Asian options are path-dependent options whose payoff depends on the average value of the underlying asset during a specific set of dates across the life of the option. Because the payoff of the Asian options depends on the average value of the underlying asset, volatility in the average value is lower than that of the plain vanilla options. Thus Asian options tend are less expensive than the comparable plain vanilla options. This makes Asian options ideal for use in hedging positions.
The averaging in Asian options can either be arithmetic or geometric. Under the Black Scholes world, Closed form expressions exist for geometric Asian options, as the product of log normal random variables is a log normal random variable, while approximations are required to price arithmetic Asian options as the sum of log normal random variables is not log normal.
In this blog post we will be pricing continuously sampled arithmetic Asian options using moment matching under the Black Scholes framework. We compare this moment matching with a Monte Carlo simulation.
The Asian Option
There are two basic forms of averages in Asian options, being arithmetic and geometric. A geometric average Asian option is easy to price because a closed-form solution is available. However the most difficult one to price is the arithmetic type, which is the most commonly used> No analytic solution exist for arithmetic Asian options. This missing solution is primarily because the arithmetic average of a set of log-normal random variables is not log-normally distributed.
Since no general analytical solution for the price of the arithmetic Asian option is known, a variety of techniques have been developed to approach the problem of its valuation. These include using partial differential equations, Monte Carlo simulation and moment matching among others. In this blog post we will be focusing on the last two approaches.
The Arithmetic Asian Option Payoff
As discussed in a previous blog post, diffusion models are continuous-time stochastic processes. In this blog post, we will be focusing on the GBM model. The SDE, under the risk-neutral measure, is given as
The payoff of the fixed strike Asian option at maturity given as
where and is the strike price. The value of a call option at time zero is then given as
where is the deterministic risk free rate.
If the distribution of was known, then calculating would be straight forward. However, given that we do not know the exact distribution of , we approximate it with another distribution by matching the first couple of moments of with the corresponding moments from another distribution.
In the next section we match the first two moments of with the fist two moments of the log-normal distribution.
The first two raw moments of are given as :
The derivation for the above two equations can be found in this paper.
Recall that if is , then will be log-normally distributed. The first two raw moments of are then given as:
Setting and and then solving the two equations simultaneously for and we have that:
This means that:
where and are the parameters of the approximate log-normal distribution.
In the following section we will asses the accuracy of this approximation relative to a Monte Carlo estimate.
In this section we provide the results that compare the Monte Carlo estimate and the moment matching estimate of the arithmetic Asian option described above. The Monte Carlo simulation was performed with the antithetic variates variance reduction technique.
The simulation was performed with the following parameters: , , , and . The simulation was performed with a different number of simulations and strike prices.
The convergence profile for the Monte Carlo estimate for the price of an arithmetic Asian option with strike 50 is shown below:
The above graph shows that the estimate becomes stable, with narrow confidence bounds, beyond 30 000 simulations.
A comparison between the Monte Carlo estimate, with 40 000 simulations, and the log-normal approximation for the arithmetic Asian Option is shown below:
|Strike||Monte Carlo Estimate||Log Normal estimate||% Difference||95% Monte Carlo Confidence bounds|
|0||95.59352||95.63202||0.0403%||[ 95.48, 95.71]|
|50||49.89697||49.93549||0.0772%||[ 49.78, 50.02]|
|90||14.97613||15.06704||0.6069%||[ 14.87, 15.08]|
|100||8.831941||8.885762||0.6094%||[ 8.75, 8.92]|
|110||4.700716||4.69511||0.1192%||[ 4.63, 4.77]|
|150||0.1835182||0.149526||18.522%||[ 0.17, 0.20]|
|200||0.0021867||0.000639||70.745%||[ 0.000874, 0.0035]|
The above table suggests that the approximation is good for at the money and deep in the money options, but is a very bad approximation for deep out of the money options. A better approximation, by matching more than two moments of , is needed for the deep out of the money options. A shifted log normal distribution may provide better relates for the deep out of the money options.
The code given below was used in this blog post.