OPTIMAL STOPPING FOR A COMPOUND

POISSON PROCESS WITH EXPONENTIAL JUMPS

E. Mordecki

Universidad de la República, Montevideo, Uruguay.

Abstract. In this paper we give the closed form solution of some optimal stopping problems for processes derived from a compound Poisson process with exponential jumps. Within the possible applications, the results can be interpreted as pricing perpetual American Options under pure jump information.

0.Introduction.

Let be given on a probability space (Ω, F, P) a Poisson process N = (N_{t})_{t≥0} with intensity λ > 0 and a sequence of independent nonnegative random variables Y = (Y_{k})_{k≥1}, with identical distribution F.

According to Feller, ([3]), the compound Poisson process X = (X_{t})_{t≥0}, is given by

(0.1)  X_{t} = ∑_{k=1}^{N_{t}} Y_{k}  t ≥ 0

Let 𝔽 = (F_{t})_{t≥0} denote the filtration generated by X, that is the minimal filtration such that X is 𝔽-adapted and satisfies the “usual” Dellacherie conditions (see Jacod and Shiryaev, [4]). τ is a stopping time (relative to 𝔽) if τ: Ω → [0, +∞] and {τ ≤ t} ∈ F_{t} for all t ≤ 0. Denote by M the class of all stopping times.

The aim of this paper is to give closed form solution to the optimal stopping problem sup_{τ∈M} E(f(S_{τ})), that is, we find a pair (τ^{∗}, s(x)), such that

(0.2)  s(x) = sup_{τ∈M} E(f(S_{τ})) = E(f(S_{τ^{∗}}))

where τ^{∗} is the optimal stopping time and s(x) is the cost function in the following four cases. The function f is given either by

f(x) = (x − K)^{+},

or

f(x) = (K − x)^{+},

with K a real constant, and the process S by

(0.3)  S_{t} = x + X_{t} − at,  x ∈ ℝ,

or

(0.4)  S_{t} = x exp{−at + X_{t}},  x > 0.

1. Main Results.

Theorem 1. Let S be given by (0.3), with F (x) = 1 − e^{−αx}, and 0 < λ < aα. Denote

(1.1)  x_{0} = \frac{λ}{α(aα − λ)} + K,  ρ = α − \frac{λ}{a},

and define

τ^{∗} = inf{t ≥ 0 : S_{t} ≥ x_{0}},
s(x) = \{ \begin{matrix}x − K, & if x ≤ x_{0}, \\ (x_{0} − K) exp\{ρ(x − x_{0})\}, & if x < x_{0}. \end{matrix}

Then the pair (τ^{∗}, s(x)) is the solution to the problem (0.2) with f(x) = (x − K)^{+}.

Theorem 2. Let S be given by (0.3). For arbitrary F denote μ = ∫_{0}^{+∞} xdF (x), that can take the value μ = +∞. Suppose that 0 < a < μλ. Denote by ρ > 0 the unique root of the equation

(1.2)  ∫_{0}^{+∞} e^{−ρy}dF (y) = 1 − \frac{a}{λ}ρ,

and

x_{0} = K − 1/ρ.

Define

τ^{∗} = inf{t ≥ 0 : S_{t} ≤ x_{0}},
s(x) = \{ \begin{matrix}K − x, & if x ≤ x_{0}, \\ (K − x_{0}) exp\{−ρ(x − x_{0})\}, & if x > x_{0}. \end{matrix}

Then the pair (τ^{∗}, s(x)) is the solution to the problem (0.2), with f(x) = (K − x)^{+}.

Remark: For X given by (0.1) with exponential jumps, that is when F (x) = 1 − e^{−αx} we have μ = \frac{1}{α} and ρ = \frac{λ}{a} − α.

Theorem 3. Let S given by (0.4). Assume that F (x) = 1 − e^{−αx} with α > 1 + \frac{λ}{a}. Denote

(1.3)  x_{0} = K \frac{a − λ/α}{a − λ/(α − 1)},  ρ = α − \frac{λ}{a}.

