EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 2299-2310 ISSN 1307-5543 – ejpam.com Published by New York Business Global European Call Option under Stochastic Interest Rate in a Fractional Brownian Motion with Transaction Cost Felipe R. Sumalpong, Jr.1,∗, Eric G. Lauron2 1 Department of Mathematics and Statistics, Faculty/Mindanao State University - Iligan Institute of Technology, Iligan City, Philippines 2 J.H Cerilles State College - Dumingag Campus, Zamboanga del Sur, Philippines Abstract. This paper deals on the valuation of European call option price in a stochastic environ- ment by employing three factors which are the stochastic model of the asset value, the stochastic interest rate and the transaction cost. We specify that our underlying asset and the stochastic in- terest rate, particularly Hull-White model, follows a fractional Brownian Motion governed by Hurst parameter H. We used the hedging and replicating technique to established the zero-coupon bond on the European option. Finally, we give a closed-form formula of the European call option price. 2020 Mathematics Subject Classifications: 62P05, 97M30 Key Words and Phrases: European Call Option, Fractional Brownian Motion, Fractional Hull- White Interest Rate Model, Transaction Cost, Hedging, Option Replication 1. Introduction The study of option pricing can be traced back to the seminal papers of Black and Scholes[1] and Merton[2]. The Black-Scholes-Merton(BSM) model is a known mathemat- ical model to evaluate the price of the option that utilizes five inputs namely as: the asset price, the strike price, the risk-free interest rates, time of expiration and the volatility. The initial equation of the BSM model was published in 1973 on the paper “The Pricing of Options and Corporate Liabilities” in Journal of Political Economy. By this model, Black, Scholes and Merton received a Nobel prize in Economics in 1997. However, there are several drawbacks of the BSM model such as the assumptions that the interest rate and the volatility rate are constant over the period of the contract does not fit the actual scenario of the market, transaction cost may not be avoided, and the evolution of the asset price does not always obey a standard Brownian motion(H = 1/2) for which Nualart[3] noted that for some stock process, the Hurst exponent H may not ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5107 Email addresses: felipejr.sumalpong@g.msuiit.edu.ph (F.Jr. Sumalpong), eric.lauron@g.msuiit.edu.ph (E. Lauron) https://www.ejpam.com 2299 © 2024 EJPAM All rights reserved. F. Sumalpong, E. Lauron / Eur. J. Pure Appl. Math, 17 (3) (2024), 2299-2310 2300 be exactly 1/2 but greater than or equal to 1/2. Hence, the standard Brownian Motion where H = 1/2 may not be a fitting stochastic process for some stock processes. In this paper, we assumed that both the asset value and the interest rate follows the fractional Brownian Motion. We specify that the asset value X(t) follows dX(t) = r(t)X(t)dt+ σXX(t)dBH 1 (t) (1) and the interest rate r(t) follows the fractional Hull-White model dr(t) = [θ(t)− ar(t)]dt+ σrdB H 2 (t) (2) where σX is the volatility of the asset price, σr is the volatility of the interest rate, and BH is a fractional Brownian motion with Hurst parameter H. With the transaction cost