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Abstract The calculation of the Asian option value has posed a great challenge to financial mathematicians as well as practitioners for the last two decades. Since there exists no analytical valuation formula to date, one has to resort to other methods to price this commonly used derivative product. One possibility is the usage of simulation approaches, which however are especially inefficient for Asian options, due to their dependence on the entire stock price trajectory. Another alternative is resorting to semi-analytical methods, based on the inversion of the option price’s Laplace transform, which however are prone to severe numerical difficulties. In this paper, we seek answer to the question whether it is possible to improve on the efficiency of the semi-analytical approach, implementing and comparing different numerical algorithms, so that they could be applied in real-life situations. We look into whether today’s superior computer environment has changed the relative strength of numerical and simulation approaches with regards to Asian option pricing. Based on a comprehensive analysis of speed and reliability, we find that the Laplace transform inversion method can be further enhanced, pushing down the prior critical value from 0.01 to 0.005 and the calculation time from 20 - 30 seconds to 3 - 4 seconds. This renders the numerical approach readily applicable for practitioners; however, we conclude that the simulation approach is a more efficient option when σ
^{2}<0.01.

Asian options are popular hedging instruments in the hands of financial risk managers, owing to their special payout structure and cost-effectiveness. With regard to exercise behavior, these options are “European”, in the sense that they can be exercised only at a pre-agreed (maturity) date; however, as opposed to (plain) vanilla products, Asian options have an “exotic” payout at maturity

where K stands for the pre-agreed strike price and ^{1}. The above equation implies that Asian options are path-dependent; moreover, since their value depends on the average of all underlying prices during the option’s lifetime, they may be considered as the quintessence of path-dependent derivatives.

The exotic nature of this option type has important financial and mathematical implications for us. From a financial (or practical) perspective, the stabilizing effect of averaging in Equation (1) makes Asian options a cheaper, yet more reliable alternative to their vanilla counterparts for financial risk managers. By taking the average of market prices for a given time period, the adverse effects of potential market manipulation and price jumps are minimized, which renders these options a cost-effective hedging instrument, especially in market environments where either the volume is low (e.g. in the corporate bond or commodity markets), or the volatility is high (e.g. in the currency and interest rate markets). As a result, according to a survey including 200 derivative-using non-financial U.S. firms conducted by Bodnar et al. [

From a mathematical (or theoretical) perspective, the valuation of Asian options is particularly challenging even in the most simple pricing models, which is partly due to path-dependence and partly because the probability distribution of averages is usually unknown. The average itself is defined as either the arithmetic or the geometric mean of the underlying asset’s prices; the former being more common on the market, while the latter having more convenient mathematical properties.

In the classical Black-Scholes framework, the value of the geometric-average Asian option can be determined just as easily (for the closed-form formula refer to Turnbull et al. [

Another approach to pricing arithmetic-average Asian options is using Monte Carlo simulation, which is a far more robust, yet computationally demanding method. Employing the geometric-average Asian option price as control variate, Fu et al. [

In this paper, we seek answer to two important questions naturally arising from the results above. Considering the development in computational power,

1) Is it possible to find a more efficient algorithm than the “Euler method” employed by Fu et al. [

2) What is an optimal daily read-replication number combination for the Monte Carlo method, with regard to both discretization and simulation error? How has the relation of the different approaches changed in the last ten years?

In Section 2, we introduce the basic principles of Asian option (and in general, financial derivative) pricing, making a clear distinction between the different valuation approaches. Then, in Section 3, we proceed with an empirical analysis, comparing the performance of pricing algorithms in MATLAB®. Finally, we make recommendations on the best valuation practices, and conclude.

In order to price Asian options, it is necessary to agree on the specific risk-neutral framework used, which is the Black-Scholes model in this paper. In this section, first we briefly recapitulate the assumptions and results of this model, then we introduce the risk-neutral valuation logic and make a distinction between the main pricing methods. Finally, we apply these to the particular valuation problem of Asian options, as well as discuss what computational difficulties arise with the different algorithms.

