J. Nig. Soc. Phys. Sci. 2 (2020) 186–196 Journal of the Nigerian Society of Physical Sciences Original Research Portfolio Strategy for an Investor with Logarithm Utility and Stochastic Interest Rate under Constant Elasticity of Variance Model Edikan E. Akpanibaha,∗, Udeme O. Inib aDepartment of Mathematics and Statistics, Federal University Otuoke, Bayelsa, Nigeria bDepartment of Mathematics and Computer Science, Niger Delta University, Bayelsa, Nigeria Abstract This paper is aim at maximizing the expected utility of an investor’s terminal wealth; to achieve this, we study the optimal portfolio strategy for an investor with logarithm utility function under constant elasticity of variance (CEV) model in the presence of stochastic interest rate. A portfolio comprising of a risk free asset and a risky asset is considered where the risk free interest rate follows the Cox- Ingersoll-Ross (CIR) model and the risky asset is modelled by CEV. Using power transformation, change of Variable and asymptotic expansion technique, an explicit solution of the optimal portfolio strategy and the Value function are obtained. Furthermore, numerical simulations are presented to study the effect of some parameters on the optimal portfolio strategy under stochastic interest rate. Keywords: Keywords: Stochastic interest rate, optimal portfolio strategy, asymptotic technique, constant elasticity of variance, logarithm utility Article History : Received: 12 April 2020 Received in revised form: 04 July 2020 Accepted for publication: 15 July 2020 Published: 01 August 2020 c©2020 Journal of the Nigerian Society of Physical Sciences. All rights reserved. Communicated by: T. Latunde 1. Introduction In the study of optimal portfolio strategy in a financial mar- ket, volatility plays a vital role in influencing the behaviour of the risky assets due to its fluctuating nature as a result of dif- ferent information available in the financial market. For an in- vestor to make relatively right choice when investing in risky assets, there is need to consider stochastic volatility models and not constant volatility in order to understand the fluctuating na- ∗Corresponding author tel. no: +2347030811189 Email address: edikanakpanibah@gmail.com (Edikan E. Akpanibah ) ture of the risky assets. One of such stochastic volatility model is the CEV model. The CEV model was developed by [1] and is an exten- sion geometric Brownian motion (GBM). According to [2], the model is capable of capturing the implied volatility skew. A lot of researchers such as ([3],[4]) studied utility maximization un- der constant elasticity model in defined contribution (DC) pen- sion scheme. In [5], optimal investment and reinsurance prob- lem of utility maximization under CEV model was studied. The authors in [2] studied optimal investment problem with taxes, dividend and transaction cost using CEV model and logarithm utility function. The optimal portfolio strategy with multiple 186 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 2 (2020) 186–196 187 contributors in a DC pension fund using Legendre transfor- mation method was studied by [6]. The authors in [7] solved the optimal portfolio problem with default risk and refund of premium clause in a DC plan; in their work, the stock market price followed the CEV model. In [8], the effect of additional voluntary contribution on the investment strategies under CEV model; they used power transformation method in solving their problem. The study of optimal portfolio with stochastic inter- est have been investigated by many authors though the risky assets were modelled by GBMs; they include [9], who studied optimal portfolio management with stochastic interest rate for a protected case of DC fund. In ([10]-[11]), stochastic interest rate model was used to obtain optimal portfolio allocation in a DC plan. Also, the authors in ([12]-[13]) considered investment strategy