Cost = cX(t)|v(t)| (3) where c is a fixed proportion of the trading amount for the asset agreed between both parties, and v(t) is the number of assets sold or bought, we formulate a European option price model on par with the BSM model. For simplicity, we assumed that the volatilities for both the asset price and the interest rate are constant, no dividend and coupon payments, and we limit our analysis to European options only. 2. Model Formulation This section presents our assumptions for the computations of the option price. The following assumptions are made: 1. Asset Price Model The asset price X(t) follows a fractional Brownian motion given by dX(t) = r(t)X(t)dt+ σXX(t)dBH 1 (t) (4) where r(t) is the total expected rate of return, σX is the volatility of the price and BH 1 (t) is a fractional Brownian Motion with Hurst parameter H. 2. Interest Rate model The risk-free interest rate r(t) follows a fractional Hull-White model given by dr(t) = [θ(t)− ar(t)]dt+ σrdB H 2 (t) (5) where θ(t) is a deterministic function of time, a is constant, σr is the volatility of the interest rate which is assumed to be constant and BH 2 (t) is a fractional Brownian Motion with Hurst parameter H. Under the fractional Brownian motion, the correlation coefficient between BH 1 (t) F. Sumalpong, E. Lauron / Eur. J. Pure Appl. Math, 17 (3) (2024), 2299-2310 2301 and BH 2 (t) for t ≥ 0 is given by Cov(BH 1 (t), BH 2 (t)) = ρ(dt)2H . (6) Moreover, the following properties are applied to the fractional Brownian Motion: E[dBH(t)] = 0, (7) E[dtdBH(t)] = 0, (8) E[dBH 1 (t)dBH 2 (t)] = ρdt2H , (9) E[(dBH(t))2] = (dt)2H , (10) E[(dt)2] = 0. (11) Lemma 1. The zero-coupon bond model with the terminal condition P (r, t;T ) = 1 can derive the following formula P (r, t;T ) = e−rB(t,T )−A(t,T ), (12) with B(t, T ) = 1 a [ 1− e−a(T−t) ] A(t, T ) = − ∫ T t θ(u)B(u, T )du+ 1 2 dt2H [∫ T t σ2rB 2(u, T )du ] where σr is the volatility of the interest rate, θ(t) is a deterministic function of time, and a is a constant assumed to be nonzero. 3. Transaction cost Transaction cost is a fixed proportion c, depending on the individual investor, of the trading amount for the asset. We have Cost = cX(t)|v(t)| where v(t) is the number of shares of the sold or bought at the price Xt. Specifically, v(t) > 0 indicates a bought share while v(t) < 0 indicates a sold shares. 4. Portfolio The portfolio is revised at time dt, where dt is a small time step from t to t+ dt. 5. Expected return of the portfolio The expected return of the portfolio Π(t) satisfies the equality E[dΠ(t)] = r(t)Π(t)dt (13) where r(t) is the interest rate. F. Sumalpong, E. Lauron / Eur. J. Pure Appl. Math, 17 (3) (2024), 2299-2310 2302 From these assumptions, we can now begin to price the zero-coupon bond of the fractional Hull-White which we will use to come up an option price. Theorem 1. Under the fractional Hull-White Interest Rate Model, the zero-coupon bond price P (r(t), t;T ) obeys the following equation: ∂P ∂t + [θ(t)− ar(t)− ψσr] ∂P ∂r + 1 2 ∂2P ∂r2 (σr) 2(dt)2H−1 − rP = 0. where r(t) is the interest rate under Hull-White model, θ(t) is a deterministic function of time, a is constant, ψ is the market price of the risk with volatility σ and σr is the volatility of the interest rate which is assumed to be constant. Proof : The