Endogenous Stochastic Arbitrage Bubbles and the Black--Scholes model
EEndogenous Stochastic Arbitrage Bubblesand the Black–Scholes model.
Mauricio Contreras G. ∗ September 22, 2020
This paper develops a model that incorporates the presence of stochastic arbitrage explicitly in theBlack–Scholes equation. Here, the arbitrage is generated by a stochastic bubble, which generalizesthe deterministic arbitrage model obtained in the literature [17]. It is considered to be a genericstochastic dynamic for the arbitrage bubble, and a generalized Black–Scholes equation is then de-rived. The resulting equation is similar to that of the stochastic volatility models, but there areno undetermined parameters as the market price of risk.The proposed theory has asymptotic behaviors that are associated with the weak and strong ar-bitrage bubble limits. For the case where the arbitrage bubble’s volatility is zero (deterministicbubble), the weak limit corresponds to the usual Black-Scholes model. The strong limit case alsogive a Black–Scholes model, but the underlying asset’s mean value replaces the interest rate. Whenthe bubble is stochastic, the theory also has weak and strong asymptotic limits that give rise tooption price dynamics that are similar to the Black–Scholes model. Explicit formulas are derivedfor Gaussian and lognormal stochastic bubbles.Consequently, the Black–Scholes model can be considered to be a “low energy” limit of a moregeneral stochastic model.Keywords: Option pricing; Black–Scholes equation; Arbitrage bubbles; Stochastic equations. ∗ Universidad Metropolitana de Ciencias de la Educación UMCE. email: [email protected] a r X i v : . [ q -f i n . M F ] S e p Introduction
Since its introduction by Fischer Black, Myron Scholes [1], and Robert C. Merton [2], the Black–Scholes (B–S) model has been widely used in financial engineering to price a derivative on equity.Several generalizations of the initial model premises have since been made. For example, some ofthese generalizations include stochastic volatility models [3]–[7]; the incorporation of jumps, whichgives rise to integrodifferential equations for the option price [8]; and, the consideration of manyassets which gives its multi-asset extension [9], [10], among others.However, one of the last assumptions of the initial model to be changed was the no-arbitragehypothesis. In effect, in the last decade, several efforts to overcome the no-arbitrage assumptionhave been made in the literature [11], [12], [13]. In addition, [14], [15], and [16] suggested that thearbitrage can be taken into account in option pricing model by changing the usual return rate ofthe B–S portfolio P from dP = rdt, (1)to dP = ( r + x ( t )) P dt, (2)where x ( t ) follows an Ornstein–Uhlenbeck process. Using these ideas, an endogenous arbitragemodel is presented in [17]. Here, equation (2) is replaced by the stochastic differential equation dP = rP dt + f ( S, t ) P dW, (3)and where the deterministic function f ( S, t ) was called an arbitrage bubble, and dW is the sameBrownian motion that is present in the underlying asset dynamics given by dS = Sµdt + SσdW. (4)In [17], using (3) and (4), the following Black–Scholes equation in the presence of an arbitragebubble is obtained ∂V∂t + 12 σ S ∂ V∂S + ( r + v ( S, t )) (cid:20) S ∂V∂S − V (cid:21) = 0 , (5)where V = V ( S, t ) and v ( S, t ) = ( r − µ )( σ − f ( S, t )) f ( S, t ) , (6)is a potential term that is equivalent to an electromagnetic potential that is induced by the arbi-trage bubble f ( S, t ) . An approximate solution of this equation for an arbitrary bubble form f ( S, t ) is given in [18] and a method to determine the bubble f from the real financial data is proposedin [19]. The resonances that appear in the model are also discussed in [20].The interacting B–S equation (5) has two limit behaviors. The first is the “weak bubble” limit f /σ << or f ≈ , in which case the potential is v ( S, t ) ≈ and (5) becomes the usual “free”Black–Scholes equation ∂V∂t + 12 σ S ∂ V∂S + r (cid:20) S ∂V∂S − V (cid:21) = 0 . (7)2he second is the “strong bubble” limit f /σ >> or f → ∞ , in which case v ( S, t ) = − ( r − µ ) (8)and equation (5) again becomes a “free” Black–Scholes equation ∂V∂t + 12 σ S ∂ V∂S + µ (cid:20) S ∂V∂S − V (cid:21) = 0 , (9)where the value of the interest rate has been changed to the mean of the underlying asset value µ .In this paper, I want to incorporate possible stochastic effects on the arbitrage bubble. Hence,instead of f being a given deterministic function, it becomes a random variable. I will explore itsconsequences on the dynamic of the option price, and I will obtain the respective weak and strongbubble limits for this case. Consider the usual underlying asset dynamics given in (4). Now, I generalize the deterministicbubble given in [17] to the stochastic case. To do that, one can assume that the arbitrage bubblesatisfies the generic stochastic differential equation df = µ f dt + Γ dW, (10)where µ = µ ( S, f, t ) , σ = σ ( S, f, t ) , µ f = µ f ( S, f, t ) and Γ = Γ(
S, f, t ) are arbitrary functionsof S , f and t , which defines the stochastic model completely. Note that for both equations (4) and(10), there is a unique Brownian motion dW . Therefore, this model is endogenous in the samesense of [17].In this case, the option price V then also becomes a function of f , so V = V ( S, f, t ) and by theItô lemma one has that dV = ∂V∂t dt + ∂V∂S dS + ∂V∂f df + 12 ∂ V∂S dS + 12 ∂ V∂f df + ∂ V∂S∂f dSdf, (11)and by replacing (4) and (10) in (11), one has that dV = L ( V ) dt + (cid:20) σS ∂V∂S + Γ ∂V∂f (cid:21) dW, (12)where L ( V ) denotes the differential operator action L ( V ) = ∂V∂t + 12 σ S ∂V∂S + 12 Γ ∂ V∂f + Sσ Γ ∂ V∂S∂f + Sµ S ∂V∂S + µ f ∂V∂f . (13)To derive the corresponding Black–Scholes equation, one must consider a portfolio P that is con-structed by a number of N V options and N S underlying assets according to P = N S S + N V V, (14)3o one has that (see [10], [9]) dP = N S dS + N V dV. (15)According to equation (3) [17], the portfolio return in the presence of a arbitrage bubble f has theform dP = P rdt + P f dW, (16)so N S dS + N V dV = P rdt + P f dW. (17)By replacing (4), (12) in (17), one obtains N S ( µSdt + σSdW )+ N V (cid:18) Ldt + σS ∂V∂S dW + Γ ∂V∂f dW (cid:19) = ( N s S + N V V ) rdt +( N s S + N V V ) f dt. (18)By equalling terms in dt and dW in this equation, one finds the system ( µ S S − Sr ) N S + ( L − rV ) N V = 0( σS − Sf ) N S + (cid:16) σS ∂V∂S + Γ ∂V∂f − V f (cid:17) N V = 0 , (19)To obtain a solution with N S (cid:54) = 0 and N V (cid:54) = 0 , the determinant associated to the matrix form ofthis system (19) must be equal to zero; that is, ( µS − Sr ) (cid:18) σS ∂V∂S + Γ ∂V∂f − V f (cid:19) − ( σS − Sf )( L − rV ) = 0 , (20)that is, ( L − rV ) = ( µS − Sr ) (cid:16) σS ∂V∂S + Γ ∂V∂f − V f (cid:17) ( σS − Sf ) . (21)Now, by replacing L ( V ) in (13) and simplifying terms, one finally ends with the following explicitBlack–Scholes equation for the option price ∂V∂t + 12 σ S ∂ V∂S + 12 Γ ∂ V∂f + Sσ Γ ∂ V∂S∂f + ( r + v ( f )) (cid:20) S ∂V∂S − V (cid:21) + (cid:18) µ f − ( µ − r )( σ − f ) Γ (cid:19) ∂V∂f = 0 . (22)where v ( f ) = ( r − µ )( σ − f ) f, (23)is the “electromagnetic” potential mentioned in [17]. Note that the (22) is the same form of theBlacks–Scholes equation for a stochastic volatility model, but without external undetermined func-tions as the market price of risk [9], [7].For the Γ = 0 case, equation (22) reduces to ∂V ( S, f, t ) ∂t + 12 σ S ∂ V ( S, f, t ) ∂S + ( r + v ( f )) (cid:20) S ∂V ( S, f, t ) ∂S − V ( S, f, t ) (cid:21) + µ f ∂V ( S, f, t ) ∂f = 0 . (24)4ere, f is, due to (10), the deterministic function dfdt = µ f ( S, f, t ) , (25)so f = f ( S, t ) and the option price becomes a function of S and t only, which is defined by V ( S, t ) = V ( S, f ( S, t ) , t ) , (26)This means that ∂V ( S, t ) ∂t = ∂V ( S, f ( S, t ) , t ) ∂f df ( S, t ) dt + ∂V ( S, f ( S, t ) , t ) ∂t , (27)that is, ∂V ( S, t ) ∂t = ∂V ( S, f ( S, t ) , t ) ∂f µ f + ∂V ( S, f ( S, t ) , t ) ∂t , (28)Consequently, in terms of V ( S, t ) , equation (24) is finally ∂V ( S, t ) ∂t + 12 σ S ∂ V ( S, t ) ∂S + ( r + v ( f )) (cid:20) S ∂V ( S, t ) ∂S − V ( S, t ) (cid:21) = 0 . (29)which is the same equation (5). Thus, the case Γ = 0 recovers the deterministic arbitrage bubblecase.In the rest of this paper, I will test the effect of the stochastic bubble on the Black–Scholessolution on analytical grounds. I will analyze two special cases: one is the Gaussian bubble, andthe other is the lognormal bubble. For these two models, one can find an analytical solution validfor some asymptotic regions in the ( S, f, t ) space.Of course, for more general models, to find solutions of equation (22) one must use numerical meth-ods [9]. Nevertheless, the analytical solutions obtained in this work can be used to test the gradeof exactitude of the numerical solutions. In a further incoming paper, I will tackle the numericalanalysis in a detailed manner and I will then compare it with the analytical solutions obtained inthe following sections. For the Gaussian bubble, one can consider that the asset’s dynamics (4) is given by the usualBlack–Scholes case; that is, µ and σ are constants. In addition, for the Gaussian bubble, the f –dynamic is given by (10) with µ f and Γ constants. In fact, these parameters represent the meanheight and the variance of the bubble.Thus, one needs to find solutions of (22), with all parameters being constant. An analytical solu-tion can be obtained that is valid in the following regions of the ( S, f, t ) space: (a) the region f ≈ , in which case v ( f ) = ( r − µ )( σ − f ) f ≈ and (22) takes the form ∂V∂t + 12 σ S ∂ V∂S + 12 Γ ∂ V∂f + Sσ Γ ∂ V∂S∂f + r (cid:20) S ∂V∂S − V (cid:21) + (cid:18) µ f − ( µ − r ) σ Γ (cid:19) ∂V∂f = 0 , (30)5nd (b) the asymptotic limit f >> σ or f → ∞ , in which case v ( f ) = ( r − µ )( σ − f ) f → − ( r − µ ) sothe asymptotic Black–Scholes equation becomes ∂V∂t + 12 σ S ∂ V∂S + 12 Γ ∂ V∂f + Sσ Γ ∂ V∂S∂f + µ (cid:20) S ∂V∂S − V (cid:21) + µ f ∂V∂f = 0 . (31)One can consider equation (30) as the “weak bubble limit” of (22), whereas (31) can be consideredas the “strong bubble limit” of (22).Instead of working directly on equations (30) and (31) to obtain the analytical solutions, onecan again consider the “full” equation (22) for the Gaussian bubble, and take the following trans-formation ¯ u = ln S − (cid:0) r − σ (cid:1) tf = ft = t. (32)This maps (22) to the following equation ∂V∂t + 12 σ ∂V∂ ¯ u + 12 Γ ∂ V∂f + σ Γ ∂ V∂ ¯ u∂f + v ( f ) ∂V∂ ¯ u + (cid:18) µ f − ( µ − r )Γ( σ − f ) (cid:19) ∂V∂f − ( r + v ( f )) V = 0 . (33)By defining V (¯ u, f, t ) = e − r ( T − t ) ψ (¯ u, f, t ) (34)one has that ∂ψ∂t + (cid:18) σ ∂ ψ∂ ¯ u + 12 Γ ∂ ψ∂f + σ Γ ∂ ψ∂ ¯ u∂f (cid:19) + v ( f ) (cid:18) ∂ψ∂ ¯ u − ψ (cid:19) + (cid:18) µ f − ( µ − r )Γ( σ − f ) (cid:19) ∂ψ∂f = 0 . (35)Now, by performing the following transformation ¯ x = (cid:16) ¯ uσ + f Γ (cid:17) − µ f t ¯ y = (cid:16) ¯ uσ − f Γ (cid:17) + µ f tτ = T − t, (36)one arrives to − ∂ψ∂τ + 12 ∂ ψ∂ ¯ x + (cid:20) σ v ( f ) −
12 ( µ − r )( σ − f ) (cid:21) ∂ψ∂ ¯ x + (cid:20) σ v ( f ) + 12 ( µ − r )( σ − f ) (cid:21) ∂ψ∂ ¯ y − v ( f ) ψ = 0 , (37)where f denotes the function f = f (¯ x, ¯ y, τ ) = Γ (cid:16) ¯ x − ¯ y − µ f Γ ( T − τ ) (cid:17) . (38)6ow, by replacing v ( f ) explicitly one has that − ∂ψ∂τ + 12 ∂ ψ∂ ¯ x + (cid:20) ( r − µ )2 σ (1 + f /σ )(1 − f /σ ) (cid:21) ∂ψ∂ ¯ x + (cid:20) − ( r − µ )2 σ (cid:21) ∂ψ∂ ¯ y − ( r − µ )(1 − f /σ ) ( f /σ ) ψ = 0 , (39)One can now consider the “weak” and “strong” limits of (39). The “weak” bubble limit, that is f /σ << , (40)will be valid in the (¯ x, ¯ y, τ ) space region for which Γ (cid:16) ¯ x − ¯ y − µ f Γ ( T − τ ) (cid:17) << σ, (41)and the equation (39) can be approximated to ( f ≈ ) − ∂ψ∂τ + 12 ∂ ψ∂ ¯ x + (cid:20) ( r − µ )2 σ (cid:21) ∂ψ∂ ¯ x − (cid:20) ( r − µ )2 σ (cid:21) ∂ψ∂ ¯ y = 0 . (42)Note that this equation also can be obtained by doing several coordinate transformations directlyto equation (30). The “strong” bubble limit, that is f /σ >> , (43)will be valid in the (¯ x, ¯ y, τ ) space region for which Γ (cid:16) ¯ x − ¯ y − µ f Γ ( T − τ ) (cid:17) >> σ, (44)so equation (39) can be approximated in this region with the limit f → ∞ , so (1 + f /σ )(1 − f /σ ) → − , (45)and v ( f ) → − ( r − µ ) , (46)so asymptotic Black–Scholes equation is − ∂ψ∂τ + 12 ∂ ψ∂ ¯ x − (cid:20) ( r − µ )2 σ (cid:21) ∂ψ∂ ¯ x − (cid:20) ( r − µ )2 σ (cid:21) ∂ψ∂ ¯ y + ( r − µ ) ψ = 0 . (47)Now by defining ψ = e ( r − µ ) τ Ψ (48)7ne can find the strong limit in terms of Ψ as − ∂ Ψ ∂t + 12 ∂ Ψ ∂ ¯ x − (cid:20) ( r − µ )2 σ (cid:21) ∂ Ψ ∂ ¯ x − (cid:20) ( r − µ )2 σ (cid:21) ∂ Ψ ∂ ¯ y = 0 . (49)Note that equation (49) has the same form of equation (42) for ψ and note also that the optionprice in the strong limit is given by V = e − r ( T − t ) ψ = e − r ( T − t ) e ( r − µ )( T − t ) Ψ = e − µ ( T − t ) Ψ . (50) f /σ ≈ − for the Gaussian