Optimal stopping and perpetual options for Lévy processes

Arrêt optimal et options perpétuelles

pour les processus de Lévy

Ernesto Mordecki

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

e-mail:  mordecki@cmat.edu.uy
http://www.cmat.edu.uy/~mordecki

Plan

1. Financial market models

2. Lévy market model

3. European and American options in B-S market 4. Generalizations of B-S market model

5. Perpetual options in Lévy markets

6. Distribution of maximum and infimum of Lévy processes

7. Closed solutions for perpetual options: mixed exponential jumps

8. Open questions and remarks

9. References 1 Financial Market Models

Two assets,

• Bonds B = {B_{t}}_{t≥0}, such that

B_{t} = B_{0}e^{rt}, B_{0} = 1.

r > 0 is the interest rate.

• Stock S = {S_{t}}_{t≥0} defined by

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

with X = {X_{t}}_{t≥0} is a random process on a stochastic basis (Ω, F, F = (F_{t})_{t≥0}, P) and

{S_{t}/B_{t}} is a martingale under P.

Black-Scholes market model.

If

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

and W = {W_{t}}_{t≥0} is a Wiener process we have the well known Black-Scholes (1973) model (B-S), with dividends rate δ. In this case S(r − δ) denotes the stock.

2 Lévy market model.

Suppose that X is a Lévy process, i.e. we add a jump process in the B-S model,

X_{t} = at + σW_{t} + J_{t}

where J = {J_{t}}_{t≥0} is certain jump process (Poisson, compund Poisson, …).

Lévy-Khinchine formula states E(e^{zX_{t}}) = exp(tψ(z))

ψ(z) = bz + \frac{1}{2}σ^{2}z^{2} + ∫_{R}(e^{zy} − 1 − zy1_{\{|y|<1\}})Π(dy),

ψ is the characteristic exponent, and the triplet (b, σ^{2}, Π) characterizes the law of the process.

Condition {S_{t}/B_{t}} is a martingale reads ψ(1) = r, and in particular E(e^{X_{t}}) must be finite.

3 European and American options in a B-S The value of the european option is given by the B-S formula

V (S_{0}, T) = E(e^{rT} (S(r)_{T} − K)^{+}) =
S_{0}P̃(S(r)_{T} > K) − Ke^{−rT} P (S(r)_{T} > K)

with

dP̃ = e^{X_{T} −rT} dP

Perpetual Options in B-S. The price of a Perpetual put option is (Mc. Kean, 1965)

sup_{τ≥0} Ee^{−rτ}(K − S(r)_{τ})^{+} = \{ \begin{matrix}K − S_{0}, & S_{0} ≤ S_{p}^{∗} \\ AS_{0}^{−α}, & S_{0} > S_{p}^{∗} \end{matrix}

where

α = \frac{2r}{σ^{2}}, S_{p}^{∗} = K \frac{α}{1 + α}, A = α^{α}(\frac{K}{1 + α})^{1+α}.

Optimal stopping time is τ^{∗} = inf{t ≥ 0: S_{t} ≤ S_{p}^{∗}}

The price of a Perpetual call option with positive dividends rate δ > 0 is (Merton, 1973)

sup_{τ≥0} Ee^{−rτ}(S(r − δ)_{τ} − K)^{+} = \{ \begin{matrix}BS_{0}^{β}, & S_{0} < S_{c}^{∗} \\ S_{0} − K, & S_{c}^{∗} ≤ S_{0} \end{matrix}

where

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

S_{c}^{∗} = K \frac{β}{β − 1}, B = β^{−β}(\frac{β − 1}{K})^{β−1}.

Optimal stopping time is

τ^{∗} = inf{t ≥ 0: S_{t} ≥ S_{c}^{∗}}

Remark: Problems with positive dividens are equivalent to the pricing of discounted options at a constant rate γ withouth dividends, i.e.

sup_{τ≥0} Ee^{−(r+γ)τ}(S(r)_{τ} − K)^{+}

and for the put,

sup_{τ≥0} Ee^{−(r+γ)τ}(K − S(r)_{τ})^{+}.

4 Generalizations of B-S

Problems:

(a) Observed stock data does not exhibit statistical properties of BS model (gaussian, independence, ...)