with interest rate of Vasicek type while the authors in ([14]-[16]), studied the optimal portfolio problem when the in- terest rate is of affine interest. From the available literatures, most of the authors that worked on optimal portfolio strategies under CEV model considered cases where the risk free interest rate is constant except for [17] who studied the optimal portfolio strategies under CEV model with stochastic interest rate. They pointed out that one of the main reasons why other authors could not combined CEV model and stochastic interest in finding the optimal portfolio strategies was in the difficulty to find a closed form solution of the optimal portfolio strategies analytically. Also they pointed out that in financial market, interest rate is not constant rather a fluctuating processes and that the interest rate volatility presents another source of risks in financial market; In other words, when this risk is not taken into consideration, we are actually undermining the risk generated by this interest rate which is crucial in affecting the prices of various assets available in fi- nancial market. In their work, they used the exponential utility to optimize the expected utility of an insurer with exponential utility function and used the Legendry transformation method and asymptotic expansion method to obtain solutions for the optimal portfolio strategies. In this work, we maximize the expected utility of an in- surer’s wealth by studying the optimal portfolio strategies of an insurer with logarithm utility function whose risky asset is model by CEV model and the risk free interest is stochastic and follows the CIR model. Furthermore, we use the power transformation, change of variable and asymptotic approach to derive an asymptotic solution of the optimal portfolio strategies and value function. Also some numerical simulations to explain our results are given. The main difference between our work and that of [17] is that we consider an investor with logarithm utility instead of exponential utility and apply power transfor- mation method and change of variable method instead of Leg- endre transformation method. 2. Preliminaries For a financial market with portfolio comprising of a risk free asset (treasury security) and a risky asset (marketable se- curity) which is open continuously for a period of T > 0 rep- resenting the expiring date of the investment. Let (Ω, F, P) be a probability space which is complete, Ω and P are real space probability measure respectively, {Br (t) , Bs (t) : t ≥ 0} is a set of Brownian motions and F the filtration representing informa- tion produced by the Brownian motions. Let the risk free asset price C (t) at time t be given as dC(t) C(t) = r (t) dt, C (0) > 0 (1) where r (t) is the short interest rate and follows the (CIR) model whose dynamics is{ dr (t) = (b − cr (t)) dt − a1 √ r (t)dBr (t) r (0) = r0 > 0 (2) and b , c and a1 are positive numbers such that the following condition holds a21 < 2b (Feller’s Condition). [17] Let S (t) be the price of the marketable security whose price process follows the CEV model which is an extension of the GMB. The CEV as earlier stated has the capability to capture the volatility skew of the marketable security unlike the GBM whose volatility is constant. From the work of [4, 6, 8, 17], the dynamic of the price process of the marketable security is given by the stochastic differential equation when t ≥ 0 as follows dS (t) S (t) = µdt + σS β (t) dBs (t) (3) where µ , σ and β are positive constant and represent the instantaneous expected rate of return, instantaneous volatility and elasticity parameter respectively see [2−4]. Also, Br (t) and Bs (t) are assumed to be correlated instantaneous correlation coefficient ρ = 1 such that dBr (t) dBs (t) = dt see [17]. Note, when β = 0 in equation (3), the model in (3) reduce to that of GMB see [2]. 