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since the beginning of financial science , stock prices , option prices and other quantities have been described by stochastic and partial differential equations . since the 1980s however , the path integral approach , created in the context of quantum mechanics by richard feynman @xcite , has been introduced to the field of finance @xcite . earlier , norbert wiener @xcite , in his studies on brownian motion and the langevin equation , used a type of functional integral that turns out to be a special case of the feynman path integral ( see also mark kac @xcite , and for a general overview see kleinert @xcite and schulman @xcite ) . the power of path - integration for finance ( @xcite,@xcite )  lies in its ability to naturally account for payoffs that are path - dependent . this makes path integration the method of choice to treat one of the most challenging types of derivatives , the path - dependent options . feynman and kleinert @xcite showed how quantum - mechanical partition functions can be approximated by an effective classical partition function , a technique which has been successfully applied to the pricing of path - dependent options ( see ref . @xcite and references therein , and refs . @xcite for recent applications ) . there exist many different types of path - dependent options . the two types which are considered in this paper are asian and barrier options . asian options are exotic path - dependent options for which the payoff depends on the average price of the underlying asset during the lifetime of the option @xcite . one distinguishes between _ average price _ and _ average strike _ asian options . the average price asian option has been treated in the context of path integrals by linetsky @xcite . the payoff of an average price is given by @xmath0 and @xmath1 for a call and put option respectively . here @xmath2 is the strike price and @xmath3 denotes the average price of the underlying asset at maturity @xmath4 . @xmath3 can either be the arithmetical or geometrical average of the asset price . average price asian options cost less than plain vanilla options . they are useful in protecting the owner from sudden short - lasting price changes in the market , for example due to order imbalances @xcite . average strike options are characterized by the following payoffs : @xmath5 and @xmath6 for a call and put option respectively , where @xmath7 is the price of the underlying asset at maturity @xmath4 . barrier options are options with an extra boundary condition . if the asset price of such an option reaches the barrier during the lifetime of the option , the option becomes worthless , otherwise the option has the same payoff as the option on which the barrier has been imposed . ( for more information on exit - time problems see ref . @xcite and the references therein )    in section [ average strike option ] we treat the geometrically averaged asian option . in section [ 1 ] the asset price propagator for this standard asian option is derived within the path integral framework in a similar fashion as in ref . @xcite for the weighted asian option . the underlying principle of this derivation is the effective classical partition function technique developed by feynman and kleinert @xcite . in section [ 2 ]  we present an alternative derivation of this propagator using a stochastic calculus approach . this propagator now allows us to price both the average price and average strike asian option . for both types of options this results in a pricing formula which is of the same form as the black - scholes formula for the plain vanilla option . our result for the option price of an average price asian option confirms the result found in the literature @xcite . for the average strike option no formula of this simplicity exists as far as we know . our derivation and analysis of this formula is presented in section [ 3 ] , where our result is checked with a monte carlo simulation . in section [ 4 ]  we impose a boundary condition on the asian option in the form of a barrier on a control process , and check whether the method used in section [ average strike option ] is still valid when this boundary condition is imposed on the propagator for the normal asian option , using the method of images . finally in section [ 5 ] we draw conclusions . the path integral propagator is used in financial science to track the probability distribution of the logreturn @xmath8 at time @xmath9 , where @xmath10 is the initial value of the underlying asset . this propagator is calculated as a weighted sum over all paths from the initial value @xmath11 at time @xmath12 to a final value @xmath13 at time @xmath14@xmath15   \,dt\right)\ ] ] the weight of a path , in the black - scholes model , is determined by the lagrangian@xmath16   = \frac{1}{2\sigma^{2}}\left [   \dot { x}-\left (   \mu-\frac{\sigma^{2}}{2}\right )   \right ]   ^{2 } \label{black - scholes lagrangiaan}\ ] ] where @xmath17 is the drift and @xmath18 is the volatility appearing in the wiener process for the logreturn @xcite .    for asian options , the payoff is a function of the average value of the asset . therefore we introduce @xmath19 as the logreturn corresponding to the average asset price at maturity @xmath4 . when @xmath3 is the geometric average of the asset price , then @xmath20 is an algebraic average . @xmath21 the key step to treat asian options within the path integral framework is to partition the set of all paths into subsets of paths , where each path in a given subset has the same average @xmath20 . summing over only these paths that have a given average @xmath20 defines the conditional propagator @xmath22 : @xmath23   \,dt\right )   \label{conditionele propagator}\ ] ] this is indeed a partitioning of the sum over all paths:@xmath24 the delta function in the sum @xmath25 over all paths picks out precisely all the paths that will have the same payoff for an asian option . the calculation of @xmath26 is straightforward ; when the delta function is rewritten as an exponential , @xmath27   + \frac{1}{t}ikx(t)\right )   dt\right )   , \ ] ] the resulting lagrangian is that of a free particle in a constant force field in 1d . the resulting integration over paths is found by standard procedures @xcite : @xmath28   ^{2}\right . \nonumber \\ &   \left . -\frac{6}{\sigma^{2}t}\left (   \bar{x}_{t}-\frac{x_{t}}{2}\right ) ^{2}\right \ }   , \label{conditionele propagator uitgerekend}\ ] ] and corresponds to the result found by kleinert @xcite and by linetsky @xcite .      the conditional propagator @xmath29 is interpreted in the framework of stochastic calculus as the joint propagator @xmath30 of @xmath31 and its average @xmath20 . the calculation of @xmath32 here is similar to the derivation presented in ref . @xcite  where this joint propagator is calculated for the vasicek model . the main point is that in a gaussian model the joint distribution of the couple @xmath33 has to be gaussian too . as a consequence this joint distribution is fully characterized by the expectation values and the variances of @xmath31 and @xmath20  and by the correlation between these two processes . the expectation value of @xmath34 is given by @xmath35 , its variance by @xmath36 and the correlation between the two processes by @xmath37 . the density function of such a gaussian process is then known to be@xmath38   ^{2}+3\left [   \bar { x}_{t}-\left (   \mu-\frac{\sigma^{2}}{2}\right )   \frac{t}{2}\right ] ^{2}\right . \nonumber \\ &   \left . \left .   -3\left [   x_{t}-\left (   \mu-\frac{\sigma^{2}}{2}\right ) t\right ]   \left [   \bar{x}_{t}-\left (   \mu-\frac{\sigma^{2}}{2}\right ) \frac{t}{2}\right ]   \right \ }   \right)\end{aligned}\ ] ] this agrees with eq . ( [ conditionele propagator uitgerekend ] ) for @xmath22 .      if the payoff at time @xmath4 of an asian option is written as @xmath39 , then the expected payoff is@xmath40   = { \displaystyle \int \limits_{-\infty}^{\infty } } dx_{t}{\displaystyle \int \limits_{-\infty}^{\infty } } d\bar{x}_{t}\text { } v_{t}^{asian}(x_{t},\bar{x}_{t})\mathcal{k}\left ( x_{t},t\ , \left \vert 0,0\right \vert \bar{x}_{t}\right ) \label{algemene vorm waarde van de aziatische optie}\ ] ] the price of the option , @xmath41 is the discounted expected payoff , @xmath42   \label{6}\ ] ] where @xmath43 is the discount ( risk - free ) interest rate . using expression ( [ algemene vorm waarde van de aziatische optie ] ) the price of any option which is dependent on the average of the underlying asset during the lifetime of the option can be calculated . we will now derive the price of an average strike geometric