OPTIMAL STRATEGIES FOR A LONG-TERM STATIC INVESTOR

OPTIMAL STRATEGIES FOR A LONG-TERM STATIC INVESTOR LINGJIONG ZHU Abstract. The optimal strategies for a long-term static investor are studied. Given a portfolio of a stock and a bond, we derive the optimal allocation of the capitols to maximize the expected long-term growth rate of a utility function of the wealth. When the bond has constant interest rate, three models for the underlying stock price processes are studied: Heston model, 3/ model and jump diffusion model. We also study the optimal strategies for a portfolio in which the stock price process follows a Black-Scholes model and the bond process has a Vasicek interest rate that is correlated to the stock price.. Introduction In this article, we are interested in the long-term optimal strategies for a static investor. The investor starts with a known initial wealth V > and the wealth at time t is denoted by V t. The investor decides what fraction of wealth α t to invest in a stock S t and the remaining α t in a bond r t, i.e. dv t ds t (.) α t + ( α t )r t dt. V t S t For a static investor, we assume that α t α is a constant between and, i.e. α, ]. We consider a hyperbolic absolute risk aversion (HARA) utility function u(c) with constant relative risk aversion coefficient γ (, ), i.e. (.) u(c) c γ γ, < γ <. We are interested in the optimal strategy to maximize the long-term growth rate, i.e. (.3) max lim α t t log Eu(V t)] max Λ(α), α if the long-term growth rate Λ(α) : lim t t log Eu(V t)] exists for any α. For the convenience of notation, let θ : γ (, ) and thus we are interested in (.4) max α lim t t log E(V t) θ ]. The optimal long-term growth rate of expected utility of wealth has been well studied in the literatures. Usually, the optimal strategy is taken to be dynamic and Date: May 4. Revised: May 4. Mathematics Subject Classification. 9B8, 9B7, 6H3. Key words and phrases. Long-Term growth rate, optimal strategies, Heston model, 3/ model, jump diffusion, Vasicek model.

LINGJIONG ZHU some dynamic programming equations are studied, see e.g. Fleming and Sheu 8]. In this article, we only concentrate on the static strategies for the simplicity. This set-up allows us to gain analytical tractability for some more sophisticated models like Heston model and 3/ model. The problem of maximizing the long-term expected utility is closely related to maximizing the probability that the wealth exeeds a given benchmark for large time horizon, i.e. max α lim t t log P (V t V e xt ), where x is a given benchmark. In a static framework, an asymptotic outperformance criterion was for example considered in Stutzer 3]. An asymptotic dynamic version of the outperformance managment criterion was developed by Pham ]. To find the optimal strategy for the long-term growth rate, the first step is to compute the limit lim t t log Eu(V t)]. Under most of the standard models, the wealth process V t has exponential growth rate and the existence of a logarithmic moment generating function plus some additional conditions can be used to obtain a large deviation principle for the probability that the wealth process outperformances a given benchmark, which is exponentially small. The connection is provided by Gärtner-Ellis theorem, see e.g. Dembo and Zeitouni 5]. For a survey on the applications of large deviations to finance, we refer to Pham ]. In this article, we study in detail the optimal strategies for a static investor investing in stocks of two stochastic volatility models, i.e. the Heston model (Section ), the 3/ model (Section 3) The Heston model, introduced by Heston 9] is a widely used stochastic volatility model. The volatility process is itself a Cox-Ross-Ingersoll process, which is an affine model and has great analytical tractability. The 3/ model is another popular model of stochastic volatility. It has been applied to interest rate modeling, e.g. Ahn and Gao ]. Carr and Sun 3] used the 3/ model to price variance swaps and Drimus 6] used it to price options on realized variance. Next, we study the optimal long-term static investment strategies when the underlying stock process follows a jump diffusion model (Section 4) assuming the alternative investment bond has constant short-rate. Finally, we study the case when the stock follows a classical Black-Scholes model while the bond has a Vasicek interest rate (Section 5). As an illustration, let us first consider a toy model. Assume that the stock price follows a geometric Brownian motion with constant drift µ > and constant volatility σ > and the bond has constant short-rate r >. We can write down a stochastic differential equation for the wealth process V t, (.5) dv t αµv t dt + ασv t db t + ( α)rv t dt, where B t is a standard Brownian motion starting at at time and therefore (.6) V t V exp (αµ + ( α)r ) } α σ t + ασb t. Hence, we can compute that (.7) E(V t ) θ ] V θ e θ(αµ+( α)r α σ )t e θ α σ t. Therefore, we are interested to maximize (.8) Λ(α) θ αµ + ( α)r ] α σ + θ α σ, α.

