Elementary Proofs on Optimal Stopping ^{1}

Ernesto Mordecki

Universidad de la República, Facultad de Ciencias Centro de Matemática, Montevideo, Uruguay

Postal Address: Facultad de Ciencias. Centro de Matemática.

Iguá 4225. CP 11400. Montevideo. Uruguay.

e-mail: mordecki@cmat.edu.uy URL: http://www.cmat.edu.uy/˜ mordecki Fax: 598-2 522 06 53

Keywords: Optimal stopping, Lévy processes, American options, Derivative pricing, Black-Scholes model. JEL: G12, MSC (1991): 60G40

Abstract

Elementary proofs of classical theorems on pricing perpetual call and put options in the standard Black-Scholes model are given. The method presented does not rely on stochastic calculus and is also applied to give prices and optimal stopping rules for perpetual call options when the stock is driven by a Lévy process with no positive jumps, and for perpetual put options for stocks driven by a Lévy process with no negative jumps

1This work was partially written at the Laboratoire de Statistique et Probabilités de l’Université Paul Sabatier, Toulouse, and benefited from helpful discussion with Walter Moreira.

Elementary Proofs on Optimal Stopping

Ernesto Mordecki ^{†} Facultad de Ciencias. Centro de Matemática. Iguá 4225. CP 11400. Montevideo. Uruguay.

December 29, 2000

1 Introduction

1.1 Consider a model of a financial market with two assets, a savings account B = {B_{t}}_{t≥0} and a stock S = {S_{t}}_{t≥0}. The evolution of B is deterministic, with

B_{t} = B_{0}e^{rt}, B_{0} = 1, r > 0,

and the stock is random, and evolves according to the formula

S_{t} = S_{0}e^{X_{t}}, S_{0} > 0,

where X = {X_{t}}_{t≥0}, the driving process, is an adapted stochastic process defined in a stochastic basis (Ω, F, F = {F_{t}}_{t≥0}, P) satisfying the usual conditions. We always assume that the process

\{\frac{S_{t}}{B_{t}} \}_{t≥0} is a martingale under P.  (1.1)

M = {τ: Ω → [0, +∞], {τ ≤ t} ∈ F_{t} for all t ≥ 0}

denotes the class of all F-stopping times.

1.2 If we are interested in pricing perpetual call and put options, possible discounted at a constant rate δ ≥ 0, we are led to solve an optimal stopping problem consisting in finding a cost function V = V (S_{0}) and an optimal stopping rule τ^{∗} that satisfy

†mordecki@cmat.edu.uy, http://www.cmat.edu.uy/˜ mordecki

V (S_{0}) = sup_{τ∈M} Ee^{−(r+δ)τ} G(S_{τ}) = Ee^{−(r+δ)τ^{∗}}G(S_{τ^{∗}}).  (1.2)

where the function G = G(S) is given by G(S) = (S − K)^{+} for call options and G(S) = (K − S)^{+} for put options. As usual, we assume that e^{−(r+δ)τ} G(S_{τ})1_{{τ=∞}} = lim sup_{t→∞} e^{−(r+δ)t}G(S_{t}).

Given a function V̄ and a stopping time τ̄ ∈ M, if we are able to verify

(A) V̄ (S_{0}) = Ee^{−(r+δ)τ̄}G(Sτ̄),

(B) V̄ (S_{0}) ≥ Ee^{−(r+δ)τ} G(S_{τ}), for all τ ∈ M

we then prove that the pair V̄ , τ̄ is the solution to the problem (1.2).

The aim of this paper is twofold. First we give elementary proofs of classical results on perpetual calls and puts in the Black-Scholes model. The proofs consist on the verification of the martingale and supermartingale property of certain auxiliary processes, in order to see (A) and (B) above, and are based on independence of increments and Jensen’s inequality respectively, not requiring Itô calculus. In this sense the proofs are elementary, and this task is carried in section 2. In section 3 we observe that the presented arguments can be extended to a more general setting, obtaining admissible prices of perpetual call options for upper semi-continuous Lévy processes, that is processes with no positive jumps, and admissible prices of perpetual put options for lower semi-continuous Lévy processes, that is, processes with no negative jumps. By admissible prices we mean that, as the market is incomplete, the martingale measure P chosen is such that X is a Lévy process under P, and prices are computed according to this measure. More general results can be obtained with different techniques as in Mordecki (2000). Section 4 summarizes the results, and concludes with a discussion of related works and references.