(b) Theoretical prices of options does not coincide with observed prices (smile phenomena)

We distinguish innovations in at least two direcctions (despite the popularity of B-S),

• Stochastic volatility models: forget independence, forget homogeneity of increments in time, preserve continuity

dS_{t} = S_{t}(μdt + σ(S_{t}, S′_{t})dW_{t})

with possible dependence of other sources of uncertainty S′

• Models with jumps: forget continuity, conserve independence and homogeneous increments

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

where X is a Lévy process.

5 Perpetual options in Lévy markets

Let X be a Lévy process, S_{t} = S_{0}e^{X_{t}} and γ > 0. Denote

V_{C}(S_{0}) = sup_{τ≥0} E(e^{−(r+γ)τ}(S_{τ} − K)^{+}).

Theorem 1 (Discounted perpetual calls) Let

M = sup_{0≤t<τ(r+γ)} X_{t}

with τ(r + γ) an exponential time with parameter r + γ, and independent of X. Then E(e^{M}) < ∞, and the optimal stopping problem has cost function

V_{C}(S_{0}) = \frac{E[S_{0}e^{M} − KE(e^{M})]^{+}}{E(e^{M})},

and optimal stopping time

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

with S_{c}^{∗} = KE(e^{M}).

Remark: This Theorem is a direct generalization of Darling, Ligget and Taylor (1972) result for random walks.

Denote now, for γ ≥ 0

V_{P} (S_{0}) = sup_{τ≥0} E(e^{−(r+γ)τ}(K − S_{τ})^{+}).

Theorem 2 (Discounted perpetual puts) Let

I = inf_{0≤t<τ(r+γ)} X_{t}

where τ(γ + r) an exponential time with parameter γ + r independent of X. Then E(e^{I}) > 0 and we have

V_{P} (S_{0}) = \frac{E[KE(e^{I}) − S_{0}e^{I}]^{+}}{E(e^{I})},

and optimal stopping time

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

where S_{p}^{∗} = KE(e^{I}).

Corollary 1 (B-S type formula) (a) If γ > 0

V_{C}(S_{0}) = S_{0}P̃(S_{0}e^{M} ≥ S_{c}^{∗}) − KP (S_{0}e^{M} ≥ S_{c}^{∗}),

and P̃ is the measure defined by

dP̃ = \frac{e^{M}}{E(e^{M})}dP.

(b) For γ ≥ 0,

V_{P} (S_{0}) = KP (S_{0}e^{I} ≤ S_{p}^{∗})) − S_{0}P̃(S_{0}e^{I} ≤ S_{p}^{∗})),

where P̃

dP̃ = \frac{e^{I}}{E^{∗}(e^{I})}dP.

6 Distribution of Maximum and infimum of LP

Theorem 3 (a) Let X be a Lévy process with σ > 0, and

Π(y) = \{ \begin{matrix}λ ∑_{k=1}^{n} a_{k}α_{k}e^{−α_{k}y}, & y > 0, \\ π(dy) & y < 0, \end{matrix}

π(dy) arbitrary, i.e. X has mixed exponential positive jumps. ψ denotes the analytical continuation to {ℜ(z) ≥ 0, z ≠ α_{k}, k = 1, …, n} of the characteristic exponent. Then M, when proper, has density

f_{M}(y) = n+1 ; ∑ ; k=1 A_{k}p_{k}e^{−p_{k}y}, y > 0,

where p_{1}, …, p_{n+1}, with

0 < p_{1} < α_{1} < p_{2} < … < α_{n} < p_{n+1}

are the real and positive roots of ψ(p) = r + γ and

A_{j} = \frac{∏_{k=1}^{n}(\frac{p_{j}}{α_{k}} − 1)}{∏_{k=1,k≠j}^{n+1}(\frac{p_{j}}{p_{k}} − 1)}, j = 1, …, n + 1.

