Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit
aa r X i v : . [ q -f i n . C P ] O c t Semi-analytic pricing of double barrier options withtime-dependent barriers and rebates at hit
Andrey Itkin and Dmitry Muravey Tandon School of Engineering, New York University, 1 Metro Tech Center, 10th floor, Brooklyn NY 11201, USA Moscow State University, Moscow, Russia
October 13, 2020 W e continue a series of papers devoted to construction of semi-analytic solu-tions for barrier options. These options are written on underlying followingsome simple one-factor diffusion model, but all the parameters of the modelas well as the barriers are time-dependent. We managed to show that these solutionsare systematically more efficient for pricing and calibration than, eg., the correspondingfinite-difference solvers. In this paper we extend this technique to pricing double barrieroptions and present two approaches to solving it: the General Integral transform methodand the Heat Potential method. Our results confirm that for double barrier options thesesemi-analytic techniques are also more efficient than the traditional numerical methodsused to solve this type of problems. Introduction
Classical problems of financial mathematics recently got new attention due to several factors. Amongthem one could mention:• Very small or even negative interest rates observed at the market, and also forced by the FederalReserve for achieving its macroeconomic goals, see, eg., (Itkin et al., 2020a) and reference therein.Therefore. financial models that allow negative rates recently redrew much attention.• Negative oil prices due to the COVID-19 pandemic and the following economic recession, see (Bou-chouev, 2020; Farrington and Cesa, 2020).• Another consequence of the COVID-19 was a huge shift to electronic trading since major optionsexchanges temporarily closed their floors, and brokers and market makers were adjusting to workingfrom home. That raised the need for real-time tools for fast calculating the option prices and Greeks,(Brogan, 2020). 1 emi-analytic pricing of double barrier options...
Those and some other aspects forced the financial society to critically reassess even simple classicalone-factor models of mathematical finance, and reanimate some of them, for instance the Ornstein-Uhlenbeck (OU) process, that traditionally have been referred to as defective/ill-posed or problematic.In (Doff, 2020) it is advocated that risk managers could even use Black-Scholes to help drive strategy.Therefore, nowadays, for instance, fast pricing of barrier options even for those simple models could beof an increasing importance. That is what this paper is devoted to as applied to double barrier options.In what follows we consider these options written on the underlying which temporal dynamics isdriven by a simple one-factor diffusion process but with time-dependent coefficients. Also, both barriersare assumed to be time dependent. Finally, when the underlying process hits any of the barriers, theCall option holder gets a rebate-at-hit (different for the upper and lower barriers), and they are also time-dependent. It is important that in this paper we consider only the underlying dynamics whose optionpricing problem by using the Feynman-Kac theorem and also some transformations could be reduced tothe heat equation. Nevertheless, to the best of our knowledge, even with this simplification a closed-formsolution of this problem is yet unknown.However, we have to mention (Mijatovic, 2010), where a similar problem was solved by using a prob-abilistic approach to obtain a decomposition of the barrier option price into the corresponding Europeanoption price minus the barrier premium for a wide class of payoff functions, barrier functions and lineardiffusions (i.e., the drift is constant and the local volatility is a function of the underlying only). For thissetting it is shown in (Mijatovic, 2010) that the barrier premium can be expressed as a sum of integralsalong the barriers of the option’s delta at the barriers, and that those deltas solve a system of Volterraintegral equations of the second kind. This is similar to the idea of the generalized integral transform(GIT) method that we use in this paper, while our setting is more general. Indeed, we allow any diffusionmodel with time-dependent coefficients and time-dependent barriers and rebates at hit subject to thecondition that the pricing partial differential equation (PDE) can be reduced to the heat equation (or,as shown in (Carr et al., 2020) to the Bessel equation). It can also be checked that the pricing PDE in(Mijatovic, 2010) by a simple change of the spatial variable can be transformed to the heat equation.Our approach advocated in this paper further extends the technique we elaborated in a series of paperswhich dealt with a similar problem for single barrier options. In (Carr and Itkin, 2020) we developed semi-analytic solutions for the barrier (perhaps, time-dependent) and American options where the underlyingstock is driven by a time-dependent OU process with a lognormal drift. This model is equivalent to thefamiliar Hull-White model in Fixed Income that was separately considered in (Itkin and Muravey, 2020).In all cases the solution was obtained by using the method of heat potentials (HP) and the GIT method.While the HP method is well-known in mathematical physics and engineering, (Tikhonov and Samarskii,1963; Friedman, 1964.; Kartashov, 2001), it is less known as applied to finance. The first use of thismethod in finance is due to (Lipton, 2002) for pricing path-dependent options with curvilinear barriers,and more recently in (Lipton and Kaushansky, 2018; Lipton and de Prado, 2020) (also see referencestherein).The GIT method is also known in physics, (Kartashov, 1999, 2001), but was unknown in finance untilthe first use in (Carr and Itkin, 2020). It is important, that it solves the problems where the underlying isdefined at the domain S ∈ [0 , y ( t )] with S being the stock price, and y ( t ) being the time-dependent barrier,however, for other domains the solution was unknown even in physics. Then in (Itkin and Muravey, 2020)the GIT solution for the first time was constructed for the domain S ∈ [ y ( t ) , ∞ ).Latter this technique was elaborated also for the CIR and CEV models, (Carr et al., 2020), and theBlack-Karasinski model, (Itkin et al., 2020a). In particular, in (Carr et al., 2020) the HP method wasfurther generalized to be capable to solving not just the heat but also the Bessel equations, and was calledthe Bessel potential (BP) method. In (Itkin et al., 2020a) the PDE is also of a special kind. It is a flavorof the time-dependent Schrödinger equation with the unsteady Morse potential (this can be obtained bythe change of variables x → − x and τ → − i τ , i = √− Page 2 of 31 emi-analytic pricing of double barrier options...
To make it rigorous, in this context a semi-analytic solution means that given a model with the time-dependent drift and volatility functions, and also with the time-dependent barriers, we obtain the barrieroption price in the explicit (analytic) form as an integral in the time t . However, this integral containsyet unknown function Ψ( t ) which solves some Volterra equation of the second kind which also obtainedin our papers. Therefore, we think that "semi-analytic" is an appropriate term. Also, in some situationsΨ( t ) can be found analytically, see eg., (Carr and Itkin, 2020; Itkin and Muravey, 2020).In addition to the explicit analytic representation of the solution, another advantage of this approachis computational speed and accuracy. As this is demonstrated in the above cited papers, our methodis more efficient than both the backward and forward finite difference (FD) methods while providingbetter accuracy and stability. To briefly explain this, let us mention that the FD method we used(and this is pretty standard) provides accuracy O ( h ) in space and O ( τ ) in time, where h, τ are thecorresponding grid steps. Since in our method the solution is represented as a time integral, it can becomputed with higher accuracy in time (eg., by using high order quadratures) , while the dependence onthe space coordinate x is explicit. Contrary, increasing the accuracy for the FD method is not easy (i.e.,it significantly increases the complexity of the method, e.g., see (Itkin, 2017)). Then the total accuracyis determined by the accuracy of solving the Volterra equation which is also determined by the orderof quadratures used to compute the integral in this equation. For instance, using Gaussian quadraturesallows small number of nodes and also high accuracy, in more detail see (Itkin and Muravey, 2020; Carret al., 2020).Also, as mentioned in (Carr et al., 2020), another advantage of our approach is computation ofoption Greeks. Since the option prices in both the HP and GIT methods are represented in closed formvia integrals, the explicit dependence of prices on the model parameters is available and transparent.Therefore, explicit representations of the option Greeks can be obtained by a simple differentiation underthe integrals. This means that the values of Greeks can be calculated simultaneously with the pricesalmost with no increase in time. This is because differentiation under the integrals slightly changes theintegrands, and these changes could be represented as changes in weights of the quadrature scheme used tonumerically compute the integrals. Since the major computational time has to be spent for computationof densities which contain special functions, they can be saved during the calculation of the prices, andthen reused for computation of Greeks.One can be curious why we need two methods - the HP and GIT, if they are used to solve the sameproblem and demonstrate the same performance. The answer is kind of elegant. As shown in (Carret al., 2020), the GIT method produces very accurate results at high strikes and maturities (i.e. wherethe option price is relatively small) in contrast to the HP method. This can be verified by looking at theexponents under the GIT solution integral which are proportional to the time τ . Contrary, when the priceis higher (short maturities, low strikes) the GIT method is slightly less accurate than the HP method,as the exponents in the HP solution integral are inversely proportional to τ . Thus, both methods arecomplementary.This situation is well investigated for the heat equation with constant coefficients. There exist tworepresentation of the solution: one - obtained by using the method of images, and the other one - by theFourier series. Despite both solutions are equal as the infinite series, their convergence properties aredifferent, (Lipton, 2002).Going back to the problem considered in this paper, we skip the explicit formulation of the model.Instead we define a wide class of models where pricing double barrier options can be translated to solvingthe heat equation with time-dependent boundaries (barriers) and time-dependent boundary conditions(rebates-at-hit). Note, that the problems considered in the above cited paper - pricing barrier andAmerican options in the time-dependent OU process, pricing barrier options in the Hull-White model,etc., also fit to this class as this is shown in the corresponding papers. Then we construct the solutionby using both the GIT and the HP methods. The latter was already shortly presented in (Itkin and Page 3 of 31 emi-analytic pricing of double barrier options...
Muravey, 2020), but for the homogeneous boundary conditions. Also, here we present full derivation ofthe explicit value of the solution spatial gradient u x at the lower x = y ( τ ) and upper x = z ( τ ) boundaries.This derivation differs from that in (Lipton and Kaushansky, 2018) (and is closer in sense to (Tikhonovand Samarskii, 1963)), but provides a similar result. Also, all the results obtained in this paper are new.The novelty of the paper is as follows. First, we construct a semi-analytical solution of the heatequation with two arbitrary moving boundaries and arbitrary time-dependent boundary conditions atthese boundaries. To the best of authors’ knowledge this problem was not solved yet.Second, various financial problems, where efficient pricing of double barrier options with rebates at hitis subject of investigation, can be reduced to this setting. As we have mentioned it already in above, theyinclude time-dependent Hull-White and OU models, the time-dependent Black-Scholes model, etc., (Carrand Itkin, 2020; Itkin and Muravey, 2020; Itkin et al., 2020a). Also, for the CIR and CEV models, wherethe pricing problem is reduced to solving the Bessel PDE with time-dependent boundaries, the latter canalso be solved in a similar manner, (Carr et al., 2020). Also, local volatility models with σ = σ ( x ) canbe also treated under this setting.Third, consider a general one-factor model dS t = µ ( t, S ) dt + σ ( t, S ) dW t , S t ( t = 0) = S . (1)where t > S t is the spot price, µ ( t, S ) is the drift, σ ( t, S ) is the volatility of the process, W t isthe standard Brownian motion under the risk-neutral measure. This model can be solved as follows. Letus split the domain of the definition of S into N intervals, and at every interval i = 1 , . . . , N approximatethe drift by a linear function of S , i.e. µ i ( t, S ) = a i ( t ) + b i ( t ) S , and the volatility - by a quadraticfunction σ i ( t, S ) = c i ( t ) + d i ( t ) S + e i ( t ) S . Then it can be shown that at every interval the correspondingpricing PDE can be transformed to the heat equation with time-dependent boundaries and the boundaryconditions. Using continuity of the solution and its gradient at every sub-boundary, this problem can besolved semi-analytically in a similar fashion. In physics this approach is called the method of multilayerheat equation, see, eg., a nice survey in (Dias). In more detail the development of this method as appliedto finance will be published elsewhere. Thus, solving (semi-analytically) the heat equation with time-dependent moving boundaries and the boundary conditions is a key element of such a method. Havingthis method in hands, pricing double barrier options for any financial model of the type Eq. (1) can bedone semi-analytically.The rest of the paper is organized as follows. Section 1 describes the double barrier pricing problem forthe time-dependent barriers and rebates at hit and shows that it can be reduced to solving inhomogeneousPDE with homogeneous boundary conditions. Section 2 describes in detail the solution of this problemby using the GIT method. We provide two alternative integral representations of the solution - one viathe Jacobi theta functions, and the other one - using the Poisson summation formula. Despite thesesolutions are equal in a sense of infinite series, their convergence properties are different. A system of theVolterra equations for the gradient of the solution at both boundaries is obtained for both representations.Section 3 provides the same development but using the HP method. The final section concludes. Let us consider a one-factor diffusion model in Eq. (1) By using a standard argument, to price optionswritten on S t as an underlying, one can apply the Feynman-Kac theorem to obtain the following partialdifferential equation (PDE) for, eg., the European Call option price ∂C∂t + 12 σ ( t, S ) ∂ C∂S + µ ( t, S ) S ∂C∂S = r ( t ) C. (2) Page 4 of 31 emi-analytic pricing of double barrier options...