2 Black-Scholes model

In this section we assume that X = {X_{t}}_{t≥0} in 1.1 is given by

X_{t} = σW_{t} + (r − \frac{σ^{2}}{2} )t

where W = {W_{t}}_{t≥0} is a Wiener process under P, and σ > 0. This corresponds to the classical Black and Scholes (1973) model.

2.1 It is well known, that the solution to (1.2) with G(S) = (S − K)^{+} and δ > 0 i.e. the pricing of a perpetual call option with discount has cost function (see Merton, (1973)),

C(S_{0}) = \{ \begin{matrix}AS_{0}^{γ} & 0 ≤ S_{0} < S_{c}^{∗}, \\ S_{0} − K & S_{c}^{∗} ≤ S_{0}, \end{matrix}  (2.1)

where

γ = \frac{1}{2} − \frac{r}{σ^{2}} + \sqrt{(\frac{1}{2} + \frac{r}{σ^{2}} )^{2} + \frac{2δ}{σ^{2}}} ,

S_{c}^{∗} = K \frac{γ}{γ − 1}, A = \frac{1}{γ^{γ}} (\frac{γ − 1}{K})^{γ−1},

and optimal stopping rule

τ_{c}^{∗} = inf{t ≥ 0: S_{t} ≥ S_{c}^{∗}}.

See also Karatzas and Shreve (1998) or Shiryaev (1999).

Let us prove this result in an elementary way. To see (A) observe that the process {e^{−(r+δ)t}S_{t}^{γ}}_{t≥0} is a martingale. In fact, for h > 0

E(e^{−(r+δ)(t+h)}S_{t+h}^{γ} − e^{−(r+δ)t}S_{t}^{γ}|F_{t})

= e^{−(r+δ)t}S_{t}^{γ}Ee^{−(r+δ)h+γσW_{h}+γ(r− \frac{σ^{2}}{2} )h} − 1

= e^{−(r+δ)t}S_{t}^{γ}(e^{−[r+δ+γ^{2} \frac{σ^{2}}{2} +γ(r− \frac{σ^{2}}{2} )]h} − 1) = 0  (2.2)

by the election of γ. We claim that

lim_{t→∞} γX_{t} − (r + δ)t = lim_{t→∞} X_{t} − (r + δ)t = −∞.  (2.3)

This follows from

E(γX_{1} − (r + δ)) < Ee^{γX_{1}−(r+δ)} = 0,

and

E(X_{1} − (r + δ)) < Ee^{X_{1}−(r+δ)} = e^{−δ} − 1 < 0

respectively. Now, if S_{0} < S_{c}^{∗} Ee^{−(r+δ)τ_{c}^{∗}}(S_{τ_{c}^{∗}} − K)^{+} = Ee^{−(r+δ)τ_{c}^{∗}}(S_{τ_{c}^{∗}} − K)^{+}1_{{τ_{c}^{∗}<∞}} =

AEe^{−(r+δ)τ_{c}^{∗}}S_{τ_{c}^{∗}}^{γ}1_{{τ_{c}^{∗}<∞}} = lim_{t→∞} AEe^{−(r+δ)(τ_{c}^{∗}∧t)}S_{(τ_{c}^{∗}∧t)}^{γ} = AS_{0}^{γ}. (2.4)

where we used (2.3). If S_{c}^{∗} ≤ S_{0} then τ_{c}^{∗} = 0 and

Ee^{−(r+δ)τ_{c}^{∗}}(S_{τ_{c}^{∗}} − K)^{+} = S_{0} − K.

In this way we proved (A).

Let us see (B). Consider for y ≥ 0 the functions v = v(y), and Φ = Φ(y) given respectively by v(y) = Ay^{γ}, and

Φ(y) = \{ \begin{matrix}y, & 0 ≤ y ≤ S_{c}^{∗} − K, \\ (\frac{y}{A})^{\frac{1}{γ}} , & S_{c}^{∗} − K < y. \end{matrix}

It is easy to verify the following three properties • Φ(v(y)) = C(y) with C(y) given in (2.1),

• Φ is concave, because γ > 1,

• Φ(αy) ≤ αΦ(y) for α ≥ 1. The process {e^{−(r+δ)t}C(S_{t})}_{t≥0} is a supermartingale. In fact, for h > 0

E(e^{−(r+δ)(t+h)}C(S_{t+h})|F_{t}) = e^{−(r+δ)(t+h)}E(Φ(v(S_{t+h}))|F_{t})

≤ e^{−(r+δ)(t+h)}Φ(E(v(S_{t+h})|F_{t})) = e^{−(r+δ)(t+h)}Φ(e^{(r+δ)h}v(S_{t})) ≤
e^{−(r+δ)t}Φ(v(S_{t})) = e^{−(r+δ)t}C(S_{t}).  (2.5)

Take now σ ∈ M.