In their original model, Black and Scholes [

・ the risk-free interest rate is constant over time (as opposed to models in which interest rates are deterministic or stochastic processes);

・ the stock price (of any firm) follows geometric Brownian motion “with a variance rate proportional to the square of the stock price”;

・ the option is a European-exercise option;

・ the stock pays no dividends (or other interim cash flows);

・ there are no transaction costs of trading or borrowing;

・ there are no limitations to trading (including the ones concerning short-selling or the trading of fractional quantities);

・ there are no limitations to taking out a loan or making a deposit at the risk-free interest rate.

Although the last few assumptions are rather technical and only ensure that the functions and processes are well-behaved, the first three assumptions form the core of the model and define very specific stock price and interest rate behavior. There are effectively three financial products in this setup:

・ a risk-free bond B,

・ a stock S, whose price follows geometric Brownian motion,

・ and a European-exercise derivative product f, whose price is a real function of the stock’s price,

which, if looked at as processes, have the respective price dynamics

where W stands for the standard Wiener process. It follows (from Itô’s lemma) that the logarithm of S is a normally distributed variate having the dynamics

Based on the assumptions above, using a replication method called dynamic delta hedging, Black and Scholes [

which is a notable result, since in many cases it can be solved by means of standard PDE methods.

Having chosen the specific price dynamics of asset prices, in theory we can calculate the expected value of the option’s payout at maturity, which, if discounted at the proper rate of return, yields the option’s value. Alas, this rate of return is unknown in most cases, moreover, it tends to change during the derivative product’s lifetime, rendering the above pricing logic unusable.

Fortunately, there exists an elegant financial mathematical trick called risk-neutral valuation, which makes it possible to discount the expected value of the option’s payout at the (known) risk-free interest rate. In what follows we briefly recapitulate the tenets of this logic, then we show how it is applied in practice.

Notation 2.1. Let

Definition 2.2. A

Definition 2.3. An investment strategy is self-financing (SF) if

Notation 2.4. Let

time. Accordingly, let

It can be shown that given the right model conditions (i.e. no arbitrage, market completeness), the dynamics of the stock price could be expressed as

with respect to a unique probability measure

Applying the martingale representation theorem, it can be shown that for any European-exercise derivative there exists a corresponding self-financing investment strategy

noting that the distribution of

Theorem 2.5. (Risk-Neutral Valuation) The present value of a European-exercise derivative equals to the expectation of the derivative price with respect to probability measure

Having described the fundamental logic of risk-neutral valuation, let us now turn to introducing the different pricing methods, which apply the above principles in practice.

Analytical methods aim at deriving a closed-form expression for the price of the particular derivative. This could be attempted by

・ either solving Equation (3) with the appropriate boundary conditions given by the payout of the derivative at expiry,

・ or calculating the risk-neutral expected value of the derivative at expiry, using the Risk-Neutral Valuation principle outlined in Section 2.2.

Whichever method we choose, the existence of a closed-form expression is not guaranteed, depending on the payout structure of the derivative. Hence, the valuation problem becomes purely mathematical and even though in some cases the derivation of a pricing formula is extremely challenging, financial mathematicians put substantial effort into it. The reason for this is that a closed-form expression for a particular financial derivative’s price yields an easy-to-calculate value, rendering analytical methods computationally superior to other (e.g. numerical or simulation) approaches. Nonetheless, apart from a few notable examples (e.g. vanilla options or geometric-average Asian options), it is not possible to derive a pricing formula. For instance, there has not been one found yet in the case of arithmetic-average Asian options.

Although a closed-form solution often proves to be elusive, sometimes it is possible to derive a complex but deterministic (semi-analytical) pricing formula. For instance, we may arrive at a complex (single/double/triple) integral form, which can be approximated by means of numerical methods (e.g. quadratures). As we explain in Section 2.3, this is the case for arithmetic-average Asian options, where only the Laplace transform of the option price could be determined in closed-form and the option price itself has to be calculated using numerical inversion techniques.

As an alternative to analytical and numerical methods, we can turn to simulation techniques known as “Monte Carlo methods”. There exist numerous approaches but the basic principle is the same (see Kalos and Whitlock [

1) Using random number generating algorithms, generate outcomes for a random variate with known distribution.

2) Applying the Law of Large Numbers, estimate the unknown expected value with the average of the generated outcomes.

3) When applicable, increase the sample size, so as to constrain the estimate into a desired confidence interval.

Monte Carlo methods are computationally intensive but easy to understand and flexible algorithms in the sense that they are compatible with a broad family of (physical, mathematical or financial) stochastic models. From the perspective of financial derivative valuation, the flexibility of Monte Carlo simulation allows for complex stock price movements (e.g. jump diffusion) and derivative payouts (e.g. path-dependency or early exercise), which are otherwise extremely hard to handle by analytical methods.

Without going into further details on the specific algorithms used in Section 3, we must point out the notoriously slow convergence of simulation methods. As stated earlier, these techniques rely heavily on the Law of Large Numbers, estimating the expected value by the average of simulated outcomes, which implies that the standard deviation of the estimate will be proportional to

For the above mentioned reasons, it is necessary to employ advanced simulation techniques so as to reduce the variance of the estimate. Without aiming at completeness, we list a few variance reduction techniques here (for a detailed description refer to Kleijnen et al. [

・ common random numbers (including antithetic variates);

・ control variates;

・ conditioning and stratified sampling;

・ importance sampling and splitting;

・ quasi (not fully random) Monte Carlo simulation.

Depending on the statistical properties of the estimated random variate, some techniques may be more suitable to a certain simulation problem than others. Based on the findings of Kemna and Vorst [

After this brief introduction to the valuation framework, let us introduce a numerical valuation method based on the work of Geman and Yor [

where

Once the call option price is calculated, we may use the put-call parity, which is valid for all European-exercise options, to arrive at the put option price.

Lemma 1 (Put-Call Parity)

where F is the “Asian forward price”, as defined in the next lemma.

Lemma 2 (Asian Forward Price) The forward price of the arithmetic-average of stock prices for a given time interval

where

Therefore, in theory we are able to price Asian options. However, solving the simple-looking pricing formula in Equation (7) hides several pitfalls haunting financial mathematicians for decades. The fundamental problem is caused by the unknown probability distribution of

The breakthrough can be credited to Geman and Yor [

where

the latter being referred to as the “normalized price” of the Asian option. Following [

The revolutionary finding of Geman and Yor [

Theorem 2.6 (Geman-Yor) The Laplace transform of the Asian option’s normalized price, defined in Equation (12), with respect to h is

where

for every

Using this result, it is possible to determine the option’s price by first calculating the above Laplace transform, then inverting it to yield the normalized option price and finally substituting into Equation (10). However, as described in Cohen [^{2}. There have been various techniques invented, based on either of the following two inversion theorems.

Theorem 2.7 (Theorem) If the Laplace transform

where

Theorem 2.8 (Post-Widder Theorem) If the Laplace transform

where

Although in contrast to the Bromwich formula the Post-Widder formula is restrained to the real line and thus looks easier to handle, its speed of convergence is notoriously slow and the techniques based on them are prone to computational difficulties, as we can see in Fu et al. [

The Euler Algorithm

Abate and Whitt [

Their final result is the algorithm which also Equation [

Algorithm 2.9 (Euler) Let us define

and

Then, the inverse Laplace transform could be approximated by the Euler series

where A, m and n are chosen parameters of the algorithm, for which Abate and Whitt [

The Talbot Algorithm

Similarly to the Euler algorithm, this algorithm is also based on the Bromwich theorem. The idea of Talbot, described in Cohen [

which is applied by Abate and Whitt [

Algorithm 2.10 (Talbot) Let us define

and

Then, the inverse Laplace transform could be approximated by the series

where M is a chosen parameter of the algorithm.