Here in case of Equities we treat S t as the stock price, then r ( t ) is the deterministic interest rate. If S t is the stochastic interest rate, then r ( t ) in the RHS of Eq. (2) should be replaced with S .The Eq. (2) should be solved subject to the terminal condition at the option maturity t = TC ( T, S ) = ( S − K ) + , (3)where K is the option strike, and some boundary conditions. Below in this paper we are concentratedonly on double barrier options with moving barriers: the lower barrier at S = L ( t ) and the upper barrierat S = H ( t ) > L ( t ), so S ∈ [ L ( t ) , H ( t )].Our main assumption in this paper is that the PDE in Eq. (2) by a series of transformations of thedependent variable C ( S, t ) U ( x, τ ) and independent variables S x ( t, S ) , t τ ( t, S ) can be reducedto the heat equation ∂U∂τ = ∂ U∂x , (4)which should be solved at the new domain x ∈ [ y ( τ ) , z ( τ )] , τ ∈ [0 , τ (0 , S )], subject to the terminalcondition U (0 , x ) = U ( x ) , (5)and the boundary conditions U ( τ, y ( τ )) = f − ( τ ) , U ( τ, z ( τ )) = f + ( τ ) . (6)Here f ± ( τ ) , y ( τ ) , z ( τ ) are some continuous functions of time τ . From the financial point of view theproblem in Eq. (4), Eq. (5), Eq. (6) (the B problem) could be viewed as a pricing problem for doublebarrier options with the moving lower y ( τ ) and upper z ( τ ) barriers and the rebates f ± ( τ ) paid at hit, i.e.when the underlying process hits either the lower or the upper barrier.Note, that many well-known financial models fit this framework. For instance, the time dependent OUprocess used in (Carr and Itkin, 2020) to model barrier and American options is such an example. Also,the time-dependent Hull-White model considered in (Itkin and Muravey, 2020) for pricing barrier optionsis another example. The number of models that fit this framework could be significantly expanded if onetransforms the original PDE in Eq. (2) to its multilayer version. This approach is discussed it detail in(Itkin et al., 2020b) and will be reported elsewhere.Below we present solution of the B problem by using two analytic methods - the GIT and HP methods.As mentioned in Introduction, the methods are complementary in a sense that despite both solutions areequal, their convergence properties are different. In particular, the GIT method is more accurate at highstrikes and maturities while the HP method - at low strikes and maturities.It is worth mentioning that the B problem is with inhomogeneous boundary conditions, hence fromthe very beginning it is useful to transform it to a similar problem but with homogeneous boundaryconditions. This could be done by the change of variables u ( τ, x ) = U ( τ, x ) − A ( τ ) − B ( τ ) x, (7) A ( τ ) = − f + ( τ ) y ( τ ) − f − ( τ ) z ( τ ) z ( τ ) − y ( τ ) , B ( τ ) = f + ( τ ) − f − ( τ ) z ( τ ) − y ( τ ) , which transforms the PDE in Eq. (2) to the inhomogeneous PDE but with the homogeneous boundaryconditions ∂u∂τ = ∂ u∂x + g ( τ, x ) , (8) g ( τ, x ) ≡ − A ′ ( τ ) − B ′ ( τ ) x, ( τ, x ) ∈ R + × [ y ( τ ) , z ( τ )] ,u (0 , x ) = U ( x ) − A (0) − B (0) x ≡ u ( x ) , u ( τ, y ( τ )) = u ( τ, z ( τ )) = 0 . Page 5 of 31 emi-analytic pricing of double barrier options...
In this section we solve the problem in Eq. (8) by using the GIT method, see (Kartashov, 1999; Carr andItkin, 2020; Itkin and Muravey, 2020; Itkin et al., 2020a) and references therein. However, as mentionedin (Kartashov, 2001), an analytic solution for the domain with two moving boundaries is yet unknown.Therefore, our solution presented in this Section is new, and it extends the results of (Carr and Itkin,2020) obtained for the domain [0 , y ( τ )].In (Carr and Itkin, 2020) the authors used the GIT proposed in (Kartashov, 1999) which is a map u ( τ, x ) ¯ u ( τ, p ) of the form ¯ u ( τ, p ) = Z y ( τ )0 u ( τ, x ) sinh( x √ p ) dx, (9)where p = a + i ω is a complex number with ℜ ( p ) ≥ β >
0, and − π < arg (cid:0) √ p (cid:1) < π . Here we proceedwith a similar idea by introducing the transform¯ u ( τ, p ) = Z z ( τ ) y ( τ ) u ( τ, x ) sinh ( p [ x − y ( τ )]) dx. (10)With a special choice of the lower boundary y ( τ ) = 0 this transform replicates that one in Eq. (9) subjectto the point that here we use the spectral parameter p instead of √ p as in (Carr and Itkin, 2020).Since the kernel of Eq. (10) is time-dependent it doesn’t make much sense to apply this transformdirectly to the inhomogeneous heat equation in Eq. (8). Therefore, we represent the image ¯ u as a differenceof two other images ¯ u = 12 (¯ u + − ¯ u − ) , ¯ u ± ( τ, p ) = Z z ( τ ) y ( τ ) u ( τ, x ) e ± p [ x − y ( τ )] dx. (11)To determine ¯ u ( τ, p ) let us multiply both parts of the first line in Eq. (8) by e ± p [ x − y ( τ )] and integrateon x . These yield Z z ( τ ) y ( τ ) ∂u ( τ, x ) ∂τ e ± p [ x − y ( τ )] dx = ∂ ¯ u ± ( τ, p ) ∂τ − u ( τ, z ( τ )) e ± pz ( τ ) z ′ ( τ ) + u ( τ, y ( τ )) e ± py ( τ ) y ′ ( τ ) (12) ± py ′ ( τ ) Z z ( τ ) y ( τ ) u ( τ, x ) e ± p [ x − y ( τ )] dx = ∂ ¯ u ± ∂τ ± py ′ ( τ )¯ u ± , Z z ( τ ) y ( τ ) ∂ u ( τ, x ) ∂x e ± p [ x − y ( τ )] dx = [Φ( τ ) − B ( τ )] e ± p [ z ( τ ) − y ( τ )] + [Ψ( τ ) + B ( τ )] + p ¯ u ± ( τ, p ) , ¯ g ± ( τ, p ) ≡ Z z ( τ ) y ( τ ) g ( τ, x ) e ± p [ x − y ( τ )] dx = B ′ ( τ ) p (cid:16) e ± p [ z ( τ ) − y ( τ )] − (cid:17) ± p h A ′ ( τ ) (cid:16) − e ± p [ z ( τ ) − y ( τ )] (cid:17) + B ′ ( τ ) (cid:16) y ( τ ) − z ( τ ) e ± p [ z ( τ ) − y ( τ )] (cid:17)i . where terms proportional to u ( τ, y ( τ ) and u ( τ, z ( τ ) vanish due to the boundary conditions in Eq. (8),and by definition Ψ( τ ) = − ∂U ( τ, x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = y ( τ ) Φ( τ ) = ∂U ( τ, x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = z ( τ ) . (13)Collecting terms in Eq. (12) yields two initial value problems, one for the function ¯ u + and the otherone - for ¯ u − ∂ ¯ u ± ( τ, p ) ∂τ + ¯ u ± h ± py ′ ( τ ) − p i = [Ψ( τ ) + B ( τ )] + [Φ( τ ) − B ( τ )] e ± p [ z ( τ ) − y ( τ )] + ¯ g ± ( τ, p ) , (14) Page 6 of 31 emi-analytic pricing of double barrier options... ¯ u ± (0 , p ) = Z z (0) y (0) u (0 , x ) e ± p [ x − y (0)] dx. Each problem in Eq. (14) (for the plus and minus signs) can be solved explicitly¯ u ± ( τ, p ) = e p τ Z z (0) y (0) u (0 , x ) e ± p [ x − y ( τ )] dx (15)+ Z τ e p ( τ − s ) h [Φ( s ) − B ( s )] e ± p [ z ( s ) − y ( τ )] + (Ψ( s ) + B ( s ) + ¯ g ± ( s, p )) e ± p [ y ( s ) − y ( τ )] i ds. Note that the last term in the second integral in Eq. (15) can be re-written in a more convenient form¯ g ± ( s, p ) e ± p [ y ( s ) − y ( τ )] = B ′ ( s ) p (cid:16) e ± p [ z ( s ) − y ( s )] − (cid:17) e ± p [ y ( s ) − y ( τ )] ± p h A ′ ( s ) (cid:16) − e ± p [ z ( s ) − y ( s )] (cid:17) e ± p [ y ( s ) − y ( τ )] + B ′ ( s ) (cid:16) y ( s ) − z ( s ) e ± p [ z ( s ) − y ( s )] (cid:17) e ± p [ y ( s ) − y ( τ )] i = B ′ ( s ) p (cid:16) e ± p [ z ( s ) − y ( τ )] − e ± p [ y ( s ) − y ( τ )] (cid:17) ± p h A ′ ( s ) (cid:16) e ± p [ y ( s ) − y ( τ )] − e ± p [ z ( s ) − y ( τ )] (cid:17) + B ′ ( s ) (cid:16) y ( s ) e ± p [ y ( s ) − y ( τ )] − z ( s ) e ± p [ z ( s ) − y ( τ )] (cid:17)i . The explicit representation for ¯ u then follows from its definition in Eq. (11)¯ u ( τ, p ) = e p τ Z z (0) y (0) u (0 , x ) sinh ( p [ x − y ( τ )]) dx (16)+ Z τ e p ( τ − s ) [[Φ( s ) − B ( s )] sinh ( p [ z ( s ) − y ( τ )]) + [Ψ( s ) + B ( s )] sinh( p [ y ( s ) − y ( τ )]) + h ( s, p )] ds,h ( s, p ) = B ′ ( s ) p [sinh( p [ z ( s ) − y ( τ )]) − sinh( p [ y ( s ) − y ( τ )])]+ 1 p (cid:2)(cid:0) A ′ ( s ) + B ′ ( s ) y ( s ) (cid:1) cosh( p [ y ( s ) − y ( τ )]) − (cid:0) A ′ ( s ) + B ′ ( s ) z ( s ) (cid:1) cosh( p [ z ( s ) − y ( τ )]) (cid:3) . General theory of the heat equation tells us that the solution at the space domain a < x < b, a, b ∈ℜ − const , can be represented as Fourier series of the form, (Polyanin, 2002)) u ( τ, x ) = ∞ X n =1 α n e − π n b − a )2 τ sin (cid:18) πn ( x − a ) b − a (cid:19) Therefore, by analogy let us look for the inverse transform of ¯ u (which actually is the solution u ( τ, x ) ofEq. (8)) to be a generalized Fourier transform of the form (Carr and Itkin (2020)) u ( τ, x ) = ∞ X n =0 A n ( τ ) sin (cid:18) πn x − y ( τ ) z ( τ ) − y ( τ ) (cid:19) , (17)where A n ( τ ) are some yet unknown Fourier coefficients (weights). Applying the direct transform inEq. (10) to the series in Eq. (17) yields¯ u ( τ, x ) = Z z ( τ ) y ( τ ) ∞ X n =1 A n ( τ ) sin (cid:18) πn x − y ( τ ) z ( τ ) − y ( τ ) (cid:19) sinh ( p [ x − y ( τ )]) dx. (18) Page 7 of 31 emi-analytic pricing of double barrier options...
Using the identity Z zy sin (cid:18) πn x − yz − y (cid:19) sinh ( p [ x − y ]) dx = ( − n +1 πn ( z − y ) sinh ( p [ z − y ]) n π + p ( z − y ) , (19)we obtain another representation for ¯ u ¯ u ( τ, x ) = 1 l ( τ ) ∞ X n =1 ( − n +1 πnA n ( τ ) sinh ( pl ( τ ))[ p + i nπ/l ( τ )] [ p − i nπ/l ( τ )] , l ( τ ) = z ( τ ) − y ( τ ) . (20)Combining Eq. (20) and Eq. (16) yields the equation for A n ( τ )1 l ( τ ) ∞ X n =1 ( − n +1 πnA n ( τ )[ p + i nπ/l ( τ )] [ p − i nπ/l ( τ )] = 1sinh ( p l ( τ )) ( e p τ Z z (0) y (0) u (0 , x ) sinh ( p [ x − y ( τ )]) dx (21)+ Z τ e p ( τ − s ) [[Φ( s ) − B ( s )] sinh ( p [ z ( s ) − y ( τ )]) + [Ψ( s ) + B ( s )] sinh ( p [ y ( s ) − y ( τ )]) + h ( s, p )] ds ) . The LHS and RHS of Eq. (21) as the functions of p are analytic in the whole complex plane domainexcept the poles p ± k = ± i πk/l ( τ ) , k = 1 , , . . . , (22)because h ( s, p ) is regular and well-behaved at p →
0. Also, as this is easy to check, these poles arecommon for the LHS and RHS of Eq. (21). For what follows we need the following residuesRes p = p ± k ∞ X n =1 p + i nπ/l ( τ )] [ p − i nπ/l ( τ )] = ± l ( τ )2i πk , Res p = p ± k p l ( τ )) = ( − k l ( τ ) . (23)The Fourier coefficients A k ( τ ) can now be found from Eq. (21) by applying contour integration on p to both sides. We integrate using the contours L + k , k = 1 , , . . . , where the integration contours look likeit is depicted in Fig. 1. Thus, we have1 l ( τ ) Z L + k ∞ X n =1 ( − n +1 πnA n ( τ )[ p + i nπ/l ( τ )] [ p − i nπ/l ( τ )] dp = Z L + k pl ( τ )) ( e p τ Z z (0) y (0) u (0 , x ) sinh ( p [ x − y ( τ )]) dx + Z τ e p ( τ − s ) [Φ( s ) sinh ( p [ z ( s ) − y ( τ )]) + Ψ( s ) sinh ( p [ y ( s ) − y ( τ )]) + h ( s, p )] ds ) dp. (24)By the Cauchy’s residue theorem each integral in Eq. (24) is equal to the sum of the correspondingresidues that can be computed with the help of Eq. (23). This yields the following formula for A k ( τ ) A k ( τ ) = 2i l ( τ ) ¯ u (cid:18) τ, i πkl ( τ ) (cid:19) . (25)With allowance for Eq. (16) this can be finally represented as A k ( τ ) = 2 l ( τ ) ( e − π k l τ ) τ Z z (0) y (0) u (0 , x ) sin (cid:18) πkl ( τ ) [ x − y ( τ )] (cid:19) dx (26)+ Z τ e − π k l τ ) ( τ − s ) (cid:20) [Φ( s ) − B ( s )] sin (cid:18) πkl ( τ ) [ z ( s ) − y ( τ )] (cid:19) Page 8 of 31 emi-analytic pricing of double barrier options... + [Ψ( s ) + B ( s )] sin (cid:18) πkl ( τ ) [ y ( s ) − y ( τ )] (cid:19) + h ( k, s, τ ) (cid:21) ds ) , with h ( k, s, τ ) = − B ′ ( s ) l ( τ ) π k (cid:20) sin (cid:18) πkl ( τ ) [ z ( s ) − y ( τ )] (cid:19) − sin (cid:18) πkl ( τ ) [ y ( s ) − y ( τ )] (cid:19)(cid:21) (27) − l ( τ ) πk " (cid:0) A ′ ( s ) + B ′ ( s ) y ( s ) (cid:1) cos (cid:18) πkl ( τ ) [ y ( s ) − y ( τ )] (cid:19) − (cid:0) A ′ ( s ) + B ′ ( s ) z ( s ) (cid:1) cos (cid:18) πkl ( τ ) [ z ( s ) − y ( τ )] (cid:19) . Keeping in mind that A ( τ ) + B ( τ ) y ( τ ) = f − ( τ ) A ( τ ) + B ( τ ) z ( τ ) = f + ( τ )we re-arrange Eq. (27) as h ( k, s, τ ) = − B ′ ( s ) l ( τ ) π k (cid:20) sin (cid:18) πkl ( τ ) [ z ( s ) − y ( τ )] (cid:19) − sin (cid:18) πkl ( τ ) [ y ( s ) − y ( τ )] (cid:19)(cid:21) (28) − l ( τ ) πk " (cid:0) ( f − ) ′ ( s ) − B ( s ) y ′ ( s ) (cid:1) cos (cid:18) πkl ( τ ) [ y ( s ) − y ( τ )] (cid:19) − (cid:16) ( f + ) ′ ( s ) − B ( s ) z ′ ( s ) (cid:17) cos (cid:18) πkl ( τ ) [ z ( s ) − y ( τ )] (cid:19) . Figure 1:
Contours of integration of Eq. (21) in the complex plane p ∈ C with poles at p ± , p ± , . . . . Re p Im p •••••• ...... p +1 p − L +1 p +2 p − L +2 p + k p − k L + k Page 9 of 31 emi-analytic pricing of double barrier options...