3. Wealth Formulations and Methodology Let Z (t) be the investor’s wealth at time t. Also, let ϕ and ϕ1 be the proportions of the investor’s wealth to be invested in risky asset and risk-free asset respectively such that ϕ1 = 1−ϕ. Since the investor’s total wealth is summation of his investments in the two assets hence the differential form of the investor’s total wealth is dZ (t) = Z (t) ( ϕ1 dC (t) C (t) + ϕ dS (t) S (t) ) (4) 187 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 2 (2020) 186–196 188 substituting equations (1) and (3) into (4), we have{ dZ(t) = Z(t)((ϕ(µ− r) + r)dt + ϕσS β (t) dBs (t)) Z (0) = Z0 (5) Next, let the investor’s utility at any given state Z at time t be given as Jϕ (t, r, s2, z) = Eϕ [ U (Z (T )) r (t) = r, S (t) = s, Z (t) = z ] (6) With boundary condition J (t, r, s2, z) = U (z) where r is the risk free interest rate and z is the wealth. The objective here is to determine the optimal portfolio strategies and the optimal value function of the investor given as ϕ∗ and J (t, r, s2, z) = supϕ Jϕ (t, r, s2, z) such that Jϕ∗ (t, r, s2, z) = J (t, r, s2, z) (7) The value function Jϕ∗ (t, r, s2, z) can be considered as a kind of utility function. From the maximum principle and Ito’s lemma, the Hamil- ton Jacobi Bellman (HJB) equation which is a nonlinear partial differential equation associated with (6) is obtained by maxi- mizing the expected utility Jϕ (t, r, s2, z) subject to the investor’s wealth in as follows Jt + µsJs + 1 2 σ 2 s2β+2 Jss +rzJz + (b − cr) Jr + 1 2 ra 2 1 +σa1 √ rsβ+1 Jsr + sup  1 2 ϕ 2σ2z2 s2β Jzz +ϕ  (µ− r) zJz +σ2zs2β+1 Jzs +σa1 √ rzsβ Jzr    = 0 (8) Differentiating (8) with respect to ϕ , we obtain the first order maximizing condition as( ϕσ2z2 s2βJzz + (µ− r) zJz + σ2zs2β+1 Jzs +σa1 √ rzsβJzr ) = 0 (9) Solving (9) for ϕ we have ϕ∗ = − [ (µ− r) Jz + σ2 s2β+1 Jzs + σa1 √ rsβJzr ] zσ2 s2βJzz (10) Where ϕ∗ is the optimal portfolio strategy of the risky asset. Substituting (10) into (8), we have Jt + µsJs + 1 2 σ 2 s2β+2 Jss + rzJz + (b − cr) Jr + 12 ra 2 1 Jrr + σa1 √ rsβ+1 Jsr + 12 [(µ−r)Jz +σ2 s2β+1 Jzs +σa1 √ rsβ Jzr ]2 σ2 s2β Jzz  = 0 (11) 4. Optimal Portfolio Strategies for an investor with Logarithm Utility In this section, we consider an investor with utility function which exhibit constant relative risk aversion (CRRA) different from the one in [17] where the investor exhibits constant ab- solute risk aversion (CARA). Since our interest here is to de- termine the optimal portfolio strategies for the investor with CRRA utility, we choose the logarithm utility function similar to the one in [2]. From [2, 14], The logarithm utility function is given as U (z) = lnz (12) Next, we conjecture a solution to (11) similar to the one in [2] with the form:{ J (t, r, s, z) = w (t, r, s) + v (t, r, s) ln z w (T, r, s) = 0, v (T, r, s) = 1, (13)  Jt = wt + vt ln z, Js = ws + vs ln z , Jss = wss + vss ln z,Jr = wr + vr ln z, Jrr = wrr + vrr ln z, Jsr = wsr + vsr ln zJsr = wsr + vsr ln z, Jzz = − vz2 , Jzr = vrz , Jzs = vsz  (14) Substituting (14) into (11), we have  vt + µsvs + 1 2 σ 2 s2β+2vss + (b − cr) vr + 1 2 ra 2 1vrr +σa1 √ rsβ+1vsr  ln z +  wt + µsws + 1 2 σ 2 s2β+2wss + (b − cr) vr + rv + 1 2 ra 2 1wrr +σa1 √ rsβ+1wsr + 12 [(µ−r)v+σ2 s2β+1 vs +σa1 √ rsβvr ]2 σ2 s2βv   = 0 (15) Splitting (15) we have( vt + µsvs + 1 2 σ 2 s2β+2vss + (b − cr) vr + 12 ra 2 1vrr + σa1 √ rsβ+1vsr ) = 0 (16)  wt + µsws + 1 2 σ 2 s2β+2wss + (b − cr) vr +rv + 12 ra 2 1wrr + σa1 √ rsβ+1wsr + 12 [(µ−r)v+σ2 s2β+1 vs +σa1 √ rsβvr ]2 σ2 s2βv  = 0 (17) Taking the boundary condition v (T, r, s) = 1 into consider- ation, we solve (16) as follows using power transformation and change variable approach. Proposition 4.1 The solution of equation (16) is given as v (t, r, s) = h (t, r, q) = 1 where h (t, r, q) = h1 (t, r, q) + √ αh2 (t, r, q) + αh3 (t, r, q) and h1 (t, r, q) = 1, h2 (t, r, q) = 0, and h3 (t, r, q) = 0 188 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 