Pricing Arithmetic Asian Options using Moment Matching

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

dS_t = S_t(r dt+ \sigma dW_t). . . (1)

The payoff of the fixed strike Asian option at maturity T given as

\max(A_T-K,0)

where A_T =\frac{\int_{0}^{T} S_t dt}{T} and K is the strike price. The value of a call option V_0 at time zero is then given as

V_0=\exp(-rT) \times E^Q[\max(A_T-K,0)]

where r is the deterministic risk free rate.

If the distribution of A_T was known, then calculating V_0 would be straight forward. However, given that we do not know the exact distribution of A_T, we approximate it with another distribution by matching the first couple of moments of A_T with the corresponding moments from another distribution.

In the next section we match the first two moments of A_T with the fist two moments of the log-normal distribution.

Moment Matching

The first two raw moments of A_T are given as :

m_1 = E[A_T] = \frac{S_0(\exp(rT)-1)}{rT}

m_2 = E[A_T^2] = \frac{2S_0^2}{T} \times \left( \frac{\exp{((2r+\sigma^2)T)}}{(r+\sigma^2)(2r+\sigma^2)}+\frac{1}{r} \left( \frac{1}{(2r+\sigma^2)} - \frac{ \exp{(rT)} }{r+\sigma^2} \right) \right)  for r>0

The derivation for the above two equations can be found in this paper.

Recall that if Z is Normal(\mu,\sigma^2), then X=exp(Z) will be log-normally distributed. The first two raw moments of X are then given as:

E[X] = \exp{(\mu+0.5\sigma^2)}

E[X^2] = \exp{(2\mu+2\sigma^2)}.

Setting E[X]=m_1 and E[X]=m_2 and then solving the two equations simultaneously for \mu and \sigma we have that:

\mu=\ln{ \left(\frac{m_1^2}{\sqrt{m_2}} \right)}

\sigma = \sqrt{ \ln{\left( \frac{m_2}{m_1^2}\right)} }

This means that:

V_0 = \exp(-rT) \times E^Q[\max(A_T-K,0)]

V_0 \approx \exp(-rT) \times E^Q[\max(\exp{(X)}-K,0)]

V_0 = \exp{(\mu-rT+0.5\sigma^2)} \Phi(\frac{ \mu-\ln K+\sigma^2}{\sigma}) -K \exp{(-rT)}\Phi (\frac{ \mu-\ln K}{\sigma})

where \mu and \sigma 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.

Simulation Study

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: r= 0.09\sigma = 0. 3, T=1S_0 = 100 and dt=1/10000.  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:

Convergence profile for the Monte Carlo price of an arithmetic  Asian Option with strike at 50.

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 A_T, 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.

Code Used

The code given below was used in this blog post.

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