Pricing perpetual American options for processes with jumps

Ernesto Mordecki http://www.cmat.edu.uy/˜ mordecki

mordecki@cmat.edu.uy

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

Plan

PART I: Classical results:

1. Brownian Motion

2. Black-Scholes financial market

3. Option pricing

4. Black-Scholes Formula

5. American options and optimal stopping

6. Perpetual options

PART II: New directions:

7. Criticism and proposals

8. Lévy processes

9. Models with jumps: Lévy markets

10. Examples of LP

11. Non-complete markets and pricing

12. General results on perpetual options

13. Closed solutions with mixed exponential jumps 14. Open questions and remarks

15. References.

PART I: Classical Results

1 Brownian Motion Brownian motion (or Wiener process) is a stochastic process W = {W_{t}}_{t≥0} such that it has

• continuous paths

• independent increments: for 0 ≤ t_{1} ≤ … ≤ t_{n}, we have that

W_{t_{1}}, W_{t_{2}} − W_{t_{1}}, …, W_{t_{n}} − W_{t_{n−1}}

are indepent random variables

• homogeneous and gaussian increments

W_{t} − W_{s} ∼ N (0, t − s).

X is Gaussian or normal (X ∼ N (μ, σ^{2})) if its probability distribution is

Φ(x) = ∫_{−∞}^{x} \frac{1}{\sqrt{2π}σ}e^{−\frac{(u−μ)^{2}}{2σ^{2}}}du

2 Black-Scholes Financial Market Model Two assets,

• Bonds B = {B_{t}}_{t≥0}, such that

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

r is the interest rate.

• Stock S = {S_{t}}_{t≥0} defined by

S_{t} = S_{0}e^{σW_{t}+(μ−\frac{1}{2}σ^{2})t} with W a Wiener process. σ is the volatility. For short S_{0} = x, B_{0} = 1.

Remark: {B_{t}} is the solution of the ODE

\frac{dB_{t}}{B_{t}} = rdt, B_{0} = 1.

Similarly, {S_{t}} is the solution of the SDE

\frac{dS_{t}}{S_{t}} = μdt + σdW_{t}, S_{0} = x.

3 Option Pricing A call option is a compromise to sell one unit of S at time τ at price K.

• If τ = T is fixed we have an european option.

• If τ is chosen by the holder we have an american option, generally τ ≤ T.

• If 0 ≤ τ ≤ +∞ we have a perpetual american option. The problem is to find the price V (x, T) of the contract, and, in the

american case the optimal exercise time τ^{∗}. In words • How much to pay for such a contract?

• In the american case: When to excercise?

4 Black Scholes Formula

Gives price for european options.

Perfect hedging: It is possible to construct a dinamic portfolio with β_{t} bonds and γ_{t} stocks, capital X_{t} = β_{t}B_{t} + γ_{t}S_{t} such that:

1. It is self-financing, i.e.

B_{t}dβ_{t} + S_{t}dγ_{t} = 0

2. It gives the exact value of the option at time T,

X_{T} = β_{T}B_{t} + γ_{T}S_{T} = (S_{T} − K)^{+}

The value of the european option was shown to be Black-Scholes Formula

V (x, T) = xΦ(x_{b}) − Ke^{−rT}Φ(x_{s})

with

x_{b} = \frac{ln(x/K) + (r + σ^{2}/2)T}{σ\sqrt{T}}

x_{s} = \frac{ln(x/K) + (r − σ^{2}/2)T}{σ\sqrt{T}}

where Φ(x) is the normal distribution.

Remarks:

1. Two main tools of stochastic calculus are employed in the proof of this result:

• Itô formula: that gives a formula for stochastic diferentials • Girsanov Theorem: Changes the measures P into P^{∗}, gives

Radom-Nykodym densities in this context.

2. There exist only one measure P^{∗}, called the risk-neutral proba-

bility measure such that

V (x, T) = E^{∗}[e^{−rT}(S_{T} − K)^{+}]

characterized by

\frac{S_{t}}{B_{t}} is a P^{∗} martingale

This means that the market is complete.

5 American options and optimal stopping

Denote by τ the excercise time (choosen now by the holder)

0 ≤ τ ≤ T

If V_{A}(x, T) is the price of the American option we have to find

• The price of the option V_{A}(x, T)

• The optimal excercise time τ^{∗} of the option.