OPTIMAL STRATEGIES FOR A LONG-TERM STATIC INVESTOR 3 It is easy to compute that (.9) Λ (α) (θµ θr) (θ θ )σ α. µ r Hence, Λ (α) if α ( θ)σ. Therefore, the optimal α is given by if µ r, (.) α µ r ( θ)σ if < µ r ( θ)σ <, if µ r ( θ)σ. The financial interpretation is clear. When µ r, it is optimal to invest in the bond only because the yield of the bond r exceeds the mean return of the stock. When µ > r, it is not always optimal in only invest in stocks. The reason is although the mean return of the stock exceeds the yield of the bond, stocks are volatile and a large volatility can decrease the expected utility of the portfolio. This is consistent with the mean-variance analysis, which says that given the mean, the investor has the incentive to minimize the variance. In general, for any wealth process V t, assume Λ(α) exists and is smooth and strictly concave, If Λ (), then the optimal α is given by α. Otherwise, Λ(α) achieves a unique maximum at some α (, ). Then the optimal α is given by (.) α if α, α if α (, ). This is the general method behind analyzing all the models in this article.. Heston Model Let us assume that the stock price follows a Heston model, namely, the stock price has a stochastic volatility which follows a Cox-Ingersoll-Ross process, ds t µs t dt + ν t S t db t, (.) dν t (γ ν t )dt + ν t dw t, where W t and B t are two standard Brownian motions and W, B t ρt, where ρ is the correlation. Assume that µ,, γ, >. The volatility process ν t is a Cox-Ingersoll-Ross process, introduced by Cox et al. 4]. We assume the Feller condition γ > holds so that ν t is always positive, see e.g. Feller 7]. The wealth process satisfies dv t αµv t dt + α ν t V t db t + ( α)rv t dt, (.) dν t (γ ν t )dt + ν t dw t. Then, we have (.3) V t V exp αµt t α ν s ds + ( α)rt + α Hence, we get (.4) Eu(V t )] θ V θ e θ(αµ+( α)r)t E t νs db s }. e θ(α R t νsdb s α R t νsds)].

4 LINGJIONG ZHU Lemma. For any ν >, (.5) lim t t log E ν ν γ γ e θ(α R t νsdb s R α t νsds)] θ α ( ρ ) + θα αρθ θαρ γ. Proof. Write B t ρw t + ρ Z t, where Z t is a standard Brownian motion independent of W t. Let Ft ν : σ(ν s, s t) be the natural sigma field of the volatility process up to time t. It is easy to compute that (.6) E νν e θ(α R t νsdb s R α t νsds)] e θ(α R t E νν E νν E E νν E νν R νsρdw t s+α νs ρ dz s R α t νsds)] e θ(α R t R νsρdw t s+α νs ρ dz s R α t νsds) Ft ν e θα R t νsρdw s+ θ α ( ρ ) R t νsds R θα t νsds] e θαρ νt+( θ α ( ρ ) θα + αρ θ) R t νsds ] e θαρ θαρ ν γt, where the last step was due to the fact that ν t ν t (γ ν s)ds + t ν s dw s. Let u(t, ν) : E νν e θαρ νt+( θ α ( ρ ) θα + αρ θ) R t ]. νsds Feynman-Kac formula implies that u(t, ν) satisfies the following partial differential equation, ( ) u u t (γ ν) ν (.7) + ν u ν + θ α ( ρ ) θα + αρ θ νu, u(, ν) e θαρ ν. Let us try u(t, ν) e A(t)+B(t)ν and it is easy to see that A(t), B(t) satisfy the following system of ordinary differential equations, A (t) γb(t), ( ) (.8) B (t) B(t) + B(t) + θ α ( ρ ) θα + αρ θ, A(), B() θαρ. We claim that there are two distinct solutions to the quadratic equation ( θ (.9) x α ( ρ ) θα x + + αρ ) θ, and B(t) converges to the smaller solution of (.9). We can compute that (.) : ( θ α ( ρ ) θα If α, then >. If α, then, (.) + αρ ) θ. ( + θ α ρ αρθ) + α (θ θ ) ( θαρ) + α (θ θ ) >, since θ (, ). Hence (.9) has two distinct solutions. B (t) is positive when B(t) is smaller than the smaller solution of (.9) or larger than the larger solution of ]]