equation can be attained by constructing a deterministic hedged portfolio that employs two coupon bonds P1(t), P2(t), that is, Π(t) = P1(t)−∆P2(t) (14) with different maturity T1 and T2 respectively. We long one share of bond P1(t) and short ∆ shares of P2(t). Taking the derivative of Equation (14), we have dΠ(t) = dP1(t)−∆dP2(t). (15) Applying Ito’s Lemma to the price function P (r(t), t;T ), the right-hand side of the equality in Equation 15 becomes, ∂P1 ∂t dt+ [θ(t)− ar(t)] ∂P1 ∂r dt+ 1 2 ∂2P1 ∂r2 ( [θ(t)− ar(t)]2(dt)2 +2[θ(t)− ar(t)]σrdtdB H 2 + (σr) 2(dBH 2 )2 ) + σr ∂P1 ∂r dBH 2 −∆σr ∂P1 ∂r dBH 2 −∆ ( ∂P2 ∂t dt+ [θ(t)− ar(t)] ∂P2 ∂r dt+ 1 2 ∂2P2 ∂r2 ( [θ(t)− ar(t)]2(dt)2 + 2[θ(t)− ar(t)]σrdtdB H 2 + (σr) 2(dBH 2 )2 )) To eliminate the risk, take ∆ = ∂P1 ∂r / ∂P2 ∂r , (16) such that ∂P2 ∂r ̸= 0. Thus, ∂P1 ∂t dt+ [θ(t)− ar(t)] ∂P1 ∂r dt+ [θ(t)− ar(t)]2(dt)2 + [θ(t)− ar(t)]σr ∂2P1 ∂r2 dtdBH 2 + (σr) 2 1 2 ∂2P1 ∂r2 (dBH 2 )2 − ( ∂P1 ∂r / ∂P2 ∂r ) ∂P2 ∂t dt− [θ(t)− ar(t)] ∂P2 ∂r ( ∂P1 ∂r / ∂P2 ∂r ) dt F. Sumalpong, E. Lauron / Eur. J. Pure Appl. Math, 17 (3) (2024), 2299-2310 2303 − 1 2 ∂2P2 ∂r2 ( ∂P1 ∂r / ∂P2 ∂r ) [θ(t)− ar(t)]2(dt)2 − [θ(t)− ar(t)]σr ∂2P2 ∂r2 ( ∂P1 ∂r / ∂P2 ∂r ) dtdBH 2 − (σr) 2 1 2 ∂2P2 ∂r2 ( ∂P1 ∂r / ∂P2 ∂r ) (dBH 2 )2. By non-arbitrage principle, ∂P1 ∂t + [θ(t)− ar(t)] ∂P1 ∂r + (σr) 2 1 2 ∂2P1 ∂r2 (dt)2H−1 − rP1(t) = ( ∂P1 ∂r / ∂P2 ∂r )[ ∂P2 ∂t + [θ(t)− ar(t)] ∂P2 ∂r + (σr) 2 1 2 ∂2P2 ∂r2 (dt)2H−1 − rP2(t) ] . This is one equation in two unknowns. However, the left-hand side is a function of T1 and the right-hand side is a function of T2. The only way for this equality to be possible is for both side to be independent of the maturity date. Thus dropping the subscripts of P and introducing the market price of the risk ψ, we have ∂P ∂t + [θ(t)− ar(t)]∂P∂r + (σr) 2 1 2 ∂2P ∂r2 (dt)2H−1 − rP (t) ∂P ∂r = ψσr. Simplifying this, the equation in the theorem can be arrived. Theorem 2. If the number of assets traded during the time interval [t, dt] is v, then E[|v|] = √ 2 π (dt)H (( ∂2V ∂X2 )2 σ2XX 2(t) + ( ∂2V ∂r∂X )2 σ2r + 2ρσXσrX(t) ∂2V ∂X2 ∂2V ∂r∂X ) 1 2 where V (t) is the option price, X(t) is the market price of the asset at time t, r(t) is the interest rate that follows the fractional Hull-White model with Hurst parameter H, σX is the volatility of the asset price, σr is the volatility of the interest rate and ρ is the correlation coefficient between the interest rate and the asset price. Proof : Suppose v is the number of asset traded during the time interval [t, dt]. At the short time t, the number of assets hold is given by ∆1 = ∂V ∂X (X, r, t). (17) F. Sumalpong, E. Lauron / Eur. J. Pure Appl. Math, 17 (3) (2024), 2299-2310 2304 After the time step dt, the number of assets held is ∆t+dt = ∂V ∂X (X + dt, r + dt, t+ dt). (18) Since the time step dt is assumed to be so small, we have dX ≃ σXX(t)dBH 1 (t) (19) and dr ≃ σrdB H 2 . (20) The number of assets traded v during the time interval [t, t+ dt] given by v = ∂2V ∂X2 σXX(t)dBH 1 (t) + ∂2V ∂r∂X σrdB H 2 (t) (21) with mean E[v] = 0, (22) and variance E[v2] = ( ∂2V ∂X2 )2 σ2XX 2(t)(dt)2H + ( ∂2V ∂r∂X )2 σ2r (dt) 2H + 2ρσXσrX(t) ∂2V ∂X2 ∂2V ∂r∂X (dt)2H . If we let the variance of v to be β2, then we have E[v2] = β2. Since v is normally distributed, the probability density function