bubble For the Gaussian case, the variable f can take negative values; thus, f could also take values near − σ . Then, when f /σ ≈ − , that is for region in the (¯ x, ¯ y, τ ) for which Γ (cid:16) ¯ x − ¯ y − µ f Γ ( T − τ ) (cid:17) ≈ − σ, (51)equation (39) reduces to − ∂ψ∂τ + 12 ∂ ψ∂ ¯ x + (cid:20) − ( r − µ )2 σ (cid:21) ∂ψ∂ ¯ y + ( r − µ )2 ψ = 0 . (52)By defining ψ = e ( r − µ )2 τ Ψ , (53)so − ∂ Ψ ∂τ + 12 ∂ Ψ ∂ ¯ x − ( r − µ )2 σ ∂ Ψ ∂ ¯ y = 0 . (54)Note that the option price in this case is V = e − rτ ψ = e − rτ e ( r − µ )2 τ Ψ = e − ( r + µ )2 τ Ψ . (55) For the lognormal bubble, the underlying S –dynamics are the same as the Gaussian but for f onetakes instead µ f = f ¯ µ f Γ = f ¯Γ , (56)where ¯ µ f and ¯Γ are constants. Consequently, (10) becomes df = ¯ µ f f dt + f ¯Γ dW. (57)In this case, both the underlying asset and the stochastic bubble have lognormal dynamics. Forthis case, equation (22) becomes ∂V∂t + 12 σ S ∂ V∂S + 12 ¯Γ f ∂ V∂f + σ ¯Γ Sf ∂ V∂S∂f + ( r + v ( f )) (cid:20) S ∂V∂S − V (cid:21) + (cid:18) ¯ µ f − ( µ − r )( σ − f ) ¯Γ (cid:19) f ∂V∂f = 0 , (58)8ow by taking the coordinate transformation ¯ u = ln S − (cid:0) r − σ (cid:1) t ¯ v = ln f − (cid:0) ¯ µ f − ¯Γ (cid:1) tt = t, (59)and defining V (¯ u, ¯ v, t ) = e − r ( T − t ) ψ (¯ u, ¯ v, t ) , (60)equation (58) maps to ∂ψ∂t + 12 σ ∂ ψ∂ ¯ u + 12 ¯Γ ∂ ψ∂ ¯ v + σ Γ ∂ ψ∂ ¯ u∂ ¯ v + v ( f ) (cid:18) ∂ψ∂ ¯ u − ψ (cid:19) − ( µ − r )( σ − f ) ¯Γ ∂ψ∂ ¯ v = 0 . (61)Now, by doing the following transformation x = (cid:0) ¯ uσ + ¯ v ¯Γ (cid:1) y = (cid:0) ¯ uσ − ¯ v ¯Γ (cid:1) τ = T − t, (62)the equation (61) gets − ∂ψ∂τ + 12 ∂ ψ∂x + (cid:18) σ v ( f ) −
12 ( µ − r )( σ − f ) (cid:19) ∂ψ∂x + (cid:18) σ v ( f ) + 12 ( µ − r )( σ − f ) (cid:19) ∂ψ∂y = 0 . (63)where f denotes the function f = f ( x, y, τ ) = e ¯Γ( x − y )+ ( ¯ µ f − ¯Γ ) ( T − τ ) . (64)By replacing v ( f ) , one finally obtains − ∂ψ∂τ + 12 ∂ ψ∂x + (cid:20) ( r − µ )2 σ (1 + f /σ )(1 − f /σ ) (cid:21) ∂ψ∂x − ( r − µ )2 σ ∂ψ∂y = 0 . (65) Note that when f /σ << , that is in the time-spatial ( x, y, τ ) region, that ¯Γ( x − y ) + (cid:18) ¯ µ f −
12 ¯Γ (cid:19) ( T − τ ) << ln σ, (66)then (1 + f /σ )(1 − f /σ ) ≈ , (67)so the Black–Scholes equation (65) can be approximated in this region by − ∂ψ∂τ + 12 ∂ ψ∂x + ( r − µ )2 σ ∂ψ∂x − ( r − µ )2 σ ∂ψ∂y = 0 . (68)9 .2 The strong bubble limit for the lognormal bubble For the case f /σ >> , that is in the time-spatial ( x, y, τ ) region, that ¯Γ( x − y ) + (cid:18) ¯ µ f −
12 ¯Γ (cid:19) ( T − τ ) >> ln σ, (69)then (1 + f /σ )(1 − f /σ ) ≈ − , (70)so the Black–Scholes equation (65) gets to the asymptotic equation − ∂ψ∂τ + 12 ∂ ψ∂x − ( r − µ )2 σ ∂ψ∂x − ( r − µ )2 σ ∂ψ∂y = 0 . (71)Note that due to (64), f can take only positive values. Therefore, there is no analog to f /σ = − case for the lognormal bubble. The asymptotic equations (42), (49), (54), (68) and (71) are particular cases of the generic equation − ∂ψ∂τ + 12 ∂ ψ∂x + α x ∂ψ∂x + α y ∂ψ∂y = 0 , (72)where α x and α y are constants. In fact, the propagator of (72) is P ( x, y, τ ) = 1 √ πτ e − ( x + αxτ )22 τ δ ( y + α y τ ) , (73)where δ ( x ) is the Dirac’s delta function. So, if Φ( x, y ) is some initial condition for equation (72),then its solution is ψ ( x, y, τ ) = (cid:90) + ∞−∞ (cid:90) + ∞−∞ P ( x − x (cid:48) , y − y (cid:48) , τ ) Φ( x (cid:48) , y (cid:48) ) dx (cid:48) dy (cid:48) , (74)that is ψ ( x, y, τ ) = (cid:90) + ∞−∞ (cid:90) + ∞−∞ √ πτ e − ( x − x (cid:48) + αxτ )22 τ δ ( y − y (cid:48) + α y τ ) Φ( x (cid:48) , y (cid:48) ) dx (cid:48) dy (cid:48) . (75) The weak and strong limits of the Gaussian bubble are given by equations (42), (49), whichgenerically can be written as − ∂ψ∂τ + 12 ∂ ψ∂ ¯ x + α ¯ x ∂ψ∂ ¯ x + α ¯ y ∂ψ∂ ¯ y = 0 . (76)10he solution (75) is then given by ψ (¯ x, ¯ y, τ ) = (cid:90) + ∞−∞ (cid:90) + ∞−∞ √ πτ e − (¯ x − ¯ x (cid:48) + α ¯ xτ )22 τ δ (¯ y − ¯ y (cid:48) + α ¯ y τ ) Φ(¯ x (cid:48) , ¯ y (cid:48) ) d ¯ x (cid:48) d ¯ y (cid:48) . (77)To perform this integral, one must invert the transformations given in Section 3 to give ¯ x and ¯ y interms of the initial variables S and f . In fact, one has that ¯ x = (cid:18) ln S − ( r − σ ) ( T − τ ) σ + f Γ (cid:19) − µ f ( T − τ )2Γ ¯ y = (cid:18) ln S − ( r − σ ) ( T − τ ) σ − f Γ (cid:19) + µ f ( T − τ )2Γ , (78)so ¯ x − ¯ x (cid:48) = 12 σ ln (cid:18) SS (cid:48) (cid:19) + 12Γ ( f − f (cid:48) ) , (79)and ¯ y − ¯ y (cid:48) = 12 σ ln (cid:18) SS (cid:48) (cid:19) −
12Γ ( f − f (cid:48) ) . (80)Also d ¯ xd ¯ y = 12 σ Γ S dSdf, (81)so equation (77) becomes ψ ( S, f, τ ) = (cid:90) + ∞ (cid:90) + ∞−∞ e − [ σ ln ( SS (cid:48) ) + ( f − f (cid:48) ) + α ¯ x τ ] √ πτ × δ (cid:20) σ ln (cid:18) SS (cid:48) (cid:19) −
12Γ ( f − f (cid:48) ) + α ¯ y τ (cid:21) Φ ( S (cid:48) , f (cid:48) ) dS (cid:48) df (cid:48) σ Γ S (cid:48) . (82)After performing the f (cid:48) integral, one gives ψ ( S, f, τ ) = (cid:90) + ∞ e − [ ln ( SS (cid:48) ) + ( α ¯ x + α ¯ y ) στ ] σ τ √ πσ τ Φ ( S (cid:48) , f ) dS (cid:48) S (cid:48) , (83)where f = f ( S, S (cid:48) , f, τ ) = f − Γ σ ln ( S/S (cid:48) ) − α ¯ y τ. (84)Two obtain an explicit analytic solution, one can consider now the case of a pure Call, for whichthe contract function Φ is Φ (
S, f ) = Φ ( S ) = max { , S − K } , (85)so ψ ( S, f, τ ) = (cid:90) + ∞ K e − [ ln ( SS (cid:48) ) + ( α ¯ x + α ¯ y ) στ ] σ τ √ πσ τ ( S (cid:48) − K ) dS (cid:48) S (cid:48) . (86)11he last integral can be performed exactly to give ψ ( S, f, τ ) = e σ ( α ¯ x + α ¯ y ) τ + σ τ S N ( d ) − E N ( d ) , (87)where d = ln( S/E ) + σ ( α ¯ x + α ¯ y ) τ + σ τσ √ τ , (88)and d = ln( S/E ) + σ ( α ¯ x + α ¯ y ) τσ √ τ . (89) For the weak limit of the Gaussian model (42), one has α ¯ x = ( r − µ )2 σ ,α ¯ y = − ( r − µ )2 σ , (90)The option price given by (34) is V ( S, f, τ ) = e − rτ ψ ( S, f, τ ) . (91)Then, due that α ¯ x + α ¯ y = 0 , (92)by using (87), (88) and (89), one finds that the option price in the weak limit of the Gaussianmodel is V ( s, f, τ ) = e − rτ · (cid:104) e σ τ S N ( d ) − E N ( d ) (cid:105) , (93)or V ( s, f, τ ) = e − ( r − σ ) τ S N ( d ) − Ee − rτ N ( d ) , (94)with d = ln( S/E ) + σ τσ √ τ , (95)and d = ln( S/E ) σ √ τ . (96) For the strong limit of the Gaussian model (49) one has α ¯ x = − ( r − µ )2 σ ,α ¯ y = − ( r − µ )2 σ , (97)12hen α ¯ x + α ¯ y = − ( r − µ ) σ , (98)so by (87), (88) and (89) the function Ψ is Ψ( S, f, τ ) = e − ( r − µ ) τ + σ τ S N ( d ) − E N ( d ) , (99)with d = ln( S/E ) + ( r − µ ) τ + σ τσ √ τ , (100)and d = ln( S/E ) + ( r − µ ) τσ √ τ . (101)The option price is given in this case by (50) V ( S, f, τ ) = e − µτ Ψ( S, f, τ ) , (102)so the option price in the strong limit of the Gaussian bubble is V ( S, f, τ ) = e − µτ (cid:104) e − ( r − µ ) τ + σ τ S N ( d ) − E N ( d ) (cid:105) , (103)or V ( S, f, τ ) = e − ( r − σ ) τ S N ( d ) − Ee − µτ N ( d ) . (104) f /σ ≈ − case for the Gaussian bubble For the case f /σ ≈ − , the dynamics are given by equation (54), which is a special case of (76),with α ¯ x = 0 ,α ¯ y = − ( r − µ )2 σ , (105)The solution is given then according to (87), (87), (89) and (55) by V = e − ( r − σ ) τ S N ( d ) − E e − ( r + µ )2 τ N ( d ) , (106)with d = ln( S/E ) − ( r − µ )2 τ + σ τσ √ τ , (107)and d = ln( S/E ) − ( r − µ )2 τσ √ τ . (108)13 .2 The solutions for the lognormal bubble The weak and strong limits of the lognormal bubble are given by equations (68), (71), which areagain of the form of equation (72), so the solution in the ( x, y, τ ) is given by (75). Now one can mapthis solution into the ( S, f, τ ) space by taking the inverse of the transformation done in Section 4.The result is x = 12 (cid:32) ln S − (cid:0) r − σ (cid:1) ( T − τ ) σ + ln f − (cid:0) ¯ u f − ¯Γ (cid:1) ( T − τ )¯Γ (cid:33) , (109) y = 12 (cid:32) ln S − (cid:0) r − σ (cid:1) ( T − τ ) σ − ln f − (cid:0) ¯ u f − ¯Γ (cid:1) ( T − τ )¯Γ (cid:33) , (110)so x − x (cid:48) = ln (cid:20)(cid:0) SS (cid:48) (cid:1) / σ (cid:16) ff (cid:48) (cid:17) / (cid:21) , (111)and y − y (cid:48) = ln (cid:20)(cid:0) SS (cid:48) (cid:1) / σ (cid:16) ff (cid:48) (cid:17) − / (cid:21) . (112)Also, one can show that dxdy = 12 σ ¯Γ Sf dS df. (113)In this way, the solution in the ( S, f, τ ) space is then ψ ( S, f, τ ) = (cid:90) ∞−∞ (cid:90) ∞−∞ √ πτ e − (cid:32) ln (cid:34) ( SS (cid:48) ) / σ (cid:18) ff (cid:48) (cid:19) / (cid:35) + αxτ (cid:33) τ × δ (cid:32) ln (cid:34)(cid:18) SS (cid:48) (cid:19) / σ (cid:18) ff (cid:48) (cid:19) − / (cid:35) + α y τ (cid:33) Φ ( S (cid:48) , f (cid:48) ) 12 σ ¯Γ S (cid:48) f (cid:48) dS (cid:48) df (cid:48) . (114)By integrating in f (cid:48) , one obtains ψ ( S, f, τ ) = (cid:90) ∞ √ πσ τ e − (cid:18) ln (cid:20) ( SS (cid:48) ) /σ (cid:21) +( αx + αy ) τ (cid:19) τ Φ ( S (cid:48) , f ) dS (cid:48) S (cid:48) , (115)where f denotes on this occasion the function f = f ( S, S (cid:48) , f ) = f (cid:18) S (cid:48) S (cid:19) ¯Γ /σ e − α y τ . (116)Note that this is the same result obtained in (83), but the form of f is different.Thus, if one considers a pure Call contract as in (85), then (115) implies that the generic so-lution for the pure Call contract is given by equations (87), (88) and (89) but with α ¯ x and α ¯ y replaced by α x and α y , respectively. 14 .2.1 The solutions for weak limit of the lognormal bubble For the weak limit, equation (68) implies that α x = ( r − µ )2 σ ,α y = − ( r − µ )2 σ , (117)so α x + α y = 0 , (118)and the solutions for the option price V are given again by equations (94), (95) and (96). For the strong limit, equation (71) implies that α x = − ( r − µ )2 σ ,α y = − ( r − µ )2 σ , (119)so α x + α y = − ( r − µ ) σ , (120)and the solutions for the option price V are given this time by equations (104), (100) and (101).Figures (5.2.2) and (5.2.2) show the behavior of the weak and strong solution V for two differentparameter sets.Figure 1: From left to right: weak solution, strong solution and both solutions for E = 10 , µ = 0 . , r = 0 . , σ = 0 . in the pure Call case. 15igure 2: From left to right: weak solution, strong solution and both solutions for E = 10 , µ = 0 . , r = 0 . , σ = 0 . in the pure Call case.Finally, one must note that all of these results are valid if µ , σ , µ f and Γ are functions of thespace–time variables ( S, f, t ) that satisfy the asymptotic behavior lim f → µ ( S, f, t ) ≈ µ lim f → σ ( S, f, t ) ≈ σ lim f → µ f ( S, f, t ) ≈ µ f lim f → Γ( S, f, t ) ≈ Γ lim f →∞ µ ( S, f, t ) ≈ µ ∞ lim f →∞ σ ( S, f, t ) ≈ σ ∞ lim f →∞ µ f ( S, f, t ) ≈ µ ∞ f lim f →∞ Γ( S, f, t ) ≈ Γ ∞ , (121)for the Gaussian Bubble or lim f → µ ( S, f, t ) ≈ µ lim f → σ ( S, f, t ) ≈ σ lim f → µ f ( S, f, t ) ≈ f µ f lim f → Γ( S, f, t ) ≈ f Γ lim f →∞ µ ( S, f, t ) ≈ µ ∞ lim f →∞ σ ( S, f, t ) ≈ σ ∞ lim f →∞ µ f ( S, f, t ) ≈ f µ ∞ f lim f →∞ Γ( S, f, t ) ≈ f Γ ∞ , (122)for the log-normal bubble. Here, µ , σ , µ f , Γ , µ ∞ , σ ∞ , µ ∞ f and Γ ∞ are constant. In this article, a stochastic model of endogenous arbitrage bubbles was developed. In this case,the arbitrage bubble satisfies a stochastic differential equation (10), and the option price is givenby the general equation (22). This equation has several interesting limit behaviors. For example,for
Γ = 0 in (22), there exist both “weak” f ≈ and “strong” f → ∞ bubble regimens. Theweak case corresponds to the usual arbitrage-free Black–Scholes model, while the strong case alsocorresponds to a Black–Scholes model where the interest rate has been changed by the mean valueof the underlying assets. 16or the case Γ (cid:54) = 0 , it has been shown that similar weak and strong bubble behaviors exist for twodifferent stochastic bubbles: the Gaussian and the lognormal bubbles. For a pure Call contractcase, the dynamic equations of these weak and stronger limits are given by equations (42), (49)and (68) and (71), respectively. The solutions of these asymptotic equations are given by equations(94) and (104), which are equivalents to the Black–Scholes solution but with Γ (cid:54) = 0 .It is interesting to note that for the Gaussian bubble case, where f can take positive and negativevalues, there exist another weak limit f /σ ≈ − , whose dynamics are given by (54) with a solutiongiven by (106). However, for the lognormal case, that limit cannot be reached because f wouldalways maintain positive according to (64).Thus, the usual Black–Scholes theory can be considered as only an asymptotic limit of a moregeneral model given by equation (22). Although the solutions studied here are limit cases of thegeneral model (22), they are by no means important. Furthermore, these solutions can test theaccuracy of the general case’s numerical solution in the different asymptotic scenarios.In a forthcoming article, I will obtain the corresponding numerical solutions of (22) and comparethem with the weak and strong limits solutions obtained in this paper. References [1] Black F and Scholes M 1973 The pricing of options and corporate liabilities
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