Ee^{−(r+δ)σ}(S_{σ} − K)^{+} = E lim_{t→∞} e^{−(r+δ)(σ∧t)}(S_{σ∧t} − K)^{+}

≤ lim inf_{t→∞} Ee^{−(r+δ)(σ∧t)}(S_{σ∧t} − K)^{+} ≤ lim inf_{t→∞} Ee^{−(r+δ)(σ∧t)}C(S_{σ∧t}) ≤ C(S_{0})

where we used (2.3). This proves (B) and concludes the proof.

2.2 We consider now the put case. For δ ≥ 0 the solution to (1.2) with G(S) = (K − S)^{+} was given in Mc. Kean (1965), see also Merton (1973), and has cost function

P (S_{0}) = \{ \begin{matrix}K − S_{0}, & 0 ≤ S_{0} ≤ S_{p}^{∗}, \\ BS_{0}^{−β}, & S_{p}^{∗} < S_{0}, \end{matrix}  (2.6)

where

β = \sqrt{(\frac{1}{2} + \frac{r}{σ^{2}} )^{2} + \frac{2δ}{σ^{2}}} − (\frac{1}{2} − \frac{r}{σ^{2}} ) > 0,

S_{p}^{∗} = K \frac{β}{β + 1}, B = β^{β}(\frac{K}{β + 1})^{β+1},

and optimal stopping rule

τ_{p}^{∗} = inf{t ≥ 0: S_{t} ≤ S_{p}^{∗}}.

See also Karatzas and Shreve (1998) or Shiryaev (1999). Let us verify (A) and (B) in order to prove this result. Denote by w = w(y), y ≥ 0 the function given by w(y) = By^{−β}. As in 2.1 the election of β allows to show that {e^{−(r+δ)t}w(S_{t})} is a martingale. Also we know that

lim_{t→∞} −βX_{t} − (r + δ)t = lim_{t→∞} X_{t} − (r + δ)t = −∞,  (2.7)

because

E(−βX_{1} − (r + δ)) < Ee^{−βX_{1}−(r+δ)} = 0,

and

E(X_{1} − (r + δ)) < Ee^{X_{1}−(r+δ)} = e^{−δ} − 1 < 0,

respectively. Also as before, when S_{0} > S_{p}^{∗}

Ee^{−(r+δ)τ_{p}^{∗}}(K − S_{τ_{p}^{∗}})^{+} = Ee^{−(r+δ)τ_{p}^{∗}}(K − S_{τ_{p}^{∗}})^{+}1_{{τ_{p}^{∗}<∞}} =

BEe^{−(r+δ)τ_{p}^{∗}}S_{τ_{p}^{∗}}^{−β}1_{{τ_{p}^{∗}<∞}} = lim_{t→∞} Ee^{−(r+δ)(τ_{p}^{∗}∧t)}v(S_{(τ_{p}^{∗}∧t)}) = BS_{0}^{−β}. (2.8)

by (2.7). If S_{0} < S_{p}^{∗}, τ_{p}^{∗} = 0 and Ee^{−(r+δ)τ_{p}^{∗}}(K −S_{τ_{p}^{∗}})^{+} = K −S_{0}, concluding (A). In order to see (B) consider the function ϕ = ϕ(y), y ≥ 0 given

ϕ(y) = \{ \begin{matrix}y, & 0 ≤ y ≤ S_{p}^{∗} − K, \\ K − (\frac{y}{B} )^{− \frac{1}{β}} , & S_{p}^{∗} − K < y. \end{matrix}

The following properties hold

• ϕ(w(y)) = P (y) with P (y) given in (2.6),

• ϕ is concave, because β > 0,

• ϕ(αy) ≤ αϕ(y) for α ≥ 1,

and allow to see, as in 2.1 that {e^{−(r+δ)t}P (S_{t})}_{t≥0} is a supermartingale. From this, for σ ∈ M we have