Implementation

By means of the above algorithms, the Laplace transform

could now be numerically inverted with respect to

We use MATLAB® for the implementation of the introduced algorithms. The codes are presented in Sections A.1 and A.2 of the Appendix. For the sake of clarity, we strive to preserve the notation presented earlier in this paper and insert as many comments into the code as possible. Without going into unnecessary technicalities, let us point out a couple of practical features and enhancements in the code:

・ Both algorithms are based on substituting back into the Laplace transform

・ A crucial step in calculating the Laplace transform presented in Equation (13) is complex integration on the

・ Based on our experience, during the implementation of the algorithms we have to work with quantities of very different magnitudes. More specifically, the integral in Equation (13) is usually of magnitude

1) Taking the (natural) logarithm of the whole expression in Equation (13) so as to exponentially reduce the magnitude of variables. Although this step does not necessarily increase precision, it definitely “earns space” in terms of magnitude.

2) Explicitly instructing MATLAB® to use the Symbolic Math Toolbox®, which allows the user to choose an arbitrary level of precision for calculations. With regard to our particular problem, we consider a precision of 25 significant digits sufficient.

Having understood the numerical approach based on the Laplace transform of the Asian option’s normalized price, we are ready to price Asian options with two algorithms. But before proceeding to the empirical analysis concerning the precision and robustness of these algorithms, let us review the simulation approach, which will serve as an ultimate check and basis of comparison for the Laplace transform method.

Keeping in mind the basics of the Monte Carlo method from the end of Section 2.2, we know that this is a highly-flexible, robust approach, suitable to various frameworks and product types. Flexibility, however, has a price taking the form of variance in the case of simulation methods, which means that the simulated price bears some uncertainty and fluctuates around the option’s true (analytical) value. Striving to cut this variance to a minimum level, we study in details how the variance reduction techniques of antithetic variates and control variates work in the case of financial derivative valuation.

Basic Method

As the name suggests, this is the simplest form of Monte Carlo method, taking the arithmetic-mean as an unbiased estimator of the expected value:

which has a variance of

where ^{th} outcome in our simulation. The random function f, whose expected value we wish to estimate is the discounted payout function of the option as suggested by the principle in Equation (5):

which, in the particular case of Asian call options, takes the form of

having an expected value

which is precisely the price of the option given in Equation (7). As we can see, it is this risk-neutral expected value that we estimate through simulation of the underlying pro- duct’s risk-neutral trajectory

which will be constructed as

where

is the risk-neutral log-return applying Equation (2) and m indicates the number of daily reads (i.e. the mesh of the equidistant time grid).

We must point out that in the case of path-dependent derivatives simulation methods have two sources of estimation error:

1) The generated trajectories (outcomes) depend on the random event

2) The generated trajectories are discretized, which means they are only approximations of the true, continuous-time trajectories required for the calculation of the path- dependent derivative payout. In other words, we substitute the integral with a sum in the payout of the Asian option and no matter how great n and thus how accurate our estimator is, we still face some discretization error. This error could be reduced by increasing m, which is the number of fragments we break up each day of interval

Method of Antithetic Variates

Keeping the logic and weak points of the basic simulation method in mind, let us see how we could improve on our estimate of the expected value. First of all, let us try and reduce the variance of the estimator. A suitable technique is the method of antithetic variates, described in Kleijnen et al. [

where ^{th} original and antithetic trajectory, respectively. This estimator has a variance of

where we use the symmetric nature of the random walk. The benefit of using this method is twofold:

1) It is apparent from the above results that as long as the original and antithetic payouts have a negative covariance, the variance of the estimator will be reduced compared to that of the basic method, while the estimator remains unbiased. As we demonstrate in Section 3, thanks to the low correlation of the original and the mirrored trajectories, the variance reduction achieved by this relatively simple method is substan- tial.