asian call option explicitly . in order to do this , expression ( [ algemene vorm waarde van de aziatische optie ] ) has to be evaluated using the payoff : @xmath44 substituting ( [ payoff ] ) in ( [ 6 ] ) yields@xmath45 where the lower boundary of the @xmath31 integration now depends on @xmath46 . when considering an average price call , the payoff ( for a call option ) is @xmath0 leading to a constant lower boundary @xmath47 for the @xmath20 integration , and the integrals are easily evaluated . in the present case however , the integration boundary is more complicated and it is more convenient to express this boundary through a heaviside function , written in its integral representation : @xmath48 now the two original integrals have been reduced to gaussians at the cost of inserting a complex term in the exponential . expression ( [ waarde van de aziatische optie 1 ] ) can be split into two terms denoted @xmath49 and @xmath50 , where@xmath51   ^{2}\right . \nonumber \\ &   \left . -\frac{6}{\sigma^{2}t}\left (   \bar{x}_{t}-\frac{x_{t}}{2}\right ) ^{2}+i\left (   x_{t}-\bar{x}_{t}\right )   \tau+x_{t}\right \}\end{aligned}\ ] ] and @xmath50 has the same form , except with @xmath20 instead of @xmath31 in the last term of the argument of the exponent . as a first step , the gaussian integrals over @xmath31 and @xmath20 are calculated , yielding@xmath52 with@xmath53\ ] ] now the integral has been reduced to a form which can be rewritten by making use of plemelj s formulae . taking into account symmetry , this reduces to@xmath54\ ] ] with@xmath55{l}\smallskip a=\dfrac{\sigma^{2}t}{6}\\ \smallskip b=\left (   \mu+\dfrac{\sigma^{2}}{2}\right )   \dfrac{t}{2}\end{array } \right.\ ] ] the first term thus becomes@xmath56   + 1\right \}\ ] ] the second term , @xmath57 is evaluated similarly , leading to @xmath58   + 1\right \ }   \right . \nonumber \\ &   \left . -\exp \left [   \left (   \mu-\frac{\sigma^{2}}{6}\right )   \frac{t}{2}\right ]   \left \ {   \operatorname{erf}\left [   \sqrt{\frac{3t}{8\sigma^{2}}}\left (   \mu-\frac{\sigma^{2}}{6}\right )   \right ]   + 1\right \ }   \right)\end{aligned}\ ] ] using the cumulative distribution function of the normal distribution @xmath59\ ] ] this can be rewritten in a more compact form as@xmath60 with the following shorthand notations@xmath55{c}\smallskip d_{1}=\sqrt{\dfrac{3t}{4\sigma^{2}}}\left (   \mu+\dfrac{\sigma^{2}}{2}\right ) \\ \smallskip d_{2}=\sqrt{\dfrac{3t}{4\sigma^{2}}}\left (   \mu-\dfrac{\sigma^{2}}{6}\right ) \end{array } \right.\ ] ] expression ( [ priceform ] ) is the analytic pricing formula for an average strike geometric asian call option , obtained in the present work with the path integral formalism . to the best of our knowledge , no pricing formula of this simplicity exists . to check this formula , we compared its results to those of a monte carlo simulation . the monte carlo scheme used is as follows @xcite : first , the evolution of the logreturn is simulated for a large number of paths . this evolution is governed by a discrete geometric brownian motion for a number of time steps . using the value for the logreturn at each time step , the average logreturn can be calculated for every path . subsequently the payoff per path can be obtained , which is then used to calculate the option price by averaging over all payoffs per path en discounting back in time . the analytical result and the monte carlo simulation agree to within a relative error of 0.3% when 500 000 samples and 100 time steps are used . this means that our analytical result lies within the error bars at every point . we also obtained the result for an average price asian option ; in contrast to the new result for the average strike option this could be compared to the existing formula @xcite , and was found to be the same . [ h ]    figure1.eps    using the propagator ( [ propagator voor aziaat met barriere ] ) the price of an asian option with a barrier @xmath78 can be calculated . the general pricing formula is given by:@xmath79 this calculation was done for an average price option : @xmath80 . the calculation , though rather cumbersome , is essentially the same as for the asian options in section [ average strike option ] . the integral over @xmath31 is a gaussian integral , and the remaining two integrals can be transformed into a standard