OPTIMAL STRATEGIES FOR A LONG-TERM STATIC INVESTOR 5 (.9). B (t) is negative if B(t) lies between the two solutions of (.9). Therefore, B(t) converges to the smaller solution of (.9) if B() θαρ is less than the larger solution of (.9). When α, B() and the large solution of (.9) equals to >. Hence, we can assume that α >. Let (.) H(x) : ( θ x α ( ρ ) θα x + + αρ ) θ. It is easy to check that ( ) θαρ (.3) H (θ θ)α <, since θ (, ) and α >. Therefore, we conclude that B() is less than the larger solution of (.9) and (.4) B(t) as t and hence (.5) t t γ A(t) t B(s)ds γ as t. Recall that E νν Hence, we conclude that (.6) Theorem. (.7) Let us define (.8) lim t t log E ν ν γ γ ( θ α ( ρ ) θα + αρ ) θ, γ θ α ( ρ ) + θα αρθ, e θ(α R t νsdb s α R t νsds) ] e θ(α R t νsdb s R α t νsds)] u(t, ν)e θαρ θαρ ν γt. θ α ( ρ ) + θα αρθ θαρ γ. Λ(α) lim t t log Eu(V t)] γ γ θ α ( ρ ) + θα αρθ θαρ γ + θαµ + θ( α)r. C : γ + θr, C : γ ( θ θ ( ρ )), 4 C : ρθ γ, 4 C 3 : 4 γ, 4 C 4 : θρ γ + θ(µ r).

6 LINGJIONG ZHU When C 4 + C C3, the optimal α and when C 4 C, the optimal α. Finally, if C C3 < C 4 < C, (.9) α where (.) α C α if α <, otherwise, C + C 4 C C 3 C C C 4 Proof. Recall that Eu(V t )] θ V θ e θ(αµ+( α)r)t E Lemma, we have ] e θ(α R t. νsdb s α R t νsds) ]. By (.) Λ(α) lim t t log Eu(V t)] γ γ θ α ( ρ ) + θα αρθ θαρ γ + θαµ + θ( α)r C α C α + C 3 + C 4 α + C. From the definition in (.8), it is clear that C, C, C, C 3 >. But C 4 may or may not be positive. It is easy to compute that (.) Λ (α) C 4 and C α C C α C α + C 3, (.3) Λ C C (C C 3 C (α) ) ((C α C ) + (C C 3 C. ))3/ On the other hand, since θ (, ), we have (.4) C C 3 C γ 4 4 γ 4 ( θ θ ( ρ )) ρ θ 4 γ 4 8 6 γ 4 8 ( θ θ ( ρ ) ρ θ ) 6 γ 4 6 (θ θ ) >. Hence, we conclude that Λ (α) < for any α, i.e. Λ(α) is strictly concave in α. Note that Λ () C 4 + C C3. If C 4 + C C3, since Λ(α) is strictly concave, the maximum must be achieved at α. Now assume that C 4 + C C3 >. When C 4 > C, it is easy to check that Λ (α) (C 4 C ) as α, and since Λ(α) is strictly concave, it yields that Λ(α) is increasing in α and the maximum is

