PERFORMANCE MEASURE OF LAPLACE TRANSFORMS FOR PRICING PATH DEPENDENT OPTIONS

This paper presents a performance measure of Laplace transforms for pricing path dependent options. We obtain a simple expression for the double transform by means of Fourier and Laplace transforms, (with respect to the logarithm of the strike and time to maturity) of the price of continuously monitored Asian options. The double transform is expressed in terms of Gamma functions only. The computation of the price requires a multivariate numerical inversion. Under jump-diffusion model, we show that the Laplace transforms of lookback options can be obtained through a recursion involving only analytical formulae for standard European call and put options. We also show that the numerical inversion can be performed with great accuracy and low computational cost. AMS Subject Classification: 11A25, 44A10, 42A38, 91G20, 91G80, 91G99


Introduction
As stock markets have become more sophisticated, so have their products.The Received: February 11, 2014 c 2014 Academic Publications, Ltd. url: www.acadpubl.eu§ Correspondence author simple buy or sell trades of the early markets have been replaced by more complex financial options and derivatives.These contracts can give investors various opportunities to tailor their deals to their investment needs.
One of the main concerns of financial derivatives is to obtain the exact values of options.For the simplest model in the case of constant coefficients, an exact pricing formula was derived by Black and Scholes-the well known Black-Scholes formula.However, in the general case of time and space dependent coefficients the exact pricing formulae are not yet known, and thus numerical solutions have been used.
Financial engineers have created various exotic products to meet the different market needs.These products are designed to meet a genuine hedging need in the market, i.e. they are "tailor-made" and to reflect a corporate treasurer's view on potential future movements in particular market variables.
A path dependent option is a contingent claim whose value depends on the sequence of prices of the underlying asset during the whole or part of the option's life rather than just the final price of the asset.Examples of path dependent options are Asian and lookback options.Asian options are path dependent options whose payoff functions depend on the average stock price over a specific period of time called the life of the option.The Asian Call option gives the holder of the option the right to buy an underlying security but under no obligation.The Asian put option gives the holder the right to sell.There are two types of Asian options with regards to the average computation: arithmetic and geometric.For the geometric Asian options there is a closedform solution to the value of these options.However, this is not the case for the arithmetic type because the arithmetic average of a set of lognormal random variables is not lognormally distributed.Until now there has been no closedform solution to the value of these types of option [21].Asian options reduce the possibility of market manipulation near the expiration date and offer a better hedge to firms with a stream of positions.Because of this, they have lower volatility and hence rendering them cheaper relative to their European counterparts.The lookback option is defined as a financial derivative whose strike price corresponds to the minimum or maximum price recorded by the underlying asset during the option's life.Lookback call(put) gives the option the right to buy(sell) an asset at its lowest(highest) price during the life of the option.Obviously, the more flexible the option, the more expensive it will be.
In the recent years the complexity of numerical computation in financial theory and practice has increased greatly, putting more demands on computation speed and efficiency.
Numerical methods are needed for pricing options in cases where analytic solutions are either unavailable or not easily obtained.They are used for a variety of purpose in finance.These include the valuation of securities, the estimation of their sensitivities, risk analysis and stress testing of portfolios.
In this paper we consider the pricing of Asian options by computing a Laplace transform with respect to time.The Fourier transform with respect to the logarithm of the strike price is used as a technique to invert Laplace transform.Such double transform is related to the characteristic function of a normal random variable.The double transform can easily be expressed in terms of gamma functions.[1] show that the double transform is easily extend to the computation of the greeks, like delta and gamma.In order to numerically invert the double transform and obtain the option price, they use a multivariate version of the Fourier-Euler algorithm.They show that this numerical inversion is highly accurate with low volatility levels compare to Monte Carlo method.
Under the standard Black-Scholes framework, the arithmetic average of prices is a sum of correlated lognormal distributions.Since the distribution of this sum does not admit a simple analytical expression, several approaches have been proposed to price Asian and lookback options [8].The Laplace transform approach was considered by H. Geman and M. Yor [6].They used a Laplace transform to price Asian option.They also exploit the relationship between the Geometric Brownian motion and the Bessel process with a stochastic time change and the additivity property of the Bessel process.Moreover, the numerical inversion of the Laplace transform given in [6] cannot be performed at low volatility levels, due to limited computer precision.Instead, the numerical inversion of the double transform can be computed with accuracy at low volatility.The logarithmic moments was considered by G. Fusai and A. Tagliani [5].
Monte Carlo method for pricing some path dependent options was considered by C. R. Nwozo and S. E. Fadugba [10].The competitive Monte Carlo methods for the pricing of Asian options was considered by B. Lapeyre and E. Temam [9].
G. Petrella and S. G. Kuo [12] used a laplace transform to price discretely monitored barrier and lookback options.L. C. G. Rogers and Z. Shi [13] transformed the problem of pricing Asian options into a means of solving a parabolic partial differential equations in two variables from the second order, but there is no analytical solution for this partial differential equation, and its numerical solution is not accurate.They also derive lower bound formula for Asian options by computing the expectation based on a zero-mean Gaussian variable.
The approach used in this paper is different from M. C. Fu et al [4] who investigated a double transform of the option price but with respect to time and strike price.They obtained a complicated expression in terms of non-standard functions, since their result is related to the Laplace transform of a lognormal variable, which does not admit an analytical expression.Moreover, their double transform proves hard to invert numerically.
In this paper, we shall consider a performance measure of the Laplace transform for pricing path dependent options namely; Asian and lookback options.