bivariate cumulative normal distribution , defined by:@xmath81=\frac{1}{2\pi \sqrt{1-\chi^{2}}}\int_{-\infty}^{a}\int_{-\infty } ^{b}\exp \left (   -\frac{1}{2\left (   1-\chi^{2}\right )   } \left (   x^{2}+y^{2}-2\chi xy\right )   \right )   dxdy \label{cumnormdistr}\ ] ] this eventually leads to the following pricing formula for an asian option with a barrier:@xmath82   } \left (   \frac{b}{s_{0y}}\right )   ^{\frac{2\left [   \frac{4}{\xi } \left (   \nu-\frac{\xi^{2}}{2}\right )   -3\frac{\rho}{\sigma}\left (   \mu -\frac{\sigma^{2}}{2}\right )   \right ]   } { \xi \left (   4 - 3\rho^{2}\right )   } } n\left (   d_{5},d_{6},-\sqrt{\frac{3}{4}}\rho \right ) \nonumber \\ &   \left . + k~e^{\frac{3}{\sigma^{2}}x_{s}\left [   \frac{2x_{s}}{t}+\left ( \mu-\frac{\sigma^{2}}{2}\right )   \right ]   } \left (   \frac{b}{s_{0y}}\right ) ^{\frac{2\left [   \frac{4}{\xi}\left (   \nu-\frac{\xi^{2}}{2}\right ) -3\frac{\rho}{\sigma}\left (   \mu-\frac{\sigma^{2}}{2}\right )   \right ]   } { \xi \left (   4 - 3\rho^{2}\right )   } } n\left (   d_{7},d_{8},-\sqrt{\frac{3}{4}}\rho \right )   \right ]   \label{optieprijs aziaat met barriere}\ ] ] where the following shorthand notations were used:@xmath83{l}d_{1}=-\dfrac{\ln \left (   \dfrac{k}{s_{0x}}\right )   -\dfrac{t}{2}\left ( \mu+\dfrac{\sigma^{2}}{6}\right )   } { \sqrt{\dfrac{\sigma^{2}t}{3}}}\\ d_{2}=\dfrac{\ln \left (   \dfrac{b}{s_{0y}}\right )   -t\left (   \nu-\dfrac{\xi ^{2}}{2}+\dfrac{\sigma \xi \rho}{2}\right )   } { \sqrt{\xi^{2}t}}\\ d_{3}=-\dfrac{\ln \left (   \dfrac{k}{s_{0x}}\right )   -\dfrac{t}{2}\left ( \mu-\dfrac{\sigma^{2}}{2}\right )   } { \sqrt{\dfrac{\sigma^{2}t}{3}}}\\ d_{4}=\dfrac{\ln \left (   \dfrac{b}{s_{0y}}\right )   -t\left (   \nu-\dfrac{\xi ^{2}}{2}\right )   } { \sqrt{\xi^{2}t}}\\ d_{5}=-\dfrac{\ln \left (   \dfrac{k}{s_{0x}}\right )   -t\left [   2\dfrac{x_{s}}{t}+\dfrac{1}{2}\left (   \mu+\dfrac{\sigma^{2}}{6}\right )   \right ]   } { \sqrt{\dfrac{\sigma^{2}t}{3}}}\\ d_{6}=\dfrac{\ln \left (   \dfrac{b}{s_{0y}}\right )   -\dfrac{t}{\sigma}\left [ 3\xi \rho \dfrac{x_{s}}{t}+\sigma \dfrac{x_{s}}{t}+\sigma \left (   \nu-\dfrac { \xi^{2}}{2}+\dfrac{\sigma \xi \rho}{2}\right )   \right ]   } { \sqrt{\xi^{2}t}}\\ d_{7}=-\dfrac{\ln \left (   \dfrac{k}{s_{0x}}\right )   -t\left [   \dfrac{2x_{s}}{t}+\dfrac{1}{2}\left (   \mu-\dfrac{\sigma^{2}}{2}\right )   \right ]   } { \sqrt{\dfrac{\sigma^{2}t}{3}}}\\ d_{8}=\dfrac{\ln \left (   \dfrac{b}{s_{0y}}\right )   -\dfrac{t}{2\sigma}\left [ \dfrac{1}{t}\left (   6\rho x_{s}\xi+2\sigma y_{s}\right )   + 2\sigma \left ( \nu-\dfrac{\xi^{2}}{2}\right )   \right ]   } { \sqrt{\xi^{2}t}}\end{array}\ ] ]      fig . ( [ versch_corr_artikel ] ) shows the option price for an asian option with a barrier as a function of the initial asset price belonging to the @xmath63 process , defined by : @xmath84 . [ h ]    figure2.eps    this figure shows that the analytical result derived in section [ derivation of the option price ] deviates from the monte carlo simulation with increasing correlation . the approximate nature of our approach can be understood as follows . the essence of the approach presented here is that to calculate the price of asian barrier options , two steps need to be taken . first , a partitioning of paths according to the average along the path must be performed , and second , the method of images must be used in order to cancel out paths which have reached the barrier . the difficulty combining these two steps , is that mirror paths have a different average than the original paths , and thus belong to a different partition . this difficulty can apparently be overcome by treating the average itself as a separate , correlated process ( as proposed in ref . this procedure , relating @xmath85 to @xmath32 , leads to the correct propagator ( and price ) in the case of a plain asian option as shown in section [ 2 ] .    