1 Introduction
Option pricing is the core content of modern finance and has fundamental meaning for global economy. By the recent announcement of Futures Industry Association trading activity in the global exchangetraded derivatives markets, in reached a record of billion contracts, where billion of them were options contracts FIA (2019)
. The value of the global derivatives markets is estimated
trillion dollars to upwards of quadrillion dollars (including so called shadow derivatives) Stankovska (2016).American option is one of the most popular financial derivatives. It is widely accepted by investors for its flexibility of exercising time (see e.g. Longstaff et al. (2001)). Barrier options are the simplest of all exotic options traded on financial markets Weron and Weron (1998). This kind of security is a standard vanilla option which begins to be valid if the price of the underlying asset hits predetermined barrier (or barriers) before the maturity. They have become increasingly popular due to the lower costs and the ability to match speculating or hedging needs more closely than their vanilla equivalents. Moreover, barrier options play an important role in managing and modeling risks in finance as well as in refining insurance products such as variable annuities and equityindexed annuities Brockman and Turtle (2003); Engelmann et al. (2006).
Over the last two decades the BlackScholes model has been increasingly attracting interest as effective tool of the options valuation. The model was of such great importance that the authors were awarded the Nobel Prize for Economics in 1997. The classical model was generalized in order to weaken its strict assumptions, allowing such features as stochastic interest Merton (1974), jumps model Merton (1976), stochastic volatility Hull and White (1987), and transactions costs Barles and Soner (1998),Davis et al. (1993). Analysis of empirical financial records indicates that the data can exhibit fat tails (see e.g. Borak et al. (2011) and the references therein). This feature has been observed in many different markets (see e.g. Borak et al. (2011) and the references therein). Such dynamics can be observed in emerging markets where the number of sellers and buyers is low. Also an interest rate often exhibits the feature of constant periods appearing  e.g. in US between and Krzyżanowski et al. (2019b). In response to the empirical evidence of fat tails, stable distribution as an alternative to the Gaussian law was proposed Fama (1968),Mandelbrot (1997). The stable distribution has found many important applications, for example in finance Rachev and Mittnik (2000), physics Gonchar et al. (2003),Janicki and Weron (1994),Mizuuchi et al. (2005) and electrical engineering Stuck and Kleiner (1974). With increasing interest of fractional calculus and nonlocal differential operators the family of fractional BlackScholes equations has emerged in the recent literature (see e.g. Krzyżanowski et al. (2019a) and references therein).
The subdiffusive BS is the generalization of the classical BS model to the cases, where the underlying assets display characteristic periods in which they stay motionless. The standard BS model does not take this phenomena into account because it assumes the asset is described by continuous Gaussian random walk. As a result of an option pricing for such underlying asset, the fair price provided by the BS model is misestimated. In order to describe this dynamics properly, the subdiffusive BS model assumes that the underlying instrument is driven by stable inverse subordinator Sato et al. (1999). Then the frequency of the constant periods appearing is dependent of subdiffusion parameter . If , the subdifussive BS is reducing to the classical model. Due to its practicality and simplicity, the standard BS model is one of the most widely used in option pricing. Although in contrast to the subdiffusive case it does not take into account the empirical property of the constant price periods in the underlying instrument dynamics. In Figure we compare sample simulation of underlying asset in classical and subdiffusive market model. Even short stagnation of a market can not be simulated by standard BS model. As a generalization of the classical BS model, its subdiffusive equivalent can be used in wide range of markets  including all cases where BS can be applied.
Since the subdiffusive BlackScholes model was proposed Magdziarz (2009) many open problems still have remained unsolved. One of them is the way of valuation American and Exotic options. In this paper we derive the Linear Complementarity Problem (LCP) system describing the fair price of an American option in subdiffusive BS model. We apply the weighted scheme of the FD method and the LongstaffSchwartz (LS) method to solve the system numerically. We compare both methods in terms of theoretical properties and practical applications. Moreover we show how to valuate wide range of barrier options in subdiffusive BS model using the FD approach.
2 Subdiffusive BlackScholes Model
The evolution of the market is taking place up to time horizon
and is contained in the probability space
. Here, is the sample space, is a filtration interpreted as the information about history of asset price and is the “objective” probability measure. The assumptions are the same as in the classical BlackScholes model Krzyżanowski (2018) with the exception that we do not have to assume the market liquidity and that the underlying instrument, instead of Geometric Brownian Motion (GBM), follows subdiffusive GBM Magdziarz (2009):where  the price of the underlying instrument, ,  drift (constant),  volatility (constant),  Brownian motion,  the inverse stable subordinator defined as Magdziarz (2009), where is the stable subordinator Sato et al. (1999), . Here is independent of for each . In Figure we compare the samples trajectories of GBM and subdifussive GBM for given stable subordinator.
Let us introduce the probability measure
(1) 
where , . As it is shown in Magdziarz (2009), is a martingale. The subdiffusive BlackScholes model is arbitragefree and incomplete Magdziarz (2009). Despite defined in is not unique, but it is the “best” martingale measure in the sense of criterion of minimal relative entropy. It means that the measure minimizes the distance to the measure Magdziarz and Gajda (2012). Between European put and call options the putcall parity holds Magdziarz (2009).
3 Selected options
In Tables and we recall the payoff functions for options considered in this article. Recall that the payoff function is the gain of the option holder at the time .

