OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT

Transcription

1 OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT SAVAS DAYANIK AND MASAHIKO EGAMI Abstract. An asset manager invests the savings of some investors in a portfolio of defaultable bonds. The manager pays the investors coupons at a constant rate and receives management fee proportional to the value of portfolio. She also has the right to walk out of the contract at any time with the net terminal value of the portfolio after the payment of investors initial funds, but is not responsible for any deficit. To control the principal losses, investors may buy from the manager a limited protection which terminates the agreement as soon as the value of the portfolio drops below a predetermined threshold. We assume that the value of the portfolio is a jump-diffusion process and find optimal termination rule of the manager with and without a protection. We also derive the indifference price of a limited protection. We describe numerical algorithms to calculate expected maximum reward and nearly optimal terminal rules for the asset manager and illustrate them on an example. The motivation comes from the collateralized debt obligations. 1. Introduction We study two optimal stopping problems of an institutional asset manager hired by ordinary investors who do not have access to certain asset classes. The investors entrust their initial funds in the amount of L to the asset manager. As long as the contract is alive, the investors receive coupon payments from the asset manager on their initial funds at a fixed rate (higher than the risk-free interest rate). In return, the asset manager collects dividend or management fee (at a fixed rate on the market value of the portfolio). At any time, the asset manager has the right to terminate the contract and to walk away with the net terminal value of the portfolio after the payment of the investors initial funds. However, she is not financially responsible for any amount of shortfall. The asset manager s first problem is to find a nonanticipative stopping rule which maximizes her expected discounted total income. Under the original contract, investors face the risk of losing all or some part of their initial funds. Suppose that the asset manager offers the investors a limited protection against this risk, in the form that the new contract will terminate as soon as the market value of the portfolio goes below a predetermined threshold. The asset manager s second problem is to find the fair price for the limited protection and the best time to terminate the contract under this additional clause. We assume that the market value X of the asset manager s portfolio follows a geometric Brownian motion subject to downward jumps which occur according to an independent Poisson process. As explained in detail in the next section, both the problems and the setting are motivated by those faced by the managers responsible for the portfolios of defaultable bonds, for example, as in Date: July 9, 21. 1

2 2 SAVAS DAYANIK AND MASAHIKO EGAMI collateralized debt obligations (CDOs). For a detailed description and the valuation of CDOs, we refer the reader to Duffie and Gârleanu 11, Goodman and Fabozzi 18, Egami and Esteghamat 13 and Hull and White 14. Briefly, a CDO is a derivative security on a portfolio of bonds, loans, or other credit risky claims. Cash flows from a collateral portfolio are divided into various quality/yield tranches which are then sold to investors. In our setting, for example, the times of the (downward) jumps in the portfolio value process can be thought as the default times of individual bonds in the portfolio. The difference between the real-world CDOs and our setting is that a CDO has a pre-determined maturity while we assume an infinite time horizon. However, a typical CDO contract has a term of 1-15 years (much longer than, for example, finite-maturity American-type stock options) and is often extendable with the investors consent. Hence our perpetuality assumption is a reasonable approximation of the reality. We believe that our analysis is also applicable in certain other financial and real-options settings with no fixed maturity, e.g., open-end mutual funds, outsourcing the maintenance of computing, printing or internet facilities in a company or in a university. To find the solutions of the asset manager s aforementioned problems, we first model them as optimal stopping problems for a suitable jump-diffusion process under a risk-neutral probability measure. By separating the jumps from the diffusion part by means of a suitable dynamic programming operator, similarly to the approach used by Dayanik, Poor, and Sezer 8 and Dayanik and Sezer 9 for the solutions of sequential statistics problems, we solve the the optimal stopping problems by means of successive approximations, which not only lead to accurate and efficient numerical algorithms but also allow us to establish concretely the form of optimal stopping strategies. Without any protection, the optimal rule of the asset manager turns out to terminate the contract if the market value of the portfolio X becomes too small or too large; i.e., as soon as X exits an interval (a, b) for some suitable constants < a < b <. In the presence of limited protection (provided to the investors by the asset manager for a fee) at some level l (a, L, it is optimal for the asset manager to terminate the contract as soon as the value X of the portfolio exits an interval (l, m) for some suitable m l, b). Namely, if the protection is binding, i.e., l (a, b), then the asset manager s optimal continuation region shrinks. In other words, investors can have limited protection only if they are also willing to give up in part from the upside-potential of their managed portfolio. Total protection (i.e, the case l = L) wipes out the upside-potential completely since the optimal strategy of the asset manager becomes stop immediately in this extreme case (i.e., l = m = L). Incidentally, a contract with a protection at some level is less valuable than an identical contract without a protection. The difference between these two values gives the fair price of the investors protection. The investors must pay this difference to the asset manager in order to compensate for the asset manager s lost potential revenues due to suboptimal termination of the contract in the presence of the protection. In other words, the asset manager will be willing to provide the protection only if the difference between the expected total revenues with and without the protection is cleared by the investors.

3 OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 3 Our model also sheds some light on the default timing problem of a single firm. Note that the lower boundary l of the optimal continuation region in the first problem s solution may be interpreted as the optimal default time of a CDO. Instead of the value of a portfolio, if X represents the market value of a firm subject to unexpected bad news (downward jumps), then the asset manager s first problem and its solution translate into the default and sale timing problem of the firm and its solution. An action (default or sale) is optimal if the value X of the firm leaves the optimal continuation region (a, b). It is optimal to default if X reaches (, a, and optimal to sell the firm if X reaches b, ). Our solution extends the work of Duffie 1, Chapter 11 who calculates (based on the paper by Leland 17) the optimal default time for a single firm whose asset value is modeled by a geometric Brownian motion. Let us also mention that optimal stopping problems (especially, pricing American-type options) for Lévy processes have been extensively studied; see, for example, Chan 3, Pham 21, Mordecki 2, 19, Boyarchenko and Levendroskiǐ 2, Kou and Wang 16 and Asmussen et al. 1. The problems are formulated in Section 2. The solutions of first and second problems are studied in Sections 3 and 4, respectively. Numerical algorithms are described in Section 5 and illustrated on a numerical example in Section The problem description Let (Ω, F, P) be a probability space hosting a Brownian motion B = {B t, t } and an independent Poisson process N = {N t, t } with the constant arrival rate λ, both adapted to some filtration F = {F t } t satisfying usual conditions. An asset manager borrows L dollars from some investors and invests in some risky asset X = {X t, t }. The process X has the dynamics (2.1) dx t X t = (µ δ)dt + σdb t y dn t λdt, t for some constants µ R, σ >, δ > and y (, 1). We denote by δ the dividend rate or the management fee received by the asset manager. Note that the absolute value of relative jump sizes are equal to y, and the jumps are downwards. Therefore, the asset price X t = X exp {(µ δ 12 ) } σ2 + λy t + σb t (1 y ) Nt, t. is a geometric Brownian motion subject to downward jumps with constant relative jump sizes. An interesting example of our setting is a portfolio of defaultable bonds as in the collateralized debt obligations. Let X t be the value of a portfolio of k defaultable bonds. After every default, the portfolio loses y percent of its market value. The default times of each bond constitutes a Poisson process with the intensity rate λ i independent of others. Therefore, defaults occur at the rate λ k i=1 λ i at the level of the portfolio. The loss ratio upon a default is the same constant y across the bonds. The defaulted bond is immediately sold at the market, and a bond with a similar default rate is bought using the sales proceeds. Under this assumption, defaults occur at the fixed rate λ because the number of bonds in the portfolio is maintained at k. Egami and Esteghamat