however , from the results shown in fig . [ versch_corr_artikel ] it is clear that this is no longer the case for an asian option with a barrier on a correlated control process . this is because the exact average of the @xmath62 process does not behave as a separate , correlated process ( the average described by this process is henceforth called the approximate average ) . this approach is exact for a plain asian option , where all paths contribute , but when a barrier is implemented using the method of images , and thus eliminating some of the paths , the following approximation is made . when the @xmath63 process hits the barrier and is thus eliminated , its corresponding @xmath62 and @xmath86 processes are eliminated as well . but the @xmath86 process considered in our derivation is only approximate , so the wrong @xmath86 paths are eliminated . the central question is whether this will lead to a difference between the distribution of contributing paths for the exact averages and the corresponding distribution for the approximate averages , when a barrier has been implemented . figure 3 shows that this is indeed the case , and that this difference increases when correlation increases . [ h ]    figure3.eps    when the correlation is zero , the paths which are eliminated for both the exact and the approximate average are randomly distributed ( because the behavior of @xmath86 has nothing to do with the behavior of @xmath63 ) , which means that both distributions remain the same gaussian as they would be without a barrier . this is the reason why our result is exact when correlation is zero . another source of approximation lies in the use of the black - scholes model which has well - known limitations @xcite . several other types of market models propose to overcome such limitations , for example by introducing additional ad hoc stochastic variables @xcite or by improving the description of the behavior of buyers / sellers @xcite . the extension of the present work to for example the heston model lies beyond the scope of this article . in this paper , we derived a closed - form pricing formula for an average price as well as an average strike geometric asian option within the path integral framework . the result for the average price asian option corresponds to that found by linetsky @xcite , using the effective classical partition function technique developed by feynman and kleinert @xcite . the result for the average strike asian option was compared to a monte carlo simulation . we found that the agreement between the numerical simulation and the analytical result for an average strike asian option is such that they coincide to within a relative error of less than 0.3 % for at least 500 000 samples and 100 time steps . furthermore , a pricing formula for an asian option with a barrier on a control process was developed . this is an asian option with the additional condition that the payoff is zero whenever the value of @xmath87the control process crosses a certain predetermined barrier . the pricing of this option was performed by constructing a new propagator which consisted of a linear combination of two propagators for a regular asian option . the resulting pricing formula is exact when the correlation is zero , and is approximate when the correlation increases . the central approximation made in our derivation , is that the process for the average logreturn @xmath86 is treated as a stochastic process , which is correlated with the process of the logreturn @xmath62 . this assumption is correct whenever all price - paths contribute to the total sum , but becomes approximate when a boundary condition is applied . the authors would like to thank dr . sven foulon and prof . karel in t hout for the fruitful discussions . this work is supported financially by the fund for scientific research - flanders , fwo project g.0125.08 , and by the special research fund of the university of antwerp bof noi ua 2007 .                                    d. lemmens , m. wouters , j. tempere , s. foulon , _ a path integral approach to closed - form option pricing formulas with applications to stochastic volatility and interest rate models , _ physical review . e 78 , 016101 ( 2008 ) .
we derive a closed - form solution for the price of an average price as well as an average strike geometric asian option , by making use of the path integral formulation . 
 our results are compared to a numerical monte carlo simulation . 
 we also develop a pricing formula for an asian option with a barrier on a control process , combining the method of images with a partitioning of the set of paths according to the average along the path . 
 this formula is exact when the correlation is zero , and is approximate when the correlation increases .