Based on perfect hedging we obtain that

V_{A}(x, T) = sup_{τ} E^{∗}[e^{−rτ}(S_{τ} − K)^{+}] = E^{∗}[e^{−rτ^{∗}}(S_{τ^{∗}} − K)^{+}]

i.e. the pricing of an american option is solved through an optimal stopping problem.

Example: An american call (withouth dividends) is equivalent in prices to the european call, and that the optimal excercise is τ_{0} = T (Merton, 1973).

General case: Non-trivial solutions, solved by:

1. Numerics on a partial differential equation with moving boundary

\frac{1}{2}σ^{2}x^{2}V_{xx}(x, t) + rxV_{x}(x, t) + V_{t}(x, t) = rV (x, t)

under condition

V (x, T) ≥ (K − x)^{+}

2. Binomial trees (discrete time-space model) and backward induction.

6 Perpetual Options: T = +∞.

A perpetual option is obtained when T = +∞. Its price gives an upper bound of finite maturity options, and approximate them if T is big. Taking T → ∞ the PDE is an ODE. The price of a Perpetual put option is (Mc. Kean, 1965)

V_{A}(x, +∞) = \{ \begin{matrix}Cx^{−α} & if & x > x^{∗} \\ K − x & if & 0 ≤ x ≤ x^{∗} \end{matrix}

where

α = \frac{2r}{σ^{2}}

and

C = α^{α}(\frac{K}{1 + α})^{1+α} x^{∗} = \frac{αK}{1 + α}

Optimal stopping time is

τ^{∗} = inf{t ≥ 0: S_{t} ≤ x^{∗}}

PART II: New directions

7 Criticism and proposals

Two observations:

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

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

We distinguish innovations in at least two direcctions (despite the popularity of BS),

• Stochastic volatility models: forget independence of increments in time, (having continuity)

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

with possibility of 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 (i.e. a stochastic process with homogeneous independent increments)

8 Lévy processes X = {X_{t}}_{t≥0} is a Lévy process if

• X has paths continuous from the right with limits from the left • X_{0} = 0, and has independent increments

• The distribution of X_{t} − X_{s} depends on the difference t − s. Lévy-Kinchine formula states

E(e^{zX_{t}}) = e^{tψ(z)},

where ψ is given by

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

where b 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. We have a triplet (b, σ, Π).

9 Models with jumps: Lévy markets Two assets, B_{t} = B_{0}e^{rt} as before, and stock given by

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

with X = {X_{t}}_{t≥0} a LP.

10 Examples

Black Scholes

X_{t} = σW_{t} + (μ − \frac{1}{2}σ^{2})t,

is a LP with triplet (μ − \frac{1}{2}σ^{2}, σ, 0). No jumps means Π = 0.

Poisson processes

Let T_{1}, T_{2}, … be a sequence of independent exponential random variables with parameter λ. Let

N_{t} = inf{k: T_{1} + T_{2} + … T_{k} ≤ t}.

N = {N_{t}}_{t≥0} is a Poisson process with parameter λ. X_{t} = cN_{t} − bt is a LP with triplet (−b, 0, Π), where

Π(dt, dx) = λdtδ_{c}(dx).

Compound Poisson process

Actuarial models:

X_{t} = bt + N_{t} ; ∑ ; k=1 Y_{k},

Y = {Y_{k}}_{k∈N} is a sequence of independent random variables with distrubution F = F (y). The triplet is (b, 0, λF (dx)).

Mixed exponential jumps

X = {X_{t}}_{t≥0} given by

X_{t} = at + σW_{t} + N_{t} ; ∑ ; i=1 Y_{k},

N = {N_{t}}_{t≥0} is a Poisson process with intensity λ + μ and Y = {Y_{k}}_{k∈N} is a sequence of i.i.d. random variables with density

h(y) = \frac{λ}{λ + μ} n ; ∑ ; k=1 a_{k}α_{k}e^{−α_{k}y}1_{\{y>0\}} + \frac{μ}{λ + μ} m ; ∑ ; k=1 b_{k}β_{k}e^{β_{k}y}1_{\{y<0\}}

for λ ≥ 0, μ ≥ 0. The process X jumps upwards at rate λ, downwards at rate μ, and the magnitude of the jumps has mixed-exponential distribution. X has triplet

(b, σ, (λ + μ)h(x)dx

Advantages: Enough rich, allows closed formulae.

11 Non-complete markets and pricing alternatives

We have the equivalences:

Complete market

Perfect hedging

One and only one martingale measure P^{∗}

Existence of rational price.