4 4 SAVAS DAYANIK AND MASAHIKO EGAMI 13 showed that the dynamics in (2.1) are a good approximation of the dynamics of the aggregate value of individual defaultable bonds when priced in the intensity-based modeling framework (see, e.g., Duffie and Singleton 12). The jump size y on the portfolio level has to be calibrated. Suppose that the asset manager pays the investors a coupon of c percent on the face value of the initial borrowing L on a continuously compounded basis. We assume c < δ. The asset manager has the right to terminate the contract at any time τ R + and receive (X τ L) +. Dividend and coupon payments to the parties cease upon the termination of the contract. Let < r < c be the risk-free interest rate, and S be the collection of all F-stopping times. The asset manager s first problem is to find her maximum expected discounted total income τ (2.2) U(x) sup E γ x e rτ (X τ L) + + e rt (δx t cl)dt, x R + τ S and a stopping time τ S which attains the supremum (if such τ exists) under the condition < r < c < δ. In (2.2), the expectation E γ is taken under the equivalent martingale measure P γ for a specified market price γ of the jump risk. In the real CDOs, the dividend payment is often subordinated to the coupon payment. But since we allow the possibility that the asset manager s net running cash flow δx t cl becomes negative, our formulation has more stringent requirement on the asset manager than a simple subordination. In the asset manager s second problem, the investors assets have limited protection. In the presence of the limited protection at level l >, the contract terminates at time τ (l, ) inf{t : X t (l, )} automatically. The asset manager wants to maximize her expected total discounted earnings as in (2.2), but now the supremum has to be taken over all F-adapted stopping times τ S which are less than or equal to τ (l, ) almost surely. 3. The solution of the asset manager s first problem In the no-arbitrage pricing framework, the value of a contract contingent on the asset X is the expectation of the total discounted payoff of the contract under some equivalent martingale measure. Since the dynamics of X in (2.1) contain jumps, there are more than one equivalent martingale measure. The restriction to F t of every equivalent martingale measure P γ in a large class admits a Radon-Nikodym derivative in the form of dp γ dη t (3.1) dp η t and = βdb t + (γ 1)dN t λdt, t (η = 1), Ft η t which has the solution η t = exp { βb t 1 2 β2 t + N t log γ (γ 1)λt }, t for some constants β R and γ >. The constants β and γ are known as the market price of the diffusion risk and the market price of the jump risk, respectively, and satisfy the drift condition (3.2) γ > and µ r + σβ λy (γ 1) =.

5 OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 5 Then the discounted value process {e (r δ)t X t : t } before the dividends are paid is a (P γ, F)- martingale; see, e.g., Pham 21, Colwell and Elliott 4, Cont and Tankov 5. Girsanov theorem implies that B γ t B t βt, t is a standard Brownian motion, and N t, t is a homogeneous Poisson process with intensity λγ independent of B γ under the new measure P γ. The dynamics of X can be rewritten as (3.3) dx t X t = µ δ + βσ λy (γ 1)dt + σdb γ t y dn t λγdt, = (r δ)dt + σdb γ t y dn t λγdt, t, where the equality µ δ + βσ λy (γ 1) = r δ follows from the drift condition in (3.2). Using Itô s rule, one can also easily verify that (3.4) X t = X exp {(r δ 12 σ2 + λγy ) t + σb γ t } (1 y ) Nt, t. The infinitesimal generator of the process X under the probability measure P γ coincides with the second order differential-difference operator (3.5) (A γ f)(x) (r δ + λγy ) x f (x) σ2 x 2 f (x) + λγf(x(1 y )) f(x) on the collection of twice-continuously differentiable functions f( ). Because {e (r δ)t X t, t } is a martingale under P γ, we have E γ x δx t e rt dt = δxe δt dt = x, and for every stopping time τ S, the strong Markov property implies that E γ x τ δx te rt dt = E γ x δx t e rt dt E γ x τ = x E γ x δx t e rt dt = x E γ x e rτ ( ) e rτ E γxτ δx s e rs ds δx τ+s e rs ds = x E γ x e rτ X τ, x R+. Because E γ x τ cle rt dt = cl r Eγ x cl r e rτ for every τ S and x R +, we can rewrite the asset manager s first problem in (2.2) as (3.6) U(x) = V (x) + x cl r, x R +, where (3.7) ( V (x) sup E γ x e rτ (X τ L) + X τ + cl τ S r ), x R +. is a discounted optimal stopping problem with the terminal reward function (3.8) h(x) (x L) + x + cl r, x R +. We fix the market price γ of jump risk, and the market price β is determined by the drift condition in (3.2). In the remainder, we shall describe the solution of the optimal stopping problem (3.7). Let T 1, T 2,... be the arrival times of process N. Observe that X Tn+1 = (1 y )X Tn+1 and ) } X Tn+t = exp {(r δ + λγy σ2 t + σ(b γ T X Tn 2 n+t Bγ T n ), t < T n+1 T n, n 1.

6 6 SAVAS DAYANIK AND MASAHIKO EGAMI Le us define for every n the standard Brownian motion B γ,n t := B γ T n+t Bγ T n, t and Poisson process T (n) k := T n+k T n, k, respectively, under P γ and the one-dimensional diffusion process (3.9) which has dynamics Y y,n t y exp {(r δ + λγy σ2 2 ) t + σb γ,n t }, t, (3.1) Y y,n = y and dy y,n t = Y y,n t (r δ + λγy )dt + σdb γ,n t, t and infinitesimal generator (under P γ x) (3.11) (A γ f)(y) = σ2 y 2 2 f (y) + (r δ + λγy )yf (y) acting on twice-continuously differentiable functions f : R + R. Then X coincides with Y X Tn,n on T n, T n+1 ) and jumps to (1 y )Y X Tn,n T n+1 T n at time T n+1 for every n ; namely, Y X Tn,n t, t < T n+1 T n, X Tn+t = (1 y )Y X Tn,n T n+1 T n, t = T n+1 T n. For n =, we shall write Y y, Y y = y exp { (r δ λγy σ 2 /2)t + σb γ t } and Y X, Y X A dynamic programming operator. Let S B denote the collection of all stopping times of the diffusion process Y X, or equivalently, Brownian motion B. Let us take any arbitrary but fixed stopping time τ S B and consider the following stopping strategy toward the solution of (3.7): (i) on {τ < T 1 } stop at time τ, (ii) on {τ T 1 }, update X at time T 1 to X T1 = (1 y )Y X T 1 and continue optimally thereon. The value of this new strategy is E γ xe rτ h(x τ )1 {τ<t1 } + e rt 1 V (X T1 )1 {τ T1 } and equals E γ x e rτ h(y X τ )1 {τ<t1 } + e rt 1 V ((1 y )Y X T 1 )1 {τ T1 } = E γ x e (r+λγ)τ h(y X τ ) + If for every bounded function w : R + R + we introduce the operator τ e (r+λγ)τ h(y X (3.12) τ ) + (Jw)(x) sup τ S B E γ x τ λγe (r+λγ)t V ((1 y )Y X t )dt. λγe (r+λγ)t w((1 y )Y X t )dt, x, then we expect that the value function V ( ) of (3.7) to be the unique fixed point of operator J; namely, V ( ) = (JV )( ), and that V ( ) is the pointwise limit of the successive approximations v (x) h(x) = (x L) + x + cl r, x, v n (x) (Jv n 1 )(x), x, n 1. Lemma 1. Let w 1, w 2 : R + R be bounded. If w 1 ( ) w 2 ( ), then (Jw 1 )( ) (Jw 2 )( ). If w( ) is nonincreasing convex function such that h( ) w( ) cl/r, then (Jw)( ) has the same properties.