Define

τ^{∗} = inf{t ≥ 0 : S_{t} ≥ x_{0}},
s(x) = \{ \begin{matrix}x − K, & if x ≥ x_{0}, \\ (x_{0} − K)(\frac{x}{x_{0}} )^{ρ}, & if 0 ≤ x < x_{0}. \end{matrix}

Then the pair (τ^{∗}, s(x)) is the solution to the problem (0.2), with f(x) = (x − K)^{+}.

Theorem 4. Let S be given by (0.4). For arbitrary F denote μ = ∫_{0}^{+∞} xdF (x), that can take the value μ = +∞. Suppose that 0 < a < λμ. Denote by ρ > 0 the unique root of the equation (1.2), and

x_{0} = K \frac{ρ}{ρ + 1}.

Define

τ^{∗} = inf{t ≥ 0 : S_{t} ≤ x_{0}},
s(x) = \{ \begin{matrix}K − x, & if 0 < x ≤ x_{0}, \\ (K − x_{0})(\frac{x}{x_{0}} )^{−ρ}, & if x > x_{0}. \end{matrix}

Then the pair (τ^{∗}, s(x)) is the solution to the problem (0.2), with f(x) = (K − x)^{+}.

Remark: For X given by (0.1) with exponential jumps, that is when F (x) = 1 − e^{−αx} we have μ = \frac{1}{α} and ρ = \frac{λ}{a} − α, as in Theorem 2.

2. Pricing American Options under Poisson-Exponential information.

Beside the probabilistic interest we are motivated by the following interpretation: Suppose that S models the evolution of the price of a risky asset in a financial market. Then, the proposed problems can be seen as the pricing of a perpetual American Option, as described for instance in Karatzas, [5] or Shiryev et al., [8].

For S given by (0.3) we are in the linear case, having an analogous of the Bachelier model, with a compound Poisson process (eventualy compensated) instead of a Wiener process.

When S is given by (0.4), we have American call and put Options, in an analogous of the Black and Scholes model (see [8]). Note that in the call case, we obtain a solution that is not differentiable at the optimal level x_{0}. This is due to the fact that the process is not stopped at x_{0}, because it jumps above this critical level, and the “smooth fitting” principle in its classical form, as described in [7] does not apply.

Other applications, for instance to investment under uncertainity can be found in Dixit and Pindyck, [1].

3.Proofs.

We begin with an auxiliary result of independent interest (compare with [6]).

Lemma 3.1. Let X be a compound Poisson process defined by (0.1). Let S = (S_{t})_{t≥0} be the solution of the stochastic differential equation

dS_{t} = (aS_{t−} + b)dt + (cS_{t−} + e)dX_{t}, S_{0} = x,

with a, b, c and e real constants. C^{∗}, the continuation region, is an open interval of the real line of the form (x_{0}, +∞), (−∞, x_{0}) or (0, x_{0}) for some x_{0} in ℝ. Let s and g denote Borel real functions, with s continuous for x ∈ R and differentiable for x ≠ x_{0}.

Denote by L the infinitesimal generator of the Markov process S, that is

(3.1)  (Lf)(x) = λ ∫_{0}^{+∞}[f(x + cxy + ey) − f(x)]dF (y) + (ax + b)f′(x)

for f in the domain of L.

Let

τ^{∗} = inf{t ≥ 0: S_{t} ∉ C^{∗}}.

Assume that the following five conditions hold:

(1) (Ls)(x) = 0 ∀x ∈ C^{∗}.

(2) (Ls)(x) ≤ 0 ∀x ≠ x_{0}.

(3) 0 ≤ g(x) ≤ s(x) ∀x ∈ ℝ.

(4) s(S_{τ^{∗}∧T ∧t}) ≤ Z P -a.s for all T ∈ M and for all t ∈ ℝ^{+}, with Z an integrable random variable, that is E|Z| < +∞.

(5) s(S_{τ^{∗}}) = g(S_{τ^{∗}}) P -a.s.

Then the pair (τ^{∗}, s) is the solution for the optimal stopping problem for the process (g(S_{t}))_{t≥0}:

s(x) = sup_{τ∈M} E(g(S_{τ})) = E(g(S_{τ^{∗}})).

