Duality and Derivative Pricing with Lévy Processes

José Fajardo^{∗} and Ernesto Mordecki^{†}

December 19, 2003

Abstract

The aim of this work is to use a duality approach to study the pricing of derivatives depending on two stocks driven by a bidimensional Lévy process. The main idea is to apply Girsanov’s Theorem for Lévy processes, in order to reduce the posed problem to the pricing of a one Lévy driven stock in an auxiliary market, baptized as “dual market”. In this way, we extend the results obtained by Gerber and Shiu (1996) for two dimensional Brownian motion. Also we examine an existing relation between prices of put and call options, of both the European and the American type. This relation, based on a change of numeraire corresponding to a change of the probability measure through Girsanov’s Theorem, is called put–call duality. It includes as a particular case, the relation known as put–call symmetry. Necessary and sufficient conditions for put–call symmetry to hold are obtained, in terms of the triplet of predictable characteristic of the Lévy process.

Key Words: Lévy processes, Optimal stopping, Girsanov’s Theorem, Dual Market Method, Derivative pricing, Symmetry. JEL Classification: G12, G13

∗IBMEC Business School, Rio de Janeiro - Brazil, e-mail: pepe@ibmecrj.br †Centro de Matemática, Facultad de Ciencias, Universidad de la República, Montevideo. Uruguay. e-mail: mordecki@cmat.edu.uy

1 Introduction

Since Margrabe’s (1978) paper, many important extensions have been carrying on to study derivatives written on two stocks. Margrabe studied the pricing of European options for the case of two non-dividend-paying stocks driven by geometric Brownian motions, to be more exactly, the pricing of the right to change one asset for another at the end of some fixed period of time obtaining closed form formulas for this problem, extending in this way the Black and Scholes pricing model. The American option pricing problem leads to the solution of an optimal stopping problem, that in general does not admit closed form solutions (see Jacka (1991)). In the perpetual case, i.e. the option has no expiration date, Gerber and Shiu (1996) obtain a closed form formula using the optional sampling theorem, assuming that stock prices are driven by geometric Brownian motions and stocks pay constant rate continuous dividends. They also study the pricing of the Perpetual Maximum Option, it is an option whose payoff is the maximum between two or more stocks and has no expiration date, and finally they study American perpetual options with more general payoffs which are homogeneous of degree one. In the present paper we consider the problem of pricing European and American type derivatives written on a two dimensional stock driven by a two dimensional Lévy processes (it can be said that the stock follows a two dimensional geometric Lévy process), with a payoff function homogeneous of an arbitrary degree. In the second part of the paper we study an existing relation between prices of put and call options, of both the European and the American type. This relation is called put–call duality. It includes as a particular case, the relation known as put–call symmetry. We suppose that the underlying stock in the market model is driven by a general Lévy processes, i.e. a stochastic process with independent and homogeneous increments, possible with discontinuous paths. In this market model, called a Lévy market, necessary and sufficient conditions for put–call symmetry to hold are obtained, in terms of the drift, the volatility, and the jump structure of the underlying log–stock price (i.e. in terms of the triplet of predictable characteristic of the Lévy process). As particular cases, we obtain the known conditions for symmetry in the lognormal jump–diffusion model introduced by Merton (1976), and examine other models of asset returns proposed in the literature. The corresponding results for stochastic volatility models, and for diffusion with jumps were obtained in Schroder (1999). The paper is organized as follows: in section 2 we describe the market model and introduce the pricing problem, illustrating with several important examples of traded derivatives. In section 3 we describe the Dual Market Method, a method which allows to reduce the two stock problem into a one stock problem. In section 4 we derive some closed form formulas based on the proposed method and known results for one-dimensional problems. In section 5 we study the put–call duality relation and finally we have the conclusions and an appendix.

2 Market Model

2.1 Multidimensional Lévy processes

Let X = (X^{1}, …, X^{d}) be a d-dimensional Lévy process defined on a stochastic basis B = (Ω, F, {F}_{t≥0}, P). This means that X is a stochastically continuous stochastic process with independent increments, such that the distribution of X_{t+s} − X_{s} does not depend on s, with P (X_{0} = 0) = 1 and trajectories continuous from the left with limits from the right. The basis B is supposed to satisfy the usual assumptions, i.e. continuity from the right and F_{0} is P complete. For z = (z_{1}, …, z_{d}) in C^{d}, when the integral is convergent (and this is always the case if z = iλ with λ in IR^{d}), Lévy-Khinchine formula states, that Ee^{zX_{t}} = exp(tψ(z)) where the function ψ is the characteristic exponent of the process, and is given by