7 OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 7 The proof easily follows from the linearity of y Y y t for every fixed t and the definition of the operator J. The next proposition guarantees the existence of unique fixed point of J. Proposition 2. For every bounded w 1, w 2 : R + R, we have Jw 1 Jw 2 λγ r+λγ w 1 w 2, where w = sup x R+ w(x) ; namely, J acts as a contraction mapping on the bounded functions. Proof. Because w 1 ( ), w 2 ( ) are bounded, (Jw 1 )( ) and (Jw 2 )( ) are finite, and for every ε and x >, there are ε-optimal stopping times τ 1 (ε, x) and τ 2 (ε, x), which may depend on ε and x, such that (Jw i )(x) ε E γ x e (r+λγ)τ i(ε,x) h(y X τ i (ε,x) ) + τi (ε,x) λγe (r+λγ)t w i ((1 y )Y X t )dt, i = 1, 2. Therefore, (Jw 1 )(x) (Jw 2 )(x) ε + w 1 w 2 λγe (r+λγ)t dt = ε + w 1 w 2 λγ r+λγ. Interchanging the roles of w 1 ( ) and w 2 ( ) gives (Jw 1 )(x) (Jw 2 )(x) ε + w 1 w 2 λγ r+λγ for every x > and ε >. Taking the supremum of both sides over x completes the proof. Lemma 3. The sequence (v n ) n of successive approximations is nondecreasing. Therefore, the pointwise limit v (x) lim n v n (x), x exists. Every v n ( ), n and v ( ) are nonincreasing, convex, and bounded between h( ) and cl/r. Lemma 3 follows from repeated applications of Lemma 1. Proposition 4 below shows that the unique fixed point of J is the uniform limit of successive approximations. Proposition 4. The limit v ( ) = lim n v n ( ) = sup n v n ( ) is the unique bounded fixed point of operator J. Moreover, v (x) v n (x) cl r ( λγ r+λγ )n for every x. Proof. Since v n ( ) v ( ) as n, and every v n ( ) is bounded from below by c r r L, and E γ τ c r e (r+λγ)t r Ldt < for every τ S B, the monotone convergence theorem implies that v (x) = sup v n (x) = sup n τ S B = sup τ S B E γ x lim n Eγ x e (r+λγ)τ h(y X τ ) + e (r+λγ)τ h(y X τ τ ) + τ λγe (r+λγ)t v ((1 y )Y X t )dt λγe (r+λγ)t v n ((1 y )Y X t )dt = (Jv )(x). Thus, v ( ) is the bounded fixed point of contraction mapping J. Lemma 3 implies v ( ) v n ( ), and v v n = Jv Jv n 1 λγ r+λγ v v n 1... ( λγ r+λγ )n cl r for every n The solution of the optimal stopping problem in (3.12). We shall next solve the optimal stopping problem Jw in (3.12) for every fixed w : R + R which satisfies the following assumption: Assumption 5. Let w : R + R be nonincreasing, convex, bounded between h( ) and cl/r, and w(+ ) = c r r L and w(+) = c r L. We shall calculate the value function (Jw)( ) and explicitly identify an optimal stopping rule. Because w( ) is bounded, we have E γ x e (r+λγ)t w((1 y )Y X t ) dt w e (r+λγ)t dt = w r + λγ < x,

8 8 SAVAS DAYANIK AND MASAHIKO EGAMI and for every stopping time τ S B, the strong Markov property of Y X at time τ implies that (Hw)(x) E γ x e (r+λγ)t w((1 y )Y X (3.13) t )dt τ = E γ x e (r+λγ)t w((1 y )Y X t )dt + E γ x e (r+λγ)τ (Hw)(Y X τ ). Therefore, E γ x τ e (r+λγ)t w((1 y )Y X t )dt = (Hw)(x) E γ xe (r+λγ)τ (Hw)(Y X τ ), and we can write the expected payoff E γ xe (r+λγ)τ h(y X τ ) + τ λγe (r+λγ)t w((1 y )Y X t )dt in (3.12) as λγ(hw)(x) + E γ x e (r+λγ)τ {h λγ(hw)} (Y X τ ) for every τ S B and x >. If we define (Gw)(x) sup E γ x e (r+λγ)τ {h λγ(hw)} (Y X (3.14) τ ), x >, τ S B then the value function in (3.12) can be calculated by (3.15) (Jw)(x) = λγ(hw)(x) + (Gw)(x), x >. Let us first calculate (Hw)( ). Let ψ( ) and ϕ( ) be, respectively, the increasing and decreasing solutions of the second order ordinary differential equation (A f)(y) (r + λγ)f(y) =, y > with boundary conditions, respectively, ψ(+) = and ϕ(+ ) =, where A is the infinitesimal generator in (3.11) of diffusion process Y X Y X,. One can easily check that (3.16) ψ(y) = y α 1 and ϕ(y) = y α for every y >, with the Wronskian (3.17) W (y) = ψ (y)ϕ(y) ψ(y)ϕ (y) = (α + α 1 )y α +α 1 1, y >, where α < α 1 are the roots of the characteristic function g(α) = σ2 2 α(α 1) + (r δ + λγy )α (r + λγ) of the above ordinary differential equation. Because both g() < and g(1) <, we have α < < 1 < α 1. Let us denote the hitting and exit times of diffusion process Y X, respectively, by and define operator (3.18) and τ a inf{t ; Y X t = a}, a >, τ ab inf{t ; Y X t (a, b)}, < a < b <, (H ab w)(x) E γ x τab e (r+λγ)t w((1 y )Y X t )dt + 1 {τab < }e (r+λγ)τ ab h(y X τ ab ) ψ a (y) ψ(y) ψ(a) ϕ(a) ϕ(y) and ϕ b(y) ϕ(y) ϕ(b) ψ(y) for every y >, ψ(b)

9 OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 9 which are, respectively, the increasing and decreasing solutions of (A f)(y) (r + λγ)f(y) =, a < y < b with boundary conditions, respectively, f(a) = and f(b) =. The Wronskian of ψ a ( ) and ϕ a ( ) becomes W ab (y) = ψ a(y)ϕ b (y) ψ a (y)ϕ b (y) = 1 ψ(a) ϕ(b) (3.19) W (y), y > ϕ(a) ϕ(b) in terms of the Wronskian W ( ) in (3.17) of ψ( ) and ϕ( ). Lemma 6. We have (i) E γ x e (r+λγ)τ a 1 {τa<τb } = ϕ b (x) ϕ b (a) for every < a x b <. (ii) E γ x e (r+λγ)τ b 1 {τa>τb } = ψ b (x) ψ b (a) for every < a x b <. (iii) E γ x e (r+λγ)τ ab h(y X τ ab )1 {τab < } = ϕ b (x) ψa(x) ϕ b (a) h(a) + ψ a(b) h(b) for every < a x b <. All three expectations are twice continuously differentiable on (a, b) and unique such solution of the ordinary differential equation (A f)(y) (r +λγ)f(y) =, y (a, b) subject to boundary conditions (i) f(a) = 1, f(b) =, (ii) f(a) =, f(b) = 1, (iii) f(a) = h(a), f(b) = h(b), respectively. Lemma 7. For every bounded function g : R + R and < a x b <, we have τab (3.2) E γ x e (r+λγ)t g(y X t )dt + 1 {τab < }e (r+λγ)τ ab h(y X τ ab ) = b a = ϕ b (x) 2ψ a (x ξ)ϕ b (x ξ)g(ξ) p 2 dξ + ϕ b(x) (ξ)w ab (ξ) ϕ b (a) h(a) + ψ a(x) ψ a (b) h(b) x a 2ψ a (ξ)g(ξ) p 2 (ξ)w ab (ξ) dξ + ψ a(x) b x 2ϕ b (ξ)g(ξ) p 2 (ξ)w ab (ξ) dξ + ϕ b(x) ϕ b (a) h(a) + ψ a(x) ψ a (b) h(b), which is twice-continuously differentiable on (a, b) and uniquely solves the boundary value problem (A f)(y) (r + λγ)f(y) + g(y) =, a < y < b with f(a) = h(a) and f(b) = h(b), where p(ξ) = σξ and q(ξ) = (r δ + λγy )ξ are the diffusion and drift coefficients of diffusion Y in (3.1). Corollary 8. We have (H ab w)(x) = ϕ b (x) x a 2ψ a (ξ)w((1 y )ξ) b p 2 dξ + ψ a (x) (ξ)w ab (ξ) x + ϕ b(x) ϕ b (a) h(a) + ψ a(x) h(b), < a x b <, ψ a (b) 2ϕ b (ξ)w((1 y )ξ) p 2 dξ (ξ)w ab (ξ) which is twice-continuously differentiable on (a, b) and uniquely solves the boundary value problem (A f)(x) (r + λγ)f(x) + w((1 y )x) =, a < x < b with f(a) = h(a) and f(b) = h(b). The proofs of Lemmas 6 and 7 can be checked by direct calculation and Itô s lemma; see also Karlin and Taylor 15, Chapter 15, and Corollary 8 immediately follows from Lemma 6 and 7. Finally, Lemma 9 follows from Corollary 8 by passing to limit as a and b because and are natural boundaries of diffusion Y X.