OPTIMAL STRATEGIES FOR A LONG-TERM STATIC INVESTOR 7 achieved at α. If C 4 C, (.5) Λ (α) C 4 C α C α + C 3 C α + C C α C α + C 3 C 4 C (C α C ) + (C 3 C C ) C α + C C α C α + C 3 (C α C ) + (C 3 C C ) (C α C ) >, C α C α + C 3 since C 3 C C >. Thus, α when C 4 C. Now assume that C C3 < C 4 < C. Then Λ () C 4 + C C3 > and Λ(α) as α. Thus, there exists a unique global maximum on (, ), given by α. So that (.6) Λ (α ) ] C 4 (C C α C ). C (C α C ) + C C 3 C C α C has the same sign as C 4 which is positive. Hence, we can solve for α and get ] (.7) α C C 3 C C + C 4 C C C4. The optimal α is given by (.8) α α if α <, otherwise. 3. 3/ Model Let us assume that the stock price follows a 3/ model, namely, ds t µs t dt + ν t S t db t, (3.) dν t ν t (γ ν t ) + ν 3/ t dw t, where B t and W t are two standard Brownian motions, which are assumed to be independent for simplicity. Therefore, the wealth process satisfies dv t αµv t dt + α ν t V t db t + ( α)rv t dt, (3.) dν t (γ ν t )dt + ν 3/ t dw t, Then, we have (3.3) V t V exp αµt t α ν s ds + ( α)rt + α Hence, we get (3.4) Eu(V t )] θ V θ e θ(αµ+( α)r)t E θ V θ e θ(αµ+( α)r)t E t νs db s }. e θ(α R t νsdb s R α t νsds)] e α (θ θ ) R t νsds].

8 LINGJIONG ZHU The volatility process ν t is not an affine process but it is still analytically tractable. The Laplace tranform of t ν sds is known, see e.g. Lewis ]. (3.5) E ν e λ R t νsds] ( ) a ( ) Γ(b a) γ γ Γ(b) ν (e γt M a, b, ) ν (e γt, ) where (3.6) a : ( + ) ( + + ) + λ ( b : + + ) + λ, and Γ( ) is the standard Gamma function and M(a, b, z) : (a) n z n n (b) n n! is the confluent hypergeometric function, also known as Kummer s function (see e.g. Abramowitz and Stegun ]), where (c) : and (c) n : c(c + ) (c + n ) for n. In our case, (3.7) λ : α (θ θ ) >, since θ (, ). We are interested in the asymptotic behavior of the Laplace γ γ transform as t. As t, ν (e γt ) and M(a, b, ν (e γt ) ). Then, it is easy to see that (3.8) lim t t log E e α (θ θ ) R t νsds] aγ ( γ + ) ( γ + ) + α (θ θ ). Hence, we conclude that (3.9) Λ(α) lim t t log Eu(V t)] θαµ + θ( α)r + γ, ( + ) ( γ + ) + α (θ θ ). It is straightforward to check that Λ (α) < and it is easy to compute that (3.) Λ (α) θ(µ r) γ (θ θ ) α ( + ). + α (θ θ ) When µ r, since Λ(α) is strictly concave, Λ(α) is decreasing for α and thus the optimal α is achieved at α. Now, assume that µ r >. When θ(µ r) γ θ θ, Λ (α) for any α and the optimal α is achieved at α. When θ(µ r) γ θ θ <, there exists a unique global maximum of Λ(α) achieved at α (, ) so that Λ (α ). Observe that if Λ (α) in (3.), then α has the same sign as µ r >. After some algebraic manipulations, we get (3.) α θ(µ r) γ (θ θ ) θ (µ r) θ θ +. Thus, α α if α < and α otherwise.