But: Non-completeness: exists more than one martingale measure. Any price is an admissible price. How to choose the measure:

• Continous processes: Föllmer and Schweizer introduced the minimal measure. Minimizes quadratic loss of a hedge.

• Lévy processes, Gerber and Shiu proposed Esscher transform, from actuarial practice. It minimizes relative entropy (Chan, 1999).

• LP with mixed-exponential jumps are stable under Esscher transforms.

How many martingale measures? If,

ψ(1) = b + \frac{1}{2}σ^{2} + ∫_{R}(e^{y} − 1 − y1_{\{|y|<1\}})Π(dy) = r,

we obtain a martingale measure. In short, we are computing admissible prices of perpetual American options under Lévy information.

12 General results on perpetual options Assumptions:

(a) The contract has no expiration date, that is, we are considering perpetual American options.

(b) The martingale measure P^{∗} chosen, is such that X: – X is a Lévy process under P^{∗},

– (S_{t}/B_{t})_{t≥0} is a martingale under P^{∗}

Black-Scholes type formula

Compute

V (S_{0}) = sup_{τ∈M} E^{∗}(e^{−(γ+r)τ}(S_{τ} − K)^{+}),

where γ > 0 is a discount factor. (Darling et al. (1972) r.w.) Proposition 1 (B-S type formula.) Consider any Lévy market and γ > 0. Perpetual call option (put similar) is

V (S_{0}) = S_{0}P̃(M ≥ log(S^{∗}/S_{0})) − KP^{∗}(M ≥ log(S^{∗}/S_{0})) where M = sup_{0≤t<τ(γ+r)} X_{t} τ(γ + r) is an exponential, S^{∗} = KE^{∗}(e^{M}), and P̃ is the measure defined by

dP̃ = \frac{e^{M}}{E^{∗}(e^{M})}dP^{∗}.

The optimal strategy is given by τ^{∗} = inf{t ≥ 0: S_{t} ≥ S^{∗}}.

13 Closed solutions with mixed-exponential jumps

Proposition 2 Let X be a LP with mixed-exponential jums. Then (a) Denote, for γ > 0,

S^{∗} = K n+1 ; ∑ ; k=1 A_{k} \frac{p_{k}}{p_{k} − 1}.

with p = (p_{1}, …, p_{n+1}), 0 < p_{1} < … < p_{n_{a}+1} the positive roots of ψ(p) = r + γ, and the coefficients A = (A_{1}, …, A_{n+1})

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,

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^{∗}}.

(b) Put: Analogous.

Remarks: When h = 0 (no jumps), we obtain Mc. Kean’s result. With h(x) = αe^{−αx} for x > 0 we obtain previous results in F&S.

14 Open questions and remarks

• How to choose the martingale measure P^{∗} in an imcomplete market. Continous trajectories: Föllmer and Schweizer introduced the minimal measure. For Lévy processes, Gerber and Shiu proposed Escher transforms. General semimartingale models: completely open question.

• Find larger classes of jump distributions with closed solutions: Erlang type distributions or, phase-type.

• Is it possible to obtain closed solutions for finite time T in american options?

15 References

Books

• Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Springer, 1998

• Merton, R.C.: Continuous Time Finance. Cambridge Oxford: Blackwell 1990

• Shiryaev, Albert N. Essentials of stochastic finance. World Scientific Publishing (1999)

• Lamberton D., Lapeyre B. Introduction to Stochastic Calculus Applied to Finance. Chapman Hall (1996)

Classical Articles

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

• Merton, R.C.: Option pricing when the underlying stock returns are discontinuous. Journal of Financial Economics 3, 125–144 (1976)

• 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)

Recent references

• Föllmer, H., Schweizer, M.: Hedging of contingent claims under incomplete information. In: Applied Stochastic Analysis (London), Stochastic Monographs 5, New York: Gordon and Breach 1991, pp 389–414

• Gerber H. U., Shiu E. S. W., Option pricing by Esscher transforms. Transactions of the Society of Actuaries. 1994. V. 46, pp. 99-191.

• Pham, H.: Optimal stopping, free boundary, and American option in a jump-diffusion model. Appl. Math. Optim. 35 no. 2, 145–164 (1997)

• Mordecki, E. (see http://www.cmat.edu.uy/ mordecki) Optimal stopping for a diffusion with jumps. Finance & Stochastics 3 (2) 227–236 (1999)