Proof. For each ω ∈ Ω we can write

s(S_{t}) − s(x) = ∫_{0}^{t} s′(S_{s−})(aS_{s−} + b)ds + ∑_{0≤s≤t} Δs(S_{s}) = ∫_{0}^{t} {s′(S_{s−})(aS_{s−} + b)+

λ ∫_{0}^{+∞}[s(S_{s−} + cS_{s−}y + ey) − s(S_{s−})]dF (y)}ds

+ ∑_{0≤s≤t} Δs(S_{s}) − λ ∫_{0}^{t}[s(S_{s−} + cS_{s−}y + ey) − s(S_{s−})]ds

= ∫_{0}^{t}(Ls)(S_{s−})ds + M_{t}.

with M = (M_{t})_{t≥0} a local martingale (a compensated sum of jumps, [4]). In view of (1) we obtain that for any T ∈ M

(3.2)  ∫_{0}^{τ^{∗}∧T ∧t}(Ls)(S_{s−})ds = 0,

and (2) gives

(3.3)  ∫_{0}^{σ}(Ls)(S_{s−})ds ≤ 0 for any σ ∈ M.

Now in view of (2)

(3.4)  M_{t} = s(S_{t}) − s(x) − ∫_{0}^{t}(Ls)(S_{s−})ds ≥ −s(x)

and as M = (M_{t})_{t≥0} is a local martingale (3.4) gives that M is a supermartingale by Fatou’s Lemma, and in consequence

(3.5)  E(M_{σ}) ≤ E(M_{0}) = 0 for any σ ∈ M.

On the other side, by (4) and (3.2)

M_{τ^{∗}∧t∧T_{n}} ≤ s(S_{τ^{∗}∧t∧T_{n}}) ≤ Z

for a localizing sequence (T_{n}) for M, that gives by dominated convergence

(3.6)  E(M_{τ^{∗}∧t}) = 0.

In conclusion (3.2) and (3.6) give

(3.7)  s(x) = E(s(S_{τ^{∗}∧t})),

and (3.3), (3.5) and (3) gives

s(x) ≥ E(s(S_{σ})) ≥ E(g(S_{σ})) for any σ ∈ M.

By (4), (5) and (3.7) we obtain

s(x) = E(s(S_{τ^{∗}})) = E(g(S_{τ^{∗}})),

and the Lemma is proved.

The proof of the Theorems reduces now to the verification of the five conditions in Lemma 3.1.

Proof of Theorem 1. First observe that

(3.8)  (Lf)(x) = λ ∫_{0}^{+∞}[f(x + y) − f(x)]dF (y) − af′(x).

We now verify conditions (1) to (5) in Lemma 3.1.

(1) For x < x_{0}, by substitution

as′(x) + λs(x) − λ ∫_{0}^{+∞}[f(x + y) − f(x)]α exp{−αy}dy =

(x_{0} − K) exp\{ρ(x − x_{0})\}(aρ + λ + \frac{λα}{ρ − α})

− λα exp\{α(x − x_{0})\}((x_{0} − K)(\frac{1}{α} + \frac{1}{ρ − α}) + \frac{1}{α^{2}} ) = 0

Because

aρ + λ + \frac{λα}{ρ − α} = (x_{0} − K)(\frac{1}{α} + \frac{1}{ρ − α}) + \frac{1}{α^{2}} = 0

as can be verified by substitution of the values of ρ and x_{0} in (1.1).

(2) If x > x_{0} we have

Ls(x) = \frac{λ}{α} − a < 0.

(3) s(x) > 0 by definition, and s(x) ≥ x − K can be verified directly if x ≥ x_{0}, and considering the tangent line to the convex function s(x), at the point x_{0} for x < x_{0},

(x_{0} − K) exp\{ρ(x − x_{0})\} ≥ (x_{0} − K) + \frac{λ}{aα}(x − x_{0}) ≥ x − K if x < x_{0}.

because \frac{λ}{aα} < 1.

(4) Take Z = s(S_{τ^{∗}}) + x_{0}. s(S_{τ^{∗}∧T ∧t}) ≤ Z follows from the fact that on the set {τ^{∗} < +∞} we have S_{τ^{∗}∧T ∧t} ≤ S_{τ^{∗}} and the function s is increasing. On the set {τ^{∗} = +∞} we have s(S_{τ^{∗}∧T ∧t}) ≤ x_{0}.