Substituting this result into Eq. (17), we obtain the solution u ( τ, x ) of the problem Eq. (8) u ( τ,x ) = 2 l ( τ ) ∞ X n =1 sin (cid:18) πn x − y ( τ ) l ( τ ) (cid:19) ( e − π n l τ ) τ Z z (0) y (0) u (0 , ξ ) sin (cid:18) πnl ( τ ) [ ξ − y ( τ )] (cid:19) dξ (29)+ Z τ e − π n l τ ) ( τ − s ) (cid:20) [Φ( s ) − B ( s )] sin (cid:18) πnl ( τ ) [ z ( s ) − y ( τ )] (cid:19) + [Ψ( s ) + B ( s )] sin (cid:18) πnl ( τ ) [ y ( s ) − y ( τ )] (cid:19) (cid:21) ds + Z τ e − π n l τ ) ( τ − s ) h ( n, s, τ ) ds ) . This expression can be further simplified, see Appendix A. Returning back to the original variable U ( τ, x ) yields the final representation U ( τ, x ) = 2 l ( τ ) ∞ X n =1 sin (cid:18) πn x − y ( τ ) l ( τ ) (cid:19) ( e − π n l τ ) τ Z z (0) y (0) U (0 , ξ ) sin (cid:18) πnl ( τ ) [ ξ − y ( τ )] (cid:19) dξ (30)+ Z τ e − π n l τ ) ( τ − s ) h Φ( s ) sin (cid:18) πnl ( τ ) [ z ( s ) − y ( τ )] (cid:19) + Ψ( s ) sin (cid:18) πnl ( τ ) [ y ( s ) − y ( τ )] (cid:19) + β ( τ, s, n ) i ds ) + F ( τ, x ) . where β ( τ, s, n ) and F ( τ, x ) are defined in Eq. (A.2) and Eq. (A.10). Also, as can be checked fromthe definition in Eq. (A.10) that at y ( τ ) < x < z ( τ ) the function F ( τ, x ) vanishes, and F ( τ, y ( τ )) = f − ( τ ) , F ( τ, z ( τ )) = f + ( τ ). Thus, Eq. (30) solves the problem in Eq. (4) with the initial condition inEq. (5) and the boundary conditions in Eq. (6).It is worth mentioning that the exact same formalism can be developed by using another integraltransform ¯ u ( τ, p ) = Z z ( τ ) y ( τ ) sinh ( p [ z ( τ ) − x ]) u ( τ, x ) dx, with the result being same as in Eq. (29). As observed in (Carr and Itkin, 2020), the sums in Eq. (29) could be expressed via the Jacobi thetafunctions of the third kind, (Mumford et al., 1983) . Using their definition θ ( z, ω ) = 1 + 2 ∞ X n =1 ω n cos (2 nz ) , (31)and the identities ∂θ ( z, ω ) ∂z = θ ′ ( z, ω ) = − ∞ X n =1 nω n sin (2 nz ) . (32)we obtain from Eq. (29)4 ∞ X n =1 e − π n l τ ) τ sin (cid:18) nπ ( x − y ( τ )) l ( τ ) (cid:19) sin (cid:18) nπ ( ξ − y ( τ )) l ( τ ) (cid:19) = θ ( φ − ( x, ξ ) , ω ) − θ ( φ + ( x, ξ ) , ω ) , (33) Which is not a surprise since it is known that the Jacobi theta functions is the solution of the heat equation with periodicboundary conditions. As applied to the problem considered in this paper, an example is a double barrier option with zerorebate at hit. Page 10 of 31 emi-analytic pricing of double barrier options... ∞ X n =1 e − π n l τ ) ( τ − s ) sin (cid:18) nπ ( x − y ( τ )) l ( τ ) (cid:19) sin (cid:18) nπ ( ξ − y ( τ )) l ( τ ) (cid:19) = θ ( φ − ( x, ξ ) , ω ) − θ ( φ + ( x, ξ ) , ω ) , ∞ X n =1 ne − π n l τ ) ( τ − s ) sin (cid:18) nπ ( x − y ( τ )) l ( τ ) (cid:19) cos (cid:18) nπ ( ξ − y ( τ )) l ( τ ) (cid:19) = − (cid:0) θ ′ ( φ − ( x, ξ ) , ω ) + θ ′ ( φ + ( x, ξ ) , ω ) (cid:1) .ω = e − π τl τ ) , ω = e − π τ − s ) l τ ) , φ − ( x, ξ ) = π ( x − ξ )2 l ( τ ) , φ + ( x, ξ ) = π ( x + ξ − y ( τ ))2 l ( τ ) . With the help of Eq. (33) the final formula for u ( τ, x ) simplifies2 l ( τ ) h U ( τ, x ) − F ( τ, x ) i = Z z (0) y (0) U (0 , ξ ) [ θ ( φ − ( x, ξ ) , ω ) − θ ( φ + ( x, ξ ) , ω )] dξ (34)+ Z τ ( (cid:2) Ψ( s ) − f − ( s ) y ′ ( s ) (cid:3) [ θ ( φ − ( x, y ( s )) , ω ) − θ ( φ + ( x, y ( s )) , ω )]+ h Φ( s ) + f + ( s ) z ′ ( s ) i [ θ ( φ − ( x, z ( s )) , ω ) − θ ( φ + ( x, z ( s )) , ω )]+ 12 h f + ( s ) (cid:2) θ ′ ( φ − ( x, z ( s )) , ω ) + θ ′ ( φ + ( x, z ( s )) , ω ) (cid:3) − f − ( s ) (cid:2) θ ′ ( φ − ( x, y ( s )) , ω ) + θ ′ ( φ + ( x, y ( s )) , ω ) (cid:3) i) ds. Note, that if rebates at hit are not paid, the boundary conditions become homogeneous, and all termsproportional to f − ( s ) = f + ( s ) = 0 in Eq. (34) disappear. Ψ( τ ) and Φ( τ ) Taking the derivative in Eq. (34) with respect to x , having in mind that according to Eq. (32) ∂θ ( φ ± ( x, ξ ) , ω ) ∂x = πl ( τ ) ∂θ ( y, ω ) ∂y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = φ ± ( x,ξ ) = πl ( τ ) θ ′ ( φ ± ( x, ξ ) , ω ) , (35) ∂ θ ( φ ± ( x, ξ ) , ω ) ∂x = π l ( τ ) ∂ θ ( y, ω ) ∂y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = φ ± ( x,ξ ) = π l ( τ ) θ ′′ ( φ ± ( x, ξ ) , ω ) , and substituting x = y ( τ ) and x = z ( τ ), we get a system of Volterra integral equations of the secondkind to determine Ψ( τ, Φ( τ ) − l ( τ ) π h Ψ( τ ) + F x ( τ, y ( τ )) i = Z z (0) y (0) U (0 , ξ ) (cid:2) θ ′ ( φ − ( y ( τ ) , ξ ) , ω ) − θ ′ ( φ + ( y ( τ ) , ξ ) , ω ) (cid:3) dξ (36)+ Z τ ( (cid:2) Ψ( s ) − f − ( s ) y ′ ( s ) (cid:3) (cid:2) θ ′ ( φ − ( y ( τ ) , y ( s )) , ω ) − θ ′ ( φ + ( y ( τ ) , y ( s )) , ω ) (cid:3) + h Φ( s ) + f + ( s ) z ′ ( s ) i (cid:2) θ ′ ( φ − ( y ( τ ) , z ( s )) , ω ) − θ ′ ( φ + ( y ( τ ) , z ( s )) , ω ) (cid:3) + 2 πl ( τ ) h f + ( s ) (cid:2) θ ′′ ( φ − ( y ( τ ) , z ( s )) , ω ) + θ ′′ ( φ + ( y ( τ ) , z ( s )) , ω ) (cid:3) − f − ( s ) (cid:2) θ ′′ ( φ − ( y ( τ ) , y ( s )) , ω ) + θ ′′ ( φ + ( y ( τ ) , y ( s )) , ω ) (cid:3) i) ds. Page 11 of 31 emi-analytic pricing of double barrier options... l ( τ ) π h Φ( τ ) + F x ( τ, z ( τ )) i = Z z (0) y (0) U (0 , ξ ) (cid:2) θ ′ ( φ − ( z ( τ ) , ξ ) , ω ) − θ ′ ( φ + ( z ( τ ) , ξ ) , ω ) (cid:3) dξ + Z τ ( (cid:2) Ψ( s ) − f − ( s ) y ′ ( s ) (cid:3) (cid:2) θ ′ ( φ − ( z ( τ ) , y ( s )) , ω ) − θ ′ ( φ + ( z ( τ ) , y ( s )) , ω ) (cid:3) + h Φ( s ) + f + ( s ) z ′ ( s ) i (cid:2) θ ′ ( φ − ( z ( τ ) , z ( s )) , ω ) − θ ′ ( φ + ( z ( τ ) , z ( s )) , ω ) (cid:3) + 2 πl ( τ ) h f + ( s ) (cid:2) θ ′′ ( φ − ( z ( τ ) , z ( s )) , ω ) + θ ′′ ( φ + ( z ( τ ) , z ( s )) , ω ) (cid:3) − f − ( s ) (cid:2) θ ′′ ( φ − ( z ( τ ) , y ( s )) , ω ) + θ ′′ ( φ + ( z ( τ ) , y ( s )) , ω ) (cid:3) i) ds. Also, since the theta function θ ( z, ω ) solves the heat equation ∂θ ( z, i t ) ∂t = 14 π ∂ θ ( z, i t ) ∂z , the second derivatives with respect to the first argument could be expressed via the first derivatives withrespect to the second argument.However, there exists a problem with the representation in Eq. (36). Indeed, using the definition of F ( τ, x ) in Eq. (A.10) it can be checked that the derivatives F x ( τ, x ) do not exist at x = y ( τ ) and x = z ( τ )as they are proportional to the Dirac Delta δ (0). Therefore, in the next Section we attack this problemagain using an alternative representation of the solution. It is known that for the fixed spatial domain x ∈ [ y ( τ ) , z ( τ )] , y ( τ ) = 0 , z ( τ ) = const there exist tworepresentations of the solution of the heat equation: one - obtained by using the method of images, andthe other one - by the Fourier series. Both solutions are equal in a sense of infinite series, but theirconvergence properties are different, see eg., (Lipton, 2002). It turns out that for a curvilinear strip wecan also obtain an alternative representation.The solution u ( τ, x ) found in Eq. (29) already has the form of the Fourier series. However, applicabilityof the method of images for the problem Eq. (8) is not transparent due to time-dependency of theboundaries. Instead, we can find an alternative representation by using the following property known asthe Poisson Summation formula, (van der Pol and Bremmer, 1950) Proposition 2.1 (Poisson Summation formula) . Let ˆ h ( ν ) be the Fourier transform of the appropriatefunction h ( x ) ˆ h ( ν ) = Z ∞−∞ h ( x ) e − π i νx dx. The following identity holds ∞ X n = −∞ h ( n ) = ∞ X k = −∞ ˆ h ( k ) . (37) Proof.