European  American 

Plain  
Knock upandin  
Knock upandout  
Knock downandin  
Knock downandout  
Knock doubleout  
Knock doublein  


European  American 

Plain  
Knock upandin  
Knock upandout  
Knock downandin  
Knock downandout  
Knock doubleout  
Knock doublein  

Here and in the rest of the paper  strike,  value of underlying instrument at time , , , , ,  barriers.
4 Valuation of American option as Free Boundary Problem
The next proposition explains why in context of American options we will proceed only with the put options.
Proposition 4.1.
If the dividend rate , then the value of American call option is equal to its European analogue. Similarly it can be shown that if , then it is not worth to realize American put before , so in this case value of American put is equal to his European equivalent.
Proof of this fact can be found for example in Krzyżanowski (2018).
We proceed with the following main result of this section
Theorem 4.1.
The fair price of an American put option in the subdiffusive BS model is equal to , where satisfies:
and is the solution of the system
(2) 
where and is Caputo fractional derivative defined as
Proof:
We consider the subdiffusive BlackScholes Equation Krzyżanowski et al. (2019a),
(3) 
with the initial condition determining put option
(4) 
At the time we can gain at least (by exercising the option) and maybe even more. It leads us to the inequality:
(5) 
After the optimal exercise moment
can not describe the value of the option. So true is the following inequality:(6) 
At each moment we decide if it is worth to use the option, or keep it. Mathematically we can describe it as:
if we realize option or
if we keep it on. This can be written as follow:
(7) 
Combining , and we get
For the sufficiently high price of the underlying instrument we will not use the option. So:
(8) 
In analogy if the price of the underlying instrument will be low we will use the option, selling the underlying instrument for :
(9) 
5 Numerical scheme for American put option
In this section we derive the numerical scheme for American put option. To do so, we will approximate limits by finite numbers and derivatives by finite differences. We will proceed for a convex combination of explicit () and implicit () discrete scheme, similarly as it was done for European options in Krzyżanowski et al. (2019a). We introduce parameter by optimization purposes  similarly as for the case , has the best properties in terms of the error and unconditional stability/convergence Krzyżanowski (2018). Instead of the continuous space , we take its discrete and finite equivalent , where ,  lower and upper boundary of the grid. We consider a uniform grid, so and , where , , , . After obtaining the discrete analogue of we will solve it recursively. As a result we will find its numerical solution , , .
We begin discretizing initial condition for put option , we get
(10) 
for .
Similarly the discrete version of boundary conditions and has the form
(11) 
for .
Discretizing we get
(12) 
for .
Here
, such that:
, such that:
, , ,
Note that the analogical scheme for the European option Krzyżanowski et al. (2019a) is
(13) 
with corresponding boundary conditions
(14) 
and initial condition for a call option
(15) 
where , .
6 Numerical schemes for Barrier options
The systems and can be used to price different types of options, only if the initialboundary conditions will be properly modified. Note that the initial condition defines type of an option (call or put), and determine style (American or European) and boundary conditions indicate it is barrier or plain option. Let us treat and not as approximations of infinite values, but as logarithm of lower and supreme barriers and defined in double barrier option. We take a logarithm because of change of variables made in Theorem and in Krzyżanowski et al. (2019a). The initial
(16) 
and boundary conditions
(17) 
where
, ,
together with is the scheme for the European double knockout call option. The same boundaryinitial conditions with create the scheme for the American double knockout call option.
Analogously, prices of oneside barrier knockout options can be obtained. Hence we have initial and boundary conditions for knockoutanddown call option
(18) 
(19) 
, , ,
and for knockoutandup call option
(20) 
(21) 
, , .
If we want to price the knockin options, it is helpful to use the fact that for fixed parameters there holds the so called inout parity
(22) 
where  the price of Vanilla (plain) option,  option prices of knockin and knockout of the same type and style.
Please note, that the value of a double knockout option for outside of the interval (but being a positive number) is equal . Analogous remark applies for onesided barrier options.
To summarize we present the way to price the considered options in Tables , and .
Style of option 
Numerical Scheme  Boundary conditions  Initial condition  Apply inout parity? 