Observe now that s(S_{τ^{∗}}) = (S_{τ^{∗}} − K)^{+} ≤ sup_{0≤t<+∞} S_{t} = x + sup_{0≤t<+∞}(X_{t} − at) From [2] we know that

(3.9)  P ( sup_{0≤t<+∞}(X_{t} − at) ≥ u) = \frac{λμ}{a} exp\{−ρu\}

so we can compute

E( sup_{0≤t<+∞}(X_{t} − at)) = \frac{λ}{α(aα − λ)}

and (4) is proved.

(5) is direct, concluding the proof of Theorem 1.

Proof of Theorem 2. As in Theorem 1, we have to verify conditions (1) to (5) in Lemma 3.1. In this case, as in Theorem 1, L is given by (3.8).

(1) If x > x_{0}, s(x) = \frac{1}{ρ} exp\{−ρ(x − x_{0})\},

(Ls)(x) = exp\{−ρ(x + y − x_{0})\}\frac{λ}{ρ}( ∫_{0}^{+∞} exp\{−ρy\}dF (y) − 1 + \frac{aρ}{λ}) = 0

by (1.2).

(2)^{1} If x < x_{0}

(Ls)(x) = −as′(x) + λ ∫_{0}^{+∞}(s(x + y) − s(x))dF (y)
≤ −as′(x_{0}) + λ ∫_{0}^{+∞}(s(x_{0} + y) − s(x_{0}))dF (y) = 0,

1We add some missing arguments in the published version.

because s′(x) = s′(x_{0}) = −1, and

s(x + y) − s(x) ≤ s(x_{0} + y) − s(x_{0})

for each y ≥ 0.

(3) s(x) > 0 is inmediate and s(x) ≥ K − x follows from the fact, that s(x) is convex and the fucnction \frac{1}{ρ} exp\{−ρ(x − x_{0})\} has tangent y = K − x at the point x = x_{0} (here applies the smooth pasting principle, because the function f(x) is decreasing and we have jumps up).

(4) In this case we have s(S_{τ^{∗}}) ≤ x_{0}.

(5) is also inmediate concluding the proof. Proof of Theorem 3. Assume first that F is arbitrary.

S_{t} = x exp{−at + X_{t}} = x exp{−at} ∏_{k=1}^{N_{t}} e^{Y_{k}}

that can be written as

S_{t} = xE( − at + ∑_{k=1}^{N_{t}}(e^{Y_{k}} − 1)),

where E is the stochastic or Doléans-Dade exponential, so S_{t} verifies the stochastic differential equation

dS_{t} = S_{t−}d(−at + ∑_{k=1}^{N_{t}}(e^{Y_{k}} − 1), S_{0} = x,

and then, it has an infinitesimal generator given by

(3.10)  (Lf)(x) = λ ∫_{0}^{+∞}[f(xe^{y}) − f(x)]dF (y) − axf′(x)

If F (x) = 1 − e^{−αx}, we transform this expresion into

(Lf)(x) = λ ∫_{0}^{+∞}[f(x(1 + y)) − f(x)]α(1 + y)^{−(α+1)}dy − axf′(x)

We now verify conditions (1) to (5) in Lemma 3.1.

(1) For x < x_{0}, by substitution

axs′(x) + λs(x) − λ ∫_{0}^{+∞} s(x + xy)α(1 + y)^{−(α+1)}dy =

(x_{0} − K)ρ(\frac{x}{x_{0}} )^{ρ}[a + \frac{λ}{ρ − α}]

+ (\frac{x}{x_{0}} )^{α}((\frac{α}{ρ − α} + \frac{α}{α − 1})x_{0} − (\frac{α}{ρ − α} + 1)K) = 0

Because

a + \frac{λ}{ρ − α} = (\frac{α}{ρ − α} + \frac{α}{α − 1})x_{0} − (\frac{α}{ρ − α} + 1)K = 0

as can be verified by substitution of the values of ρ and x_{0} in (1.3).

(2) If x > x_{0} we have

Ls(x) = x(\frac{λ}{α − 1} − a) < 0.

