Pricing American Call Options by the Black-Scholes Equation with a Nonlinear Volatility Function

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1 Pricing American Call Options by the Black-Scholes Equation with a Nonlinear Volatility Function Maria do Rosário Grossinho, Yaser Faghan Kord Daniel Ševčovič February 0, 018 arxiv: v [q-fin.cp] 17 Feb 018 Abstract In this paper we analyze a nonlinear generalization of the Black-Scholes equation for pricing American style call option in which the volatility may depend on the underlying asset price and the Gamma of the option. We propose a novel method of pricing American style call options by means of transformation of the free boundary problem for a nonlinear Black-Scholes equation into the so-called Gamma variational inequality with the new variable depending on the Gamma of the option. We apply a modified projective successive over-relation method in order to construct an effective numerical scheme for discretization of the Gamma variational inequality. Finally, we present several computational examples for the nonlinear Black-Scholes equation for pricing American style call option under presence of variable transaction costs. Keywords and phrases: American option pricing, nonlinear Black-Scholes equation, variable transaction costs, PSOR method Mathematics Subject Classification: 35K15 35K55 90A09 91B8 1 Introduction In the financial market, the price of a European style option can be computed from a solution to the well-known Black Scholes linear parabolic equation derived by Black and Scholes in [4]. Recall that a European call option gives its owner the right but not obligation to purchase an underlying asset at the expiration price E at the expiration time T. In this paper, we consider American style options which, as it is known, can be exercised anytime t in the time interval [0, T ]. The classical linear Black Scholes model was derived under several restrictive assumptions, namely no transaction costs, frictionless, liquid and complete market, etc. However, we need more realistic models in the market data analysis in order to overcome these restriction of the classical Black-Scholes theory. One of the first nonlinear models taking into account transaction costs is the jumping volatility model by Avellaneda, Lévy and Instituto Superior de Economia e Gestão and CEMAPRE, Universidade de Lisboa, Portugal. mrg@iseg.ulisboa.pt, yaser.kord@yahoo.com Dept. Applied Mathematics & Statistics, Comenius University, Bratislava, Slovakia. sevcovic@fmph.uniba.sk 1

2 Paras []. The nonlinearity of the original Black-Scholes model can also arise from the feedback and illiquid market effects due to large traders choosing given stock-trading strategies (Schönbucher and Wilmott [0], Frey and Patie [8], Frey and Stremme [9]), imperfect replication and investors preferences (Barles and Soner [3]), risk from unprotected portfolio (Kratka [17], Jandačka and Ševčovič [1]). In this paper we are concerned with a new nonlinear model derived recently by Ševčovič and Žitňanská [5] for pricing call or put options in the presence of variable transaction costs. The model generalizes the well-known Leland model with constant transaction costs (c.f. [19], [10]) and the Amster et al. model [1] with linearly decreasing transaction costs. It leads to the generalized Black-Scholes equation with the nonlinear volatility function ˆσ which depends on the product H = S S V of the underlying asset price S and the second derivative (Gamma) of the option price V. t V + 1 ˆσ(S SV ) S SV + (r q)s S V rv = 0, (1) where r, q 0 are the interest rate and the dividend yield, respectively. The price V (t, S) of a call option is given by a solution to the nonlinear parabolic equation (1) depending on the underlying stock price S > 0 at the time t [0, T ] subjected to the terminal pay-off diagram V (T, S) = (S E) +, where T > 0 is the time of maturity and E > 0 is the exercise price. For European style call options various numerical methods for solving the fully nonlinear parabolic equation (1) were proposed and analyzed by Ďuriš et al. in [6]. In [3] and [5] Ševčovič, Jandačka and Žitňanská investigated a new transformation technique (referred to as the Gamma transformation). They showed that the fully nonlinear parabolic equation (1) can be transformed to a quasilinear porous-media type of a parabolic equation τ H uβ(h) u β(h) (r q) u H + qh = 0 () for the transformed quantity H(τ, u) = S S V (t, S) where τ = T t, u = ln(s/e) and β(h) = 1 ˆσ(H) H. In a series of papers [13, 14, 15, 16] Koleva and Vulkov investigated the transformed Gamma equation (1) for pricing European style of call and put options. The derived the second order positivity preserving and splitting based numerical methods for solving (1). Our goal is to study American style call options. Their prices can be computed by means of the generalized Black-Scholes equation with the nonlinear volatility function (1). If the volatility function is constant then it is well known that American options can be priced by means of a solution to a linear complementarity problem (cf. Kwok [18]). Similarly, for the nonlinear volatility model, one can construct a nonlinear complementarity problem involving the variational inequality for the left-hand side of (1) and the inequality V (t, S) (S E) +. However, due to the fully nonlinear character of the differential operator in (1), the direct computation of the nonlinear complementarity becomes harder and unstable. Therefore, we propose a novel approach and reformulate the nonlinear complementarity problem in terms of a new transformed variable for which