2) Since we use every random number twice, we only need to generate n/2 outcomes instead of the original n ones, which means that the method of antithetic variates is (almost) twice as efficient as the basic method.

Method of Control Variates

As demonstrated through the above example of antithetic variates, it is possible to keep the estimator unbiased, while significantly reducing its variance. As it turns out, it is possible to further improve on our simulation technique by using control variates instead of antithetic variates (for further details refer to Kleijnen et al. [

where c is a properly chosen3 constant, ^{th} outcomes of the original and control variates in our simulation, respectively. It is easy to see that the variance will be

Kemna and Vorst [

1) It reduces the simulation error of our original estimator introduced in the basic method, even more so than the method of antithetic variates4.

2) It effectively eliminates some of the discretization error inherent in the simulation technique by using a “discretization adjusted” control variate.

Before proceeding with the empirical analysis of our valuation framework in the next section, let us make a few notes with regard to the practical implementation of the simulation methods discussed above. Firstly, we perform the simulation following the algorithms presented in this section. Although the true variances and covariances are unknown, they could be estimated from the sample. Secondly, the greatest challenge during the simulation is to avoid running out of memory. Thanks to the capacity of modern computers, this is usually not an issue when simulating only the endpoints of trajectories as in the case of European-exercise vanilla options. However, when the entire trajectory needs to be generated and stored in the memory (e.g. in the case of path-dependent derivatives), we run out of resources alarmingly fast. Without going into further technical details (e.g. issues with “memory paging”), what we do to avoid this eventuality is break each trajectory into small enough sections that the generated log-return matrix fits into the memory. Then, we roll over the trajectory, calculating the (geometric) mean step by step, efficiently reusing the available memory space again and again as shown in

After the introduction of the different approaches to Asian option pricing, now we take a more practical perspective and compare the efficiency and computational characteristics of the methods described in Section 2.3 by analyzing outputs from MATLAB®. The computations are performed under Windows 7 operation system by means of an Intel® Core i7 2.2 GHz CPU 8 GB RAM hardware.

First, we examine how the two semi-analytical (i.e. Euler and Talbot) algorithms fare compared to each other, then we contrast simulation outputs for different reading frequencies and sample sizes. Then, we compare the semi-analytical and the simulation approaches and make suggestions on their appropriate usage in real life situations.

At first glance the semi-analytical algorithms presented in the previous section might look awkward but in effect they are very efficient for certain parameter ranges of the Asian call option price

For the sake of simplicity, let us fix ^{6} as suggested by Abate and Whitt [

・ The Euler algorithm is more robust than the Talbot algorithm with respect to both calculation time and parameter levels. In general, setting

・ The Talbot algorithm is more reliable than the Euler algorithm in the sense that if m is set sufficiently high, then the precise Asian option value can be calculated even for small values of

・ When

By repeating the above calculation test with different combination of option parameters, we conclude that it is only the value of

Laplace transform inversion approach. An important question to be answered though is why the chosen numerical algorithms have convergence issues when

・ In the case of the Euler algorithm, the Laplace transform oscillates and quickly tends to zero as k (i.e. the locus in the complex plane) changes, which limits the effect of the precision parameter, there being no reason to use

・ In the case of the Talbot algorithm, the Laplace transform shows some oscillating nature again as k (i.e. the locus in the complex plane) changes but remains zero outside a given range. The problem here is not that the transform diminishes to magnitudes which are hard to handle numerically but that this specific location of oscillation is unknown, moreover, it shifts along the k axis towards infinity as

Although with the Euler algorithm we quickly run into a “dead-end street”, since the embedded integral cannot be calculated numerically to an arbitrary precision, the deformation of the Bromwich-contour in case of the Talbot algorithm seems to leave some space for improvement. If we could (either numerically or analytically) pinpoint the location of oscillation without having to calculate all the zeros before, we could significantly improve on the algorithm’s performance.