The Double Transform for Pricing Asian Option
This section presents the double transform for pricing Asian option.

The Laplace Transform for the Asian Option
We begin with the assumption that the risk-neutral process for the underlying asset is given by a stochastic differential equation.
where W t is a Brownian motion or wiener process, r is the interest rate, t is the time and σ is the volatility.Under this condition, in order to price continuously monitored Asian option, we need the probability density function of the random variable S i.e.
The payoff of a fixed strike Asian option is given by The case of floating strike Asian options which is characterized by a payoff max S 0 At t − S t , 0 can be dealt with by using the parity result in [7].The presence of a continuous dividend yield q can be taken into account in order to replace r by (r − q) and the spot price by S 0 e −qt .If the interest rate or volatility is not constant, then the pricing of the Asian option becomes more difficult.
We obtain the price of the Asian option by computing the discounted expected value: where E 0 is the expected value under the risk-neutral probability measure and J = ( K S 0 )t.In order to compute this expectation, we first use the scaling property of the Brownian motion to express A t as where where f D is the density function of the random variable D (v) h ; J ≡ K 4P σ 2 ; and t ≡ 4h σ 2 .After a final change of variable, w = ln x, we are interested in the function: where k = ln P .Note that we have used the fact that the density law of the logarithm of a random variable is related to the density of the same random variable by the relationship: We shall compute the analytical expression of the double transform c(k, h) for Laplace and Fourier with respect to h and k respectively.Following [4], we multiply ( 8) by an exponentially decaying function e −a f k , c(k, h) becomes square integrable in k over the negative axis.Therefore, we replace the function c(k, h) by c(k, h; a f ), where c(k, h; a f ) ≡ c(k, h)e −a f k , a f > 0, and we compute the double of c(k, h; a f ) : where and Γ(.) is the gamma function of complex argument and µ 2 = 2λ + ν 2 .Also, we can obtain the delta and gamma of the Asian option respectively.After some algebra we have that: Equation ( 12) is called the delta of the Asian option.We also obtain the gamma of the Asian option from the gamma function This reduces to Equation ( 13) is the gamma of the Asian option.

The delta and gamma of the Asian option can also be recovered by numerically inverting their double transforms:
where A(γ, λ) and B(γ, λ) are a simple rescaling of the function C(γ, λ) given below respectively:

Numerical Inversion for Asian Option
To obtain the function c(k, h) by the double numerical inversion, we begin with the price of the Asian option given by The numerical Inversion of the double transform in (11) can be performed by resorting to the multivariate version of the Fourier-Euler algorithm given by where From (17) we see that the two-dimensional formula is also the iterated onedimensional formulas.In particular, if l 1 = l 2 = 1, then the expression within the braces in ( 17) is regarded as one-dimensional Euler algorithm in [1] with the one dimensional transform replaced by the two-dimensional transform f .Given the transform C(γ, λ), we first compute the Fourier inverse with respect to γ numerically.Then we invert the Laplace transform with respect to λ by using the numerical univariate inversion formula.Let L −1 and F −1 denote respectively the Laplace and Fourier inverses, then the function c(k, h) gives; Using the Fourier inversion formula, we obtain; Given that |C(γ + ia f , λ)| is integrable, in this case the trapezoidal rule is exact [1].Then, if we discretize the inversion integral by a step size ∆ f , we have that: If we set ∆ f = π k and a f = By substituting λ = a i + iw in (20) and by the means of the Bromwich contour for the inversion of the Laplace transform, where a i is at the right of the largest singularity of the function C(γ, λ).we have: Equation ( 19) can be approximated again using the trapezoidal rule with step size ∆ f = π h and by setting a i = g i 2h , with g i such that a i is greater than the right-most singularities, this yields c(k, h) ≈ e 0.5(g This is an inversion formula.M. Craddock [2] discussed the sources of error and how it can be controlled by g f and g i in (22).He also obtained through numerical experiments for these two parameters as g f = g i = 18.4.In this work Euler transformation will be used since it gives a much faster convergence for infinite sums [1,2].Specifically, the Euler sum provides an estimate E(m, n) of the series and The use of the Euler algorithm requires (m + n) evaluation of the complex function a j .In particular, Fourier and Laplace inversions require (m f +n f )(m i + n i ) evaluations of the double transform.The computational cost of the inversion is directly related to this product.In order to avoid numerical difficulties in the computation of the binomial coefficient in the Euler algorithm, we set where the choice of m f and m i has to be tuned according to the volatility level.