3 the differential operator has the form of a quasilinear parabolic operator appearing in the left-hand side of (). In order to apply the Gamma transformation for American style options we derive the nonlinear complementarity problem for the transformed variable H and we solve the variational problem by means of the modified projected successive over relaxation method (cf. Kwok [18]). Using this method we compute American style call option prices for the Black-Scholes nonlinear model for pricing call options in the presence of variable transaction costs. The paper is organized as follows. In section, we present a nonlinear option pricing model under variable transaction costs. Section 3 is devoted to the transformation of the free boundary problem (variational inequalities) into the so-called Gamma variational inequality. In section 4, we present a reformulation of the problem with PSOR method applying efficient numerical scheme for the Gamma equation based on finite volume method. Finally, in section 5, we show numerical experiments for the option price of the transformed problem. Nonlinear Black-Scholes equation for pricing option in the presence of variable transaction costs In the original Black-Scholes theory continuous hedging of the portfolio including underlying stocks and options is allowed. In the presence of transaction costs for purchasing and selling the underlying stock, this continuous feature may lead to an infinite number of transaction costs yielding unbounded total transaction costs. One of the basic nonlinear models including transaction costs is the Leland model [19] for option pricing in which the possibility of rearranging portfolio at discrete time can be relaxed. Recall that, in the derivation of the Leland model [10, 11, 19], it is assumed that the investor follows the delta hedging strategy in which the number δ of bought/sold underlying assets depends on the delta of the option, i.e. δ = S V. Then, applying selffinancing portfolio arguments, one can derive the extended version of the Black Scholes equation t V + (r q)s S V + 1 σ S SV r T C S = 0. (3) Here the transaction cost measure r T C is given by r T C = E[ T C], (4) S t where T C is the change in transaction costs during the time interval t. If C 0 represents a percentage of the cost of the sale and purchase of a share relative to the price S then T C = 1 CS δ where δ is the number of bought ( δ > 0) or sold ( δ < 0) underlying assets during the time interval t. The parameter C > 0 measuring transaction costs per unit of the underlying asset can be either constant or it may depend on the number of transaction, i. e. C = C( δ ). Furthermore, assuming the underlying asset follows the geometric Brownian motion ds = µsdt+σsdw it can be shown that δ = S V σs S V Φ t where Φ N(0, 1) 3

4 is normally distributed random variable. Hence r T C = 1 E[C(α Φ )α Φ ], (5) t where α := σs S V t (c.f. [4], [1]). Next we recall a notion of the mean value modification of the transaction cost function introduced by Ševčovič and Žitňanská in [5]. Definition 1 [5, Definition 1] Let C = C(ξ), C : R + 0 R, be a transaction costs function. The integral transformation C : R + 0 R of the function C defined as follows: π C(ξ) = E[C(ξ Φ ) Φ ] = C(ξx)x e x / dx, (6) is called the mean value modification of the transaction costs function. Here Φ is the random variable with a standardized normal distribution, i.e., Φ N(0, 1)..1 Constant Transaction Costs - Leland s model In the case when the transaction cost measure C = C 0 > 0 is constant, then using the fact that E[ Φ ] =, we can express r π T C in (5) as follows: r T C = 1 σ SLe S V. Here C 0 is the constant parameter and Le = 0 C 0 π σ t Inserting r T C into (3) we obtain the Leland equation: > 0 is the so-called Leland number. t V + (r q)s S V + 1 ˆσ(S SV ) S SV rv = 0, (7) with the diffusion term ˆσ(S S V ) = σ (1 Le sgn( S V )) = σ (1 Le sgn(s S V )) given by the Leland model (c.f. [10, 11, 19]).. Non-increasing Transaction Costs Function Following Amster et al. function: [1] we can consider a linear non-increasing transaction costs C(ξ) = C 0 κξ, where ξ 0, (8) Here κ 0 is the rate measuring the change of the transaction costs and C 0 is positive constant parameter. The mean value modification function of the Amster model et al. is as follows: C(ξ) = C 0 π/κξ where ξ 0, (9) where κ and C 0 are the same as in relation (8). In the real market C(ξ) has to be non-negative but the function (8) may attain negative values provided the transaction amount δ = β C π 0/κ. 4