of v is given as f(v) = 1 β √ 2π e −(v)2 2β2 . (23) Hence, we have E[|v|] = ∫ +∞ −∞ |v|f(v)dv = ∫ +∞ −∞ |v| 1 β √ 2π e −(v)2 2β2 dv By letting u = −(v)2 2β2 , we have dv = −β2du v . Substituting we have E[|v|] = √ 2 π (dt)H × [( ∂2V ∂X2 )2 σ2XX 2(t) F. Sumalpong, E. Lauron / Eur. J. Pure Appl. Math, 17 (3) (2024), 2299-2310 2305 + ( ∂2V ∂r∂X )2 σ2r + 2ρσXσrX(t) ∂2V ∂X2 ∂2V ∂r∂X ] . Theorem 3. Using the Hull-White Interest rate model and with transaction cost in a fractional Brownian Motion, the value of the option is modeled as: ∂V ∂t + 1 2 ∂2V ∂X2 σ2X(X(t))2(dt)2H−1 + 1 2 ∂2V ∂r2 σ2r (dt) 2H−1 + ∂V ∂X rX(t) + ∂2V ∂X∂r σrσXX(t)ρ(t)(dt)2H−1 + ∂V ∂r [θ(t)− ar(t)− ψσr] − rV (t) + cX(t) √ 2 π (dt)H × [( ∂2V ∂X2 )2 σ2XX 2(t) + ( ∂2V ∂r∂X )2 σ2r +2ρσXσrX(t) ∂2V ∂X2 ∂2V ∂r∂X ] 1 2 = 0. where V (t) is the option price, X(t) is the market value of the asset at time t, r(t) is the interest rate, σX is the volatility of the asset, σr is the volatility of the interest rate, c is fixed proportion of the trading amount for the asset agreed by both parties, and H is the Hurst parameter. Proof : Let V (t) = V (t,X(t), r(t)) be the option price. Define the portfolio Π(t) = V (t)−∆1X(t)−∆2P (t) where ∆1,∆2 are the respective shares of the asset price X(t) and the zero-coupon bond P (t). The value of the change of portfolio at time [t, t+ dt] with transaction cost is now dΠ(t) = V (t)−∆1dX(t)−∆2dP (t) + c|v(t)|X(t). Taking the expectation we have E[dΠ(t)] = E [V (t)]− E [∆1dX(t)]− E [∆2dP (t)] + cX(t)E [|v(t)|] . By non-arbitrage principle, E [V (t)]− E [∆1dX(t)]− E [∆2dP (t)]− rV (t)dt+∆1rX(t)dt +∆2rP (t)dt+ cX(t)E [|v(t)|] = 0 To eliminate the risk, take ∆1 = ∂V ∂X (24) F. Sumalpong, E. Lauron / Eur. J. Pure Appl. Math, 17 (3) (2024), 2299-2310 2306 and ∆2 = ∂V ∂r / ∂P ∂r . (25) Hence, ∂V ∂t + 1 2 ∂2V ∂X2 σ2X(X(t))2(dt)2H−1 + 1 2 ∂2V ∂r2 σ2r (dt) 2H−1 + ∂V ∂X rX(t) + ∂2V ∂X∂r σrσXX(t)ρ(t)(dt)2H−1 + ∂V ∂r [θ(t)− ar(t)− ψσr] − rV (t) + cX(t)E [|v(t)|] = 0. Employing the transaction cost, we have can derived the equation in the theorem. 3. The Model Theorem 4. Based on the price model in Theorem 3 under a fractional Brownian motion, the closed form formula for the European call option price is given by VC(X, r, t) = XN(d1)−KP (r, t, T )N(d2) (26) where d1 = d2 +M d2 = ln X KP (r,t;T ) −M N σ̂2 = σ2X + σ22B 2 + 2ρσXσ2B B = 1 P ∂P ∂r M = √√√√2 ∫ T t [ 1 2 (ds)2H−1σ̂2 + c(ds)H−1 √ 2 π σ̃ ] ds N = √√√√2 ∫ T t [ 1 2 (ds)2H−1σ̃2 + c(ds)H−1 √ 2 π σ̃ ] ds Proof : Theorem 3 can be solved by the transformation of independent variables y = X P (r, t;T ) (27) and a new unknown function denoted as V̂ (y, t) = V (X, r, t) P (r, t;T ) . (28) F. Sumalpong, E. Lauron / Eur. J. Pure Appl. Math, 17 (3) (2024), 2299-2310 2307 We have the following computations, ∂V ∂t = V̂ ∂P ∂t + P ∂V̂ ∂t − y ∂P ∂t ∂V̂ ∂y , ∂V ∂r = V̂ ∂P ∂r − y ∂P ∂r ∂V̂ ∂r , ∂V ∂X = ∂V̂ ∂y , ∂2V ∂r2 = V̂ ∂2P ∂r2 − y ∂V̂ ∂y ∂2P ∂r2 − y2 ∂2V̂ ∂y2 1 P ( ∂P ∂r )2 , ∂2V ∂r∂X = −y∂ 2V̂ ∂y2 1 P ∂P ∂r , ∂2V ∂X2 = 1 P ∂2V̂ ∂y2 . Substituting these equations gives, ∂V̂ ∂t + [ 1 2 y2(dt)2H−1∂ 2V̂ ∂y2 ][ σ2X + 1 P 2 ( ∂P ∂r )2 (σr) 2 − 2 P ∂P ∂r σrσXρ(t) ] + cy2 √ 2 π (dt)H−1 ∣∣∣∣∣∂2V̂∂y2 ∣∣∣∣∣× [ σ2X + 1 P 2 ( ∂P ∂r )2 σ2r −2ρσXσr ( 1 P )( ∂P ∂r )]1/2 = 0 Taking another transformation by letting z = ln y. Hence we have, ∂V̂ ∂y = ∂V̂ ∂z 1 y (29) and ∂2V̂ ∂y2 = ( 1 y )2 ( ∂2V̂ ∂z2 − ∂V̂ ∂z ) . Furthermore, let B = 1 P ∂P ∂r and σ̃2 = σ2X + σ2rB 2 + 2ρσXσrB. Therefore, we have ∂V̂ ∂t + [ 1 2 (dt)2H−1σ̃2 + c(dt)H−1 √ 2 π ] × ( ∂2V̂ ∂z2 − ∂V̂ ∂z ) = 0 F. Sumalpong, E. Lauron / Eur. J. Pure Appl. Math, 17 (3) (2024), 2299-2310 2308 Finally we take the following transformation. Let V̂ (z, t) = µ(η, τ), where η = z + α(t) and τ = γ(t). At the expiry date T of the contract, α(T ) = γ(t) = 0. So we have the following calculations, ∂V̂ ∂t = ∂µ ∂η α′(t) + ∂µ ∂τ γ′(t) (30) and ∂V̂ ∂z = ∂µ ∂η , (31) ∂2V̂ ∂z2 = ∂2µ ∂η2 . (32) We let α′(t) = 1 2 (dt)2H−1σ̃2 + c(dt)H−1 √ (2/π)σ̃ and γ′(t) = − [ 1 2 (dt)2H−1σ̃2 + c(dt)H−1 √ (2/π)σ̃ ] . Substituting these, we have ∂µ ∂η α′(t) + ∂µ ∂τ (−α′(t)) + α′(t) [ ∂2µ ∂η2 − ∂µ ∂η ] = 0. Hence, this can be reduced further to ∂µ ∂τ − ∂2µ ∂η2 = 0 (33) with the initial condition µ(η, T ) = (eη −K)+ . (34) The solution for this equation is µ(η, τ) = 1 2 √ πτ ∫ +∞ −∞ µ0(ξ)e −(η−ξ)2/4τdξ (35) Now for eξ −K ≥ 0, ξ ≥ lnK. (36) Hence, the integration domain can be equivalently [−∞, lnK]. Hence, we can write the solution as µ(η, τ) = 1 2 √ πτ ∫ +∞ lnK (eξ −K)e−(η−ξ)2/4τdξ = 1 2 √ πτ ∫ +∞ lnK eξe−(η−ξ)2/4τdξ − 1 2 √ πτ ∫ +∞ lnK Ke−(η−ξ)2/4τdξ F. Sumalpong, E. Lauron / Eur. J. Pure Appl. Math, 17 (3) (2024), 2299-2310 2309 Let ξ = η + √ 2τω, then we can have ω = ξ − η√ 2τ (37) and dξ = √ 2τdω. (38) Therefore, µ(η, τ) = 1√ 2π eη+τ ∫ η−lnK+2τ√ 2τ −∞ e− ζ2 2 dζ − 1√ 2π K ∫ η−lnK√ 2τ −∞ e−ω2/2dω. This can be written as µ(η, τ) = eη+τN (d1)−KN (d2) (39) where d1 = η − lnK + 2τ√ 2τ (40) d2 = η − lnK√ 2τ (41) and N(d1) = 1√ 2π ∫ d1 −∞ e −1 2 ζ2dζ (42) N(d2) = 1√ 2π ∫ d2 −∞ e −1 2 ω2 dω. (43) By inverse change of variables, the formula in the theorem can be derived. Corollary 5. Without the transaction cost and under a fractional Brownian motion, the closed form formula for the European call option price VC(X, r, t) is given by VC(X, r, t) = X(t)N(d1)−KP (r, t, T )N(d2) (44) where d1 = d2 + √ σ̃2 ∫ T t (ds)2H , d2 = ln X KP (r,t;T ) − 1 2 σ̃ 2 ∫ T t (ds)2H√ σ̃2 ∫ T t (ds)2H , σ̃2 = σ2X + σ22B 2 + 2ρσXσ2B, B = 1 P ∂P ∂r , REFERENCES 2310 N(d) = 1√ 2π ∫ d −∞ e −1 2 y2dy. 4. Conclusion and Recommendations This paper presents an extension of the BSMmodel for European call option introduced by Black, Scholes and Merton by considering a stochastic rate under a fractional Brownian Motion instead of a constant rate and adding a transaction cost. A closed-form formula for European call option is derived under risk-neutral measure using replication techniques. Also, the fractional Brownian motion employs the flexible dependence of the increments of the evolution of the prices making it more closer to the real world scenario. For further studies, a similar formula can be derived also that employs stochastic volatilities for both asset value and interest rate. Moreover, this pricing can be extended further by considering geometric fractional Brownian Motion. References [1] F Black and M Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81:637–654, 1973. [2] R Merton. Theory of rational option pricing. Journal of Economics and Management Science, 4:141–183, 1973. [3] D Nualart. Fractional brownian motion: stochastic calculus and applications. European Mathematical Society, 3:141–162, 2006.