The Double Transform for the Lookback Options
This section presents the Laplace transform for pricing lookback options.

The Laplace Transform for the Lookback Options
In a discrete time setting the minimum(maximum) of the asset price will be determined at discrete monitoring points.We assume that the monitoring points are equally spaced in time.More precisely, consider the asset value S t , monitored in the interval [0, T ] at a sequence of equally spaced monitoring points, 0 ≡ t 0 < t 1 < ... < t m ≡ T .Let X i = log , where X i is the return between t i−1 and t i and S k = S t k = S k−1 e X k = S 0 e (X 1 ,X 2 ,...,X k ) , k = 0, 1, 2, ..., m.We consider the maxima and minima of the asset price only at the monitoring points as, Where t j−1 ≤ t < t j , standard finance theory gives the values of the floating lookback call and put options respectively at any t ∈ [0, T ] as where r is the risk-free interest rate and E * is the expectation under the riskneutral measure (the measure could be specified by arbitrage arguments for the Brownian model or by equilibrium arguments for general models).In the same way, at any time t ∈ [0, T ], for the fixed strike call and put we have respectively The price of a lookback put option under a given risk-neutral measure is given by To compute the value of E * [M 0,T | F t ] we consider any time t ∈ [t j−1,t j ], with Then from (36), (35) becomes Since A(1, t) can be computed using (36) and X j,m , we only need to compute the second term in the right hand side of (35).If t = t j is a monitoring point and S t j ≤ M 0,t j−1 , that is, whenever the previous maximum of the asset price is less than the value at the jth monitoring point (and can therefore be ignored), then the second term in the RHS of (35) is zero.However, in general, when either t is not a monitoring point or t = t j but S t j < M 0,t j−1 , it is necessary to compute the second term in the RHS of (35).For this purpose, following the Laplace transform of a call option price introduced by M. C. Fu et al [4] Φ 0,T (λ) = E ∞ 0 e −λk C(K, T, S, r, σ)dk The above equation can be written as The Corollary 1 presents the three greeks (delta, gamma and vega) of the Laplace transform for pricing Lookback put option.
Corollary 1. Laplace Transform for Pricing Lookback Put Option At any time t j−1 ≤ t < t j , with 1 ≤ j ≤ m, we have where Then from (51) we have the following greeks; Similarly, the Corollary 2 presents the three greeks (delta, gamma and vega) of the Laplace transform for pricing Lookback call option.
Corollary 2. Laplace Transform for Pricing Lookback Call Option At any time t j−1 ≤ t < t j , with 1 ≥ j ≥ m, we have Then from (55) we have the following greeks; (58) where σ is the volatility parameter and L −1 is the Laplace inversion with respect to ω.
Corollaries 3 and 4 below present some special cases of the Laplace transform for the price of a fixed strike lookback put and call options.

Corollary 3. Laplace Transform for the Price of a Fixed Strike Lookback Put Option
At any time t j−1 ≤ t < t j , m ≤ j ≤ 1, the price of a fixed strike lookback put option is given by

Corollary 4. Laplace Transform for the Price of a Fixed Strike Lookback Call Option
At any time t j−1 ≤ t < t j , j ≤ 1, the price of a fixed strike lookback call option is given by

Numerical Inversion for Lookback Option
The Euler algorithm has gained a lot of popularity in queueing and network analysis due to its simplicity of implementation, speed and high accuracy.In finance, M. C Madan et al [4] used it to price continuous Asian options by inverting the Laplace transform of the form given by [6], where The average to time t is denoted by a t , and the Laplace transform of C (ν) (., q) in the first parameter is given by We present below the Laplace inversion for floating lookback call and put options.