Ee^{−(r+δ)σ}(K − S_{σ})^{+} = E lim_{t→∞} e^{−(r+δ)(σ∧t}(K − S_{σ∧t})^{+}

≤ lim inf_{t→∞} Ee^{−(r+δ)(σ∧t)}(K − S_{σ∧t})^{+}

≤ lim inf_{t→∞} Ee^{−(r+δ)(σ∧t)}P (S_{σ∧t}) ≤ P (S_{0})

by (2.7), proving (B) and completing the proof.

3 Perpetual options for semi-continuous Lévy process

We now see that the proofs above remain in force in a more general setting. For a call option we consider a stock driven by a Lévy process without positive jumps, i.e. an upper semi-continuous Lévy process. In the put case the driving process has no negative jumps, i.e. it is lower semicontinuous. For general reference on Lévy processes see Jacod and Shiryaev (1987), Skorokhod (1991), Bertoin (1996) or Sato (1999). Lévy-Khinchine formula states, for a Lévy process X = {X_{t}}_{t≥0}

Ee^{iμX_{t}} = exp \{t[iμa − \frac{1}{2}σ^{2}μ^{2} + ∫_{R}(e^{iμy} − 1 − iμy1_{\{|y|<1\}})Π(dy)]\}, (3.1)

where a and σ ≥ 0 are real constants, and Π is a positive measure on R−{0} such that ∫ (1 ∧ y^{2})Π(dy) < +∞, called the Lévy measure. The triplet of parameters (a, σ^{2}, Π) completely determine the law of the process.

3.1 Call perpetual options for upper semi-continuous Lévy processes

Let X = {X_{t}}_{t≥0} be a Lévy process with no positive jumps. This is equivalent to say that the measure Π in (3.1) is supported on the set (−∞, 0). We exclude from consideration the case of the negative of a subordinator, i.e. an a.s. non-increasing process. If λ ≥ 0 we introduce the Laplace exponent κ = κ(λ) of X

κ(λ) = aλ + \frac{1}{2}σ^{2}λ^{2} + ∫_{−∞}^{0}(e^{λy} − 1 − λy1_{\{−1<y<0\}})Π(dy),  (3.2)

that satisfies Ee^{λX_{t}} = e^{tκ(λ)}. It is known that κ(0) = 0, κ is convex, and lim_{λ→∞} κ(λ) = ∞. The martingale condition (1.1) reads κ(1) = r, so for any δ > 0 there exists γ > 1 such that κ(γ) = r + δ.

Theorem 3.1 Consider the model of a financial market of 1.1 with driving process X, a Lévy process with no positive jumps, triplet (a, σ^{2}, Π) and exponent (3.2). Then, the solution to the problem (1.2) with G(S) = (S −K)^{+} and δ > 0 has cost function

C(S_{0}) = \{ \begin{matrix}AS_{0}^{γ} & 0 ≤ S_{0} < S_{c}^{∗}, \\ S_{0} − K & S_{c}^{∗} ≤ S_{0}, \end{matrix}

where γ > 1 is such that κ(γ) = r + δ, S_{c}^{∗} = K \frac{γ}{γ−1}, A = \frac{1}{γ^{γ}} (\frac{γ−1}{K})^{γ−1}, and optimal stopping rule

τ_{c}^{∗} = inf{t ≥ 0: S_{t} ≥ S_{c}^{∗}}.

Proof. We follow the proof in 2.1. {e^{−(r+δ)t}S_{t}^{γ}}_{t≥0} is a martingale, as

E(e^{−(r+δ)(t+h)}S_{t+h}^{γ} − e^{−(r+δ)t}S_{t}^{γ}|F_{t}) =

e^{−(r+δ)t}S_{t}^{γ}(e^{(−r−δ+κ(γ))h}) − 1) = 0.

The rest follows exactly as before, taking into account that on {τ_{c}^{∗} < ∞}, we have

(S_{τ_{c}^{∗}} − K)^{+} = AS_{τ_{c}^{∗}}^{γ},

because the process has no positive jumps.