2 (2020) 186–196 189 Proof Assume{ v (t, r, s) = h (t, r, q) , q = s−2β h (T, r, s) = 1 } (18) Then vt = ht, vs = −2βs−2β−1hq , vss = 2β (2β + 1) s−2β−2hq + 4β2 s−4β−2hqq, vr = hr, vrr = hrr, vrs = −2βs−2β−1hrq  (19) Substituting (19) into (16), we have[ ht − 2µβqhq + σ2β (2β + 1) hq + 2β2σ2qhqq + (b − cr) hr + 1 2 ra 2 1hrr − 2βσa1 √ r √ qhrq ] = 0 (20) We can rewrite (20) as (A + B + C) h = 0 (21) Where A = [ (b − cr) hr + 1 2 ra21hrr ] h (22) B = [ ht + β ( σ2 (2β + 1) − 2µq ) hq + 2β 2σ2qhqq ] h (23) C = [ −2βσa1 √ r √ qhrq ] h (24) Next we follow the approach in [17] by applying the asymp- totic expansion method to solve the problem in (21). Assume that the volatility follows a slow fluctuating pro- cess, we attempt to determine an asymptotic solution for (21) by using the slow-fluctuating process rα to replace r in (2), such that 0 < α � 1 is a small positive integer: drα (t) = (b − crα (t)) dt − a1 √ rα (t)dBr (t) (25) Substituting (25) into (21) and also replacing b − cr by α (b − cr) and √ r by √ α √ r, we will have( αA + B + √ αC ) hα = 0 (26) Next, we conjecture a solution for (26) as follows hα (t, r, q) = h1 (t, r, q) + √ αh2 (t, r, q) + αh3 (t, r, q) (27) Substituting (27) into (26) ( αA + B + √ αC ) ( h1 (t, r, q) +√αh2 (t, r, q) +αh3 (t, r, q) ) = 0 Simplifying the above equation, we arrive at ( Bh1 (t, r, q) + [ Bh2 (t, r, q) + Ch1 (t, r, q) ]√ α + [ Ah1 (t, r, q) + Bh3 (t, r, q) + Ch2 (t, r, q) ] α ) = 0 This implies that{ Bh1 (t, r, q) = 0 h1 (T, r, s) = 1 (28) { Bh2 (t, r, q) + Ch1 (t, r, q) = 0 h1 (T, r, s) = 1 , h2 (T, r, s) = 0 (29) { Ah1 (t, r, q) + Bh3 (t, r, q) + Ch2 (t, r, q) h1 (T, r, s) = 1, h2 (T, r, s) = 0 h3 (T, r, s) = 0 (30) To obtain the solution of (27), we solve (28), (29) and (30). Recall from (22), (23) and (24), equation (28), (29) and (30) can be expressed as h1t + β ( σ2 (2β + 1) − 2µq ) h1q + 2β2σ2qh1qq = 0 h1 (T, r, q) = 1 (31)  h2t + β ( σ2 (2β + 1) − 2µq ) h2q +2β2σ2qh2qq − 2βσa1 √ r √ qh1rq = 0 h2 (T, r, q) = 0 (32)  h3t + β ( σ2 (2β + 1) − 2µq ) h3q + 2β2σ2qh3qq + (b − cr) h1r + 1 2 ra 2 1h1rr − 2βσa1 √ r √ qh2rq h3 (T, r, q) = 0 (33) Let h1 (t, r, q) = D0 (t, r) + qD1 (t, r) (34) And( h1t = D0t + qD1t, h1q = D1, h1qq = 0 ) (35) Substituting (35) in (31), we have D0t + βσ 2 (2β + 1) D1 = 0, D0 (T, r) = 1 (36) D1t − 2µβD1 = 0, D1 (T, r) = 0 (37) Solving (37), we have D1 (t, r) = 0 (38) Putting (38) into (36) solving it, we have D0 (t, r) = 1 (39) Hence from (34) h1 (t, r, q) = 1 (40) Next, we solve (32), by assuming a solution of the form h2 (t, r, q) =  D2 (t, r) + q 12 D3 (t, r) + qD4 (t, r) +q 3 2 D5 (t, r)  (41) 189 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 2 (2020) 186–196 190 And h2t = D2t + q 1 2 D3t + qD4t + q 3 2 D5t , h2q = 1 2 q − 1 2 D3 + D4 + 3 2 q 1 2 D5 h2qq = − 1 4 q − 3 2 D3 + 3 4 q − 1 2 D5, h1rq = 0  (42) Substituting (42) into (32), we have D2t + βσ 2 (2β + 1) D4 = 0, D2 (T, r) = 0 (43) D4t − 2µβD4 = 0, D4 (T, r) = 0 (44) D5t − 3µβD5 = 0, D5 (T, r) = 0 (45) D3t −µβD3 + 3 2 βσ2 (3β + 1) D5 = 0, D3 (T, r) = 0 (46) Solving (43), (44), (45) and (46) with their boundary con- ditions, we have D2 (t, r) = 0, D3 (t, r) = 0, D4 (t, r) = 0, D5 (t, r) = 0 hence from (41) h2 (t, r, q) = 0 (47) Next , we attempt to solve (33), by assuming a solution of the form h3 (t, r, q) =  D6 (t, r) + q 12 D7 (t, r) + qD8 (t, r) +q 3 2 D9 (t, r) + q2 D9 (t, r)  (48)  h3t = D6t + q 1 2 D7t + qD8t + q 3 2 D9t + q2 D10t , h3q = 1 2 q − 1 2 D7 + D8 + 3 2 q 1 2 D9 + 2qD10 h2qq = − 1 4 q − 3 2 D7 + 3 4 q − 1 2 D9 + 2D10h1r = 0, h1rr = 0, h2rq = 0  (49) Substituting into (33), we have D6t + βσ 2 (2β + 1) D8 = 0, D6 (T, r) = 0 (50) D7t −µβD7 + 3 2 βσ2 (3β + 1) D9 = 0, D7 (T, r) = 0 (51) D8t − 2µβD8 + 2βσ 2 (4β + 1) D10 = 0, D8 (T, r) = 0(52) D9t − 3µβD9 = 0, D9 (T, r) = 0 (53) D10t − 4µβD10 = 0, D10 (T, r) = 0 (54) Solving (50), (51), (52), (53) and (54), we obtain D6 (t, r) = 