OPTIMAL STRATEGIES FOR A LONG-TERM STATIC INVESTOR 9 We summarise our results in the following theorem. Theorem 3. (3.) Λ(α) lim t t log Eu(V t)] θαµ + θ( α)r + γ ( + ) ( γ + ) + α (θ θ ). θ θ The optimal α is given by α if µ r and α if θ(µ r) γ and if µ > r and θ(µ r) γ θ θ <, the optimal α is given by (3.3) α where (3.4) α α if α <, otherwise, θ(µ r) γ (θ θ ) θ (µ r) θ θ +. Remark 4. For simplicity, we only considered the case when ρ. Indeed, for general ρ, the joint Fourier-Laplace transform of the logarithm of the spot price, i.e. log(s t /S ) and the total integrated variance, i.e. t ν sds is also known in the close-form, see e.g. Carr and Sun 3] and our methods can still be applied to obtain the optimal strategy α. But the computations would be more involved. 4. Jump Diffusion Model Let us assume that the stock price follows a jump diffusion model. More precisely, (4.) ds t µs t dt + σs t db t + S t dj t, where J t N t i (Y i ), where Y i are i.i.d. random variables distributed on (, ) with a smooth and bounded probability density function and Y i are independent of N t which is a standard Poisson process with intensity λ >. We further assume that EY ] <. The wealth process satisfies (4.) dv t αµv t dt + ασv t db t + αv t dj t + ( α)rv t dt. Then, we have (4.3) V t V e αµt+( α)rt+ασbt α σ t+ P N t i log(α(yi )+). Therefore, (4.4) Λ(α) θαµ + ( α)r] + (θ θ)α σ + λ(e(α(y ) + ) θ ] ). Remark 5. (i) If Y y is a positive constant, then (4.5) Λ(α) θαµ + ( α)r] + (θ θ)α σ + λ((α(y ) + ) θ ). (4.6) (ii) If Y is exponentially distributed with parameter ρ >, then E(α(Y ) + ) θ ] (αy + α) θ e ρy ρdy ( ( )) e ρ( α ) (α/ρ) θ Γ θ +, ρ α,

LINGJIONG ZHU where Γ(s, x) : x ts e t dt is an upper incomplete Gamma function. It is easy to compute that (4.7) Λ (α) θ(µ r) + (θ θ)σ α + λθe (α(y ) + ) θ (Y ) ], and (4.8) Λ (α) (θ θ)σ + λθ(θ )E (α(y ) + ) θ (Y ) ] <, since θ (, ) and α, ]. In the expression of Λ(α), (4.9) E(α(Y ) + ) θ ] E(α Y + ) θ ] E(α(Y + ) + ) θ ] E(α(Y + ) + )] α(ey ] + ) +, since θ (, ) and α(y + ) + a.s. The coefficient of α term in Λ(α) is (θ θ)σ which is negative. Thus, Λ(α) as α. Recall that Λ(α) is strictly concave. Therefore, if Λ () θ(µ r) + λθ(ey ] ), then, the optimal α is achieved at α. If Λ () θ(µ r) + λθ(ey ] ) >, then, there exists a unique α (, ) so that Λ (α ). In this case, the optimal α is given by (4.) α if α, α if α (, ). We summarize our conclusions in the following theorem. Theorem 6. (4.) Λ(α) θαµ + ( α)r] + (θ θ)α σ + λ(e(α(y ) + ) θ ] ). When θ(µ r) + λθ(ey ] ), α. Otherwise, (4.) α if α, α if α (, ), where α is the unique positive solution to (4.3) θ(µ r) + (θ θ)σ α + λθe (α(y ) + ) θ (Y ) ]. 5. Black-Scholes Model with Vasicek Interest Rate Let us assume that the stock price follows a Black-Scholes model with constant drift µ and volatility σ and the interest rate r t follows a Vasicek model. The Vasicek model is a standard interest rate model, introduced by Vasicek 4]. The wealth process satisfies the following stochastic differential equation. (5.) dv t αµv t dt + ασv t db t + ( α)r t V t dt, where B t is a standard Brownian motion starting at at time and (5.) dr t (γ r t )dt + dw t, where W t is a standard Brownian motion so that W, B t ρt, where ρ is the correlation.