See (van der Pol and Bremmer, 1950).Applying Eq. (37) to the functions h ( x ) = e − π x cos ( πxα ) , ˆ h ( ν ) = Z ∞−∞ e − π x β − π i νx cos ( πxα ) dx, Page 12 of 31 emi-analytic pricing of double barrier options... h ( x ) = xe − π x β sin ( πxα ) , ˆ h ( ν ) = Z ∞−∞ xe − π x β − π i νx sin ( πxα ) dx, we obtain the following identities ∞ X n = −∞ e − π n β cos ( πnα ) = s βπ e − α β ∞ X n = −∞ e − n β cosh (2 nαβ ) (38)= s β π ∞ X n = −∞ h e − β (2 n − α ) + e − β (2 n + α ) i = 2 s β π ∞ X n = −∞ e − β (2 n + α ) , ∞ X n = −∞ πne − π n β sin ( πnα ) = β / √ π ∞ X n = −∞ e − β (2 n + α ) h α + 2 n + ( α − n ) e αβn i = β / √ π ∞ X n = −∞ h e − β (2 n + α ) ( α + 2 n ) + e − β (2 n − α ) ( α − n ) i = 2 β / √ π ∞ X n = −∞ e − β (2 n + α ) ( α + 2 n ) . Since each summand in Eq. (30) can be represented in the form of the LHS of Eq. (38), by using asimple trigonometric formula for the product of sines, we immediately arrive at another form of U ( τ, x ),see Appendix B U ( τ, x ) = ∞ X n = −∞ ( Z z (0) y (0) U (0 , ξ )Υ n ( x, τ | ξ, dξ + Z τ h Φ( s ) + f + ( s ) z ′ ( s ) i Υ n ( x, τ | z ( s ) , s ) ds, + Z τ (cid:2) Ψ( s ) − f − ( s ) y ′ ( s ) (cid:3) Υ n ( x, τ | y ( s ) , s ) ds (39)+ Z τ f − ( s )Λ n ( x, τ | y ( s ) , s ) − f + ( s )Λ n ( x, τ | z ( s ) , s ) ds ) , Υ n ( x, τ | ξ, s ) = 12 p π ( τ − s ) " e − (2 nl ( τ )+ x − ξ )24( τ − s ) − e − (2 nl ( τ )+ x + ξ − y ( τ ))24( τ − s ) , Λ n ( x, τ | ξ, s ) = x − ξ + 2 nl ( τ )4 p π ( τ − s ) e − (2 nl ( τ )+ x − ξ )24( τ − s ) + x + ξ − y ( τ ) + 2 nl ( τ )4 p π ( τ − s ) e − (2 nl ( τ )+ x + ξ − y ( τ ))24( τ − s ) . Note that the Fourier series in these expressions usually converge rapidly when n grows. Similarly,taking the derivative of this series on x provides a convenient way of calculating the correspondingderivative ∂U ( τ,x ) ∂x , (DLMF). Ψ( τ ) and Φ( τ ) In Section 2.4 we managed to obtain two alternative representations of the solution of the problem, bothin a semi-analytical form. These solutions, however, depend on two yet unknown functions gradientsΨ( τ ) , Φ( τ ) that can be found by solving a system of two Volterra equations of the second kind. Theseequations are obtained by taking the derivative in Eq. (30) or Eq. (39) with respect to x and substituting x = y ( τ ) and x = z ( τ ) into thus found expressions. However, at least formally there exist a problem withmaking the last step, because at these boundaries some integrals in the system of the Volterra equationswill contain singularities. Below we describe the resolution of these problems. Page 13 of 31 emi-analytic pricing of double barrier options...
Let us again consider Eq. (39). It is easy to see that the functions ∂ Υ n ( x,τ | ξ,s ) ∂x , ∂ Λ n ( x,τ | ξ,s ) ∂x are regularonly if n = 0 , x ∈ [ y ( τ ) , z ( τ )] , ξ ∈ [ y ( s ) , z ( s )] , s → τ . At n = 0 functions ∂ Υ ( x,τ | y ( s ) ,s ) ∂x , ∂ Λ ( x,τ | y ( s ) ,s ) ∂x havea singularity when s → τ, x → y ( τ ), and functions ∂ Υ ( x,τ | z ( s ) ,s ) ∂x , ∂ Λ ( x,τ | z ( s ) ,s ) ∂x - when s → τ, x → z ( τ ).Since the functions ∂ Υ ( x,τ | y ( s ) ,s∂x , ∂ Υ ( x,τ | z ( s ) ,s∂x can be represented as a sum of double-layer potentialswith a negative sign, the limiting valueslim x → y ( τ )+0 Z τ ξ ( s ) ∂ Υ ( x, τ | y ( s ) , s ) ∂x ds, lim x → z ( τ ) − Z τ ξ ( s ) ∂ Υ ( x, τ | z ( s ) , s ) ∂x ds can be computed similar to Eq. (B.8).Applying Eq. (73) to the limits corresponding to ∂ Λ ( x,τ | y ( s ) ,s ) ∂x , ∂ Λ ( x,τ | z ( s ) ,s ) ∂x yieldslim x → y ( τ )+0 Z τ f − ( s ) ∂ Λ ( x, τ | y ( s ) , s ) ∂x ds (40)= − f − ( τ ) √ πτ + Z τ f − ( s ) e − ( y ( τ ) − y ( s ))24( τ − s ) − f − ( τ )2 p π ( τ − s ) ds − Z τ f − ( s ) ( y ( τ ) − y ( s )) e − ( y ( τ ) − y ( s ))24( τ − s ) p π ( τ − s ) ds, lim x → z ( τ ) − Z τ f + ( s ) ∂ Λ ( x, τ | z ( s ) , s ) ∂x ds = − f + ( τ ) √ πτ + Z τ f + ( s ) e − ( z ( τ ) − z ( s ))24( τ − s ) − f + ( τ )2 p π ( τ − s ) ds − Z τ f + ( s ) ( z ( τ ) − z ( s )) e − ( z ( τ ) − z ( s ))24( τ − s ) p π ( τ − s ) ds. Finally, taking the derivative of Eq. (39) on x , setting x = y ( τ ) and x = z ( τ ), and using these expressions,we obtain the following system of the Volterra equations of the second kind for the unknown functionsΨ( τ ) , Φ( τ ) − Ψ( τ ) = − f − ( τ ) √ πτ + Z τ f − ( s ) e − ( y ( τ ) − y ( s ))24( τ − s ) h y ′ ( s )( y ( τ ) − y ( s ))) − ( y ( τ ) − y ( s )) τ − s ) i − f − ( τ )2 p π ( τ − s ) ds (41) − Z τ Ψ( s ) y ( τ ) − y ( s )2 p π ( τ − s ) e − ( y ( τ ) − y ( s ))24( τ − s ) ds + Z z (0) y (0) U (0 , ξ ) υ − ( τ | ξ, dξ + Z τ (cid:16)h Φ( s ) + f + ( s ) z ′ ( s ) i υ − ( τ | z ( s ) , s ) + (cid:2) Ψ( s ) − f − ( s ) y ′ ( s ) (cid:3) υ − ( τ | y ( s ) , s ) (cid:17) ds + Z τ (cid:16) f − ( s ) λ − ( τ, | y ( s ) , s ) − f + ( s ) λ − ( τ, | z ( s ) , s ) (cid:17) ds, Φ( τ ) = f + ( τ ) √ πτ − Z τ f + ( s ) e − ( z ( τ ) − z ( s ))24( τ − s ) h z ′ ( s )( z ( τ ) − z ( s )) − ( z ( τ ) − z ( s )) τ − s ) i − f + ( τ )2 p π ( τ − s ) ds − Z τ Φ( s ) z ( τ ) − z ( s )2 p π ( τ − s ) e − ( z ( τ ) − z ( s ))24( τ − s ) ds + Z z (0) y (0) U (0 , ξ ) υ + ( τ | ξ, dξ + Z τ (cid:16)h Φ( s ) + f + ( s ) z ′ ( s ) i υ +0 ( τ | z ( s ) , s ) + (cid:2) Ψ( s ) − f − ( s ) y ′ ( s ) (cid:3) υ + ( τ | y ( s ) , s ) ds (cid:17) + Z τ (cid:16) f − ( s ) λ + ( τ | y ( s ) , s ) − f + ( s ) λ +0 ( τ | s ) (cid:17) ds. Here υ n ( τ | ξ, s ) = − y ( τ ) − ξ + 2 nl ( τ )2 p π ( τ − s ) e − ( y ( τ ) − ξ +2 nl ( τ ))24( τ − s ) , (42) Page 14 of 31 emi-analytic pricing of double barrier options... λ n ( τ | ξ, s ) = e − ( y ( τ ) − ξ +2 nl ( τ ))24( τ − s ) p π ( τ − s ) " − ( y ( τ ) − ξ + 2 nl ( τ )) τ − s ) ,υ − ( τ | ξ, s ) = ∞ X n = −∞ υ n ( τ | ξ, s ) , υ + ( τ | ξ, s ) = ∞ X n = −∞ υ n + ( τ | ξ, s ) ,υ − ( τ | s ) = ∞ X n = −∞ n =0 υ n ( τ | y ( s ) , s ) , υ +0 ( τ | s ) = ∞ X n = −∞ n =0 υ n + ( τ | z ( s ) , s ) ,λ − ( τ | ξ, s ) = ∞ X n = −∞ λ n ( τ | ξ, s ) , λ + ( τ | ξ, s ) = ∞ X n = −∞ λ n + ( τ | ξ, s ) ,λ − ( τ | s ) = ∞ X n = −∞ n =0 λ n ( τ | y ( s ) , s ) , λ +0 ( τ | s ) = ∞ X n = −∞ n =0 λ n + ( τ | z ( s ) , s ) . It is worth emphasizing that all summands in Eq. (41) are regular. The integrals with respect to the timein the first two lines have weak (integrable) singularities, while other summands are regular.This system can be further simplified by using Eq. (71) and reduction to the Lebesgue-Stieltjesintegrals − Ψ( τ ) = Z z (0) y (0) U (0 , ξ ) υ − ( τ | ξ, dξ (43) − f − ( τ ) √ πτ + Z τ f − ( s ) − f − ( τ )2 p π ( τ − s ) ds + Z τ h f − ( s ) d (cid:0) η − ( τ | y ( s ) , s ) (cid:1) − f + ( s ) d (cid:0) η − ( τ | z ( s ) , s ) (cid:1)i − Z τ Ψ( s ) y ( τ ) − y ( s )2 p π ( τ − s ) e − ( y ( τ ) − y ( s ))24( τ − s ) ds + Z τ h Φ( s ) υ − ( τ | z ( s ) , s ) + Ψ( s ) υ − ( τ | s ) i ds Φ( τ ) = Z z (0) y (0) U (0 , ξ ) υ + ( τ | ξ, dξ + f + ( τ ) √ πτ − Z τ f + ( s ) − f + ( τ )2 p π ( τ − s ) ds + Z τ h f − ( s ) d (cid:16) η + ( τ | y ( s ) , s ) (cid:17) − f + ( s ) d (cid:16) η + ( τ | z ( s ) , s ) (cid:17)i − Z τ Φ( s ) z ( τ ) − z ( s )2 p π ( τ − s ) e − ( z ( τ ) − z ( s ))24( τ − s ) ds + Z τ h Φ( s ) υ +0 ( τ | s ) + Ψ( s ) υ + ( τ | y ( s ) , s ) i ds. Here the following notation is used η − ( τ | ξ, s ) = − δ ξ,y ( s ) p π ( τ − s ) + 1 p π ( τ − s ) ∞ X n = −∞ e − ( y ( τ ) − ξ +2 nl ( τ ))24( τ − s ) , (44) η + ( τ | ξ, s ) = − δ ξ,z ( s ) p π ( τ − s ) + 1 p π ( τ − s ) ∞ X n = −∞ e − ( y ( τ ) − ξ +(2 n +1) l ( τ ))24( τ − s ) ,υ − ( τ | ξ, s ) = − y ( τ ) − ξ + 2 nl ( τ )2 p π ( τ − s ) e − ( y ( τ ) − ξ +2 nl ( τ ))24( τ − s ) ,υ + ( τ | ξ, s ) = − y ( τ ) − ξ + (2 n + 1) l ( τ )2 p π ( τ − s ) e − ( y ( τ ) − ξ +(2 n +1) l ( τ ))24( τ − s ) , where δ ξ,x is the Kronecker symbol.The functions υ, η have the following propertieslim s → τ υ − ( τ | s ) = 0 , lim s → τ υ − ( τ | z ( s ) , s ) = 0 , Page 15 of 31 emi-analytic pricing of double barrier options... lim s → τ υ + ( τ | y ( s ) , s ) = 0 , lim s → τ υ +0 ( τ | s ) = 0 , lim s → τ η − ( τ | y ( s ) , s ) = 0 , lim s → τ η − ( τ | z ( s ) , s ) = 0 , lim s → τ η + ( τ | y ( s ) , s ) = 0 , lim s → τ η + ( τ | z ( s ) , s ) = 0 . Again, using the Poisson summation formula yields a few alternative representations of the functions η ± ( τ | ξ, s ) and υ ± ( τ | ξ, s ) via the Fourier series η − ( τ | ξ, s ) = − y ( s ) − ξ p π ( τ − s ) + 1 l ( τ ) " ∞ X n =1 e − π n l τ ) ( τ − s ) cos (cid:18) πn ( ξ − y ( τ )) l ( τ ) (cid:19) , (45) η + ( τ | ξ, s ) = − ξ − z ( s ) p π ( τ − s ) + 1 l ( τ ) " ∞ X n =1 e − π n l τ ) ( τ − s ) ( − n cos (cid:18) πn ( ξ − y ( τ )) l ( τ ) (cid:19) ,υ − ( τ | ξ, s ) = 2 πl ( τ ) ∞ X n =1 ne − π n l τ ) ( τ − s ) sin (cid:18) πn ( ξ − y ( τ )) l ( τ ) (cid:19) ,υ + ( τ | ξ, s ) = 2 πl ( τ ) ∞ X n =1 ne − π n l τ ) ( τ − s ) ( − n sin (cid:18) πn ( ξ − y ( τ )) l ( τ ) (cid:19) . Finally, using Eq. (31) and Eq. (33), we obtain another representation of Eq. (45) in terms of theJacobi theta function θ ( z, ω ) η − ( τ | ξ, s ) = − y ( s ) − ξ p π ( τ − s ) + 1 l ( τ ) θ ( φ − ( ξ, y ( τ )) , ω ) , (46) η + ( τ | ξ, s ) = − ξ − z ( s ) p π ( τ − s ) + 1 l ( τ ) θ ( φ − ( ξ + l ( τ ) , y ( τ )) , ω ) ,υ − ( τ | ξ, s ) = − π l ( τ ) θ ′ ( φ − ( ξ, y ( τ )) , ω ) ,υ + ( τ | ξ, s ) = − π l ( τ ) θ ′ ( φ − ( ξ + l ( τ ) , y ( τ )) , ω ) . The formulas Eq. (44) and Eq. (45) are complementary. Since the exponents in Eq. (46) are propor-tional to the difference τ − s , the Fourier series Eq. (46) converge fast if τ − s is large. Contrary, theexponents in Eq. (44) are inversely proportional to τ − s . Therefore, the series Eq. (44) converge fast if τ − s is small. Similar to Section 2, the HP method, (Tikhonov and Samarskii, 1963; Friedman, 1964.; Kartashov, 2001),can be used to price double barrier options by solving the problem in Eq. (8). The idea was first proposedand developed in (Itkin and Muravey, 2020) and is a generalization of the standard HP method for thecase of two moving boundaries. Note, that to the best of authors’ knowledge, yet the closed form (or evensemi-closed form) solution of this problem was not known in physics, even not mentioning finance. Belowwe explain our approach paying attention to all intermediate details as the behavior of the solution atthe boundaries is not trivial.Following the main idea of the HP method, let us search for the solution of the B problem in Eq. (4)Eq. (6), Eq. (5) in the form U ( τ, x ) = q ( τ, x ) + 12 √ πτ Z z (0) y (0) U (0 , x ′ ) e − ( x − x ′ )24 τ dx ′ , (47) Page 16 of 31 emi-analytic pricing of double barrier options... so function q ( τ, x ) solves a problem with the homogeneous initial condition ∂q ( τ, x ) ∂τ = ∂ q ( τ, x ) ∂x , (48) q (0 , x ) = 0 , y (0) < x < z (0) ,q ( τ, y ( τ )) = φ ( τ ) ≡ f − ( τ ) − √ πτ Z z (0) y (0) u (0 , x ′ ) e − ( y ( τ ) − x ′ )24 τ dx ′ ,q ( τ, z ( τ )) = ψ ( τ ) ≡ f + ( τ ) − √ πτ Z z (0) y (0) u (0 , x ′ ) e − ( z ( τ ) − x ′ )24 τ dx ′ . In (Itkin and Muravey, 2020) it is proposed to search for the solution of Eq. (48) in the form of ageneralized heat potential q ( x, τ ) = 14 √ π Z τ p ( τ − k ) ( x − y ( k ))Ω( k ) e − ( x − y ( k ))24( τ − k ) + ( x − z ( k ))Θ( k ) e − ( x − z ( k ))24( τ − k ) ! dk, (49)where Ω( k ) , Θ( k ) are the heat potential densities. In other words, the solution is represented as a sumof two heat potentials: one corresponds to the lower barrier, and the other one - to the upper barrier.It is easy to check, that each such a potential solves the heat equation in Eq. (48), see (Tikhonov andSamarskii, 1963) as the derivative with respect to τ of the RHS of Eq. (49) can be pulled into the integralsince the value of both integrands at k = τ vanishes.To find the unknown functions Ω( k ) , Θ( k ) one can substitute into Eq. (49) the values x = y ( τ ) and x = z ( τ ), and get a system of two integral equations that the functions Ω( k ) , Θ( k ) solve. However, itis well-known, (Tikhonov and Samarskii, 1963), that these integrals at x → y ( τ ) and x → z ( τ ) have adiscontinuity, but with the finite value at x = y ( τ ) ± x = z ( τ ) ±
0. To investigate this discontinuityin more detail and derive the value of heat potential integral at the boundary x → y ( τ ) ±
0, we considera problem similar to Eq. (48) L q ( τ, x ) = 0 , ( x, τ ) ∈ Ω : [ y ( τ ) , ∞ ) × R + , (50) q (0 , x ) = 0 , y (0) < x < ∞ ,q ( τ, y ( τ )) = χ ( τ ) , q ( τ, x ) (cid:12)(cid:12)(cid:12) x →∞ = 0 . with the operator L defined as L = − ∂∂τ + σ ∂ ∂x , (51)where σ = const . Using the HP method, the solution of this problem can be expressed as q ( τ, x ) = Z τ Ω( k ) x − y ( k )4 σ p π ( τ − k ) e − ( x − y ( k ))24 σ τ − k ) dk, (52)where Ω( τ ) is the heat potential density, and y ( τ ) is a smooth curve (the moving boundary). Our aimbelow is to derive the value of this integral at x → y ( τ ) ±
0, and the gradient ∂q ( τ, x ) /∂x in the samelimit, namely ϕ ( τ ) = lim x → y ( τ ) ± q ( τ, x ) , ψ ( τ ) = lim x → y ( τ ) ± ∂q ( τ, x ) ∂x . (53) Page 17 of 31 emi-analytic pricing of double barrier options... ϕ ( t ) This result is obtained, eg., in (Tikhonov and Samarskii, 1963). Consider a function W ( τ, x ) = 2 σ φ ( t ) W ( τ, x ) = Z τ Ω( k ) x − y ( k )2 σ p π ( τ − k ) e − ( y ( τ ) − y ( k ))24 σ τ − k ) dk. (54)Also consider an auxiliary integral˜ V ( τ, x ) = Z τ y ′ ( k )Ω( k ) σ p π ( τ − k ) e − ( x − y ( k ))24 σ τ − k ) dk. (55)Assume that y ( k ) is differentiable. As shown in (Tikhonov and Samarskii, 1963), ˜ V ( τ, x ) is continuousalong the curve x = y ( τ ) because it converges uniformly and y ′ ( k ) is bounded, while W ( τ, x ) is discon-tinuous. To show this, first assume that Ω( τ ) = Ω = const . Then the difference W − ˜ V , where thesub-index means that we use Φ instead of Φ( τ ) in the definitions Eq. (54), Eq. (55), can be calculateddirectly with the change of variables k a = ( x − y ( k )) / (2 σ √ τ − k ) W − ˜ V = 12 σ √ π Z τ Ω e − ( x − y ( k ))24 σ τ − k ) (cid:20) x − y ( k )( τ − k ) / − y ′ ( k )( τ − k ) / (cid:21) dk = Ω √ π Z ζ + ζ − e − a da, (56) ζ − = x − y (0)2 σ √ τ , ζ + = ∞ , x > y ( τ ) , , x = y ( τ ) , −∞ , x < y ( τ ) . Accordingly, at, say x → y ( τ ) + 0 we obtain[ W ( τ, y ( τ ) + 0) − W ( τ, y ( τ ))] − h ˜ V ( τ, y ( τ ) + 0) − ˜ V ( τ, y ( τ )) i = Ω √ π Z ∞ e − a da = Ω . (57)Since the function ˜ V is continuous, the expression in the second square brackets in Eq. (57) vanishes,and so W ( τ, y ( τ ) + 0) − W ( τ, y ( τ )) = Ω . (58)If Ω( τ ) is not constant, then W ( τ, x ) = W ( τ, x ) − Z τ x − y ( k )2 σ p π ( τ − k ) e − ( x − y ( k ))24 σ τ − k ) [Ω( τ ) − Ω( k )] dk. (59)We assume that the boundary y ( τ ) and the potential density Ω( k ) are differentiable functions of theirarguments, i.e., at least C . Then the integral in Eq. (59) has the same singularity as the function ˜ V ( τ, x ),converges uniformly, and thus is a continuous function on the curve x = y ( τ ). This implies that W ( τ, x + 0) − W ( τ, x ) = W ( τ, x + 0) − W ( τ, x ) = Ω( τ ) , (60)and, in particular, this is true for x = y ( τ ). In a similar way one can show that W ( τ, x −
0) = W ( τ, x ) − Ω( τ ) , (61)Combining these results together, we obtain the final formula for ϕ ( t ) ϕ ( τ ) = ± Ω( τ )2 σ + Z τ Ω( k ) y ( τ ) − y ( k )4 σ p π ( τ − k ) e − ( y ( τ ) − y ( k ))24 σ τ − k ) dk. (62) Page 18 of 31 emi-analytic pricing of double barrier options... ψ ( t ) Using the definition of q ( τ, x ) in Eq. (52) we need an explicit formula for ψ ( τ ) = lim x → y ( τ ) ± ∂q ( τ, x ) ∂x = lim x → y ( τ ) ± ∂∂x Z τ Ω( k ) x − y ( k )4 σ p π ( τ − k ) e − ( x − y ( k ))24 σ τ − k ) dk. (63)However, as shown in Section 3.1, this integral is discontinuous at x → y ( τ ) (this is an improperRiemann integral of second kind). Hence, we cannot compute ψ ( τ ) directly by taking derivative of q ( τ, x )with respect to x .Therefore, to proceed let us represent this integral as Z τ Ω( k ) x − y ( k )4 σ p π ( τ − k ) e − ( x − y ( k ))24 σ τ − k ) dk = Ω( τ ) Z τ x − y ( k )4 σ p π ( τ − k ) e − ( x − y ( k ))24 σ τ − k ) dk (64)+ Z τ [Ω( k ) − Ω( τ )] x − y ( k )4 σ p π ( τ − k ) e − ( x − y ( k ))24 σ τ − k ) dk = I + I . We showed in Section 3.1 that the second integral in Eq. (64) has the same singularity as the function˜ V ( τ, x ), converges uniformly, and thus is a continuous function on the curve x = y ( τ ). Then, it is acontinuous function for x ∈ ℜ . Thus, by the standard theorem of integral calculus we can differentiatethis integral by parameter x , and the result is continuous in x , (Butuzov and Butuzova, 2016)lim x → y ( τ ) ± ∂∂x Z τ [Ω( k ) − Ω( τ )] x − y ( k )4 σ p π ( τ − k ) e − ( x − y ( k ))24 σ τ − k ) dk (65)= lim x → y ( τ ) ± Z τ [Ω( k ) − Ω( τ )] e − ( x − y ( k ))24 σ τ − k ) σ p π ( τ − k ) − ( x − y ( k )) σ ( τ − k ) ! dk = Z τ [Ω( k ) − Ω( τ )] e − ( y ( τ ) − y ( k ))24 σ τ − k ) σ p π ( τ − k ) − ( y ( τ ) − y ( k )) σ ( τ − k ) ! dk. As far as the first integral I in Eq. (64) is concerned, it was already considered in Section 3.1, andis denoted as W ( τ, x ) / σ in Eq. (56). Since the integral on a in the RHS of Eq. (56) can be computedexplicitly, we have W − ˜ V = Ω √ π Z ζ + ζ − e − a da = Ω Erfc (cid:16) x − y (0)2 σ √ τ (cid:17) , x > y ( τ ) , − Erf (cid:16) x − y (0)2 σ √ τ (cid:17) , x = y ( τ ) , − Erfc (cid:16) − x − y (0)2 σ √ τ (cid:17) , x < y ( τ ) . (66)Also, recall that the function ˜ V ( τ, x ) is the continuous function along the curve x = y ( τ ) as y ′ ( τ ) isbounded, and the integral converges uniformly. Therefore ∂W ∂x = ∂ ˜ V ∂x − Ω Λ( τ, x ) , (67)Λ( τ, x ) = σ √ πτ e − ( x − y (0))24 πσ , x > y ( τ ) , σ √ πτ e − ( x − y (0))24 πσ , x < y ( τ ) . Thus, Λ( τ, y ( τ ) −
0) = Λ( τ, y ( τ ) + 0), hence the function Λ( τ, x ) is differentiable at this point. Thisimplies ∂W ∂x = − Ω Z τ y ′ ( k ) x − y ( k )2 σ p π ( τ − k ) e − ( x − y ( k ))24 σ τ − k ) dk − Ω σ √ πτ e − ( x − y (0))24 σ τ . (68) Page 19 of 31 emi-analytic pricing of double barrier options...