10 1 SAVAS DAYANIK AND MASAHIKO EGAMI Lemma 9. For every x >, we have (Hw)(x) E γ x e (r+λγ)t w((1 y )Y X t )dt = lim (H abw)(x) a,b = ϕ(x) x 2ψ(ξ)w((1 y )ξ) p 2 dξ + ψ(x) (ξ)w (ξ) x 2ϕ(ξ)w((1 y )ξ) p 2 dξ, (ξ)w (ξ) which is twice-continuously differentiable on R + and satisfies the ordinary differential equation (A f)(x) (r + λγ)f(x) + w((1 y )x) =. Using the potential theoretic direct methods of Dayanik and Karatzas 7 and Dayanik 6, we shall now solve the optimal stopping problem (Gw)( ) (3.14) with payoff function (h λγ(hw))(x) = (x L) + x + cl x ϕ(x) r λγ = (x L) + x+ cl r 2λγ σ 2 (α 1 α ) x α dξ + ψ(x) 2ψ(ξ)w((1 y )ξ) 2ϕ(ξ)w((1 y )ξ) p 2 (ξ)w (ξ) x p 2 dξ (ξ)w (ξ) x ξ 1 α w((1 y )ξ)dξ+x α 1 ξ 1 α 1 w((1 y )ξ)dξ, where ψ(x) = x α 1, ϕ(x) = x α, p 2 (ξ) = σ 2 ξ 2, W (ξ) = ψ (ξ)ϕ(ξ) ψ(ξ)ϕ (ξ) = (α 1 α )ξ α +α 1 1. We observe that (Hw)(x) = E γ x e (r+λγ)t w((1 y )Y X t )dt cl r e (r+λγ)t dt = <. Hence, (h λγ(hw))( ) is bounded, and because ψ(+ ) = ϕ(+) = +, we have cl r(r+λγ) lim sup x (h λγ(hw)) + (x) ϕ(x) = and lim sup x x (h λγ(hw)) + (x) ψ(x) By Propositions 5.1 and 5.13 of Dayanik and Karatzas 7, value function (Gw)( ) is finite; the set =. (3.21) Γw {x > ; (Gw)(x) = (h λγ(hw))(x)} = {x > ; (Jw)(x) = h(x)} is the optimal stopping region, and (3.22) τw inf{t ; Y X t Γw} is an optimal stopping time for (3.14) and for (3.12) because of (3.15). According to Proposition 5.12 of Dayanik and Karatzas 7, we have (Gw)(x) = ϕ(x)(mw)(f (x)), x, and Γw = F 1 ({ζ > ; (Mw)(ζ) = (Lw)(ζ)}), where F (x) ψ(x)/ϕ(x) and (Mw)( ) is the smallest nonnegative concave majorant on R + of h λγ(hw) F 1 (ζ), ζ >, (3.23) (Lw)(ζ) ϕ, ζ =. To describe explicitly the form of the smallest nonnegative concave majorant (M w)( ) of (Lw)( ), we shall firstly identify a few useful properties of function (Lw)( ). Because Y X X Y 1 by (3.9) and w( ) is bounded, the bounded convergence theorem implies that lim (Hw)(x) = x Eγ 1 e (r+λγ)t lim w((1 y )xyt 1 )dt = w(+ ) x r + λγ cl r,

11 OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 11 and lim x (h λγ(hw))(x) = lim x ((x L) + x + cl r (3.24) Note also that (Lw) (ζ) = d dζ (h λγ(hw))(x) (Lw)(+ ) = lim = +. x ϕ(x) ( h λγ(hw) ϕ ) 1 F 1 (ζ) = F c r λγ(hw)(x)) r+λγ L >. Therefore, ( h λγ(hw) ϕ ) F 1 (ζ). Because F ( ) is strictly increasing, we have F >. Because w( ) is nonincreasing, the mapping x E γ x e (r+λγ)t w((1 y )Y X t )dt = E γ x e (r+λγ)t w((1 y )X Yt 1 )dt is decreasing. Then for x > L, because h( ) cl/r is constant, the mapping x ( h λγ(hw) ϕ )(x) is increasing. For every < x < L, we can calculate explicitly that 1 F ( h λγ(hw) ϕ ) (x) = x α 1 ( α ) cl α 1 α r (1 α )x λγ( α )α 1 x α 1 ξ 1 α 1 w((1 y )ξ)dξ, r + λγ and because lim x x α 1 x ξ 1 α 1 w((1 y )ξ)dξ = w(+) α 1 ( h λγ(hw) lim x 1 F ϕ x and α 1 > 1, we have ) (x) = +. Let us also study the sign of the second derivative (Lw) ( ). For every x L, Dayanik and Karatzas 7, page 192 show that (3.25) (Lw) (F (x)) = and ϕ( ), p 2 ( ), W ( ), F ( ) are positive. Therefore, 2ϕ(x) p 2 (x)w (x)f (x) (A (r + λγ))(k λγ(hw))(x) sgn(lw) (F (x)) = sgn(a (r + λγ))(h λγ(hw))(x). Recall from Lemma 9 that (A (r + λγ))(hw)(x) = w((1 y )x) and because h(x) = ( x + cl r )1 {x<l} + (c r)l r 1 {x>l}, we have (A (r + λγ))(h λγ(hw))(x) = λγ(1 y )x (r + λγ) cl r + λγw((1 y )x) 1 {x L} (c r)l + λγw((1 y )x) (r + λγ) 1 r {x>l}. Note that lim x (A (r + λγ))(h λγ(hw))(x) = cl < and lim x (A (r + λγ))(h λγ(hw))(x) = (c r)l <. Note also that x (A (r + λγ))(h λγ(hw))(x) is convex and continuous on x (, L) and x (L, ). Therefore, (A (r + λγ))(h λγ(hw))(x) is strictly negative in some open neighborhoods of and +, and in the complement of their unions, whose closure contains L if it is not empty, it is nonnegative. Therefore, (3.25) implies that (Lw)(ζ) is strictly concave in some neighborhood of ζ = and ζ =, and in the complement of their unions, whose closure contains F (L) if it is not empty, this function is convex. Earlier we also showed that ζ (Lw)(ζ) is increasing at every ζ > F (L) and (Lw)(+ ) = (Lw) (+) = +. Moreover, (Lw) (F (L) ) (Lw) (F (L)+) = L1 α 1 α 1 α < ;

12 12 SAVAS DAYANIK AND MASAHIKO EGAMI (Mw)(ζ) (Mw)(ζ) (Lw)(ζ) (Lw)(ζ) ζ 1 w F (L) ζ 2 w ζ ζ 1 w F (L) ζ 2 w ζ concave concave concave convex concave Figure 1. The sketches of two possible forms of (Lw)( ) and their smallest nonnegative concave majorants (M w)( ). namely, (Lw) (F (L) ) < (Lw) (F (L)+). Two possible forms of ζ (Lw)(ζ) and their smallest nonnegative concave majorants ζ (M w)(ζ) are depicted by two pictures of Figure 1. The properties of the mapping ζ (Lw)(ζ) imply that there are unique numbers < ζ 1 w < F (L) < ζ 2 w < such that (Lw) (ζ 1 w) = (Lw)(ζ 2w) (Lw)(ζ 1 w) ζ 2 w ζ 1 w = (Lw) (ζ 2 w), and the smallest nonnegative concave majorant (Mw)( ) of (Lw)( ) on (, ζ 1 w ζ 2 w, ) coincides with (Lw)( ), and on (ζ 1 w, ζ 2 w) with the straight-line that majorizes (Lw)( ) everywhere on R + and is tangent to (Lw)( ) exactly at ζ = ζ 1 w and ζ 2 w; see Figure 1. More precisely, (Lw)(ζ), (Mw)(ζ) = ζ 2 w ζ ζ 2 w ζ 1 w (Lw)(ζ 1w) + ζ ζ 1w ζ 2 w ζ 1 w (Lw)(ζ 2w), ζ (, ζ 1 w ζ 2 w, ), ζ (ζ 1 w, ζ 2 w). Let us define x 1 w F 1 (ζ 1 w) and x 2 w F 1 (ζ 2 w). Then by Proposition 5.12 of Dayanik and Karatzas 7, the value function of the optimal stopping problem in (3.14) equals (3.26) (Gw)(x) = ϕ(x)(mw)(f (x)) (h λγ(hw))(x), x (, x 1 w x 2 w, ), = (x 2 w) α 1 α x α 1 α (x 2 w) α 1 α (x1 w) α 1 α (h λγ(hw))(x 1 w) + xα 1 α (x 1 w) α 1 α (x 2 w) α 1 α (x1 w) α 1 α (h λγ(hw))(x 2 w), x (x 1 w, x 2 w).