Plain  No  
Knock upandin  Yes  
Knock upandout  No  
Knock downandin  Yes  
Knock downandout  No  
Knock doubleout  No  
Knock doublein  Yes 
To price the European put options we can firstly compute their call equivalents and then apply the PutCall parity. We can also use other initial conditions than in Table
Style of option 
Numerical Scheme  Boundary conditions  Initial condition  Apply inout parity? 

Plain  No  
Knock upandin  Yes  
Knock upandout  No  
Knock downandin  Yes  
Knock downandout  No  
Knock doubleout  No  
Knock doublein  Yes 
where
(23) 
for (double options),
(24) 
for (knockdown options),
(25) 
for , (knockup options).
If there is no dividend (the case we consider in this paper), the American call is equal its European equivalent so Table 3 holds also for American call options. For American put options we have,
Style of option 
Numerical Scheme  Boundary conditions  Initial condition  Apply inout parity? 

Plain  No  
Knock upandin  Yes  
Knock upandout  No  
Knock downandin  Yes  
Knock downandout  No  
Knock doubleout  No  
Knock doublein  Yes 
Note, that for each type of barrier option, the definition of is different.
In Figures , , we compare the fair prices given for different values of for American put, European downandout call and European double knockout call option respectively.
7 Finite difference method
In this section we show consistency and give the condition providing stability/convergence of the numerical schemes considered in this paper. It is important, because if the scheme is not stable/convergent the FD method can not be used. We also present the optimal choice of parameter in terms of conservation of unconditional stability/convergence and minimization of potential numerical error. The unconditional stability/convergence is the property that numerical scheme is stable/convergent independently of and Krzyżanowski et al. (2019a).
Theorem 7.1.
For and , the truncation error of the numerical scheme satisfies
Moreover if