5 C Ξ,C Ξ Figure 1: A piecewise linear transaction costs function with C 0 = 0.0, κ = 1, ξ = 0.01, ξ + = 0.0 and its mean value modification C(ξ) (dashed line).3 Piecewise Decreasing Transaction Costs Function Next we want to propose a more realistic example of non-constant transaction costs function and then their relevant mean value modification C(ξ). In a stylized financial market the transaction costs function should not reach the negative value. In this part we introduce a realistic example of transaction cost (can be seen in Ševčovič and Žitňanská [5]). The advantage of this linear decreasing function is the excluding of the negative values of such a function. Definition A piecewise linear decreasing transaction costs function is given C 0, if 0 ξ < ξ, C(ξ) = C 0 κ(ξ ξ ), if ξ ξ ξ +, C 0, if ξ ξ +. (10) where ξ ξ + are given positive constants and as well as κ, C 0 are assumed to be positive. This transaction costs function seems to be more close to reality at which it pays the amount C 0 for the small volume of traded assets, when the traded stocks volume is higher, there is a discount for that and when the trades are very large, it just pays a small constant C 0. For better understanding, in Fig. 1 we show the graphs of both relevant transaction costs function and its mean value function with the known parameter values. Proposition 1 [5, Eq. (4)] Let C 0, κ be the positive constants, then for the piecewise linear function (10) the modified mean value transaction costs function is given by C(ξ) = C 0 κξ ξ + ξ ξ ξ e u / du, for ξ 0. (11) Applying integration by parts we can simply deduce the following function (see Žitňanská and Ševčovič [5]). There is a bound for this mean value transaction costs function C(ξ). 5

6 Proposition [5, Proposition.] Let C 0 be positive in Definition (). Then the modification transaction costs function in (11) verifies C 0 C(ξ) C 0 (1) and lim ξ C(ξ) = lim ξ C(ξ) = C 0. (13) Proposition 3 [5, Proposition.1] Assume that C : R + 0 R is a measurable and bounded transaction costs function. Then the price of the option based on the variable transaction costs is given by the solution of the following nonlinear Black Scholes PDE t V + (r q)s S V + 1 ˆσ(S SV ) S SV rv = 0, (14) where the nonlinear diffusion coefficient ˆσ is ˆσ(S SV ) = σ (1 π C(σS S V ) t) sgn(s S V ) σ. (15) t 3 Transformation of the free boundary problem to the Gamma variational inequality In this section we investigate the transformation method of the free boundary problem (variational inequalities) into the so-called Gamma variational inequality. In the context of European style option the transformation method to the Gamma equation was proposed and analyzed by Jandačka and Ševčovič [1]. Indeed, let us consider the generalized nonlinear Black-Scholes equation for European option pricing of the form t V + (r q)s S V + Sβ(S SV ) rv = 0, S > 0, t (0, T ), (16) where β(h) = 1 ˆσ(H) H. Then, making the change of variables u = ln( S ) and τ = T t and computing the second E derivative of the equation (16) with respect to u, we derive the so-called Gamma equation, given by τ H u β(h) uβ(h) (r q) u H + qh = 0 (17) More details can be found in Ševčovič and Žitňanská [5]. Lemma 1 [5, Proposition 3.1, Remark 3.1] Let us consider the call option with the pay-off diagram V (T, S) = (S E) +. Then the function H(τ, u) = S S V (t, S) where u = ln( S ) and τ = T t is a solution to (17) subject to the Dirac initial condition E H(0, x) = δ(x) if and only if is a solution to (16). V (t, S) = + (S Ee u ) + H(τ, u)du 6

7 3.1 American style options The advantage of American style over European style option contracts is the flexibility that they offer. The owner of American option has the right to exercise the contract earlier than the expiration date T of the contract. More precisely, in mathematical modeling of American options, unlike European style options, there is the possibility of early exercising the contract at some time t [0, T ) prior to the maturity time T. It is fairly saying that the most of the derivative contracts traded in the financial markets are of the American style. An American call option is the contract that gives the right but not the obligation to buy the underlying asset at the strike price E anytime t [0, T ] prior to the expiration time t = T. As in the case of European style options, we are interested in knowing the fair option premium at the starting point t = 0 of contracting. In the case of an American call option the challenge is to find the price of the option V (t, S) at the time t [0, T ] having in view the possible gain if exercising it at that time t. Comparing an American style contract with the European one the relation between them is as follows: V A (t, S) V E (t, S) = (S E) +, S 0, t [0, T ]. (18) For the American call option, if the price of the option V (t, S) at anytime prior to the maturity T is lower than its payoff function (S E) + then the policy is to purchase the option and exercise it immediately as we are allowed for these type of contracts. But in this case there would be an arbitrage opportunity for the holder of the option. With respect to the highly demand for trading such an option, the market will increase its price to a value higher or equal to the payoff function and, then, the arbitrage opportunity will be removed. Assuming that the American call option on the underlying stock is paying the dividend yield q > 0, then the price of the American call option satisfies V A (t, S) > V E (t, S), S > 0, t [0, T ). (19) It is well-known that pricing an American call option on an underlying stock paying continuous dividend yield q > 0 leads to a free boundary problem. In addition to a function V (t, S), we need to find the early exercise boundary function S f (t) with respect to time t [0, T ]. Furthermore, we note that the function S f (t) has the following properties: If S f (t) > S for t [0, T ] then V (t, S) > (S E) +. If S f (t) S for t [0, T ] then V (t, S) = (S E) +. In the last decades many authors analyzed the free boundary position function S f. In [1] Stamicar, Ševčovič and Chadam derived accurate approximation to the early exercise position for times t close to expiry T for the Black-Scholes model with constant volatility (see also [7], [], [6]). The method has been generalized for the nonlinear Black-Scholes model by Ševčovič in [3]. Remark 1 Following Kwok [18] (see also [4]) we can also formulate the free boundary problem for pricing the American call option. It consists of finding a function V (t, S) and 7

8 the early exercise boundary function that solve the Black-Scholes PDE on a time depending domain: t V + 1 ˆσ(S SV ) S SV + (r q)s S V rv = 0, 0 < S < S f (t), (0) V (T, S) = (S E) +, (1) V (t, S f (t)) = S f (t) E, S V (t, S f (t)) = 1, V (t, 0) = 0. () 3. Transformation of the variational inequality In the presence of transaction costs for buying and selling the underlying stock, we face the nonlinear problem in which we transformed the arising free boundary problem for pricing the American call option into the so-called Gamma variational inequality for the new variable H = S S V. Lemma Let V (t, S) be a given function. Let u = ln( S ), τ = T t and define the E function Y (τ, u) Y (τ, u) = t V + (r q)s S V + Sβ(S SV ) rv. Then τ H + u β(h) + uβ(h) + (r q) u H qh = 1 E e u [ uy u Y ], where H(τ, u) = S S V (t, S). P r o o f. By differentiating the function Y with respect to the variable u and using the fact u = S S, we obtain u Y = t (S S V ) + S(β + u β) + (r q)sh qs S V where S = Ee u. Furthermore, since uy = t (S S V +S SV )+(r q)s(h + u H)+S(β + u β)+s( xβ + u β) qs S V qh, then uy u Y = Ee u Ψ[H], (3) where Ψ[H] := τ H + u β(h) + uβ(h) + (r q) u H qh. Remark For the particular case Y = 0, we conclude that the function V (t, S) is a solution to the European style call option satisfying the nonlinear Black-Scholes equation (14) if and only if the function H(τ, u) is a solution to the so-called Gamma equation τ H + u β(h) + uβ(h) + (r q) u H qh = 0, subject to the initial condition H(x, 0) = δ(x), where δ is the Dirac function (cf. [3], [5]). 8