13 OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 13 The optimal stopping region in (3.21) becomes Γw = {x > ; (Gw)(x) = (h λγ)(hw)(x)} = (, x 1 w x 2 w, ), and the optimal stopping time in (3.22) becomes τw = inf{t ; Y X t (, x 1 w x 2 w, )}. Proposition 1. The value function x (Gw)( ) of (3.14) is continuously differentiable on R + and twice-continuously differentiable on R + \ {x 1 w, x 2 w}. Moreover, (Gw)( ) satisfies (i) (A (r + λγ))(gw)(x) =, x (x 1 w, x 2 w), (ii) (Gw)(x) > h(x) λγ(hw)(x), x (x 1 w, x 2 w), (iii) (A (r + λγ))(gw)(x) <, x (, x 1 w) (x 2 w, ), (iv) (Gw)(x) = h(x) λγ(hw)(x), x (, x 1 w x 2 w, ). The differentiability of (Gw)( ) is clear from (3.26). The variational inequalities can be verified directly. For (iii) note that, if x (, x 1 w) (x 2 w, ), then sgn{(a (r + λγ))(gw)(x)} = sgn{(a (r + λγ))(h λγ(hw))(x)} = sgn{(lw) (F (x))} <. Because (Hw)( ) is twice-continuously differentiable and (A (r+λγ)(hw))(x) = w((1 y )x) for every x > by Proposition 9, Proposition 1 and (3.15) lead directly to the next proposition. Proposition 11. The value function x (Jw)( ) of (3.12) is continuously differentiable on R + and twice-continuously differentiable on R + \ {x 1 w, x 2 w}. Moreover, (Jw)( ) satisfies (i) (A (r + λγ))(jw)(x) + λγw((1 y )x) =, x (x 1 w, x 2 w), (ii) (Jw)(x) > h(x), x (x 1 w, x 2 w), (iii) (A (r + λγ))(jw)(x) + λγw((1 y )x) <, x (, x 1 w) (x 2 w, ), (iv) (Jw)(x) = h(x), x (, x 1 w x 2 w, ). By Lemma 3, every v n ( ), n and v ( ) are nonincreasing, convex, and bounded between h( ) and cl/r. Moreover, by using induction on n, we can easily show that v n (+) = cl/r and v n (+ ) = (c r)l/r for every n {, 1,..., }. Therefore, Proposition 11, applied to w = v, and Proposition 4 directly lead to the next theorem. Theorem 12. The function x v (x) = (Jv )(x) is continuously differentiable on R + and twice-continuously differentiable on R + \ {x 1 v, x 2 v } and satisfies the variational inequalities (i) (A (r + λγ))v (x) + λγv ((1 y )x) =, x (x 1 v, x 2 v ), (ii) v (x) > h(x), x (x 1 v, x 2 v ), (iii) (A (r + λγ))v (x) + λγv ((1 y )x) <, x (, x 1 v ) (x 2 v, ), (iv) v (x) = h(x), x (, x 1 v x 2 v, ),

14 14 SAVAS DAYANIK AND MASAHIKO EGAMI which can be expressed in terms of the generator A γ in (3.5) of the jump-diffusion process X as (i) (A γ r)v (x) =, x (x 1 v, x 2 v ), (ii) v (x) > h(x), x (x 1 v, x 2 v ), (iii) (A γ r)v (x) <, x (, x 1 v ) (x 2 v, ), (iv) v (x) = h(x), x (, x 1 v x 2 v, ). The next theorem identifies the value function and an optimal stopping time for the optimal stopping problem in (3.7). For every w : R + R satisfying Assumption 5 let us denote by τw the stopping time of jump-diffusion process X defined by τw inf{t ; X t (, x 1 w x 2 w, )}. Theorem 13. For every x R +, we have V (x) = v (x) = E γ x e r τv h(x τv ), and τv is an optimal stopping time for (3.7). Proof. Let τ ab = inf{t ; X t (, a b, )} for every < a < b <. By Itô s rule, we have t τ τab e r(t τ τ ab) v (X t τ τab ) = v (X ) + e rs (A γ r)v (X s )ds t τ τab t τ τab + e rs v (X s )σx s dbs γ + e rs v ((1 y )X s ) v (X s )(dn s λγds) for every t, τ S, and < a < b <. Because v ( ) and v ( ) are continuous and bounded on every compact subinterval of (, ), both stochastic integrals are square-integrable martingales, and taking expectations of both sides gives (3.27) E γ xe r(t τ τ ab) v (X t τ τab ) = v (x) + E γ x t τ τ ab e rs (A γ r)v (X s )ds. Because (A γ r)v ( ) and v ( ) h( ) by the variational inequalities of Theorem 12, we have E γ e r(t τ τ ab) v (X t τ τab ) v (x) for every t, τ S, and < a < b <. Because lim a,b τ ab = a.s. and h( ) is continuous and bounded, we can take limits of both sides as t, a, b and use the bounded convergence theorem to get E γ e rτ v (X τ ) v (x) for every τ S. Taking supremum over all τ S gives V (x) = sup τ S E γ e rτ v (X τ ) v (x). In order to show the reverse inequality, we replace in (3.27) τ and τ ab with τv. (A γ r)v (x) = for every x (x 1 v, x 2 v ) by Theorem 12 (i), we have E γ xe r(t τv ) v (X t τv ) = v (x) + E γ x t τv e rs (A γ r)v (X s )ds = v (x) Because for every t. Since v is bounded and continuous, taking limits as t and the bounded convergence theorem gives v (x) = E γ xe r τv v (X τv ) = E γ xe r τv h(x τv ) V (x) by Theorem 12 (iv), which completes the proof.

15 OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 15 Proposition 14. The optimal stopping regions Γv n = {x > ; (Jv n )(x) h(x)} = (, x 1 v n x 2 v n, ), n {, 1,..., } are decreasing; namely, Γv Γv 1... Γv, and < x 1 v... x 1 v 1 x 1 v L x 2 v x 2 v 1... x 2 v <. Moreover, x 1 v = lim n x 1 v n and x 2 v = lim n x 2 v n. The proof follows from the monotonicity of operator J and that v n (x) v (x) as n uniformly in x >. The next proposition and its corollary identify the optimal expected reward and nearly optimal stopping strategies for the asset manager in the first problem. Proposition 15. For all n, we have v (x) E γ xe r τvn h(x τvn) + cl r ( r+λγ )n+1. Hence, for every ε > and n such that cl r ( λ r+λγ )n+1 ε, the stopping time τv n is ε-optimal for (3.7). Proof. Recall that τv n = inf{t ; X t Γv n } = inf{t ; X t (, x 1 v n x 2 v n, ). If we replace τ and τ ab in (3.27) with τv n, then for every t we obtain E γ xe r(t τvn) v (X t τvn) = v (x) + E γ x t τv n e rs (A γ r)v (X s )ds = v (x), because, for every < t < τv n we have X t (x 1 v n, x 2 v n ) (x 1 v, x 2 v ), at every element x of which (A γ r)v (x) equals according to 12 (i). Because v ( ) is continuous and bounded, taking limits as t and the bounded convergence theorem give v (x) = E γ xe r τvn v (X τvn). By Proposition 4, v (x) E γ x e r τvn( v n+1 (X τvn) + cl r + cl r ( λγ r + λγ ( λγ ) n+1 ) E γ x r + λγ ) n+1 = E γ x e r τvn( h(x τvn) e r τvn( ) (Jv n )(X τvn) λ ) + cl r ( λγ ) n+1, r + λγ because (Jv n )( ) = h( ) on Γv n X τvn on { τv n < }. Corollary 16. The maximum expected reward of the asset manager is given by U(x) = x cl r + V (x) = x cl r + v (x) for every x. The stopping rule τv is optimal, and τv n is ε-optimal for every ε > and n such that cl r ( λγ r+λγ )n+1 < ε, in the sense that for every x > U(x) = E γ x U(x) ε E γ x e r τv (X τv L) + + e r τvn (X τvn L) + + τv τvn e rt (δx t cl)dt, e rt (δx t cl)dt. 4. The solution of the asset manager s second problem In the asset manager s second problem, the investors assets have limited protection. In the presence of the limited protection at level l >, the contract terminates at time τ l, inf{t : X t / (l, )} automatically. The asset manager wants to maximize her expected total discounted earnings as in (2.2), but now the supremum has to be taken over all stopping times τ S which