or

or , and the inequality
(26) 
holds, then the scheme with boundaryinitial conditions is stable and convergent. We obtain the same result for all considered in this article knockout options, since the initialboundary conditions have no influence for consistencestabilityconvergence analysis. Indeed, if the boundary conditions are known (fixed) values then the numerical error corresponding to these conditions is equal , similarly as in the case of European call option Krzyżanowski et al. (2019a).
The proof is the same as for theorems , and of Krzyżanowski et al. (2019a).
We recall the observation from Krzyżanowski et al. (2019a) that the optimal choice of for given
is such that .
Then the lowest boundary for an error is achieved without loosing the unconditional stability/convergence. For
the stability/convergence is not provided. In Figure the relation between the fair price of American put and is presented. The real price of the option
is close to . The jump presented in the figure is the result of the increasing error. It is the consequence of lack of the stability.
7.1 LongstaffSchwartz method
The LongstaffSchwartz method is one of the most popular approaches for valuing American/Bermudan options and their Asian equivalents Korn et al. (2010); Longstaff and Schwartz (2001). Moreover it has an important applications in solving dynamic investment portfolio problems and in American/Bermuda style swaptions valuation (see e.g. Longstaff et al. (2001) and references therein). All these applications have an important meaning in finance. Only for notional amount of interest rate swaps outstanding at the end of the losses caused by wrong exercise strategies were estimated on billions of dollars Longstaff et al. (2001).
The main idea of this method is the use of least squares to estimate the conditional expected payoff to the option holder from continuation. This strategy allows to find the value of an option with the optimal exercise time (i.e. the moment where exercising option is the most profitable, if there are more than one such moments we choose the lowest of them). The method was introduced for the classical BS model but it can be extended for many other cases Longstaff and Schwartz (2001). In this paper we focus on American option, but note that the same method can be used to price Bermudan and Americanstyle Asian options. The LS algorithm can be found in Appendix.
Note that the inverse stable process is not Markovian so the expected value in could be taken not only under current but also previous states Longstaff and Schwartz (2001). Such proceeding could increase precision but will cost significant gain of running time. We decided to simplify the algorithm considering the expected value in only by the current state. Interesting could be problem of optimal choice of the set of states  e.g. using statistical background.
The LS method has its limitations. As it is shown in Mostovyi (2013), for continuous underlying process and small values of the method is unstable. The reason is the illcondition of the underlying regression problem Mostovyi (2013). The analytical formulas indicating regime where the method is stable and where is not are still unknown. This fact limitates the range of possible applications of the algorithm. Note that the error in LS method could be of different origin. The first is produced by discretization the continuous stochastic processes into nodes and assumption that the American option can be exercise only at these points. The second is related to Monte Carlo method i.e. that we estimate the expected value by the mean of size . The next possible origin of the error is coming from approximation of the conditional expected value by the average of basis functions. The last possible type of the error is produced for nonMarkovian underlying processes. Since these processes have memory, the expected value should be conditioned not only under the current state but also under the whole history of the underlying asset. Since in the algorithm the stochastic process is considered only at discrete nodes, even conditioning by current and all previous states produce an error. In contrast to LS, the FD method produces an error coming only from discretization of the variables (and approximation of infinities by and ), therefore its stability/convergence is easier to analyze.
7.2 Numerical examples
We compare both methods presented in this paper in pricing American put option. Simulations are made for , , , , , , , , , and different values of and .
Figure presents a comparison between methods FD and LS for particular choices of . We see, that as increases, the results of both methods are closer (because LS algorithm is based on the Monte Carlo, oscillations are visible). As gets higher, the LS result is closer to the FD output. Also the time of computation is visibly higher for lower values of and . In Figure there is a comparison between FD and LS in estimating the fair price of the American put option for “small” and “big” parameter . Also the relation between running time of both methods and is provided. In both figures the FD is the reference method. The LS method requires to generate (and save) paths of and that is memory/speed expensive. As decreases, more time for generating is required. For , , so the dynamics of is ”faster” than its classical equivalent (). Thus, the errors caused by approximating the nonMarkovian process by Markovian approach can accumulate with errors caused by illcondition of the regression problem. Even for ”big” the LS method is running visibly longer and has lower precision than the FD. The advantage of LS is finding sample of optimal exercise times, which is not provided by the FD. We can conclude that the presented LS algorithm could not work correctly for ”small” (in particular if ). For small the method is inefficient. The use of the method should be considered only if there is a need of optimal exercise strategy. For computing the fair price of American option the FD approach is recommended.
8 Summary
In this paper:

We have derived the system of equations describing the fair price of American put option in subdiffusive BS model.

We have introduced the weighted numerical scheme for this system.

We have given condition under which the discrete scheme is stable and convergent. We have given the order of convergence.

We have given the formula for the optimal choice of discretization parameter in dependence of subdiffusion parameter . Such numerical scheme has the lowest numerical error without loosing unconditionally stability/convergence.

We have shown how to modify previous results for valuing wide range of barrier options in frame of the same model.

We have applied the LongstaffSchwartz method for the subdiffusive BS model. This method is worse than the FD method in terms of speed and precision of computation. Moreover for small values of this method is unstable so can not be used in many different cases. By the other hand with the fair price of an option, the LS method finds also the optimal exercise strategy, what is not provided for FD method.

We have presented some numerical examples to illustrate introduced theory.
Acknowledgements
This research was partially supported by NCN Sonata Bis 9 grant nr 2019/34/E/ST1/00360.
Appendix A LongstaffSchwartz Algorithm
The method uses a dynamic programming to find the optimal stopping time and Monte Carlo to approximate the fair price of an option. We start assuming the exercise time is equal . Going backwards to , we replace by each moment we find where is better to exercise. Let us denote as the fair price of an American option with the payoff function and underlying asset . It is easy to conclude that
Let us divide the interval into subintervals (of the same length) using the grid , moreover we introduce
(27) 
where is the optimal exercise moment in , . The interpretation of the function is the expected profit from keeping the option up to time . For each trajectory we will proceed using the following algorithm
In other words at each grid point from we compare profit from keeping and exercise the option and then decide is it more profitable to exercise the option or keep it on. In the algorithm above . The key question is how to estimate values for . To do so, Longstaff and Schwartz proposed to use least squares regression. This can be done since the conditional expectation is an element of space, so it can be represented using its infinite countable orthonormal basis. To proceed with the computations, the finite set of such basis elements should be chosen. For the simulations we choose 3 first elements of Laguerre polynomials , , . Note that the early exercise at can be profitable only if , i.e. if option is in the money. The whole LS algorithm for the subdiffusive case will look as follow:
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