9 Lemma 3 If we assume the asymptotic behavior then we have + lim Y (τ, u) = 0 and lim u u e u u Y (τ, u) = 0, (S Ee u ) + Ψ[H](τ, u)du = Y (τ, u) u=ln(s/e) t V + (r q)s S V + Sβ(S SV ) rv. P r o o f. Using Lemma and equation (3) we can express the term + (S Eeu ) + Ψ[H](τ, u)du as follows: + (S Ee u ) + 1 E e u [ uy u Y ]du = 1 E = 1 E ln(s/e) ln(s/e) (Se u E)[ uy u Y ]du [ Se u u Y (Se u E) u Y du ] + [(Se u E) u Y ] ln(s/e) }{{} = 1 E + 0 E u Y du = Y (τ, u) u=ln(s/e) = t V + (r q)s S V + Sβ(S SV ) rv. Theorem 1 The function V (t, S) is a solution to the nonlinear complementarity problem (NLCP): V (t, S) g(s) and t V + (r q)s S V + Sβ(S SV ) rv 0 ( t V + (r q)s S V + Sβ(S SV ) rv ) (V g) = 0 for any S > 0 and t [0, T ] where g(s) (S E) + if and only if the following Gamma variational inequality and complementarity constraint: (S Ee u ) + [ Ψ[H](τ, u)]du 0, (4) (S Ee u ) + H(τ, u)du g(s), (5) ( + ) (S Ee u ) + Ψ[H](τ, u))du (S Ee u ) + H(τ, u)du g(s) = 0, (6) hold for any S 0 and τ [0, T ]. P r o o f. It directly follows by applying Lemma and Lemma 3. 9

10 Remark 3 For calculating V (T, S) in Theorem 1 we use the fact that H(0, u) = H(u), u R, where H(u) := δ(u) is the Dirac delta function such that + δ(u)du = 1, + δ(u u 0 )φ(u)du = φ(u 0 ), for any continuous function φ. In what follows, we will approximate the initial Dirac δ-function can be approximated as follows: H(x, 0) f(d)/(ˆσ τ ), where 0 < τ 1 is a sufficiently small parameter, f(d) is the PDF function of the normal distribution, that is: f(d) = e d / / π and d = (x + (r q σ /)τ ) /σ τ. Note that this approximation follows from observation that for a solution of the linear Black Scholes equation with a constant volatility σ > 0 at the time T τ close to expiry T the value H lin (x, τ ) = S S V lin (S, T τ ) is given by H lin (x, τ ) = f(d)/(ˆσ τ ). Moreover, H lin (., τ ) δ(.) as τ 0 in the sense of distributions. 4 Solving the Gamma variational inequality by the PSOR method According to Theorem 1, the American call option problem can be rewritten in terms of the function H(τ, u) in the form of the Gamma variational inequality (4) (5) with the complementarity constraint (6). In order to apply the Projected Successive Over Relaxation method (PSOR) (c.f. Kwok [18]) to the variational inequalities (4) (5), we need first to discretize the nonlinear operator Ψ: Ψ[H] τ H (r q) u H u β(h) uβ(h) + qh. (7) In the next, we follow the paper by Ševčovič and Žitňanská [5] in order to derive an efficient numerical scheme for solving the Gamma variational inequality for a general form of the function β(h) including the special case of the variable transaction costs model. 4.1 Numerical scheme for the Gamma variational inequality The proposed numerical discretization is based on the finite volume method. Assume that the spatial interval belongs to u ( L, L) for sufficiently large L > 0 where the time interval [0, T ] is uniformly divided with a time step k = T m into discrete points τ j = jk for j = 1,,, m. Furthermore, we divide the spatial interval [ L, L] into a uniform mesh of discrete points u i = ih where i = n,, n with a spatial step h = L n. The discretization of the operator Ψ[H] leads to a tridiagonal matrix multiplied by the vector H j = (H j n+1,, H j n 1) R n 1. More precisely, the vector Ψ[H] j at the time level τ j is given by Ψ[H] j = (A j H j d j ) where the (n 1) (n 1) matrix A j 10

11 has the form with coefficients of the form: b j n+1 c j n a j A j n+ b j n+ c j n+. = a j n b j n c j n 0 0 a j n 1 b j n 1 a j i = k h β (H j 1 i 1 ) + k (r q), h c j i = k h β (H j 1 i ) k (r q), h b j i = (1 + kq) (a j i + cj i ), (8) and d j i = Hj 1 i + k ( β(h j 1 i ) β(h j 1 i 1 h )). Finally, using numerical integration the variational inequality (4) (5) can be discretized as follows: n V (S, T τ j ) = h (S Ee u i ) + H j i, j = 1,,, m. (9) i= n Then, the full space-time discretized version of the problem (4) (5) is given by h Let us assume that n (S Ee u i ) [ ] + (A j H j ) i d j i 0, (30) i= n h n (S Ee u i ) + H j i g(s) (S E) +. (31) i= n P li = h[max(s l Ee u i, 0)] = h[max(ee v l Ee u i, 0)] (3) where v l = u l+1 + u l 1, for l = n,, n. Remark 4 The matrix P = (P li ) is invertible. 4. Applying the PSOR method In this section, our aim is to solve the problem (30) (31) by means of the PSOR method. Then, according to (3), we can rewrite (30) (31) for the American call option in this form P AH P d P H g (P AH P d) i (P H g) i = 0, for all i, 11