16 16 SAVAS DAYANIK AND MASAHIKO EGAMI are less than or equal to τ l, almost surely. Namely, we would like to solve the problem (4.1) U l (x) sup τ S E γ x e r( τ l, τ) (X τl, τ L) + + τl, τ e rt (δx t cl)dt, x R +. If l < x 1 v, then U l (x) = U(x) = E γ xe r( τv ) (X τl, v L) + + τv e rt (δx t cl)dt for every x >. On the one hand, because for every τ S, τv τ also belongs to S, we have U l (x) U(x). On the other hand, because l x 1 v, we have τv = τ l, τv a.s. and U l (x) E γ x = E γ x e r( τ l, τv ) (X τl, τv L) + + e r τv (X τv L) + + τv Therefore, U l (x) = U(x) for every x > if l x 1 v. τl, τv e rt (δx t cl)dt e rt (δx t cl)dt = U(x) for every x. Assumption 17. In the remainder, we shall assume that the protection level l satisfies the inequalities x 1 v < l L. The strong Markov property of X can be used to similarly show that (4.2) U l (x) = x cl r + V l(x), x, where (4.3) V l (x) sup E γ x e r( τ l, τ) h(x τl, τ ), x > τ S is the discounted optimal stopping problem for the stopped jump-diffusion process X τl, t, t with the same terminal payoff function h( ) as in (3.8). Let us define stopping time τ l, inf{t ; Y X t / (l, )} of diffusion process Y X and the operator (4.4) (J l w)(x) sup E γ x e rτ h(x τl, τ )1 {τl, τ<t 1 } + e rt 1 w(x T1 )1 {τl, τ T 1 } τ S B = sup τ S B E γ x e (r+λγ)(τ l, τ) h(y X τ l, τ ) + τl, τ λγe (r+λγ)t w((1 y )Y X t )dt, x. We expect that V l ( ) = (JV l )( ); namely, that V l ( ) is one of the fixed points of operator J l. We can find one of the fixed points of J l by taking limit of successive approximations defined by v l, (x) h(x) (x L) + x + cl r, x >, v l,n (x) (J l v l,n 1 )(x), x >, n 1. The results of previous section can be adapted to the new problem, and we state only the differences.

17 OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 17 (M l w)(ζ) (Mw)(ζ) (Mw)(ζ) (Lw)(ζ) (M l w)(ζ) (Lw)(ζ) F (l) F (l) ζ l,1 w ζ l,2 w ζ l,1 w ζ 1 w F (L) ζ l,2 w ζ 2 w ζ ζ 1 w F (L) ζ 2 w ζ Figure 2. The sketches of (Lw)( ) and (M l w)( ). On the left, because F (l) ζ 1 w, ζ l,1 w ζ 1 w and ζ l,2 w ζ 2 w and (M l w)( ) (Mw)( ). On the right, because ζ 1 w < F (l) F (L), ζ 1 w < F (l) = ζ l,1 w < ζ l,2 w < ζ 2 w, and (M l w)(x) < (Mw)(x) for every x (ζ 1 w, ζ 2 w). Lemma 18. Let w 1, w 2 : R + R be bounded. If w 1 ( ) w 2 ( ), then (J l w 1 )( ) (J l w 2 )( ). If w( ) is nonincreasing and convex such that h( ) w( ) cl/r, then (J l w)( ) has the same properties. Proposition 19. For every bounded w 1, w 2 : R + R, we have J l w 1 J l w 2 namely, J l acts as a contraction mapping on the collection of bounded functions. Lemma 2. The sequence (v l,n ) n of successive approximations is increasing. pointwise limit v l, (x) = lim n v l,n (x), x > exists. nonincreasing, convex, and bounded between h( ) and cl/r. λγ r+λγ w 1 w 2 ; Therefore, the Every v l,n ( ), n and v l, ( ) are Proposition 21. The limit v l, ( ) = lim n v l,n ( ) = sup n v l,n is the unique bounded fixed point of J l. Moreover, v l, (x) v l,n (x) cl r ( λγ r+λγ )n for every x > and n. Let w : R + R be a function as in Assumption 5. Then (4.5) (J l w)(x) = λγ(hw)(x) + (G l w)(x), x >, where (G l w)( ) is the value function of the discounted optimal stopping problem (4.6) (G l w)(x) sup E γ x e (r+λγ)τl, τ {h λγ(hw)} (Y X τ l, τ ), x >, τ S B for the stopped diffusion process Y X τ l,, t at stopping time τ l,. We obviously have (G l w)(x) = h(x) for every x (, l. If the initial state X of Y X τ l,, t is in (l, ), then l becomes an absorbing left-boundary for the stopped process Y X τ l,, t.

18 18 SAVAS DAYANIK AND MASAHIKO EGAMI Let (M l w)( ) be the smallest concave majorant on F (l), ) of (Lw)( ) defined by (3.23) and equal on (, F (l)) identically to (Lw)( ). Then by Proposition 5.5 of Dayanik and Karatzas 7 (G l w)(x) = ϕ(x)(m l w)(f (x)), x > and Γ l w = F 1 ({ζ > ; (M l w)(ζ) = (Lw)(ζ)}) are value function and optimal stopping region for (4.6). The analysis of the shape of (Lw)( ) prior to Figure 1 implies that there are unique numbers < ζ l,1 w < F (L) < ζ l,2 w < such that (Lw) (ζ l,1 w) = (Lw)(ζ l,2w) (Lw)(ζ l,1 w) = (Lw) (ζ l,2 w) ζ l,2 w ζ l,1 w if F (l) ζ 1 w, namely, ζ l,1 w ζ 1 w and ζ l,2 w ζ 2 w ζ l,1 w = l and (Lw)(ζ l,2w) (Lw)(ζ l,1 w) ζ l,2 w ζ l,1 w = (Lw) (ζ l,2 w) if F (l) > ζ 1 w, and (M l w)(ζ) = (Lw)(ζ), ζ l,2 w ζ ζ l,2 w ζ l,1 w (Lw)(ζ l,1w) + ζ ζ l,1w ζ l,2 w ζ l,1 w (Lw)(ζ l,2w), ζ (, ζ l,1 w ζ l,2 w, ), ζ (ζ l,1 w, ζ l,2 w). Let us define x l,1 w = F 1 (ζ l,1 w) and x l,2 w = F 1 (ζ l,2 w). Then the value function equals (4.7) (G l w)(x) = ϕ(x)(m l w)(f (x)) (h λγ(hw))(x), x (, x l,1 w x l,2 w, ), = (x l,2 w) α 1 α x α 1 α (x l,2 w) α 1 α (xl,1 w) α 1 α (h λγ(hw))(x l,1 w) + x α 1 α (x l,1 w) α 1 α (x l,2 w) α 1 α (xl,1 w) α 1 α (h λγ(hw))(x l,2 w), and the optimal stopping time becomes x (x l,1 w, x l,2 w) (4.8) Γ l w = {x > ; (G l w)(x) = (h λγ(hw))(x)} = (, x l,1 w x l,2 w, ), and an optimal stopping time is given by (4.9) τ l w inf{x > ; Y X t Γ l w} = inf{x > ; Y X t (, x l,1 w x l,2 w, )} for the problem in (4.6). A direct verification together with the chain of equalities sgn{(a (r + λγ))(g l w)(x)} = sgn{(a (r + λγ))(h λγ(hw))(x)} = sgn{(lw) (F (x))} < for every x (l, x l,1 w) (x l,2 w, ) from Dayanik and Karatzas 7, page 192 proves the next proposition. Proposition 22. The value function x (G l w)(x) is continuously differentiable on l, ) and twice-continuously differentiable on l, ) \ {x l,1 w, x l,2 w}. The function (G l w)(x), x l, )