Laplace Inversion for Floating Lookback Put Option
Consider the Laplace transform of the form We rescale the constant C ≤ min(S t , M 0,t j−1 ) to get The purpose of introducing the arbitrary constant, C is to make sure that the Laplace inversion will not be evaluated at extreme points for a wide range of model parameters.When t is a monitoring point, we set C = min(S t , 0.99M 0,t j−1 ).Thus the function f (N, S t ) is confined to the positive real line, because N = log M 0,t j−1 C ≥ 0. We refer to the one-sided version (one dimension) of the Euler algorithm.Using N 1 = N 2 = 40 iterations to compute the partial sums ensures extremely accurate results.If t is not a monitoring point with t j−1 ≤ t < t j , we have that f (M 0,t j−1 ; S t ≤ Ce N P (X t,t j ≤ N )), since Y t j,m ≤ X t,t j and log St C ≥ 0, where N = log M 0,t j−1 C . We can then apply the results in [12] for a plain vanilla put option.We choose N = βA with if the resulting C satisfies C ≤ S t , which holds in all numerical cases.In the double exponential jump diffusion model we need to be sure the characteristic function of X t,t j has no point of catastrophe, to avoid discontinuity we set We set the number of iterations to N 1 = N 2 = 150 in all cases to ensure high accuracy in the inversion.

Laplace Inversion for Floating Lookback Call Option
Similarly, we are going to invert the Laplace transform of the form We rescale the constant C to get ).Thus the function g(N, S t ) is confined to the positive real line, because n = − log Cm 0,t j−1 ≥ 0. We refer to the one-sided version of the Euler algorithm.Using N 1 = N 2 = 40 iterations to compute the partial sums ensures extremely accurate results.If t is not monitoring point with H t j,m < X t,t j , we have g(n; S t ≤ S t E * (e Xt,t j − e −n )I {Xt,t j −n} .We can then apply the results in [12] for a plain vanilla put option.We choose n = βB with In the double exponential jump diffusion model we need to be sure that the characteristic function of X t,t j has no singular point, to avoid this we set We set the number of iterations to N 1 = N 2 = 150 in all cases to ensure high accuracy in the inversion.

Numerical Experiments
This section presents some numerical experiments of Laplace transform for pricing Asian and lookback options The parameter settings and the results generated are shown in Tables 1 and 2 below.
Example 2: We compare the performance of Laplace transform for pricing Asian call option with other methods such as [1], [10], [15] and [18] with the parameters: K = 2, t = 0, where the values of r, σ and T vary.The comparative result analysis is shown in Table 4 below.The comparative results analysis of Laplace transform and Monte Carlo methods for M = 110 and M = 120 are shown in Tables 5 and 6 below respectively.

Discussion of the Results
In Tables 1, 2 and 3, we considered how to select the values of m f and m i in the Euler algorithm in order to achieve a given accuracy and how they affect the estimate in the pricing of Asian option.We can also see from the Tables 1 and 2 below that as the volatility increases, the optimal values of m f and m i decrease quickly and consequently the computational time required for estimating the option price decreases.In Table 3, we compared the Laplace transform method with other approximations namely lognormal density derived by E. Levy [11], Crank Nicolson finite difference method with 3000 spatial and time grids.Also Crank Nicolson Finite difference method with upper bound by L. C. G. Rogers and et al [13], we can also see that the Laplace transform performs better in pricing Asian Options.From Table 4, the range of prices obtained with the Laplace transform inversion method for pricing Asian call option are very close to outputs given by the Euler numerical approach.From Tables 5 and 6, we can see that Laplace transform performs better than its counterpart Monte Carlo method when pricing the floating lookback put option.Our numerical analysis demonstrates that the Laplace Transform is accurate and perform very well for pricing path dependent options.

Conclusion
In this paper, we have considered a performance measure of Laplace transform for pricing path dependent options namely "Asian and Lookback options" against some existing approximation techniques.We also discussed the numerical inversion and obtained very accurate results, in particular for the difficult volatility levels.The outputs from our algorithm present accurate results in comparison to different numerical approximation methods available in the literatures and the numerical inversion in [6], Laplace transform seems likely to be quick if the implemented initial parameters do not take extreme values.One limitation of the Laplace transform for pricing lookback option is that the European call and put prices have to be computed accurately and fast, preferably by using analytical formula to reduce errors and to increase the speed in computing the recursions.The method could also be extended for pricing other derivatives, whose values are function of the joint distribution of the terminal asset value and its discretely monitored maximum (or minimum) throughout the lifetime of the option, such as partial lookback options.The method of Laplace transform leads to good results and is moreover less time consuming than the Euler method for example.The above results can be obtained using Mathematica.

Example 1 :
We consider the pricing of Asian option using the following parameters: 135, n i = m i + 15, m f = 55, n f = m f + 15, g i = g f = 22.4

Table 1 :
Accuracy Desired and Parameters of the Euler Algorithm

Table 2 :
The Parameters of the Euler Algorithm and Asian Option Prices

Table 3 :
The Comparative Result Analysis of Asian Option Pricing Models

Table 5 :
The Comparison of the Laplace Transform, (LT) and Monte Carlo Method, (MCM) for pricing Floating Lookback Put Option with M = 110

Table 6 :
The Comparison of the Laplace Transform (LT) and Monte Carlo Method, (MCM) for Pricing Floating Lookback Put Option with M = 120