12 Β H H Figure : A graph of the function β(h) related to the piecewise linear decreasing transaction costs function (see [1]). where A = A j, g i = (S i E) + and H = H j. This nonlinear complementarity problem can be solved by the PSOR algorithm, given by the following iterative scheme: 1. for k = 0 set v j,k = v j 1,. until k k max repeat: 3. set v j = v j,k+1, l>i { } v j,k+1 i = max v j,k i + ω(t j,k+1 i v j,k i ), g i, t j,k+1 i = 1 à ii ( l<i à il v j,k+1 l à il v j,k l for i = n,, n and j = 1,,, m, where v j = P H j, d j = P d j and à = P Aj P 1. Here ω [1, ] is a relaxation parameter which can be tuned in order to speed up convergence process. Finally, using the value H j = P 1 v j and equation (9), we can evaluate the option price. + d j i ), 5 Numerical experiments In this section, we focus our attention on numerical experiments for computing American style call option prices based on the nonlinear Black-Scholes equation that includes a piecewise linear decreasing transaction costs function. In Fig., we show the corresponding function β(h) given by β(h) = σ ( 1 π C(σ H ) t) sgn(h) σ H, t 1

13 Table 1: Model and numerical parameter values for calculation of numerical experiments. Model parameters Numerical parameters C 0 = 0.0 m=00, 800 κ = 0.3, ξ = 0.05ξ + = 0.1 n=50, 500 t = 1/61 h=0.01 σ = 0.3 τ = r = 0.011, q = k = T/m T = 1, E = 50 L =.5 where C is the mean value modification of the transaction costs function. The parameters C 0, κ, ξ ±, t characterizing the nonlinear piecewise linear variable transaction cost function and other model parameters are given in Table 1. Here t is the time interval between two consecutive portfolio rearrangements, the maturity time T, the historical volatility σ, the dividend yield q, the strike price E and r is the risk free interest rate. The small parameter 0 < τ 1 represents the smoothing parameter for approximation of the Dirac δ function (see Remark 3). For the given numerical parameters in Table 1, we computed option values V vtc for several underlying asset prices S. The prices were calculated by numerical solutions for both Bid and Ask option prices in Table. The Bid price V Bidvtc is compared to the price V BinMin computed by means of the binomial tree method (cf. [18]) with the lower volatility ˆσ min = σ 1 (1 C 0 π σ ), whereas the upper bound price V t BinMax corresponds to the solution with the higher constant volatility ˆσ max = σ 1 (1 C 0 π Similarly, as well as for the Ask price V Askvtc the lower bound V BinMin corresponds to the solution of the binomial tree method with the lower volatility ˆσ min = σ (1 + 1 C 0 π σ ), whereas the upper bound V t BinMax higher constant volatility ˆσ max = σ (1 + C 0 π 1 σ ). t σ t ). corresponds to the solution with the Remark 5 In the case of a European style option, it can be shown analytically by using the parabolic comparison principle that V σmin (S, t) V vtc (t, S) V σmax (t, S), S > 0, t [0, T ]. For more details we refer to [5]. For the case of American style options, these inequalities can be observed in Table. In Table 3, we present a comparison of results obtained by our method based on the Gamma equation in which we considered constant volatilities σ min and σ max and those obtained by the well-known method based on binomial trees for American style call options (cf. [18]). The difference in the prices is in the order of the mesh size h = L/n. In Fig. 3 we present the free boundary function S f (t) obtained by our method with variable transaction costs function for bid option value compared to the binomial trees with σ min, σ max in which parameter values are given by E = 50, σ = 0.3, r = 0.011, q = 0.008, T = 1. In Fig. 4 we plot the graphs of the solutions V vtc (t, S) at t = 0 for both bid 13