19 OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 19 solves the variational inequalities (i) (A (r + λγ))(g l w)(x) =, x (x l,1 w, x l,2 w), (ii) (G l w)(x) > h(x) λγ(hw)(x), x (x l,1 w, x l,2 w), (iii) (A (r + λγ))(g l w)(x) <, x (l, x l,1 w) (x l,2 w, ), (iv) (G l w)(x) = h(x) λγ(hw)(x), x l, x l,1 w x l,2 w, ). Because (J l w)(x) = λγ(hw)(x) + (G l w)(x) for every x >, (Hw)( ) is twice-continuously differentiable, and (A (r + λγ))(hw)(x) = w((1 y )x) for every x >, the next proposition immediately follows from Proposition 22. Proposition 23. The value function x (J l w)(x) in (4.4) is continuously differentiable on l, ), twice-continuously differentiable on l, ) \ {x l,1 w, x l,2 w}, and satisfies (i) (A (r + λγ))(j l w)(x) + λγw((1 y )x) =, x (x l,1 w, x l,2 w), (ii) (J l w)(x) > h(x), x (x l,1 w, x l,2 w), (iii) (A (r + λγ))(j l w)(x) + λγw((1 y )x) <, x (l, x l,1 w) (x l,2 w, ), (iv) (J l w)(x) = h(x), x l, x l,1 w x l,2 w, ). As in the asset manager s first problem, the successive approximations v l,n ( ), n and their limit v l, ( ) satisfy Assumption 5. Therefore, Propositions 21 and 23 lead to the next theorem. Theorem 24. The function x v l, (x) = (Jv )(x) is continuously differentiable on l, ), twice-continuously differentiable on l, ) \ {x l,1, x l,2 } and satisfies the variational inequalities (i) (A (r + λγ))v l, (x) + λγv l, ((1 y )x) =, x (x l,1 v l,, x l,2 v l, ), (ii) v l, (x) > h(x), x (x l,1 v l,, x l,2 v l, ), (iii) (A (r + λγ))v l, (x) + λγv l, ((1 y )x) <, x (l, x l,1 v l, ) (x l,2 v l,, ), (iv) v l, (x) = h(x), x l, x l,1 v l, x l,2 v l,, ), which can be expressed in terms of the generator A γ in (3.5) of the jump-diffusion process X as (i) (A γ r)v l, (x) =, x (x l,1 v l,, x l,2 v l, ), (ii) v l, (x) > h(x), x (x l,1 v l,, x l,2 v l, ), (iii) (A γ r)v l, (x) <, x (l, x l,1 v l, ) (x l,2 v l,, ), (iv) v l, (x) = h(x), x l, x l,1 v l, x l,2 v l,, ). Note again that the second part follows from the first part and from the equality (A (r + λγ))v l, (x) + λγv l, ((1 y )x) = (A γ r)v l, (x) for every x (l, ) \ {x l,1 v l,, x l,2 v l, }. By the next theorem, optimal stopping time for asset manager s second problem is of the form τ l w inf{t ; X t (, x l,1 w x l,2 w, )}.

20 2 SAVAS DAYANIK AND MASAHIKO EGAMI Theorem 25. For every x R +, we have V l (x) = v l, (x) = E γ x e r τ l v l, h(x τl v l, ), and τ l v l, is an optimal stopping time for (4.3). Since v l, (x) = h(x) = V (x) for every x (, l, Theorem 25 has to be proved on (l, ), which can be done as in the proof of Theorem 13 but with localizing stopping rules τ lb for b > l. The proof of the next proposition is similar to that of Proposition 14. Proposition 26. The optimal stopping regions Γ l v l,n = {x > ; (Jv l,n )(x) h(x)} = (, x l,1 v l,n x l,2 v l,n, ), n {, 1,..., } are decreasing; namely, Γ l v l, Γ l v l,1... Γ l v l,, and < x l,1 v l,... x l,1 v l,1 x l,1 v l, L x l,2 v l, x l,2 v l,1... x l,2 v l, <. Moreover, x l,1 v l, = lim n x l,1 v l,n and x l,2 v l, = lim n x l,2 v l,n. Proposition 27. For every n, we have v l, (x) E γ xe r τ lv l,n h(x τl v l,n ) + cl r ( Hence, for every ε > and n such that cl r ( r+λγ )n+1 ε, τ l v l,n is ε-optimal for (4.3). λ λ r+λγ )n+1. The proof is similar to that of Proposition 15 if we replace localizing stopping times τ ab with τ lb. Finally, Corollary 28 identifies the maximum expected reward and nearly optimal stopping strategies of the asset manager for the second problem. Corollary 28. The maximum expected reward of the asset manager is given by U l (x) = x cl r + V l (x) = x cl r + v l, (x) for every x. The stopping rule τ l v l, is optimal, and τ l v l,n is ε-optimal for every ε > and n such that cl r ( λγ r+λγ )n+1 < ε, in the sense that for every x > U l (x) = E γ x U l (x) ε E γ x e r τ lv l, (X τl v l, L) + + e r τ lv l,n (X τl v l,n L) + + τl v l, τl v l,n e rt (δx t cl)dt, e rt (δx t cl)dt. We expect that the value of the limited protection at level l to increase as l decreases. We also expect that the asset manager quits early as the protection limit l increases to L. This expectations are validated later, and they are backed up by the findings of the next lemma. Lemma 29. Let w : R + R be as in Assumption 5. Suppose that < l < u < L. Then (i) (M l w)( ) (M u w)( ) on R +, (ii) < ζ l,1 w < ζ u,1 w < F (L) < ζ u,2 w < ζ l,2 w <, (iii) (J l w)( ) (J u w)( ) on R +, (iv) < x l,1 w < x u,1 w < L < x u,2 w < x l,2 w <. Recall that (M l w)( ) and (M u w)( ) coincide, respectively, on (, F (l) and (, F (u) with (Lw)( ) and on (F (l), ) and (F (u), ) with the smallest nonnegative concave majorants of (Lw)( ), respectively, over (F (l), ) and (F (u), ). Therefore, (i) and (ii) of Lemma 29 immediately follow; see Figure 2. Finally, (iii) and (iv) follow from (i) and (ii) by the relation (4.5): (J l w)(x) =

21 OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 21 (Mw)(ζ) (M l w)(ζ) (M u w)(ζ) F (u) ζ u,1 w F (l) ζ l,1 w (Lw)(ζ) ζ u,2 w ζ l,2 w ζ 1 w F (L) ζ 2 w ζ Figure 3. Comparison of (M l w)( ) and (M u w)( ) for ζ 1 w < F (l) < F (u) < F (L). Observe that ζ 1 w < ζ l,1 w < ζ u,1 w < F (L) < ζ u,2 w < ζ l,2 w and (M l w)( ) (M u w)( ). λγ(hw)(x) + (G l w)(x) = λγ(hw)(x) + ϕ(x)(m l w)(f (x)) for every x; x l,1 w = F 1 (ζ l,1 w), x l,2 w = F 1 (ζ l,2 w), and that F ( ) is strictly increasing. Proposition 3 shows that demanding higher portfolio insurance or limiting more severely the downward risks or losses also limits the upward potential and reduces the total value of the portfolio. Proposition 3. For every < l < u < L, we have (i) v l,n (x) v u,n (x), x R +, n {, 1,..., }, (ii) U l (x) U u (x), x R +, (iii) < x l,1 v l,n x u,1 v u,n < L < x u,2 v u,n x l,2 v l,n <. Proof. Note first that v l, (x) = h(x) = v u, (x) for every x R +. Suppose that for some n we have v l,n ( ) v u,n ( ) on R +. Then by Lemmas 18 and 29 (iii), v l,n+1 ( ) = (J l v l,n )( ) (J l v u,n )( ) (J u v u,n )( ) = v u,n+1 ( ). Therefore, for every n, we have v l,n ( ) v u,n ( ) and v l, ( ) = lim n v l,n ( ) lim n v u,n ( ) = v u, ( ), which proves (i). By (4.2), U l (x) = x cl r + v l, (x) x cl r + v u, (x) = U u (x) for every x >, and (ii) follows. Finally, (4.8) and (i) imply (, x l,1 v l, x l,1 v l,, ) = Γ l v l, = {x > ; (J l v l, )(x) h(x)} = {x > ; v l, (x) h(x)} {x > ; v u, (x) h(x)} = {x > ; (J u v u, )(x) h(x)} = (, x u,1 v u, x u,1 v u,, ). Hence, < x l,1 v l, x u,1 v u, < L < x u,2 v u, x l,2 v l, <. Similarly, (, x l,1 v l,n x l,1 v l,n, ) = Γ l v l,n = {x > ; (J l v l,n )(x) h(x)} = {x > ; v l,n+1 (x) h(x)} {x > ; v u,n+1 (x) h(x)} = {x > ; (J u v u,n )(x) h(x)} = (, x u,1 v u,n x u,1 v u,n, ), which implies < x l,1 v l,n x u,1 v u,n < L < x u,2 v u,n x l,2 v l,n < for every finite n.

