The Asian Options

1. The Asian Options Chapter 6: ADVANCED OPTION PRICING MODEL

2. The Asian Options • 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.

3. The Asian Options -- Introduction • They 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 arithmetic property of these options under which the lognormal assumptions fall apart - 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.

4. Formulations • The payoff of geometric Asian options is given as: The payoff of arithmetic Asian options is given as:

5. Formulations (Cont’d) • Define SA as the average price and ST the terminal price and X the strike price. • The payoff functions for Asian options are given as: For an average price Asian: 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.

6. Geometric Asian Options

7. Pricing – Geometric Closed Form (Kemna Vorst) 1990 • 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 • The solutions to the geometric averaging Asian calls and puts are given as: and,

8. Pricing – Geometric Closed Form (Kemna Vorst) 1990 (Cont’d) 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 q is the dividend yield.

9. Arithmetic Asian Options

10. Arithmetic Rate Approximation • As there are no closed form solutions to arithmetic averages due to the 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. • Levy puts forward another analytical approximation which is suggested to give more accurate results than the TW approximation.

11. Arithmetic Rate Approximation (Turnbull & Wakeman) 1991 • The analytical approximations for a call and a put under TW by matching cumulants of distribution are given as: where where T2 is the time to maturity from time zero and X is the strike price. For simplicity, we assume the averaging just begun, i.e., t = 0.

12. Arithmetic Rate Approximation TW (Con’t) Since TW average options are valued based on matching moments, let the first moment be M1 and the second moment be M2. For q≠r, The adjusted volatility is given as: By assuming the average asset price is lognormal, we can value the Asian options like an option on a futures contract with

13. Arithmetic Rate Approximation TW (Con’t) • Suppose the average option is not newly-issued. • T1: period of time with prices already been observed • T2: future time to maturity • : average of already observed prices • SA : average asset price over the remaining part of the averaging period

14. Arithmetic Rate Approximation TW (Con’t) • The payoff from an average price call is Hence, where • Therefore, the Asian option can be valued as before by changing X to X* and multiply the previous result by T2/(T1+T2) for X* >0. • For X* ≤0, the option turns to a forward contract because SA-X*>0. Its value is

15. Other Models • 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.

16. Continuous Asian Options (Jan Vercer) • The price of an Asian option can be found by solving a partial differential equation (PDE). PDE in two space dimensions (see Ingersoll (1987)) is prone to oscillatory solutions. Ingersoll also observed that the two-dimensional PDE for a floating strike Asian option can be reduced to a one-dimensional PDE. Rogers and Shi (1995) have formulated a one-dimensional PDE that can model both floating and fixed strike Asian options. However this one-dimensional PDE is difficult to solve numerically since the diffusion term is very small for values of interest on the finite difference grid. • Vercer provides a simpler and unifying approach for pricing Asian options, for both discrete and continuous arithmetic average. The resulting one-dimensional PDE for the price of the Asian option can be easily implemented to give extremely fast and accurate results. Moreover, this approach easily incorporates cases of continuous or discrete dividends.

17. Continuous Asian Options (Jan Vercer) • Suppose that the underlying asset evolves under the risk neutral measure according to the equation where r is the interest rate, q is a continuous dividend yield, and  is the volatility of the underlying asset. • Denote the trading strategy by t, the number of shares held at time t. If we take the strategy

18. Continuous Asian Options (Cont’d) and let the wealth evolve according to the following self-financing strategy with the initial wealth After some simplification, we have

19. Relationship to Asian Options • Using this idea that we can replicate the stock price average by self-financing trading in the stock, we can apply this fact to pricing Asian options. The general payoff of the Asian option could be written as • Because of the Asian Put-Call parity and the martingale property of stock and options it is enough to compute the value of the Asian option with the payoff When K1 = 0, we have the fixed strike Asian call option, when K2 = 0, we have the floating strike Asian put.

20. Relationship to Asian Options (Cont’d) • In order to replicate such option, we start with the initial wealth and follow the self-financing strategy It turns out that and the payoff of the option is then

21. Change of Numeraire • We can use the change of numeraire technique to reduce dimensionality of the problem by defining • According to Ito’s lemma, where is a Brownian motion under the numeraire measure. The price of Asian call option could be written as

22. PDE and Finite Difference Method • If we introduce where Zt is a process with the initial condition then the price of the option is It is easy to show that the function u satisfies the following partial differential equation This unconditionally stable equation could be easily solved numerically by the finite difference method.

23. Arithmetic Rate Approximation (Monte Carlo Simulation) • 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. • Kemna & Vorst (1990) 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.

24. Monte Carlo Simulation – Variance Reduction • 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 utilizing 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.

25. Antithetic Variables -- Theory • The task is to estimate the expected value = E[X] of the random variable X • The idea: make two different runs such that “small” values of the observed variable in one run are replaced by “large” values of this variable in the other run and vice versa, and use the average over the two runs as the estimator • Let X be the observed value of the studied variable in one experiment Let Y be the result from another experiment having the same distribution as X Then Z = 1/2(X + Y ) is a random variable, which has the same expectation as X = E[Z] The variance of Z V[Z] = 1/4(V[X] + V[Y ] + 2Cov[X; Y ])=1/2(V[X] + Cov[X; Y ]) If X and Y are negatively correlated, Cov[X; Y ] < 0, then the variance of Z is smaller than in the case where X and Y are independent (1/2V[X])

26. Antithetic Variables – Monte Carlo Simulations • Let the cdf of X be F(x) and let the samples of X be generated by the inverse transform method X = F-1(U) where U ~ U(0; 1) (uniform distribution). Then Y , which is generated using the formula Y = F-1(1 - U) obeys the same distribution as X. When U ~ U(0; 1) then also 1 - U ~ U(0,1). It is clear that X and Y are negatively correlated – if U happens to be small (0) then 1 - U takes a large value (1) – X and Y take values at the opposite ends of the range X and Y constitute a antithetic pair of random variables • Example Let X ~ Exp(1), i.e. X = -log U Y = -log (1 - U) One can show that in this case Cov[X; Y ] = 1 -2/6 = -0.645 Variance is reduced by 64 % in comparison with the use of independent random number sequences (and costs nothing!)