3.2 Put perpetual options for lower semi-continuous Lévy processes

Let X = {X_{t}}_{t≥0} be a Lévy process with no negative jumps. The measure Π in (3.1) is supported now on the set (0, ∞). We exclude the case of a subordinator, i.e. an a.s. increasing process. The Laplace exponent is now defined for λ ≤ 0 by

κ(λ) = aλ + \frac{1}{2}σ^{2}λ^{2} + ∫_{0}^{∞}(e^{λy} − 1 − λy1_{\{0<y<1\}})Π(dy),  (3.3)

and satisfies Ee^{λX_{t}} = e^{tκ(λ)}. Observe that the martingale condition (1.1) requires in this case an extra assumption on the jumps, i.e. Ee^{X_{1}} = e^{r} < ∞. The finiteness of the exponential moment is ensured by the condition ∫_{0}^{∞} e^{y}Π(dy) < ∞. As κ(0) = 0, κ is convex, and lim_{λ→−∞} κ(λ) = ∞ we obtain that for any δ ≥ 0 there exists β > 0 such that κ(−β) = r + δ.

Theorem 3.2 Consider the model of a financial market of 1.1 with driving process X, a Lévy process with no negative jumps, triplet (a, σ^{2}, Π) and exponent (3.3). Then, the solution to the problem (1.2) with G(S) = (K − S)^{+} and δ ≥ 0 has cost function

P (S_{0}) = \{ \begin{matrix}K − S_{0} & 0 ≤ S_{0} ≤ S_{p}^{∗}, \\ BS_{0}^{−β} & S_{p}^{∗} < S_{0}, \end{matrix}

where β > 0 is such that κ(−β) = r + δ, S_{p}^{∗} = K \frac{β}{β+1}, B = β^{β}(\frac{K}{β+1})^{β+1}, and optimal stopping rule

τ_{p}^{∗} = inf{t ≥ 0: S_{t} ≤ S_{p}^{∗}}.

Proof. We follow the proof in 2.2. {e^{−(r+δ)t}w(S_{t})} is a martingale in view of κ(−β) = r + δ. (2.8) follows because on the set {τ_{p}^{∗} < ∞} we have (K − S_{τ_{p}^{∗}})^{+} = BS_{τ_{c}^{∗}}^{−β}, as the process has no negative jumps. The rest follows exactly.

4 Conclusion

This article presents new proofs of classical theorems on pricing call and put perpetual American options in the Black-Scholes model, with a savings account B = {B_{t}}_{t≥0} and a stock S = {S_{t}}_{t≥0}. The proofs consist in the verification of the martingale and supermartingale properties, for process of the type {f(S_{t})}_{t≥0} for certain auxiliary functions f. The martingale property is established elementary, by independence of increments. The supermartingale property is verified through Jensen’s inequality. No stochastic calculus is involved. By observing that the overshot when the process reaches a fixed level is null for semi-continuous Lévy processes, upper for call options, and lower for put options, the results are extended to these cases. More general results based on different techniques can be found in Mordecki (2000) and the references therein.

References

[1] Bertoin, J. (1996): Lévy Processes. Cambridge: Cambridge University Press.

[2] Black, R. Scholes, M. (1973): The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659.

[3] Jacod, J., Shiryaev, A.N. (1987): Limit Theorems for Stochastic Processes. Berlin Heidelberg: Springer.

[4] Karatzas, I., Shreve, S.E. (1998): Methods of Mathematical Finance. New York Berlin Heidelberg: Springer.

[5] Mc Kean, Jr. H.P. (1965): Appendix: A free boundary problem for the heat equation arising from a problem in Mathematical Economics. Industrial Management Review, 6 (spring), 32–39.

[6] Merton, R.C. (1973): Theory of rational option pricing, Bell J. Econom. Manag. Sci., 4, 141–183.

[7] Mordecki, E. (2000): Optimal stopping and perpetual options for Lévy processes. Pre-Mat 36. http://kolmogorov.cmat.edu.uy/˜ mordecki/publications.html

[8] Sato, Ken-iti (1999): Lévy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics, 68. Cambridge: Cambridge University Press.

[9] Shiryaev, A. N. (1999): Essentials of stochastic finance. Facts, models, theory. Advanced Series on Statistical Science and Applied Probability, 3: World Scientific Publishing Co., Inc., River Edge, NJ.

[10] Skorokhod, A. V. (1991): Random processes with independent increments. Dordrecht: Kluwer Academic Publishers.