Moreover, we consider it an achievement that we could significantly reduce calculation times compared to the ones produced by Fu et al. [

In the previous section we investigate how the two numerical algorithms perform for different combinations of option parameters. Before making a comparison with the simulation approach, let us look into the latter on its own and examine the three Monte Carlo algorithms introduced in Section 2.3.2. As explained earlier, in the case of path- dependent derivatives (such as Asian options) simulation techniques have two governing parameters:

・ n, which is the number of trajectories simulated,

・ m, which is the number of fragments each day is partitioned into,

where n determines the simulation error, whereas m determines the discretization error. Striving to isolate the effects of these parameters as much as possible, first we examine what level of m leads to a satisfactory approximation of the continuous Asian option price, and only then will we turn to adjusting (lowering) n in order to attain an estimate with desirable precision. Just as in the case of numerical methods, we constrain our analysis to ATM (

However, before proceeding with the fine-tuning of the above parameters and measuring the performance of the different simulation algorithms, let us have a word on what level of error we consider satisfactory. In order to make a later comparison between simulation and numerical methods possible, we aim at the same (i.e. four-decimal) valuation precision. To achieve this, we need to set the simulation parameters so that the confidence interval^{7} around the estimated option value is sufficiently small to ensure the desired precision. Since the exact probability distribution of the Asian option price is unknown, let us suffice with Chebyshev’s inequality and give an approximation for the confidence interval by

which implies that by choosing

Keeping in mind the findings about simulation precision, let us determine what level of m ensures a good approximation of the underlying product’s continuous trajectory. If we set

・ The control variate method has lower (discretization) error than the other two methods by about one order of magnitude.

・ There is a rapid decrease in discretization error when using

・ Interestingly, the antithetic variate method does not reduce but rather amplifies discretization error. On a second thought, this phenomenon is not surprising at all, as in the case of the antithetic variate method each generated random number is used twice, so as to cut down on simulation error and time. However, this method also entails that any discretization error in a given simulated trajectory is emphasized (duplicated) as there are less independent trajectories used in the simulation process.

・ By taking a closer look at the absolute errors for different levels of m, we might notice patterns, such as peaks or troughs, especially in case of the first two methods. This is not a coincidence, as we use the same random number sequences for each valuation, which means that longer simulations (i.e. where T is greater) use the same random numbers as their shorter counterparts, and thus any discretization error will also be preserved. This “slowly decaying” nature of the estimation error is very useful, making it easier for us to identify those simulation methods which effectively reduce discretization error. For instance, it is very hard to discern patterns in the chart of the control variate method, which indicates that this method does exactly what it professes, namely that it reduces discretization error inherent in the simulation of the arithmetic-average by offsetting it with the one inherent in the simulation of the geometric-average.

From the above results, we can conclude that it is best to choose the control variate method, as it effectively reduces discretization error. Also, it is unnecessary to set m greater than 10, as creating more refined trajectories significantly increases simulation time but does not attain a proportionate discretization error reduction. Hence, in the rest of our work we set

From a perspective of discretization error reduction, the method of control variates is clearly the best performing one. Now, let us make a comparison between simulation methods based on the simulation error of their outputs, measured by the standard deviation of the estimate.

The estimated standard errors produced by each simulation method (using a given number of trajectories) are shown on a log-log scale in

・ The antithetic method achieves only a slight (about 15% - 20%) reduction in simulation variance compared to the basic method.

・ The control variate method achieves a substantial (about 90% - 95%) reduction in simulation variance compared to the basic method. The variance reduction is so efficient that as few as

・ The standard error from regular and antithetic methods grows only at a rate ~

The question naturally arises whether we quantify the variance reduction by the antithetic and the control variate methods. For this, we need to estimate the correlation of antithetic variates and control variates, respectively. As we can see in

We know from Equations (15) and (16) that

and (for a variance-optimal

provided

we can quantify the variance reduction. In the case of the antithetic method,

From the results presented we may conclude that the control variate method effectively reduces both discretization error and simulation error even for

Having compared the different numeric and simulation methods separately and having identified the optimal parameter values for each, let us compare the two approaches in the search for an answer to our second research question. Namely, given today’s computational capacity what method (algorithm) should we use for pricing Asian options?