ψ(z) = (a, z) + \frac{1}{2}(z, Σz) + ∫_{IR^{d}} (e^{(z,y)} − 1 − (z, y)1_{\{|y|≤1\}})Π(dy),  (1)

where a = (a_{1}, …, a_{d}) is a vector in IR^{d}, Π is a positive measure defined on IR^{d} \{0} such that ∫_{IR^{d}}(|y|^{2} ∧1)Π(dy) is finite, and Σ = ((s_{ij})) is a symmetric nonnegative definite matrix, that can always be written as Σ = A′A (where ′ denotes transposition) for some matrix A. The triplet (a, Σ, Π) completely determines the law of the process X. Particular interest has the case when α = ∫_{IR^{d}} Π(dy) is finite, i.e. X is a diffusion with jumps. Introducing F by Π(dy) = αF (dy), Lévy-Khinchine formula is (changing the value of a if necessary)

ψ(z) = (a, z) + \frac{1}{2}(z, Σz) + ∫_{IR^{d}} (e^{(z,y)} − 1)Π(dy),  (2)

and the process X = {X_{t}}_{t≥0} can be represented by

X_{t} = at + AW_{t} + ∑_{k=1}^{N_{t}} Y_{k},

where W is a standard d-dimensional Brownian motion, N = {N_{t}}_{t≥0} is a Poisson process with parameter α, and {Y_{k}}_{k≥1} is a sequence of independent d-dimensional random vectors with identical distribution F (dy). Another important case is when the coordinates of X are independent processes. This happens if and only if Σ is a diagonal matrix (and A can be chosen to be diagonal also) and the measure Π has support on the set {x ∈ IR^{d}: ∏_{k=1}^{d} x_{k} = 0}, (i.e. it is concentrated on the union of the coordinate axes, see E 12.10 in Sato (1999)). In this case ψ(z) = ∑_{k=1}^{d} ψ_{k}(z_{k}), where ψ_{k} is the characteristic exponent of the k-coordinate of X, given by

ψ_{k}(z_{k}) = a_{k}z_{k} + \frac{1}{2}s_{kk}z_{k}^{2} + ∫_{IR} (e^{z_{k}y} − 1 − z_{k}y1_{\{|y|≤1\}})Π_{k}(dy),

where Π_{k}(A) = ∫_{{x∈IR^{d}: x_{k}∈A}} Π(dx).

2.2 Market and Problem

Consider a market model with three assets (S^{1}, S^{2}, S^{3}) given by

S_{t}^{1} = e^{X_{t}^{1}}, S_{t}^{2} = S_{0}^{2}e^{X_{t}^{2}}, S_{t}^{3} = S_{0}^{3}e^{X_{t}^{3}}  (3)

where (X^{1}, X^{2}, X^{3}) is a three dimensional Lévy process, and for simplicity, and without loss of generality we take S_{0}^{1} = 1. The first asset is the bond and is usually deterministic. Randomness in the bond {S_{t}^{1}}_{t≥0} allows to consider more general situations, as for example the pricing problem of a derivative written in a foreign currency, referred as Quanto option. Consider a function:

f: (0, ∞) × (0, ∞) → IR

homogeneous of an arbitrary degree α; i.e. for any λ > 0 and for all positive x, y

f(λx, λy) = λ^{α}f(x, y).

In the above market a derivative contract with payoff given by

Φ_{t} = f(S_{t}^{2}, S_{t}^{3})

is introduced. Assuming that we are under a risk neutral martingale measure, thats to say, \frac{S^{k}}{S^{1}} (k = 2, 3) are P -martingales, i.e. P is an equivalent martingale measure (EMM)^{1}, we want to price the derivative contract just introduced. In the European case, the problem reduces to the computation of

E_{T} = E(S_{0}^{2}, S_{0}^{3}, T) = E [e^{−X_{T}^{1}} f(S_{0}^{2}e^{X_{T}^{2}} , S_{0}^{3}e^{X_{T}^{3}} )]  (4)

In the American case, if M_{T} denotes the class of stopping times up to time T, i.e:

M_{T} = {τ : 0 ≤ τ ≤ T, τ stopping time}

for the finite horizon case, putting T = ∞ for the perpetual case, the problem of pricing the American type derivative introduced consists in solving an optimal stopping problem, more precisely, in finding the value function A_{T} and an optimal stopping time τ^{∗} in M_{T} such that

A_{T} = A(S_{0}^{2}, S_{0}^{3}, T) = sup_{τ∈M_{T}} E [e^{−X_{τ}^{1}}f(S_{0}^{2}e^{X_{τ}^{2}}, S_{0}^{3}e^{X_{τ^{3}}^{3}})]
= E [e^{−X_{τ^{∗}}^{1}}f(S_{0}^{2}e^{X_{τ^{∗}}^{2}}, S_{0}^{3}e^{X_{τ^{∗}}^{3}})] .