0, D7 (t, r) = 0, D8 (t, r) = 0, D9 (t, r) = 0, D10 (t, r) = 0 Hence from (48) h3 (t, r, q) (55) Therefore, from (27), we have hα (t, r, q) = h1 (t, r, q) + √ αh2 (t, r, q) + αh3 (t, r, q) = 1 Hence from (18), v (t, r, s) = 1 Proposition 4.2 The solution of equation (17) is given as w (t, r, s) = f1 (t, r, q) + √ α f2 (t, r, q) + α f3 (t, r, q) Where  f1 (t, r, q) = [ (2β+1)(µ−r)2 8βµ + r ] (T − t) + ( q (µ−r) 2 4µβσ2 − (2β+1)(µ−r)2 8βµ2 ) [ 1 − e2µβ(t−T ) ] f2 (t, r, q) = K2 (t, r) + q 1 2 K3 (t, r) + qK4 (t, r) f3 (t, r, q) = K6 (t, r) + q 1 2 K7 (t, r) + qK8 (t, r)  and  K2 (t, r) = [ (2β+1)(µ−r)2 8βµ + r ] (T − t) − (2β+1)(µ−r)2 8βµ2 [ 1 − e2µβ(t−T ) ] K3 (t, r) = a1 √ r(µ−r) σβµ2 [ 1 − 2eµβ(t−T ) + e2µβ(t−T ) ] K4 (t, r) = (µ−r)2 4µβσ2 [ 1 − e2µβ(t−T ) ] K6 (t, r) =   (2β+1)(µ−r)(b−cr)[1−e2µβ(t−T ) ] 8βσ2µ2 + (2β+1)(µ−r)2 4σ2µ + r r(2β+1)(µ−r)(b−cr) 4βµ2 + ra21 (2β+1)(µ−2r) 8βµ2 + (µ−3r)a21 2µ2  (T − t) +  r(2β+1)(µ−r)(b−cr) 4µ − (b−cr) 2 + ra21 (2β+1)(µ−2r) 8µ  (t2 − T 2) r(2β+1)(µ−r)(b−cr) 8β2µ3 + (2β+1)(µ−r)(ra21 +µ−r) 8βσ2µ2 + ra21 (2β+1)(µ−2r) 8µ + (µ−3r)a21 4βµ3  ( 1 − e2µβ(t−T ) ) (µ−3r)a21 4βµ3 ( 1 − eµβ(t−T ) )  K7 (t, r) = a1 √ r(µ−r) σβµ2 [ 1 − 2eµβ(t−T ) + e2µβ(t−T ) ] K8 (t, r) =  (µ−r)2 4µβσ2 [ 1 − e2µβ(t−T ) ] +  ra21 −2 (b − cr) (µ− r) 4µβσ2   12µβ [ 1 − e2µβ(t−T ) ] + (t − T ) e2µβ(t−T )    Proof Substitute (t, r, s) = 1, vs = 0, vr = 0 into (17), we have wt + µsws + 12 σ2 s2β+2wss + (b − cr) wr + r + 12 ra21wrr +σa1 √ rsβ+1wsr + 1 2 (µ−r)2 σ2 s2β  = 0(56) w (T, r, s) = 0 Assume{ w (t, r, s) = f (t, r, q) , q = s−2β f (T, r, q) = 0 (57) Then wt = ft, ws = −2βs−2β−1 fq , wss = 2β (2β + 1) s−2β−2 fq + 4β2 s−4β−2 fqq, wr = fr, wrr = frr, wrs = −2βs−2β−1 frq (58) 190 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 2 (2020) 186–196 191 Substituting (58) into (56), we have ft − 2µβq fq + σ2β (2β + 1) fq + 12 (µ−r)2 q σ2 + r + 2β2σ2q fqq + (b − cr) fr + 12 ra 2 1 frr − 2βσa1 √ r √ q frq  = 0 (59) We can rewrite (59) as (E + F + G) f = 0 (60) Where E = [ (b − cr) fr + 1 2 ra21 frr ] f (61) F =  ft + β ( σ2 (2β + 1) − 2µq ) fq +2β2σ2q fqq + 1 2 (µ−r)2 q σ2 + r  f (62) G = [ −2βσa1 √ r √ q frq ] f (63) Similarly, we apply the same approach used in solving (21) to determine the solution of (60) as follows drα (t) = (b − crα (t)) dt − a1 √ rα (t)dBr (t) (64) Substituting (64) into (60) and also replacing b − cr by α (b − cr) and √ r by √ α √ r, we will have( αE + F + √ αG ) fα = 0 (65) Next, we conjecture a solution for (65) as follows fα (t, r, q) = f1 (t, r, q) + √ α f2 (t, r, q) + α f3 (t, r, q) (66) Substituting (66) into (65) and simplifying it, we have F f1 (t, r, q) + [ F f2 (t, r, q) + G f1 (t, r, q) ]√ α + [ E f1 (t, r, q) + F f3 (t, r, q) + G f2 (t, r, q) ] α = 0 This implies that{ F f1 (t, r, q) = 0 f1 (T, r, s) = 0 (67) { F f2 (t, r, q) + G f1 (t, r, q) = 0 f1 (T, r, s) = f2 (T, r, s) = 0 (68) { E f1 (t, r, q) + F f3 (t, r, q) + G f2 (t, r, q) f1 (T, r, s) = f2 (T, r, s) = f3 (T, r, s) = 0 (69) To obtain the solution of (66), we solve (67), (68) and (69). Recall from (61), (62) and (63), equation (67), (68) and (69) can be expressed as f1t + β ( σ2 (2β + 1) − 2µq ) f1q + 2β2σ2q f1qq + 12 (µ−r)2 q σ2 + r = 0 f1 (T, r, q) = 1 (70)  f2t + β ( σ2 (2β + 1) − 2µq ) f2q + 2β2σ2q f2qq + 12 (µ−r)2 q σ2 + r − 2βσa1 √ r √ q f1rq = 0 f2 (T, r, q) = 0 (71)  f3t + β ( σ2 (2β + 1) − 2µq ) f3q + 2β2σ2q f3qq + 12 (µ−r)2 q σ2 + r + (b − cr) f1r + 1 2 ra 2 1 f1rr − 2βσa1 √ r √ q f2rq f3 (T, r, q) = 0 (72) Let f1 (t, r, q) = K0 (t, r) + qK1 (t, r) (73) and( f1t = K0t + qK1t, f1q = K1, f1qq = 0 ) (74) Substituting (74) in (70), we have K0t + βσ 2 (2β + 1) K1 + r = 0, K0 (T, r) = 0 (75) K1t − 2µβK1 + 1 2 (µ− r)2 σ2 = 0, K1 (T, r) = 0 (76) Solving (76), we have K1 (t, r) = (µ− r)2 4µβσ2 [ 1 − e2µβ(t−T ) ] (77) Putting (77) into (75) solving it, we have K0 (t, r) =  [ (2β+1)(µ−r)2 8βµ + r ] (T − t) − (2β+1)(µ−r)2 8βµ2 [ 1 − e2µβ(t−T ) ]  (78) Hence from (34) f1 (t, r, q) =  [ (2β+1)(µ−r)2 8βµ + r ] (T − t) + ( q (µ−r) 2 4µβσ2 − (2β+1)(µ−r)2 8βµ2 ) [ 1 − e2µβ(t−T ) ] (79) Next, we solve (71), by assuming a solution of the form f2 (t, r, q) = K2 (t, r)+q 1 2 K3 (t, r)+qK4 (t, r)+q 3 2 K5 (t, r)(80) and f2t = K2t + q 1 2 K3t + qK4t + q 3 2 K5t , f2q = 1 2 q − 1 2 K3 + K4 + 3 2 q 1 2 K5 f2qq = − 1 4 q − 3 2 K3 + 3 4 q − 1 2 K5, K1r = (µ−r) 2µβσ2 [ e2µβ(t−T ) − 1 ]  (81) Substituting (81) into (71), we have K2t + βσ2 (2β + 1) K4 + r = 0, K2 (T, r) = 0( K3t −µβK3 + 3 2 βσ 2 (3β + 1) K5 −2βσa1 √ rK1r = 0 ) , K3 (T, r) = 0 K4t − 2µβK4 + (µ−r)2 2σ2 = 0, K4 (T, r) = 0 K5t − 3µβK5 = 0, K5 (T, r) = 0  (82) 191 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 2 (2020) 186–196 192 Solving (82) with their boundary conditions, we have K2 (t, r) =  [ (2β+1)(µ−r)2 8βµ + r ] (T − t) − (2β+1)(µ−r)2 8βµ2 [ 1 − e2µβ(t−T ) ]  K3 (t, r) = a1 √ r(µ−r) σβµ2 [ 1 − 2eµβ(t−T ) + e2µβ(t−T ) ] K4 (t, r) = (µ−r)2 4µβσ2 [ 1 − e2µβ(t−T ) ] K5 (t, r) = 0  (83) Hence from (80) f2 (t, r, q) =  K2 (t, r) + q 12 K3 (t, r) + qK4 (t, r) +q 3 2 K5 (t, r)  (84) Where K2, K3, K4, and K5 are given in equation (83) Next, we attempt to solve (72), by assuming a solution of the form f3 (t, r, q) =  K6 (t, r) + q 12 K7 (t, r) + qK8 (t, r) +q 3 2 K9 (t, r) + q2 K10 (t, r)  (85)  f3t = K6t + q 1 2 K7t + qK8t + q 3 2 K9t + q2 K10t , f3q = 1 2 q − 1 2 K7 + K8 + 3 2 q 1 2 K9 + 2qK10 f3qq = − 1 4 q − 3 2 K7 + 3 4 q − 1 2 K9 + 2K10, f1r = K0r + qK1r, f1rr = K0rr + qK1rr f2rq = 1 2 q − 1 2 K3r + K4r + 3 2 q 1 2 K5r  (86) Substituting into (72), we have ( K6t + βσ2 (2β + 1) K8 + r + (b − cr) K0r −βσa1 √ rK3r + 1 2 ra 2 1 K0rr ) = 0 K6 (T, r) = 0 K7t −µβK7 + 3 2 βσ 2 (3β + 1) K9 − 2βσa1 √ rK4r = 0, K7 (T, r) = 0 K8t + 2βσ2 (4β + 1) K10 + (b − cr) K1r −2µβK8 − 3βσa1 √ rK9r + 1 2 ra 2 1 K1rr + (µ−r)2 2σ2  = 0 K8 (T, r) = 0 K9t − 3µβK9 = 0, K9 (T, r) = 0 K10t − 4µβK10 = 0, K10 (T, r) = 0  (87) Solving (87), we obtain K6 (t, r) =   (2β+1)(µ−r)(b−cr)[1−e2µβ(t−T ) ] 8βσ2µ2 + r(2β+1)(µ−r)(b−cr) 4βµ2 + (2β+1)(µ−r)2 4σ2µ + r + ra21 (2β+1)(µ−2r) 8βµ2 + (µ−3r)a21 2µ2  (T − t) +  r(2β+1)(µ−r)(b−cr) 4µ − (b−cr) 2 + ra21 (2β+1)(µ−2r) 8µ  (t2 − T 2) r(2β+1)(µ−r)(b−cr) 8β2µ3 + (2β+1)(µ−r)(ra21 +µ−r) 8βσ2µ2 + ra21 (2β+1)(µ−2r) 8µ + (µ−3r)a21 4βµ3  ( 1 − e2µβ(t−T ) ) (µ−3r)a21 4βµ3 ( 1 − eµβ(t−T ) )  K7 (t, r) = a1 √ r(µ−r) σβµ2 [ 1 − 2eµβ(t−T ) + e2µβ(t−T ) ] K8 (t, r) =  (µ−r)2 4µβσ2 [ 1 − e2µβ(t−T ) ] + ra21−2(b−cr)(µ−r) 4µβσ2  12µβ [ 1 − e2µβ(t−T ) ] + (t − T ) e2µβ(t−T )   K9 (t, r) = 0 K10 (t, r) = 0  (88) Hence from (85) f3 (t, r, q) =  K6 (t, r) + q 12 K7 (t, r) + qK8 (t, r) +q 3 2 K9 (t, r) + q2 K10 (t, r)  (89) With K6, K7 , K8, K9 , K10 given in (88) Therefore, from (66), we have fα (t, r, q) = f1 (t, r, q) + √ α f2 (t, r, q) + α f3 (t, r, q) With f1 (t, r, q) , f2 (t, r, q) and f3 (t, r, q) given in equation (79), (84) and (89) respectively. Hence from (57), w (t, r, s) = f1 (t, r, q) + √ α f2 (t, r, q) + α f3 (t, r, q) Proposition 4.3 The optimal portfolio strategies and optimal value function are given as ϕ∗ = (µ− r) σ2 s2β (90) ϕ∗1 = (µ− r) σ2 s2β (91) And J (t, r, s, z) = ( ln z + f1 (t, r, q) + √ α f2 (t, r, q) +α f3 (t, r, q) ) (92) where  f1 (t, r, q) =  [ (2β+1)(µ−r)2 8βµ + r ] (T − t) +  q (µ−r) 2 4µβσ2 − (2β+1)(µ−r)2 8βµ2  [1 − e2µβ(t−T )]  f2 (t, r, q) = K2 (t, r) + q 1 2 K3 (t, r) + qK4 (t, r) f3 (t, r, q) = K6 (t, r) + q 1 2 K7 (t, r) + qK8 (t, r)  and 192 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 2 (2020) 186–196 193  K2 (t, r) =  [ (2β+1)(µ−r)2 8βµ + r ] (T − t) − (2β+1)(µ−r)2 8βµ2 [ 1 − e2µβ(t−T ) ]  K3 (t, r) =  a1√r(µ−r)σβµ2  1 − 2eµβ(t−T ) +e2µβ(t−T )   K4 (t, r) = (µ−r)2 4µβσ2 [ 1 − e2µβ(t−T ) ] K6 (t, r) =   (2β+1)(µ−r)(b−cr) [ 1−e2µβ(t−T ) ] 8βσ2µ2 + (2β+1)(µ−r)2 4σ2µ + r r(2β+1)(µ−r)(b−cr) 4βµ2 + ra21 (2β+1)(µ−2r) 8βµ2 + (µ−3r)a21 2µ2  (T − t) +  r(2β+1)(µ−r)(b−cr) 4µ − (b−cr) 2 + ra21 (2β+1)(µ−2r) 8µ  (t2 − T 2) r(2β+1)(µ−r)(b−cr) 8β2µ3 + (2β+1)(µ−r) ( ra21 +µ−r ) 8βσ2µ2 + ra21 (2β+1)(µ−2r) 8µ + (µ−3r)a21 4βµ3  ( 1 − e2µβ(t−T ) ) (µ−3r)a21 4βµ3 ( 1 − eµβ(t−T ) )  K7 (t, r) = a1 √ r(µ−r) σβµ2 [ 1 − 2eµβ(t−T ) + e2µβ(t−T ) ] K8 (t, r) =  (µ−r)2 4µβσ2 [ 1 − e2µβ(t−T ) ] + ra21−2(b−cr)(µ−r) 4µβσ2  12µβ [ 1 − e2µβ(t−T ) ] + (t − T ) e2µβ(t−T )    Proof 1. recall that equation (11) is given as ϕ∗ = − [(µ−r)Jz +σ2 s2β+1 Jzs +σa1 √ rsβ Jzr ] zσ2 s2β Jzz . From proposition 1 and equation (14) v (t, r, q) = 1 Jz = 1 z , Jzz = − 1 z2 , Jzr = vr z = 0 , Jzs = vs z = 0 Substituting the above equation into (11), we have ϕ∗ = − [ (µ− r) 1z ] zσ2 s2β ( − 1 z2 ) = (µ− r) 1z σ2 s2β 1z = (µ− r) σ2 s2β 2. recall that from equation (13), J (t, r, s, z) = w (t, r, s) + v (t, r, s) ln z From proposition 1 and 2, the proof is completed. Remark 4.1 From proposition 4.3, we observed that our result is similar to the one in [3] but the difference between our result and theirs is that their interest rate is a constant while ours is stochastic. 5. Numerical Simulations Here, the numerical simulations are used to study the effect of parameters on the portfolio strategies under logarithm utility. To achieve this, the following data will be use unless stated otherwise: σ = 1, β = −1, µ = 0.4, r (0) = 0.05, S (0) = 1.5, T = 3 6. Discussion Figure 1, present a simulation of optimal portfolio strate- gies against time. The graph shows that the investor will invest more in marketable security and gradually increases investment in treasury security to balance with the marketable security and as expiry date draws closer; there is a continuous decrease in investment in risky asset and a continuous increase in that of risk free asset. This is so because investors like avoiding risk toward the end of investment especially for highly risky assets. Figure 2, shows a plot of optimal portfolio strategies against time with different values of the elasticity parameter β. The graph shows that as the elasticity parameter decreases, the in- surer is more scared to invest in marketable security as expira- tion date approaches. Furthermore, we observed a very sharp decline when β = −2, showing how volatile the risky asset can be hence discouraging for investors with high risk aversion co- efficient but when β = 0, the decline is almost unnoticeable, showing that the risky asset is not volatile. This is shows that geometric Brownian motions i.e when β = 0 may lead an in- vestor astray while taking investment decisions. Figure 3, present a simulation of optimal portfolio strategies against time with different values of µ; the graph shows that as µ increases, the investment strategy for the marketable security decreases continuously while that of treasury security increases continuously with time. The reason being the interest rate is not constant. Furthermore, we observe that as expiration date of in- vestment draws closer, the risk-free interest may increase faster than µ , thereby making µ− r < 0. Hence a drastic decrease in optimal portfolio strategy of marketable security. Figure 4, shows a simulation of the optimal portfolio strate- gies against time with different values of σ, it is observed that as σ increases the optimal investment in marketable security decreases while that of risk-free asset increases. Recall that σ is the instantaneous volatility representing the risk coefficient of marketable security. Therefore, for risk averse investors, bigger σ , implies less investment in marketable security. 