must be made is that at any point in time on the tree, the value of the option is dependent upon the average of the price that the path has taken. Given the 'averaging' nature of Asian options, we must then determine a minimum and maximum range at each node (dependent on the path which the underlying asset has taken). The problem is that as the number of nodes on a tree grows, so does the number of averages which must be taken, particularly in the central nodes - this is because the number of averages to be taken is exponentially related to the number of possible asset paths.
Hull & White (1993) attempts to solve this problem by adding a state variable to the tree nodes and approximation is undertaken with interpolation techniques in backward induction. The binomial tree can therefore be set up as:
1. The minimum and maximum averages at each time node at determined:
Where u denotes the size of the up move and d denotes the size of a down move. i is the number of nodes.
2. The approximate average is calculated at each time node.
3. Payoff for each approximate average is determined via means of linear interpolation
4. Discount backwards towards the first time node:
Where k = 1, ..., A. and A is the first time node. and denote up and down probabilities of the binomial tree.
However, there is the problem that convergence is not guaranteed.
Chalasani, Jha & Varikooty (1997) provides a method for the computation of the lower bounds of the Asian option with reasonable accuracy. Basing their work off Rogers & Shi (1995), the authors use a modified choice of random variable z used to estimate the conditional expectation of the option payoff. An improved lower bounds is given, which can be programmed using a binomial lattice.
Various papers documenting the use of finite differences to solve Asian options have been published since the early-mid 90s. Rogers & Shi (1995) present a method using a one dimensional PDE which can be solved using finite differences - however, their method is prone to problems associated with the diffusion term, particularly with lower volatilities and short time to maturities. Andreasen (1998) expand on Rogers & Shi's model by using a change of numeraire to solve the price of Asian options numerically.
Example of finite differences methods and the application towards Asian options can also be found in Tavella & Randall (2000).
Various methods using Monte Carlo simulation (MCS) have been developed to price arithmetic Asian options. The aforementioned analytical approximations by TW, Levy and Curran can all be computed using a simulation method. Monte Carlo simulation can give relatively accurate prices for option values, and in the case of Asian options, which are highly path dependent, this method is particularly useful.
In section 1, we gave a geometric closed form solution to Asian options originally presented by Kemna & Vorst (1990). The authors also present a solution for pricing arithmetic rate options using Monte Carlo simulation and the geometric closed form method (1990 as a control variate.
The control variate technique can be used to find more accurate analytical solutions to a derivative price if there is a similar derivative with a known analytic solution. With this in mind, MCS is then undertaken on the two derivatives in parallel.
Given the price of the geometric Asian, we can price the arithmetic Asian by considering the equation:
Where is the estimated value of the arithmetic Asian through simulation,is the simulated value of the geometric Asian, and is the exact value of the geometric Asian given above.
8) Monte Carlo / Quasi-Monte Carlo
While we have highlighted that MCS methods are used in various Asian option pricing methods, the general idea is that they are not particularly effective in pricing Asian options in terms of computational speed. However, by utilising control variates or antithetic variable techniques, the accuracy of MCS methods may be improved - for example, by using the closed form geometric average rate formula as a control variate.
QMCS using quasi-random number sequences can also help with the accuracy, however it is shown in numerous papers that convergence is not particularly stable, especially with exotic Asian options and is not an effective pricing method.
One of the biggest problems in finding a solution to arithmetic rate Asian options is that the many models do not address the error bound associated with solving the pricing problem. such as those given in Milevsky & Posner (1998, 1998) - who look to solve the problem using Gamma and Johnson distributions. Geman & Yor (1993), applies a Laplace transform and numerical inversion, but again does not address the error bounds. Shaw (2000) and Linetsky (2002) also use Laplace transforms to price Asian options and reach a reasonable degree of accuracy.
Carverhill & Clewlow (1990) make use of Fourier transform techniques. Zhang (2001) presents a semi-analytical approach which is shown to be fast and stable by means of a singularity removal technique and then numerically solving the resultant PDE.
Ju (2002) produces an analytical approximation to price Asian options by assuming that even though the weighted average of lognormal variables is not lognormal, we are still able to approximate the weighted average by a lognormal variable if the first two moments of moments are true.
Average Rate Options
Average Strike Options
Asian American Options
Asian Basket Options
Asian Spread Options
Fixed Strike Asian
Floating Strike Asian
Forward Starting Asian Options
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