We have seen that the Euler and Talbot algorithms are powerful tools for pricing Asian options, though being rather sensitive to low values of

・ the Euler algorithm to calculate the Asian option price to the desired four-decimal precision;

n | |
---|---|

<0.025 | 10000 |

<0.05 | 100000 |

<0.1 | 1000000 |

・ the control variate algorithm to run a simulation for a given n, showing each curve only until the desired precision can be maintained.

Let us note that while the time required by numeric algorithms is fundamentally a function of

Based on the figures, we claim that the control variate algorithm generally beats numerical algorithms in terms of calculation time for usual values of

By maintaining a practical approach to the valuation problem in this paper, we pay special attention to the implementation of the numeric algorithms and to the specific computational issues that arose in MATLAB®. In Section 3, we test the two semi-analytical (numerical) methods and the three simulation methods for different option parameter settings, striving to answer the questions how the efficiency of the different algorithms as well as their relative performance have changed due to the technological development of the last decade.

Pointing out the importance of

Based on the simulation results in Section 3.2 we confirm that the method of using the geometric-average Asian option price as control variate is an extremely efficient one, rendering traditional Monte Carlo methods useless in the particular valuation problem of arithmetic-average Asian options. Indeed, whereas for the latter methods the generation of 100 000 trajectories is not sufficient to price the option with the desired (four- digit) accuracy, the control variate method requires only 10,000 trajectories to fulfill our requirements, taking fractions of a second to compute the Asian option price.

Hence, we can experience an interesting dichotomy between numerical and simulation methods: one approach tends to excel when the other shows inferior performance and vice versa, depending on the value of

Regarding the comparison of the numerical and simulation approaches let us make two subtle but crucial remarks. Firstly, we cannot emphasize enough the geometric- average Asian option price’s importance in this particular valuation problem. Had it not been for this control variate’s fortunate availability, the simulation approach would have performed far worse than in this special case. Thus, before waving aside the Laplace transform inversion methods for their notorious sensitivity to small values of

Secondly, let us point out the Laplace transform method’s weakness, namely that it is limited by the assumptions of the Black-Scholes model. Presumably the strongest of these tenets is the constant volatility assumption, the relaxation of which could fundamentally alter the Laplace transform formula derived by Geman and Yor [

optimal method | |
---|---|

<0.025 | MC with control variate ( |

<0.1 | Euler algorithm ( |

>0.1 | Talbot algorithm ( |

the assumption of geometric Brownian motion is relaxed (e.g. log-returns are modeled by the more general Lévy processes), the validity of the numerical results is questionable. In contrast, the flexibility of simulation methods makes it possible to adjust the original valuation framework to a wider range of assumptions, thus enabling market participants to use more realistic stock processes for Asian option pricing.

Horvath, A. and Medvegyev, P. (2016) Pricing Asian Options: A Comparison of Numerical and Si- mulation Approaches Twenty Years Later. Journal of Mathematical Finance, 6, 810- 841. http://dx.doi.org/10.4236/jmf.2016.65056

1 function x = Euler(n, P, S, K, T, sigma)

2 %% Recommended parameters by Abate-Whitt (1995).

3 m = 11;

4 A = sym(18.4);

5

6 %% Parameters of the normalized option price.

7 h = sym((T*sigmaˆ2)/4);

8 q = sym((K*T*sigmaˆ2)/(4*S));

9 nu = sym(−2*log(P)/(sigmaˆ2*T) − 1);

10

11 %% Lambda calculation (Euler algorithm).

12 arg = sym(zeros((n + m) + 1, 1));

13 for k = 0:(n + m)

14 arg(1 + k) = (A + 2*k*pi*1i)/(2*h);

15 end

16

17 if sum(single(abs(arg)) < max(0, 2*(single(nu) + 1))) ˜= 0

18 x = −999; % An easy to identify numeric error output.