22 22 SAVAS DAYANIK AND MASAHIKO EGAMI 5. Numerical algorithms The following algorithm describes how one can calculate maximum expected reward and optimal stopping strategy of the asset manager in the first problem. Initialization. Set n = 1, v (x) = h(x), x. Calculate ϕ(x) = x α, ψ(x) = x α 1, F (x) = ψ(x)/ϕ(x) = x α 1 x α, where (5.1) α,1 = (r δ + λγy σ2 2 ) (r δ + λγy σ2 2 )2 + 2σ 2 (r + λγ) σ 2, α < α 1. Step 1. Calculate h λγ(hv n ) F 1 (ζ), ζ >, (Lv n )(ζ) = ϕ, ζ =. Step 2. Calculate the critical boundaries ζ 1 v n < F (L) < ζ 2 v n, which are unique solutions of (Lv n ) (ζ 1 v n ) = (Lv n)(ζ 2 v n ) (Lv n )(ζ 2 v n ) ζ 1 v n ζ 1 v n = (Lv n ) (ζ 2 v n ), and the smallest nonnegative concave majorant (Mv n )( ) of (Lv n )( ) on R + by (Lv n )(ζ), ζ (, ζ 1 v n ζ 2 v n, ), ζ 2 v n ζ (Mv n )(ζ) = ζ 2 v n ζ 1 v n (Lv n)(ζ 1 v n ) + ζ ζ ζ (ζ 1 v n, ζ 2 v n ). 1v n ζ 2 v n ζ 1 v n (Lv n)(ζ 2 v n ), Step 3. Calculate x 1 v n = F 1 (ζ 1 v n ), x 2 v n = F 1 (ζ 2 v n ), and (h λγ(hv n ))(x), x (, x 1 v n x 2 v n, ), (Gv n )(ζ) = (x 2 v n ) α 1 α x α 1 α (x 2 v n ) α 1 α (x1 v n ) α 1 α (h λγ(hv n ))(x 1 v n ) + x α 1 α (x 1 v n ) α 1 α (x 2 v n ) α 1 α (x1 v n ) α 1 α (h λγ(hv n ))(x 2 v n ), Step 4. Calculate v n+1 (x) = λγ(hv n )(x) + (Gv n )(x) for every x >. x (x 1 v n, x 2 v n ). Step 5. If some stopping criterion has not yet been satisfied (for example, the uniform bound cl r ( λγ r+λγ )n+1 on v v n has not yet been reduced below some desired error level), then set n to n + 1 and got to Step 1, otherwise stop. Outcome. After the algorithm terminates with v n+1, x 1 v n, and x 2 v n, (i) we have x cl r + v n(x) U(x) x cl r + v n(x) + cl r ( λγ r+λγ )n for every x >, (ii) the stopping time τv n = inf{t ; X t (x 1 v n, x 2 v n )} is ε-optimal for every ε > cl r ( λγ U(x) cl r r+λγ )n for the portfolio manager s first problem; namely, for every x > ( ) λγ n τvn E γ x e r τvn (X τvn L) + + r + λγ e rt (δx t cl)dt U(x).

23 OPTIMAL STOPPING PROBLEMS FOR ASSET MANAGEMENT 23 The following algorithm calculates the maximum expected reward and optimal stopping strategy of the asset manager in the second problem with portfolio protection level set at some < l < L. Initialization. Set n = 1, v l, = h(x), x >. Calculate ϕ(x) = x α, ψ(x) = x α 1, F (x) = ψ(x)/ϕ(x) = x α 1 x α, where α and α 1 are as in (5.1). Step 1. Calculate h λγ(hv l,n ) F 1 (ζ), ζ >, (Lv l,n )(ζ) = ϕ, ζ =. Step 2. Calculate (M l v l, )( ), which equals (Lv l,n )( ) on (, F (l) and coincides on (F (l), ) with the smallest nonnegative concave majorant of the restriction of (Lv l,n )( ) to F (l), ). Let < ζ l,1 v l,n < F (L) < ζ l,2 v l,n be the endpoints of interval {ζ; (M l v l, )(ζ) > (Lv l,n )(ζ). Then (Lv l,n )(ζ), ζ (, ζ l,1 v l,n ζ l,2 v l,n, ), (M l v l,n )(ζ) = ζ l,2 v l,n ζ ζ l,2 v l,n ζ l,1 v l,n (Lv l,n)(ζ l,1 v l,n ) + ζ ζ l,1 v l,n ζ l,2 v l,n ζ l,1 v l,n (Lv l,n)(ζ l,2 v l,n ), ζ (ζ l,1 v l,n, ζ l,2 v l,n ). Step 3. Calculate x l,1 v l,n = F 1 (ζ l,1 v l,n ), x l,2 v l,n = F 1 (ζ l,2 v l,n ), and (Gv l,n )(ζ) = (h λγ(hv l,n ))(x), x (, x l,1 v l,n x l,2 v l,n, ), (x l,2 v l,n ) α 1 α x α 1 α (x l,2 v l,n ) α 1 α (xl,1 v l,n ) α 1 α (h λγ(hv l,n ))(x l,1 v l,n ) + x α 1 α (x l,1 v l,n ) α 1 α (x l,2 v l,n ) α 1 α (xl,1 v l,n ) α 1 α (h λγ(hv l,n ))(x l,2 v l,n ), Step 4. Calculate v l,n+1 (x) = λγ(hv l,n )(x) + (Gv l,n )(x) for every x >. x (x l,1 v l,n, x l,2 v l,n ). Step 5. If some stopping criterion has not yet been satisfied (for example, the uniform bound cl r ( λγ r+λγ )n+1 ) on v l, v l,n has not yet been reduced below some desired error level), then set n to n + 1 and got to Step 1, otherwise stop. Outcome. After the algorithm terminates with v l,n+1, x l,1 v l,n, and x l,2 v l,n, (i) we have x cl r + v l,n(x) U l (x) x cl r + v l,n(x) + cl r ( λγ r+λγ )n for every x >, (ii) the stopping time τ l v l,n = inf{t ; X t (x l,1 v l,n, x l,2 v l,n )} is ε-optimal for every ε > cl r ( λγ r+λγ )n for the portfolio manager s second problem; namely, for every x > U l (x) cl ( ) λγ n τl E γ x e r τ lv l,n (X τl v r r + λγ l,n L) + v l,n + e rt (δx t cl)dt U l (x). 6. Numerical illustration For illustration, we take L = 1, σ =.275, r =.3, c =.5, δ =.8, λγ =.1, y =.3. Observe that < r < c < δ. We obtain α =.391 and α 1 = We implemented the numerical algorithms of Section 5 in R in order to use readily available routines to calculate the smallest nonnegative concave majorants of functions. We have used gcmlcm function from the R