As it was mentioned, the function ˜ V ( τ, x ) is continuous over the curve x = y ( τ ). However, itsderivative with respect to x at x = y ( τ ) in Eq. (67) has a form of the RHS in Eq. (54). Therefore,according to the result of Section 3.1, in the limit x → y ( τ ), again using Eq. (62), we obtainlim x → y ( τ ) ± ∂W ∂x = ∓ Ω y ′ ( τ ) σ − Ω Z τ y ′ ( k ) y ( τ ) − y ( k )2 σ p π ( τ − k ) e − ( y ( τ ) − y ( k ))24 σ τ − k ) dk − Ω σ √ πτ e − ( y ( τ ) − y (0))24 σ τ . (69)Combining Eq. (65) and Eq. (69) together yields the final result ψ ( τ ) = Z τ Ω( k ) e − ( y ( τ ) − y ( k ))24 σ τ − k ) σ p π ( τ − k ) − ( y ( τ ) − y ( k )) σ ( τ − k ) ! dk − Ω( τ ) f ( τ ) , (70) f ( τ ) = ± y ′ ( τ )2 σ + 12 σ √ πτ e − ( y ( τ ) − y (0))24 σ τ + Z τ e − ( y ( τ ) − y ( k ))24 σ τ − k ) σ p π ( τ − k ) ( y ′ ( k )[ y ( τ ) − y ( k )] σ − ( y ( τ ) − y ( k )) σ ( τ − k ) ) dk. Thus, we proved that the derivative ∂q ( τ, x ) /∂x is also discontinuous at x = y ( τ ), and obtained itsexplicit representation in Eq. (70). Note, that this derivative should not be confused with the normal(directional) derivative of u ( τ, x ) which is continuous at x = y ( τ ). Indeed, the function q , as defined inEq. (52), is the double layer heat potential. The claim that this derivative is continuous is commonlyreferred as the Lyapunov-Tauber theorem of classic potential theory, see (Lyapunov, 1949), and (Guinter,1967; Quaife, 2011; Costabel, 1990; Kristensson, 2009) and references therein for the extension to themulti-dimensional case.It is worth mentioning, that the formula for f ( τ ) can be further simplified. Indeed d e − ( y ( τ ) − y ( k ))24 σ τ − k ) √ τ − k = e − ( y ( τ ) − y ( k ))24 σ τ − k ) p ( τ − k ) − e − ( y ( τ ) − y ( k ))24 σ τ − k ) √ τ − k − y ′ ( k )( y ( τ ) − y ( k ))2 σ ( τ − k ) + ( y ( τ ) − y ( k )) σ ( τ − k ) ! dk = e − ( y ( τ ) − y ( k ))24 σ τ − k ) p ( τ − k ) y ′ ( k )( y ( τ ) − y ( k )) σ ( τ − k ) − ( y ( τ ) − y ( k )) σ ( τ − k ) ! dk. Therefore, e − ( y ( τ ) − y ( k ))24 σ τ − k ) p ( τ − k ) y ′ ( k )( y ( τ ) − y ( k )) σ ( τ − k ) − ( y ( τ ) − y ( k )) σ ( τ − k ) ! dk = d e − ( y ( τ ) − y ( k ))24 σ τ − k ) − √ τ − k + dk p ( τ − k ) . (71)Plugging this expression into the formula for f ( τ ) and integrating yields an alternative representationfor f ( τ ) f ( τ ) = ± y ′ ( τ )2 σ + 12 σ √ πτ + Z τ dk σ p π ( τ − k ) , (72)and for ψ ( τ ), respectively ψ ( τ ) = − Ω( τ ) (cid:18) σ √ πτ ± y ′ ( τ )2 σ (cid:19) + Z τ Ω( k ) e − ( y ( τ ) − y ( k ))24 σ τ − k ) − Ω( τ )4 σ p π ( τ − k ) dk (73) Page 20 of 31 emi-analytic pricing of double barrier options... − Z τ Ω( k ) ( y ( τ ) − y ( k )) e − ( y ( τ ) − y ( k ))24 σ τ − k ) σ p π ( τ − k ) dk. The last formula for the particular case σ = 1 / √ With allowance for the representation obtained in Eq. (62), by substituting the limiting values x → y ( τ )and x → z ( τ ) into Eq. (49), we obtain a system of two integral equation for functions Ω( τ ) , Θ( τ )2 φ ( τ ) = Ω( τ ) + 12 √ π Z τ Ω( k ) y ( τ ) − y ( k )( τ − k ) / e − ( y ( τ ) − y ( k ))24( τ − k ) + Θ( k ) y ( τ ) − z ( k )( τ − k ) / e − ( y ( τ ) − z ( k ))24( τ − k ) ! dk, (74)2 ψ ( τ ) = − Θ( τ ) + 12 √ π Z τ Ω( k ) z ( τ ) − y ( k )( τ − k ) / e − ( z ( τ ) − y ( k ))24( τ − k ) + Θ( k ) z ( τ ) − z ( k )( τ − k ) / e − ( z ( τ ) − z ( k ))24( τ − k ) ! dk. Each equation in this system is a Volterra equation of the second kind. The system can be solved, eg.,by the Variational Iteration Method (VIM), see (Wazwaz, 2011) with a linear complexity by using theFast Gaussian Transform. Once this is done, the solution of our double barrier problem is found.It is interesting that the representation of the solution gradient in Eq. (73) provides connection betweenthe GIT and HP methods. Indeed, by definition in Eq. (13) and also using Eq. (7), Eq. (47)Ψ( τ ) = − ∂U ( τ, x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = y ( τ ) (75)= − ∂q ( τ, x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = y ( τ )+0 + 14 √ πτ Z z (0) y (0) U (0 , x ′ )( y ( τ ) − x ′ ) e − ( y ( τ ) − x ′ )24 τ dx ′ , Φ( τ ) = ∂U ( τ, x ) ∂τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = z ( τ ) = ∂q ( τ, x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = z ( τ ) − + 14 √ πτ Z z (0) y (0) U (0 , x ′ )( z ( τ ) − x ′ ) e − ( z ( τ ) − x ′ )24 τ dx ′ . Therefore, once the pair Ω( τ ) , Θ( τ ) is known, the other pair Ψ( τ ) , Φ( τ ) can be obtained explicitlyfrom Eq. (75). The opposite is also true, i.e., once the pair Ψ( τ ) , Φ( τ ) is known, the heat potentialdensities Ω( τ ) , Θ( τ ) can be found by solving this system of Volterra equations of the second kind. Thus,both the GIT and HP methods are interchangeable. But as was mentioned in Introduction, despite bothsolutions are equal, their convergence properties are different. In this paper we extend the technique of semi-analytic (or semi-closed form) solutions, developed in (Carrand Itkin, 2020; Itkin and Muravey, 2020; Carr et al., 2020; Itkin et al., 2020a; Lipton and Kaushansky,2018; Lipton and de Prado, 2020), to pricing double barrier options and present two approaches to solvingit: the General Integral transform method and the Heat Potential method. By semi-analytic solution wemean that first, we need to solve a system of two linear Volterra equations of the second kind, and thenthe option price is represented as a one-dimensional integral.
Page 21 of 31 emi-analytic pricing of double barrier options...
Therefore, perhaps the main point is about efficiency and robustness of the proposed approach. Asshown in (Carr and Itkin, 2020; Itkin and Muravey, 2020; Carr et al., 2020; Itkin et al., 2020a), from thecomputational point of view the solution proposed by using the same technique for pricing single barrieroptions under various models with time-dependent barriers is very efficient and, at least theoretically, ofthe same complexity, or even faster than the forward finite-difference (FD) method. On the other hand,our approach provides high accuracy in computing the options prices, as this is regulated by quadraturerule used to discretize the integral kernel in Eq. (36) Eq. (41)), or in Eq. (74). Therefore, the accuracyof the method in x space can be easily increased by using high order quadratures. For instance, usingthe Simpson instead of the trapezoid rule doesn’t affect the complexity of our method but increases theaccuracy, while increasing the accuracy for the FD method is not easy (i.e., it significantly increases thecomplexity of the method, e.g., see (Itkin, 2017)).As applied to pricing double barrier options - the problem considered in this paper, the differenceis that instead of a single Volterra equation of the second kind we now have to solve a system of twoequations, either in Eq. (36) Eq. (41)), or in Eq. (74). This can be done in the same way as for thesingle barrier problem. The Volterra equation is solved by discretizing the kernel of the integral in timeusing some quadrature rule which yields a system of linear equations with respect to the discrete valuesof Ψ( τ ) , Φ( τ ). It can be checked that the matrix of this system is of the form M = A BC D ! , where A, D are lower triangular matrices with ones on the main diagonal, and
B, C are lower triangularmatrices with zeros on the main diagonal. Therefore, this system can be solved by a simple Gausselimination method (by a set of algebraic multiplications and additions) with complexity O (2 N ) where N is the number of the discretization points in τ for Ψ( τ ) , Φ( τ ). Alternatively, when using Eq. (74) orEq. (41), since the kernel is proportional to Gaussians, the discrete sum approximating the integral canbe computed with linear complexity O (2 N ) using the Fast Gauss Transform, see eg., (Spivak et al., 2010).Once the vectors Ψ( τ ) , Φ( τ ) (for the GIT method), or Ω( τ ) , Θ( τ ) (for the HP method) are found, theycan be substituted into Eq. (34) or Eq. (39) for the GIT method), or into Eq. (49) (for the HP method).Then the final solution is obtained by computing the integral(s) numerically. Various numerical examplesillustrating this technique for a single barrier pricing problem can be found in (Carr and Itkin, 2020;Itkin and Muravey, 2020; Carr et al., 2020; Itkin et al., 2020a). Also, those examples demonstrate thatcomputationally our method is more efficient than both the backward and even forward FD methods (ifone uses them to solve this kind of problems), while providing better accuracy and stability.Somebody could be a bit confused of this terminology, since despite the solution is found explicitlyas an integral, the latter depends of the unknown function of time Ψ( τ ). In support of this terminology,we can mention that the solution is definitely of a closed form on variable x . On variable τ the integrandexplicitly depends on yet unknown function Ψ( τ ) which solves the Volterra integral equation of thesecond kind. However, this equation can be solved with no iterations. Indeed, after the function Ψ( τ ) isdiscretized on some grid in τ (so now it is represented by a finite vector ψ ), the integral equation reducesto the linear equation for ψ , with the matrix being low triangular. Thus, the solution can be immediatelyobtained by a simple Gauss elimination with no iterations. Therefore, this is explicit and as such, thesolution is given by a series of algebraic operations (substitutions). The finer is the grid, the closer is thesolution to the exact one.Also, we can make a reference to Lipton and de Prado (2020); Carr et al. (2020) where the phrase"semi-closed" was used verbatim. And in Lipton et al. (2019); Lipton and Kaushansky (2018) it is called as"semi-analytical" solution. Going back in time, in Kartashov and Lyubov (1974); Kartashov (1999, 2001)both GIT and HP methods are claimed as analytical. One can also look at Tikhonov and Samarskii (1963), Page 22 of 31 emi-analytic pricing of double barrier options... page 533, subsection 2, which from the very beginning says, "Heat potentials are a convenient analyticaldevice for solving boundary-value problems". Therefore, we think this terminology is appropriate.Also, as mentioned in (Carr et al., 2020), another advantage of the approach advocated in this paper iscomputation of option Greeks. Indeed, in both the HP and GIT methods the option prices are representedin an explicit analytic form on x (via the integrals on τ and the auxiliary variable ξ ). This means thatan explicit dependence of the option prices on the model parameters is available and transparent. Thus,explicit representations of the option Greeks can be obtained by a simple differentiation under the integrals.This means that the Greek values can be computed simultaneously with the option prices with almostno additional increase in the elapsed time. This is possible because differentiation under the integralsslightly changes the integrands, while these changes could be represented as changes in weights of thequadrature scheme used to compute the integrals.Also, the integrands in the integral representation of the solution could be treated as a product ofsome density function and weights. The major computational time is spent for computing the densities asthey contain special functions. However, once computed the results can be saved during the calculation ofprices, and then reused when computing the Greeks. Therefore, computing Greeks can be done very fast.This is also true eg., for Vega and other Greeks that cannot be computed by the FD method togetherwith prices and require a separate run of the FD machinery. Here we don’t have such a problem asdifferentiation of the integral representation with respect to the model parameters is done analytically.Finally, as mentioned in (Itkin and Muravey, 2020), the GIT and HP methods are complementary.In more detail, this means the following. Our experiments showed that performance of both the GITand HP methods is same. However, the GIT method produces more accurate results at high strikes andmaturities (i.e. where the option price is relatively small) in contrast to the HP method which is moreaccurate at short maturities and low strikes. For the CIR and CEV models this behavior was explainedin (Carr et al., 2020), and for the Hull-White model - in (Itkin and Muravey, 2020). Briefly, for the heatequation that we consider in this paper, the exponents in both the HP and GIT integrals are inverselyproportional to τ . However, the GIT integrals contain a difference of two exponents (see the definitionof Υ n ( x, τ | ξ, s ) in Eq. (39) which becomes small at large τ . On contrary, the HP exponent in Eq. (49)tends to 1 at large τ . Therefore, the convergence properties of two methods are different at large τ .This situation is well known for the heat equation with constant coefficients. There exist two represen-tation of the solution: one - obtained by using the method of images, and the other one - by the Fourierseries. Despite both solutions are equal as the infinite series, their convergence properties are different. Acknowledgments
We are grateful to Alex Lipton for some fruitful discussions. Dmitry Muravey acknowledges support bythe Russian Science Foundation under the Grant number 20-68-47030.
References
I. Bouchouev. Negative oil prices put spotlight on investors.
Risk.net , 2020.R. Brogan. Options traders adapt to electronic markets in pandemic, 2020. URL https://flextrade.com/options-traders-adapt-to-electronic-markets-in-pandemic/ .V.F. Butuzov and M.V. Butuzova.
Integrals depending on parameters . Moscow State University, Moscow,2016. in Russian.P. Carr and A. Itkin. Semi-closed form solutions for barrier and American options written on a time-dependent Ornstein Uhlenbeck process, March 2020. Arxiv:2003.08853.
Page 23 of 31 emi-analytic pricing of double barrier options...
P. Carr, A. Itkin, and D. Muravey. Semi-closed form prices of barrier options in the time-dependent cevand cir models.
Journal of Derivatives , 28(1):26–50, 2020.M. Costabel. Boundary integral operators for the heat equation.