19 return

20 end

21

22 try

23 %% Substitution into the Laplace transform.

24 mu = sqrt(2*arg + nuˆ2);

25 alpha = (mu + nu)/2;

26 beta = (mu − nu)/2;

27

28 integrand = @(alpha, beta, q, u) u.ˆ(double(beta) − 2).* ∙∙∙

29 (1 − u).ˆ(double(alpha) + 1).*exp(−u/(2*double(q)));

30 a = sym(integral(@(u)integrand(alpha, beta, q, u), 0, 1, ∙∙∙

31 'ArrayValued', true, 'AbsTol', 0, 'RelTol', 1));

32 a = log(a) + log(2*q)*(1-beta) − log(2.*arg) ∙∙∙

33 −log(alpha + 1) − mfun('lnGAMMA', double(beta));

34 a = real(exp(a));

35

36 %% Summation (Euler algorithm).

37 s = (exp(A/2)/h).*((−1).ˆ(0:(n + m)))’.*a;

38 s(1) = s(1)/2;

39 for k = 1:(n+m)

40 s(1 + k) = s(1 + k) + s(k);

41 end

42

43 x = sym(0);

44 for k = 0:m

45 x = x + nchoosek(m, k)*2ˆ(−m)*s(1 + (n + k));

46 end

47

48 x = x*P*(S*4)/(T*sigmaˆ2);

49 x = double(x);

50

51 catch

52 x = double(−999);

53 end

54

55 end

A.2. Talbot Algorithm1 function x = Talbot(M, P, S, K, T, sigma)

2 %% Parameters of the normalized option price

3 h = sym((T*sigmaˆ2)/4);

4 q = sym((K*T*sigmaˆ2)/(4*S));

5 nu = sym(−2*log(P)/(sigmaˆ2*T) − 1);

6

7 %% Lambda calculation (Talbot algorithm).

8 d = sym(zeros(M,1));

9 d(1) = 2*M/5;

10 for k = 1:(M − 1)

11 d(k + 1) = 2*k*pi/5 * (cot(k*pi/M) + 1i);

12 end

13

14 g = sym(zeros(M,1));

15 g(1) = 1/2 * exp(d(1));

16 for k = 1:(M − 1)

17 g(k + 1) = (1 + 1i*(k*pi/M)*(1 + (cot(k*pi/M))ˆ2) ∙∙∙

18 −1i*cot(k*pi/M))*exp(d(k+1));

19 end

20

21 arg = d/h;

22

23 if sum(single(abs(arg)) < max(0, 2*(single(nu) + 1))) ˜= 0

24 x = −999; % An easy to identify numeric error output.

25 return

26 end

27

28 try

29 %% Substitution into the Laplace transform.

30 mu = sqrt(2*arg + nuˆ2);

31 alpha = (mu + nu)/2;

32 beta = (mu − nu)/2;

33

34 integrand = @(alpha, beta, q, u) u.ˆ(double(beta) − 2).* ∙∙∙

35 (1 − u).ˆ(double(alpha) + 1).*exp(−u/(2*double(q)));

36 a = sym(integral(@(u)integrand(alpha,beta,q,u), 0, 1, ∙∙∙

37 'ArrayValued', true, 'AbsTol', 0, 'RelTol', 100));

38 a = log(a) + log(2*q) * (1-beta) − log(2.*arg) ∙∙∙

39 −log(alpha + 1) ? mfun ('lnGAMMA', double(beta)) + log(g);

40 a = real(exp(a));

41

42 %% Summation (Talbot algorithm).

43 x = sym(0);

44 for k = 0:(M − 1)

45 x = x + a(k + 1);

46 end

47

48 x = x*2/(5*h)*P*(S*4)/(T*sigmaˆ2);

49 x = double(x);

50

51 catch

52 x = double(−999);

53 end

54

55 end

A.3. Behavior of the Laplace Transform