14 Table : Bid (top table) and Ask (bottom table) American call option prices V Bidvtc and V Askvtc obtained from the numerical solution of the nonlinear model with variable transaction costs for different meshes. Comparison to the option prices V BinMin and V BinMax computed by means of binomial trees for constant volatilities σ min and σ max. n = 50, m = 00 n = 500, m = 800 S V BinMin V Bidvtc V BinMax S V BinMin V Bidvtc V BinMax n = 50, m = 00 n = 500, m = 800 S V BinMin V Askvtc V BinMax S V BinMin V Askvtc V BinMax Sf t BinMin Gamma BinMax Figure 3: The early exercise boundary function S f (t), t [0, T ], computed for the model with variable transaction costs (dashed line Gamma) and comparison with early exercise boundary computed by means of binomial trees with constant volatilities σ min (bottom curve) and σ max (top curve). 14

15 1 10 V(S,t) Payoff Gamma BinMin BinMax S 1 V(S,t) Payoff Gamma BinMin BinMax S Figure 4: The American Bid (top) and Ask (bottom) call option price V (t, S) at t = 0 computed by means of the nonlinear Black-scholes model with variable transaction costs with the mesh size n = 500, m = 800 in comparison to solutions V σmin, V σmax calculated by the binomial trees with constant volatilities σ min and σ max. and ask prices. We also plot the prices obtained by the binomial tree method with the constant lower volatility σ min and the higher volatility σ max, respectively. 6 Conclusions In this paper we investigated a nonlinear generalization of the Black-Scholes equation for pricing American style call options assuming variable transaction costs for trading the underlying assets. In this way, we presented a model that addresses a more realistic financial framework than the classical Black-Scholes model. From the mathematical point of view, we studied a problem that consists of a fully nonlinear parabolic equation in which the nonlinear diffusion coefficient may depend on the second derivative of the option price. Furthermore, for the American call option we transformed the nonlinear complementarity problem into the so called Gamma variational inequality. We solved the Gamma variational inequality by means of the PSOR method and presented an effective numerical scheme for discretization the Gamma variational inequality. Then, we performed numerical computations for the model with variable transaction costs and compared the results with lower and upper bounds computed by means of the binomial tree method for constant volatilities. Finally, we presented a comparison between the respective early exercise boundary functions. Acknowledgements: 15

16 Table 3: Ask call option values V Askvtc of the numerical solution of the model under constant volatilities σ = σ min (left) and σ = σ max (right) and comparison to the prices computed by the Binomial tree method (with n = 100 and n = 00 nodes, respectively. σ = σ min σ = σ max n = 50, m = 00 n = 500, m = 800 n = 50, m = 00 n = 500, m = 800 S V Askvtc V BinMin V Askvtc V BinMin S V Askvtc V BinMax V Askvtc V BinMax This research was supported by the European Union in the FP7-PEOPLE-01-ITN project STRIKE - Novel Methods in Computational Finance (304617), the project CEMAPRE MULTI/00491 financed by FCT/MEC through national funds and the Slovak research Agency Project VEGA 1/0780/15. References [1] Amster, P., Averbuj, C. G., Mariani, M. C., and Rial, D.: A Black Scholes option pricing model with transaction costs. J. Math. Anal. Appl., 303, (005). [] Avellaneda, M., and Paras, A.: Dynamic Hedging Portfolios for Derivative Securities in the Presence of Large Transaction Costs. Applied Mathematical Finance., 1, (1994). [3] Barles, G., and Soner, H. M.: Option Pricing with transaction costs and a nonlinear Black Scholes equation. Finance Stochast.,, (1998). [4] Black, F., and Scholes, M.: The pricing of options and corporate liabilities. J. Political Economy, 81, (1973). [5] Dewynne, J. N., Howison, S. D., Rupf, J., and Wilmott, P.: Some mathematical results in the pricing of American options. Euro. J. Appl. Math., 4, (1993). [6] Ďuriš, K., Shih-Hau Tan, Choi-Hong Lai, and Ševčovič, D.: Comparison of the analytical approximation formula and Newton s method for solving a class of nonlinear Black-Scholes parabolic equations, Computational Methods in Applied Mathematics 16(1) 016, [7] Evans, J.D., R. Kuske, and J.B. Keller: American options on assets with dividends near expiry, Mathematical Finance, 1(3), (00). 16

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