Integral Equations and Operator Theory ,13(4):498–552, 1990.C.J. Dias. A method of recursive images to solve transient heat diffusionin multilayer materials. 85:1075–1083.DLMF.
NIST Digital Library of Mathematical Functions . http://dlmf.nist.gov/, Release 1.0.28 of 2020-09-15. URL http://dlmf.nist.gov/ . F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I.Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain,eds.R. Doff. Valuing scenarios with real option pricing.
Risk.net , August 2020.S. Farrington and M. Cesa. Podcast: Kaminski and ronn on negative oil and options pricing.
Risk.net ,May 2020.A. Friedman.
Partial Differential Equations of Parabolic Type . Prentice-Hall, New Jersey„ 1964.I.S. Gradshtein and I.M. Ryzhik.
Table of Integrals, Series, and Products . Elsevier, 2007.N.M. Guinter.
Potential Theory and Its Applications to Basic Problems of MathematicalPhysics . FrederickUngar, New York, 1967.A. Itkin.
Pricing Derivatives Under Lévy Models. Modern Finite-Difference and Pseudo-DifferentialOperators Approach. , volume 12 of
Pseudo-Differential Operators . Birkhauser, 2017.A. Itkin and D. Muravey. Semi-closed form prices of barrier options in the Hull-White model, April 2020.Arxiv:2004.09591.A. Itkin, A. Lipton, and D. Muravey. From the black-karasinski to the verhulst model to accommodatethe unconventional fed’s policy, June 2020a. URL https://arxiv.org/abs/2006.11976 .A. Itkin, A. Lipton, and D. Muravey. Multilayer heat equations: application to finance. in preparation,2020b.E. M. Kartashov. Analytical methods for solution of non-stationary heat conductance boundary problemsin domains with moving boundaries.
Izvestiya RAS, Energetika , (5):133–185, 1999.E.M. Kartashov.
Analytical Methods in the Theory of Heat Conduction in Solids . Vysshaya Shkola,Moscow, 2001.E.M. Kartashov and B. Ya Lyubov. Analytical methods in the theory of heat conduction in solids.
Izv.Akad. Nauk SSSR, Energ. Trans. , (6):83–111, 1974.G. Kristensson. Jump conditions for single and doublelayer potentials, 2009. file:///C:/AndreyItkin/MyFinance/FinPapers/BK/liter/JumpConditions.pdf .A. Lipton. The vol smile problem.
Risk , pages 61–65, February 2002.A. Lipton and M.L. de Prado. A closed-form solution for optimal mean-reverting trading strategies, 2020.available at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3534445 . Page 24 of 31 emi-analytic pricing of double barrier options...
A. Lipton and V. Kaushansky. On the first hitting time density of an ornstein-uhlenbeck process, October2018. URL https://arxiv.org/pdf/1810.02390.pdf .A. Lipton, V. Kaushansky, and C. Reisinger. Semi-analytical solution of a McKean-Vlasov equation withfeedback through hitting boundary.
Euro. Jnl of Applied Mathematics , pages 1–34, 2019.A.M. Lyapunov.
Works on the theory of potential . Technical and Theoretical State Publishing House,Moscow - Leningrad, 1949. in Russian.A. Mijatovic. Local time and the pricing of time-dependent barrier options.
Finance and Stochastics , 14(1):13–48, 2010.D. Mumford, C. Musiliand M. Nori, E. Previato, and M. Stillman.
Tata Lectures on Theta . Progress inMathematics. Birkhäuser Boston, 1983. ISBN 9780817631093.A.D. Polyanin.
Handbook of linear partial differential equations for engineers and scientists . Chapman& Hall/CRC, 2002.B. Quaife.
Fast Integral Equation Methods for the Modified Helmholtz Equation . PhD thesis, Universityof Calgary, 2011.M. Spivak, S.K. Veerapaneni, and L. Greengard. The fast generalized gauss transform.
SIAM Journalon Scientific Computing , 32(5):3092–3107, 2010.A.N. Tikhonov and A.A. Samarskii.
Equations of mathematical physics . Pergamon Press, Oxford, 1963.B. van der Pol and H. Bremmer.
Operational calculus based on the two- sided Laplace integral . CambridgeUniversity Press, Cambridge, UK, 1950.A. M. Wazwaz.
Linear and Nonlinear Integral Equations . Higher Education Press, Beijing and Springer-Verlag GmbH Berlin Heidelberg, 2011.
Appendices
A Simplification of Eq. (29)
To simplify Eq. (29) we proceed by integrating by parts the last integral in Eq. (29) Z τ e − π n l τ ) ( τ − s ) h ( n, s, τ ) ds = − B ( τ ) l ( τ ) π n (cid:20) sin (cid:18) πnl ( τ ) [ z ( τ ) − y ( τ )] (cid:19) − sin (cid:18) πnl ( τ ) [ y ( τ ) − y ( τ )] (cid:19)(cid:21) + B (0) l ( τ ) π k e − π n l τ ) τ (cid:20) sin (cid:18) πnl ( τ ) [ z (0) − y ( τ )] (cid:19) − sin (cid:18) πnl ( τ ) [ y (0) − y ( τ )] (cid:19)(cid:21) − l ( τ ) πn " f − ( τ ) cos (cid:18) πnl ( τ ) [ y ( τ ) − y ( τ )] (cid:19) − f + ( τ ) cos (cid:18) πnl ( τ ) [ z ( τ ) − y ( τ )] (cid:19) + l ( τ ) πn e − π n l τ ) τ " f − (0) cos (cid:18) πnl ( τ ) [ y (0) − y ( τ )] (cid:19) − f + (0) cos (cid:18) πnl ( τ ) [ z (0) − y ( τ )] (cid:19) Page 25 of 31 emi-analytic pricing of double barrier options... + l ( τ ) π n Z τ B ( s ) e − π n l τ ) ( τ − s ) π n l ( τ ) (cid:20) sin (cid:18) πnl ( τ ) [ z ( s ) − y ( τ )] (cid:19) − sin (cid:18) πnl ( τ ) [ y ( s ) − y ( τ )] (cid:19)(cid:21) + πnl ( τ ) (cid:20) z ′ ( s ) cos (cid:18) πnl ( τ ) [ z ( s ) − y ( τ )] (cid:19) − y ′ ( s ) cos (cid:18) πnl ( τ ) [ y ( s ) − y ( τ )] (cid:19)(cid:21) ! ds + l ( τ ) πn Z τ f − ( s ) e − π n l τ ) ( τ − s ) π n l ( τ ) cos (cid:18) πnl ( τ ) [ y ( s ) − y ( τ )] (cid:19) − πnl ( τ ) y ′ ( s ) sin (cid:18) πnl ( τ ) [ y ( s ) − y ( τ )] (cid:19) ! ds − l ( τ ) πn Z τ f + ( s ) e − π n l τ ) ( τ − s ) π n l ( τ ) cos (cid:18) πnl ( τ ) [ y ( s ) − y ( τ )] (cid:19) − πnl ( τ ) z ′ ( s ) sin (cid:18) πnl ( τ ) [ y ( s ) − y ( τ )] (cid:19) ! ds + l ( τ ) πn Z τ B ( s ) e − π n l τ ) ( τ − s ) " y ′ ( s ) cos (cid:18) πnl ( τ ) [ y ( s ) − y ( τ )] (cid:19) − z ′ ( s ) cos (cid:18) πnl ( τ ) [ z ( s ) − y ( τ )] (cid:19) ds, or Z τ e − π n l τ ) ( τ − s ) h ( n, s, τ ) ds = l ( τ ) πn " ( − n f + ( τ ) − f − ( τ ) + α ( τ, n ) e − π n l τ ) τ + Z τ e − π n l τ ) ( τ − s ) β ( τ, s, n ) ds + Z τ e − π n l τ ) ( τ − s ) B ( s ) (cid:20) sin (cid:18) πnl ( τ ) [ z ( s ) − y ( τ )] (cid:19) − sin (cid:18) πnl ( τ ) [ y ( s ) − y ( τ )] (cid:19)(cid:21) ds, (A.1)where α ( τ, n ) = B (0) l ( τ ) π n (cid:20) sin (cid:18) πnl ( τ ) [ z (0) − y ( τ )] (cid:19) − sin (cid:18) πnl ( τ ) [ y (0) − y ( τ )] (cid:19)(cid:21) (A.2)+ l ( τ ) πn e − π n l τ ) τ " f − (0) cos (cid:18) πnl ( τ ) [ y (0) − y ( τ )] (cid:19) − f + (0) cos (cid:18) πnl ( τ ) [ z (0) − y ( τ )] (cid:19) ,β ( τ, s, n ) = f − ( s ) πnl ( τ ) cos (cid:18) πnl ( τ ) [ y ( s ) − y ( τ )] (cid:19) − y ′ ( s ) sin (cid:18) πnl ( τ ) [ y ( s ) − y ( τ )] (cid:19) ! − f + ( s ) πnl ( τ ) cos (cid:18) πnl ( τ ) [ z ( s ) − y ( τ )] (cid:19) − z ′ ( s ) sin (cid:18) πnl ( τ ) [ z ( s ) − y ( τ )] (cid:19) ! . Now we can transform the whole term2 l ( τ ) ∞ X n =1 sin (cid:18) πn x − y ( τ ) l ( τ ) (cid:19) Z τ e − π n l τ ) ( τ − s ) h ( n, s, τ ) ds, (A.3)which appears in Eq. (29). For doing that, first let us consider the integral Z z (0) y (0) u (0 , ξ ) sin (cid:18) πnl ( τ ) [ ξ − y ( τ )] (cid:19) dξ, (A.4)which is also a part of the RHS in Eq. (29). Recalling that by definition in Eq. (8) u (0 , x ) = U (0 , x ) − A (0) − B (0) x , and applying another identity Z z (0) y (0) [ A (0) + B (0) ξ ] sin (cid:18) πnl ( τ ) [ ξ − y ( τ )] (cid:19) dξ = l ( τ ) π n ( πn ( A (0) + B (0) y (0)) cos (cid:18) πn ( y (0) − y ( τ ) l ( τ ) (cid:19) − πn [ A (0) + B (0) z (0)] cos (cid:18) πn ( z (0) − y ( τ ) l ( τ ) (cid:19) Page 26 of 31 emi-analytic pricing of double barrier options... + B (0) l ( τ ) (cid:20) sin (cid:18) πn ( z (0) − y ( τ ) l ( τ ) (cid:19) − sin (cid:18) πn ( y (0) − y ( τ ) l ( τ ) (cid:21)(cid:19) ) , we obtain Z z (0) y (0) u (0 , ξ ) sin (cid:18) πnl ( τ ) [ ξ − y ( τ )] (cid:19) dξ = Z z (0) y (0) U (0 , ξ ) sin (cid:18) πnl ( τ ) [ ξ − y ( τ )] (cid:19) dξ − α ( τ, n ) . (A.5)Therefore, the terms proportional to α ( τ, n ) in Eq. (29) are cancelling out. Also, substituting Eq. (A.9)into Eq. (29) and moving the RHS of Eq. (A.9) into the LHS of Eq. (29) results in the change of u ( τ, x )to U ( τ, x ) in the LHS, and cancelling out the terms proportional to B ( s ). Finally, introducing the newfunction F ( τ, x ) F ( τ, x ) = A ( τ ) + B ( τ ) x − π ∞ X n =1 ( − n − f + ( τ ) + f − ( τ ) n sin (cid:18) πnl ( τ ) [ x − y ( τ )] (cid:19) (A.6)we obtain the representation of U ( τ, x ) U ( τ, x ) = 2 l ( τ ) ∞ X n =1 sin (cid:18) πn x − y ( τ ) l ( τ ) (cid:19) ( e − π n l τ ) τ Z z (0) y (0) U (0 , ξ ) sin (cid:18) πnl ( τ ) [ ξ − y ( τ )] (cid:19) dξ (A.7)+ Z τ e − π n l τ ) ( τ − s ) h Φ( s ) sin (cid:18) πnl ( τ ) [ z ( s ) − y ( τ )] (cid:19) + Ψ( s ) sin (cid:18) πnl ( τ ) [ y ( s ) − y ( τ )] (cid:19) + β ( τ, s, n ) i ds ) + F ( τ, x ) . Further, using the well-known identities, (Gradshtein and Ryzhik, 2007) ∞ X k =1 sin kxk = π − x , < x < π, ∞ X k =1 ( − k − sin kxk = x , < x < π, (A.8)yields the following relationship ∞ X n =1 πn " ( − n − f + ( τ ) + f − ( τ ) sin (cid:18) πn x − y ( τ ) l ( τ ) (cid:19) = 2 π ( πf + ( τ )2 x − y ( τ ) l ( τ ) + f − ( τ )2 (cid:20) π − π x − y ( τ ) l ( τ ) (cid:21) ) = f + ( τ ) − f − ( τ ) l ( τ ) x + f + ( τ ) y ( τ ) − f − ( τ ) z ( τ ) l ( τ ) = − [ A ( τ ) + B ( τ ) x ] , x ∈ ( y ( τ ) , z ( τ )) . (A.9)With the help of Eq. (A.9) we arrive at another formula for F ( τ, x ): F ( τ, x ) = f − ( τ ) , x = y ( τ ) , , x ∈ ( y ( τ ) , z ( τ )) ,f + ( τ ) , x = z ( τ ) . (A.10)Combining Eq. (A.7) and Eq. (A.10) together, and taking into account that the Fourier series inEq. (A.7) is equal to zero if x = y ( τ ) or x = z ( τ ), yields U ( τ, x ) = f − ( τ ) , x = y ( τ ) , ˜ U ( τ, x ) , x ∈ ( y ( τ ) , z ( τ )) ,f + ( τ ) , x = z ( τ ) . (A.11) Page 27 of 31 emi-analytic pricing of double barrier options...