(3) s(x) > 0 by definition, and s(x) ≥ x − K is direct when x ≥ x_{0}. When x < x_{0}, it can be verified considering the tangent line to the convex function s(x), at the point x_{0}, giving

s(x) ≥ (x_{0} − K) + \frac{λ}{a(α − 1)}(x − x_{0}) ≥ (x − K) if 0 ≤ x ≤ x_{0},

that gives the result because \frac{λ}{a(α−1)} < 1.

(4) Take Z = s(S_{τ^{∗}}) + x_{0}. s(S_{τ^{∗}∧T ∧t}) ≤ Z follows from the fact that on the set {τ^{∗} < +∞} we have S_{τ^{∗}∧T ∧t} ≤ S_{τ^{∗}} and the function s is increasing. On the set {τ^{∗} = +∞} we have S_{τ^{∗}∧T ∧t} ≤ x_{0}.

Now, as in the proof of Theorem 1, taking into account (3.9), and ρ > 1 we verify E(Z) < +∞.

E( sup_{0≤t<+∞} exp(X_{t} − at)) = \frac{λ}{aα} \frac{1}{ρ − 1} < +∞

and (4) is proved.

(5) is direct, concluding the proof of Theorem 3.

Proof of Theorem 4. As in Theorem 1, we have to verify conditions (1) to (5) in Lemma 3.1.

In this case, as in Theorem 3, L is given by (3.10).

(1) If x > x_{0}, s(x) = (K − x_{0})(\frac{x}{x_{0}} )^{−ρ}

(Ls)(x) = λ ∫_{0}^{+∞} [(K − x_{0})(\frac{xe^{y}}{x_{0}} )^{−ρ} − (\frac{x}{x_{0}} )^{−ρ}]dF (y)

= (K − x_{0})(\frac{x}{x_{0}} )−ρ[λ(E(e^{−ρy}) − 1) − aρ] = 0,

by the definition of ρ.

(2)^{2} If 0 < x < x_{0}

(Ls)(x) = −axs′(x) + λ ∫_{0}^{+∞}(s(xe^{y}) − s(x))dF (y)

≤ x(a + λ ∫_{0}^{+∞} \frac{s(x_{0}e^{y}) − s(x_{0})}{x_{0}}dF (y)) = 0

2Idem pg. 6

because s′(x) = −1 and, by convexity for all y ≥ 0

\frac{s(xe^{y}) − s(x)}{x} ≤ \frac{s(x_{0}e^{y}) − s(x_{0})}{x_{0}}.

(3) s(x) > 0 is inmediate and s(x) ≥ K − x follows from the fact, that s(x) is convex and the function (K −x_{0})(\frac{x}{x_{0}} )^{−ρ} has tangent y = K −x at the point x = x_{0} (here applies the smooth pasting principle, because the function f(x) is decreasing and we have jumps up).

(4) In this case we have s(S_{τ^{∗}}) ≤ x_{0}.

(5) is also inmediate concluding the proof. Acknowledgements. The author is indebted to A.N. Shiryaev for introducing him in this subject, and to D.O. Kramkov and E. Accinelli for helpful discussion.

References

1. Dixit A.K., Pindyck R.S., Investment under Uncertainty, Princeton University Press, Princeton, New York, 1994.

2. Embrechts P., Klüppelberg C., Some Aspects of Insurance Mathematics, Theory of Probability and Applications 38 (1993), 374–416.

3. Feller W., An Introduction to Probability Theory and its Application, II, J. Wiley, New York-London-Sydney, 1966.

4. Jacod J. and Shiryaev A.N., Limit Theorems for Stochastic Processes, Springer, 1987.

5. Karatzas I, On the pricing of American Options, Applied Math. Optimization 17 (1988), 37–60.

6. Kramkov D.O. and Mordecki E., Optimal Stopping Rules for Poisson Processes (1995); Preprint.

7. Shiryaev A.N., Optimal Stopping Rules, Springer, 1978.

8. Shiryaev A.N., Kabanov Y. M., Kramkov D.O., Melnikov A.V., On the pricing of option of European and American types, II: Continuous time., Theory of Probability and Applications, 39 (1994), 80-129..

Facultad de Ingeniería, IMERL, C.C. 30, Montevideo, Uruguay.

e-mail: mordecki@cmat.edu.uy