OPTIMAL STRATEGIES FOR A LONG-TERM STATIC INVESTOR Therefore, the wealth process is given by (5.3) V t V e αµt+ασbt α σ t+( α) R t rsds. Lemma 7. For any r r >, (5.4) lim t t log E r r ( θ( α) γ e θ(ασbt+( α) R t rsds)] + θασρ ) ( + θ( α) + θασρ ) θασγρ + θ α σ ( ρ ). Proof. Write B t ρw t + ρ Z t, where Z t is a standard Brownian motion independent of B t and W t. Therefore, we have (5.5) E rr E rr E rr E rr e θ(ασbt+( α) R t rsds)] e θ(ασρwt+ασ ρ Z R t t+( α) rsds)] e θ α σ ( ρ )t+θ(ασρw R t t+( α) rsds)] e θ( α) R t θασρ rsds+ r t+ θασρ where the last line uses the fact that W t rt r R t rsds] e θασρr θασγρ t+ θ α σ ( ρ )t, γ t + t r sds. Let u(t, r) : E rr e θ( α) R t θασρ rsds+ r t+ θασρ R t ] rsds. Then, u(t, r) satisfies the following partial differential equation, (5.6) u t u (γ r) r + u u(, r) e θασρ r. r + Let us try u(t, r) e A(t)+B(t)r. Then, we get (5.7) ( θ( α) + θασρ A (t) γb(t) + B(t), B (t) B(t) + θ( α) + θασρ A(), B() θασρ., ) ru, It is not hard to see that B(t) θ( α) + θασρ as t and therefore (5.8) A(t) t γ t t B s ds + t ( θ( α) γ t + θασρ B(s) ds ) + ( θ( α) + θασρ ),

LINGJIONG ZHU as t. Hence, we conclude that (5.9) Theorem 8. (5.) lim t t log E r r ( θ( α) γ t e θ(ασbt+( α) R t rsds)] + θασρ ) ( + θ( α) + θασρ ) θασγρ + θ α σ ( ρ ). Λ(α) lim t log Eu(V t)] ( θ( α) γ + θασρ ) + θασγρ ( θ( α) + θασρ ) + θ α σ ( ρ ) + θαµ θα σ. When θ θσρ + σ θ σ, the optimal α is given by (5.) α if γ + θ (θ )σ + µ, if γ + θ < (θ )σ + µ. Otherwise, the optimal α is given by if α, (5.) α α if α (, ), if α, where (5.3) α θ θ γθ + θµ + σρ θ + θσρ σ θ θ + σ Proof. Recall that V t V e αµt+ασbt α σ t+( α) R t rsds. Applying Lemma 7, we have (5.4) Λ(α) lim t log Eu(V t)] ( θ( α) γ + θασρ ) + θασγρ t ]. ( θ( α) + θασρ ) + θ α σ ( ρ ) + θαµ θα σ. Thus, Λ(α) is a quadratic function of α. If the coefficient of α in Λ(α) is nonnegative, i.e. θ (5.5) θ θσρ ] + σ θ σ, then Λ(α) is convex in α and the optimal α is achieved at either α or α. Indeed, one can compute that (5.6) Λ() γθ + θ,

OPTIMAL STRATEGIES FOR A LONG-TERM STATIC INVESTOR 3 and (5.7) Λ() (θ θ)σ + θµ. Therefore, when θ θσρ + σ θ σ, (5.8) α if γ + θ (θ )σ + µ, if γ + θ < (θ )σ + µ. If the coefficient of α in Λ(α) is negative, the function Λ(α) has a unique maximum at some α (, ) and if α, (5.9) α α if α (, ), if α, where (5.) α θ θ γθ + θµ + σρ θ + θσρ σ θ θ + σ ]. 6. Concluding Remarks In this article, we studied the optimal long-term strategy for a static investor for the Heston model, the 3/ model, the jump diffusion model and the Black-Scholes model with Vasicek interest rate. It will be interesting to generalize our results to the multivariate case, i.e. when the investor can invest in a basket of stocks S (i) t, i d, and the wealth process is given by (6.) dv t V t d i ds (i) t α i S (i) t + ( ) d α i r t dt. One can also study the Heston model, the 3/ model, and the jump diffusion model with stochastic interest rate. In Section 5, the interest rate is assumed to follow the Vasicek model. A drawback of the Vasicek model is that the process can go negative with positive probability. Our analysis in Section 5 cannot be directly applied to the Cox-Ingersoll-Ross interest rate unless one assumes that ρ. This can be left for the future investigations. i Acknowledgements The author also wishes to thank an annonymous referee for very careful readings of the manuscript and helpful suggestions that greatly improved the paper. The author is supported by NSF grant DMS-947, DARPA grant and MacCracken Fellowship at New York University.

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