(b) Let X be a Lévy process with σ > 0, and

Π(y) = \{ \begin{matrix}π(dy) & y > 0, \\ μ ∑_{k=1}^{n} b_{k}β_{k}e^{β_{k}y}dy & y < 0, \end{matrix}

π(dy) is arbitrary. ψ denotes now the analytical continuation to {ℜ(z) ≤ 0, z ≠ −β_{k}, k = 1, …, n} of the characteristic exponent. Then I has density

f_{I}(y) = n+1 ; ∑ ; k=1 B_{k}q_{k}e^{−q_{k}y}, y < 0,

where −q_{n+1} < −q_{n} < … < −q_{1} < 0 are the real and negative roots of ψ(p) = r + γ, and the coefficients are given by

B_{j} = \frac{∏_{k=1}^{n}(\frac{q_{j}}{β_{k}} − 1)}{∏_{k=1,k≠j}^{n+1}(\frac{q_{j}}{q_{k}} − 1)}, j = 1, …, n + 1,

7 Closed solutions: mixed exponential jumps

Plug in previous formulae to obtain

Theorem 4 (Perpetual Call) Let X be a LP with positive mixed exponential jums. Denote, for γ > 0,

S_{c}^{∗} = K n+1 ; ∑ ; k=1 A_{k} \frac{p_{k}}{p_{k} − 1},

with 0 < p_{1} < … < p_{n+1} the positive roots of ψ(p) = r + γ, and the coefficients A_{1}, …, A_{n+1} as before. The price of a perpetual call is

V_{c}(S_{0}) = \{ \begin{matrix}∑_{k=1}^{n+1} \frac{A_{k}}{p_{k}−1}(\frac{S_{0}}{S_{c}^{∗}})^{p_{k}} & S_{0} ≤ S_{c}^{∗}, \\ S_{0} − K & S_{0} > S_{c}^{∗}. \end{matrix}

The optimal exercise is given by the stopping time

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

Remarks:

• When Π = 0 (no jumps), we obtain Merton’s (1973) result.

• With h(x) = αe^{−αx} for x > 0 and π = 0 we obtain our previous results in Finance and Stochastics (1999).

Theorem 5 (Perpetual Put) Let X be a LP with negative mixed exponential jums. Denote, for γ ≥ 0,

S_{p}^{∗} = K n+1 ; ∑ ; k=1 B_{k} \frac{q_{k}}{q_{k} + 1},

with −q_{1}, …, −q_{n+1}, the negative roots of ψ(p) = r + γ, and the coefficients B_{1}, …, B_{n+1} as before. The price of a perpetual put is

V_{P} (S_{0}) = \{ \begin{matrix}K − S_{0} & S_{0} ≤ S_{p}^{∗}, \\ ∑_{k=1}^{n+1} \frac{B_{k}}{q_{k}+1}(\frac{S_{0}}{S_{p}^{∗}})^{q_{k}} & S_{0} > S_{p}^{∗}. \end{matrix}

The optimal exercise is given by the stopping time

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

Remarks:

• When Π = 0 (no jumps), we obtain Mc Kean’s (1965) result.

• With h(x) = 0 and π = λF, F a distribution we recover X.L. Zang’s (1994) results.

8 Open questions and remarks

(a) How to choose the martingale measure P in an incomplete market. For Lévy processes, Gerber and Shiu (1994) proposed Escher transforms.

(b) X_{t} = at + σW_{t} + ∑_{i=1}^{N_{t}} Y_{i} with {Y_{i}} i.i.d.r.v. twosided mixed exponentially distributed, gives closed prices for call and put, is enough rich, and remains with the same type distribution (changing parameters) under Esscher transforms.

  (c) Details at http://www.cmat.edu.uy/~mordecki

9 References

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

Darling, D.A., Ligget, T. Taylor, H.M.: Optimal stopping for partial sums. Ann. Math. Stat. 43, 1363–1368 (1972)

Gerber, H.U., Shiu, E.S.W.: Option pricing by Esscher transforms (with discussion) Trans. Soc. Actuaries 46 99–191. (1994)

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

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

Mordecki, E. Optimal stopping for a diffusion with jumps. Finance & Stochastics 3 (2) 227–236 (1999)

Mordecki, E.: Ruin probabilities for Lévy processes with mixed exponential negative jumps (1999). Submitted. WP-Premat 99/28.

Mordecki, E.: Optimal stopping and perpetual options for Lévy processes (2000). Submitted, WP-Premat, 2000/36.

Zhang, X.L.: Méthodes numériques pour le calcul des options américaines dans des modèles de diffusion avec sauts. Paris: Thèse de doctorat de l’Ecole National des Ponts et Chausséss 1994