7. Conclusion This paper considers optimal portfolio strategy for an in- surer with logarithm utility using CEV to model the risky asset in the presence of stochastic interest rate. Investment in trea- sury security and marketable security were considered such that the risk free interest rate follows the CIR model. The asymp- totic solutions of the the optimal portfolio strategies and it value function was found using power transformation, change of Vari- able and asymptotic expansion technique. Furthermore, we present some numerical simulations to study the effect of some parameters on the optimal portfolio strategy under stochastic interest rate. 193 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 2 (2020) 186–196 194 Figure 1. Time evolution of optimal portfolio strategies. Figure 2. Time evolution of optimal portfolio strategy with different β. 194 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 2 (2020) 186–196 195 Figure 3. Time evolution of optimal portfolio strategy with different µ. Figure 4. Time evolution of optimal portfolio strategy with different σ. 195 Akpanibah & Ini / J. Nig. Soc. Phys. Sci. 2 (2020) 186–196 196 References [1] J. C. Cox & S. A. Ross, “The valuation of options for alternative stochastic processes”, Journal of financial economics 3 (1976) 145 [2] D. Li, X. Rong & H. Zhao, “Optimal investment problem with taxes, dividends and transaction costs under the constant elasticity of variance model”, Transaction on Mathematics 12 (2013) 243. [3] J. Xiao, Z. Hong & C. Qin. “The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts”, Insurance 40 (2007) 302. [4] J. Gao, “Optimal portfolios for DC pension plan under a CEV model”, Insurance Mathematics and Economics 44 (2009) 479. [5] M. Gu, Y. Yang, S. Li & J. Zhang, “Constant elasticity of variance model for proportional reinsurance and investment strategies”, Insurance: Mathe- matics and Economics 46 (2010) 580. [6] B. O. Osu, E. E. Akpanibah & B. I. Oruh, “Optimal investment strategies for defined contribution (DC) pension fund with multiple contributors via Legendre transform and dual theory”, International journal of pure and ap- plied researches 2 (2017) 97. [7] D. Li, X. Rong, H. Zhao & B. Yi, “Equilibrium investment strategy for DC pension plan with default risk and return of premiums clauses under CEV model”, Insurance 72 (2017) 6. [8] E. E. Akpanibah, & O. Ogheneoro, “Optimal Portfolio Selection in a DC Pension with Multiple Contributors and the Impact of Stochastic Addi- tional Voluntary Contribution on the Optimal Investment Strategy”, Inter- national journal of mathematical and computational sciences 12 (2018) 14. [9] J. F. Boulier, S. Huang & G. Taillard, “Optimal management under stochas- tic interest rates: the case of a protected defined contribution pension fund”, Insurance 28 (2001) 173. [10] G. Deelstra M. Grasselli & P. F. Koehl, “Optimal investment strategies in the presence of a minimum guarantee”, Insurance 33 (2003) 189. [11] J. Gao, “Stochastic optimal control of DC pension funds”, Insurance 42 (2008) 1159. [12] P. Battocchio & F. Menoncin, “Optimal pension management in a stochas- tic framework”, Insurance 34 (2004) 79. [13] A. J. G. Cairns, D. Blake & K. Dowd, “Stochastic lifestyling: optimal dynamic asset allocation for defined contribution pension plans”, Journal of Economic Dynamics & Control 30 (2006) 843. [14] C. Zhang & X Rong, “Optimal investment strategies for DC pension with a stochastic salary under affine interest rate model”. Hindawi Publishing Corporation, 2013 (2013) http://dx.doi.org/10.1155/2013/297875http://dx.doi.org/10.1155/2013/297875 [15] E. E. Akpanibah , B. O. Osu, K. N. C. Njoku & E. O. Akak, “Optimiza- tion of Wealth Investment Strategies for a DC Pension Fund with Stochas- tic Salary and Extra Contributions”, International Journal of Partial Diff. Equations and Application 5 (2017) 33. [16] K.N. C Njoku, B. O. Osu, E. E. Akpanibah & R. N. Ujumadu, “Effect of Extra Contribution on Stochastic Optimal Investment Strategies for DC Pension with Stochastic Salary under the Affine Interest Rate Model”, Jour- nal of Mathematical Finance 7 (2017) 821. [17] Y. He & P. Chen, “Optimal Investment Strategy under the CEV Model with Stochastic Interest Rate”, Mathematical Problems in Engineering 2020 (2020) 1 196