Here the function ˜ U ( τ, x ) : ( y ( τ ) , z ( τ )) × R + → R is defined as follows˜ U ( τ, x ) = 2 l ( τ ) ∞ X n =1 sin (cid:18) πn x − y ( τ ) l ( τ ) (cid:19) ( e − π n l τ ) τ Z z (0) y (0) U (0 , ξ ) sin (cid:18) πnl ( τ ) [ ξ − y ( τ )] (cid:19) dξ (A.12)+ Z τ e − π n l τ ) ( τ − s ) h Φ( s ) sin (cid:18) πnl ( τ ) [ z ( s ) − y ( τ )] (cid:19) + Ψ( s ) sin (cid:18) πnl ( τ ) [ y ( s ) − y ( τ )] (cid:19) + β ( τ, s, n ) i ds ) . Note, that for the derivative ∂F ( τ,x ) ∂x we have ∂F ( τ, x ) ∂x = B ( τ ) − l ( τ ) ( f + ( τ ) ∞ X n =1 ( − n − cos (cid:18) πnl ( τ ) [ x − y ( τ )] (cid:19) + f − ( τ ) ∞ X n =1 cos (cid:18) πnl ( τ ) [ x − y ( τ )] (cid:19)) = f + ( τ ) − f − ( τ ) l ( τ ) − l ( τ ) ( f + ( τ ) ∞ X n =1 ( − n − cos (cid:18) πnl ( τ ) [ x − y ( τ )] (cid:19) + f − ( τ ) ∞ X n =1 cos (cid:18) πnl ( τ ) [ x − y ( τ )] (cid:19)) = 2 l ( τ ) ( f + ( τ ) "
12 + ∞ X n =1 ( − n cos (cid:18) πnl ( τ ) [ x − y ( τ )] (cid:19) − f − ( τ ) "
12 + ∞ X n =1 cos (cid:18) πnl ( τ ) [ x − y ( τ )] (cid:19) . Applying well known representations for the Dirac delta function δ ( x ) δ ( z ( τ ) − x ) = 2 l ( τ ) "
12 + ∞ X n =1 ( − n cos (cid:18) πnl ( τ ) [ x − y ( τ )] (cid:19) ,δ ( x − y ( τ )) = 2 l ( τ ) "
12 + ∞ X n =1 cos (cid:18) πnl ( τ ) [ x − y ( τ )] (cid:19) yields the following formula for the derivative of F ( τ, x ) ∂F ( τ, x ) ∂x = f + ( τ ) δ ( x − z ( τ )) − f − ( τ ) δ ( x − y ( τ )) . (A.13)Thus, this derivative is defined only in the sense of distributions. B Transformation of Eq. (30) to Eq. (39) . Applying a product-to-sum trigonometric identities to Eq. (A.12) yields˜ U ( τ, x ) = 1 l ( τ ) ∞ X n =1 ( e − π n l τ ) τ Z z (0) y (0) U (0 , ξ ) (cid:20) cos (cid:18) πnl ( τ ) [ x − ξ ] (cid:19) − cos (cid:18) πnl ( τ ) [ x + ξ − y ( τ )] (cid:19)(cid:21) dξ (B.1)+ Z τ e − π n l τ ) ( τ − s ) h Φ( s ) + f + ( s ) z ′ ( s ) i (cid:20) cos (cid:18) πnl ( τ ) [ x − z ( s )] (cid:19) − cos (cid:18) πnl ( τ ) [ x + z ( s ) − y ( τ )] (cid:19)(cid:21) ds + Z τ e − π n l τ ) ( τ − s ) (cid:2) Ψ( s ) − f − ( s ) y ′ ( s ) (cid:3) (cid:20) cos (cid:18) πnl ( τ ) [ x − y ( s )] (cid:19) − cos (cid:18) πnl ( τ ) [ x + y ( s ) − y ( τ )] (cid:19)(cid:21) ds + πnl ( τ ) Z τ e − π n l τ ) ( τ − s ) f − ( s ) (cid:20) sin (cid:18) πnl ( τ ) [ x − y ( s )] (cid:19) + sin (cid:18) πnl ( τ ) [ x + y ( s ) − y ( τ )] (cid:19)(cid:21) ds − πnl ( τ ) Z τ e − π n l τ ) ( τ − s ) f + ( s ) (cid:20) sin (cid:18) πnl ( τ ) [ x − z ( s )] (cid:19) + sin (cid:18) πnl ( τ ) [ x + z ( s ) − y ( τ )] (cid:19)(cid:21) ds ) , Page 28 of 31 emi-analytic pricing of double barrier options...
Since the functions h ( n ) = e − βn cos ( αn ) , h ( n ) = ne − βn sin ( αn )are even, h (0) = 0, and in the first three lines of Eq. (B.1) we have a difference of cosines, so at n = 0the difference vanishes, the series in Eq. (B.1) can be slightly modified by replacing ∞ X n =1 h i ( n ) = 12 ∞ X n = −∞ h i ( n ) , i = 1 , . Now applying formulas Eq. (38) to Eq. (B.1) and using α = x − ξl ( τ ) , β = τl ( τ ) , s β π = l ( τ ) √ πτ , β n + α ) = l ( τ )4 τ (cid:18) n + x − ξl ( τ ) (cid:19) = ( x − ξ + 2 nl ( τ )) τ , we obtain the following identities12 l ( τ ) ∞ X n = −∞ e − π n l τ ) ( τ − s ) cos (cid:18) πnl ( τ ) [ x − ξ ] (cid:19) = 12 p π ( τ − s ) ∞ X n = −∞ e − ( x − ξ +2 nl ( τ ))24( τ − s ) (B.2)12 l ( τ ) ∞ X n = −∞ e − π n l τ ) ( τ − s ) πnl ( τ ) sin (cid:18) πnl ( τ ) [ x − ξ ] (cid:19) = 14 p π ( τ − s ) ∞ X n = −∞ ( x − ξ + 2 nl ( τ )) e − ( x − ξ +2 nl ( τ ))24( τ − s ) . Observe that each term in Eq. (B.1) can be represented as one of the series in Eq. (B.2). Therefore,assuming x ∈ ( y ( τ ) , z ( τ )), we immediately arrive at the alternative representation for ˜ U ( τ, x )˜ U ( τ, x ) = ∞ X n = −∞ ( Z z (0) y (0) U (0 , ξ )Υ n ( x, τ | ξ, dξ + Z τ h Φ( s ) + f + ( s ) z ′ ( s ) i Υ n ( x, τ | z ( s ) , s ) ds, + Z τ (cid:2) Ψ( s ) − f − ( s ) y ′ ( s ) (cid:3) Υ n ( x, τ | y ( s ) , s ) ds (B.3)+ Z τ f − ( s )Λ n ( x, τ | y ( s ) , s ) − f + ( s )Λ n ( x, τ | z ( s ) , s ) ds ) + F ( τ, x ) , Υ n ( x, τ | ξ, s ) = 12 p π ( τ − s ) " e − (2 nl ( τ )+ x − ξ )24( τ − s ) − e − (2 nl ( τ )+ x + ξ − y ( τ ))24( τ − s ) , Λ n ( x, τ | ξ, s ) = x − ξ + 2 nl ( τ )4 p π ( τ − s ) e − (2 nl ( τ )+ x − ξ )24( τ − s ) + x + ξ − y ( τ ) + 2 nl ( τ )4 p π ( τ − s ) e − (2 nl ( τ )+ x + ξ − y ( τ ))24( τ − s ) . B.1 The limiting values x → y ( τ ) and x → z ( τ ) in Eq. (B.3) The Eq. (B.3) provides an alternative representation of the solution ˜ U ( τ, x ) of the heat equation in Eq. (4)with the initial condition in Eq. (5) and the boundary conditions in Eq. (6) at the time-dependent domain x ∈ ( y ( τ ) , z ( τ )) in terms of the Fourier series. In this section we show that the function ˜ U can beanalytically continued to the boundary points y ( τ ) and z ( τ ), andlim x → y ( τ )+0 ˜ U ( τ, x ) = f − ( τ ) , lim x → z ( τ ) − ˜ U ( τ, x ) = f + ( τ ) . (B.4)It is easy to check that the functions Υ n ( x, τ | ξ, s ) and Λ n ( x, τ | ξ, s ) are regular only if n = 0 , x ∈ [ y ( τ ) , z ( τ )] , ξ ∈ [ y ( s ) , z ( s )] , s → τ . In this case the following identities holdlim s → τ Υ n ( x, τ | ξ, s ) = 0 , lim s → τ Λ n ( x, τ | ξ, s ) = 0 , n = 0 . (B.5) Page 29 of 31 emi-analytic pricing of double barrier options... At n = 0 functions Υ ( x, τ | y ( s ) , s ) and Λ ( x, τ | y ( s ) , s ) have a singularity when s → τ, x → y ( τ ), andfunctions Υ ( x, τ | z ( s ) , s ) and Λ ( x, τ | z ( s ) , s ) - when s → τ, x → z ( τ ). Note, that the singularity of Υ is integrable and so weak. Therefore, when calculating a corresponding limit of both parts in Eq. (B.3),for the regular terms we can switch the order of the integration and limit operators, and then use thefollowing propertieslim x → y ( τ )+0 ∞ X n = −∞ Υ n ( x, τ | ξ, s ) = 0 , lim x → z ( τ ) − ∞ X n = −∞ Υ n ( x, τ | ξ, s ) = 0 , (B.6)lim x → y ( τ )+0 X n = −∞ n =0 Λ n ( x, τ | ξ, s ) = 0 , lim x → z ( τ ) − X n = −∞ n =0 Λ n ( x, τ | ξ, s ) = 0 , lim x → y ( τ )+0 Λ ( x, τ | z ( s ) , s ) = 0 , lim x → z ( τ ) − Λ ( x, τ | y ( s ) , s ) = 0 , to obtain lim x → y ( τ )+0 ˜ U ( τ, x ) = lim x → y ( τ )+0 Z τ f − ( s )Λ ( x, τ | y ( s ) , s ) (B.7)lim x → z ( τ ) − ˜ U ( τ, x ) = − lim x → z ( τ ) − Z τ f + ( s )Λ ( x, τ | z ( s ) , s ) . To proceed we need the notion of heat potentials and the results obtained in Section 3 (see also(Tikhonov and Samarskii, 1963)). It can be shown that the functions Λ ( x, τ | y ( s ) , s ) and Λ ( x, τ | z ( s ) , s )can be represented as a sum of double layer heat potentials. Therefore, we can evaluate the limitsEq. (B.7) with the help of Eq. (53), Eq. (62).In more detail, according to Eq. (B.7) in the explicit form the limits of ˜ U ( τ, x ) read˜ U ( τ, y ( τ )) = lim x → y ( τ )+0 Z τ f − ( s ) " x − y ( s )4 p π ( τ − s ) e − ( x − y ( s ))24( τ − s ) + x − y ( τ ) + y ( s )4 p π ( τ − s ) e − ( x − y ( τ )+ y ( s ))24( τ − s ) ds, ˜ U ( τ, z ( τ )) = − lim x → z ( τ ) − Z τ f + ( s ) " x − z ( s )4 p π ( τ − s ) e − ( x − z ( s ))24( τ − s ) + x − z ( τ ) + z ( s )4 p π ( τ − s ) e − ( x − z ( τ )+ z ( s ))24( τ − s ) ds. This can also be re-written in the form˜ U ( τ, y ( τ )) = lim x → y ( τ )+0 Z τ f − ( s ) x − y ( s )4 p π ( τ − s ) e − ( x − y ( s ))24( τ − s ) ds (B.8) − lim y ( τ ) − x → y ( τ ) − Z τ f − ( s ) 2 y ( τ ) − x − y ( s )4 p π ( τ − s ) e − ( x − y ( τ )+ y ( s ))24( τ − s ) ds, ˜ U ( τ, z ( τ )) = − lim x → z ( τ ) − Z τ f + ( s ) x − z ( s )4 p π ( τ − s ) e − ( x − z ( s ))24( τ − s ) ds + lim z ( τ ) − x → z ( τ )+0 Z τ f + ( s ) 2 z ( τ ) − x − z ( s )4 p π ( τ − s ) e − ( x − z ( τ )+ z ( s ))24( τ − s ) ds. Using Eq. (62), these expressions can be transformed to˜ U ( τ, y ( τ )) = f − ( τ )2 + f − ( τ )2 = f − ( τ ) , ˜ U ( τ, z ( τ )) = f + ( τ )2 + f + ( τ )2 = f + ( τ ) . Since ˜ U ( τ, x ) has the same limits as the boundary values of U ( τ, x ), and at x ∈ ( y ( τ ) , z ( τ )) we have˜ U ( τ, x ) = U ( τ, x ), Eq. (B.3) allows an alternative form U ( τ, x ) = ∞ X n = −∞ ( Z z (0) y (0) U (0 , ξ )Υ n ( x, τ | ξ, dξ + Z τ h Φ( s ) + f + ( s ) z ′ ( s ) i Υ n ( x, τ | z ( s ) , s ) ds, Page 30 of 31 emi-analytic pricing of double barrier options... + Z τ (cid:2) Ψ( s ) − f − ( s ) y ′ ( s ) (cid:3) Υ n ( x, τ | y ( s ) , s ) ds + Z τ f − ( s )Λ n ( x, τ | y ( s ) , s ) − f + ( s )Λ n ( x, τ | z ( s ) , s ) ds ) ..