2.3 Examples of Derivatives

In what follows we introduce some relevant derivatives as particular cases of the problem described. 2.3.1 Option to Default. Consider the derivative which has the payoff

f(x, y) = min{x, y}

if X^{1} = rt, then the value of the Option to Default a promise S_{T}^{1} backed by a collateral guarantee S_{T}^{2} , at the time T would be:

D = E [e^{−rT} min{S_{T}^{1} , S_{T}^{2} }]

2.3.2 Margrabe’s Options. Consider the following cases:

a) f(x, y) = max{x, y}, called the Maximum Option, b) f(x, y) = |x − y|, the Symmetric Option,

1See appendix

c) f(x, y) = min{(x − y)^{+}, ky}, the Option with Proportional Cap.

2.3.3 Swap Options. Consider

f(x, y) = (x − y)^{+},

obtaining the option to exchange one risky asset for another. 2.3.1Quanto Options. Consider

f(x, y) = (x − ky)^{+},

and take S_{t}^{2} = 1, then

E_{T} = Ee^{X_{T}^{1}} (S_{T}^{1} − k)^{+}

where e^{X_{T}^{1}} is the spot exchange rate ( foreign units/domestic units) and S_{T}^{1} is the foreign stock in foreign currency. Then we have the price of an option to exchange one foreign currency for another. 2.3.4 Equity-Linked Foreign Exchange Option (ELF-X Option). Take

S = S^{1} : foreign stock in foreign currency

and Q is the spot exchange rate. We use foreign market risk measure, then an ELF-X is an investment that combines a currency option with an equity forward. The owner has the option to buy S_{t} with domestic currency which can be converted from foreign currency using a previously stipulated strike exchange rate R (domestic currency/foreign currency). The payoff is:

Φ_{t} = S_{t}(1 − RQ_{t})^{+}

Then take S_{t}^{2} = 0 and f(x, y) = (y − Rx)^{+}. 2.3.5 Vanilla Options. Take

X_{t}^{1} = rt,

then in the call case we have

f(x, y) = (x − ky)^{+}

and

f(x, y) = (ky − x)^{+}

in the put case, with S_{t}^{3} = S_{0}^{3}e^{X_{t}} and S_{t}^{2} = 1.

3 Dual Market method

The main idea to solve the posed problems is the following: make a change of measure through Girsanov’s Theorem for Lévy processes, in order to reduce the original problems to a pricing problems for an auxiliary derivative written on one Lévy driven stock in an auxiliary market with deterministic interest rate. This method was used in Shepp and Shiryaev (1994) and Kramkov and Mordecki (1994) with the purpose of pricing American perpetual options with path dependent payoffs. It was employed by Aloisio and De Deus (1997) to consider the pricing of swaps, and is strongly related with the election of the numéraire (see Geman et al. (1995)). This auxiliary market will be called the Dual Market. More precisely, observe that

e^{−X_{t}^{1}}f(S_{0}^{2}e^{X_{t}^{2}}, S_{0}^{3}e^{X_{t}^{3}}) = e^{−X_{t}^{1}+αX_{t}^{3}}f(S_{0}^{2}e^{X_{t}^{2}−X_{t}^{3}}, S_{0}^{3}),

let ρ = − log Ee^{−X_{1}^{1}+αX_{1}^{3}}, that we assume finite. The process

Z_{t} = e^{−X_{t}^{1}+αX_{t}^{3}+ρt}  (5)

is a density process (i.e. a positive martingale starting at Z_{0} = 1) that allow us to introduce a new measure, the dual martingale measure, P̃ by its restrictions to each F_{t} by the formula

\frac{dP̃_{t}}{dP_{t}} = Z_{t}.

Denote now by X̃_{t} = X_{t}^{2} − X_{t}^{3}, and S_{t} = S_{0}^{2}eX̃_{t}. Finally, let

F (x) = f(x, S_{0}^{3}).

With the introduced notations, under the change of measure we obtain

E_{T} = Ẽ [e^{−ρT} F (S_{T})]

A_{T} = sup_{τ∈M_{T}} Ẽ [e^{−ρτ}F (S_{τ})]

The following step is to determine the law of the process X under the auxiliar probability measure P̃.

Lemma 3.1. Let X be a Lévy process on IR^{d} with characteristic exponent given in (1). Let u and v be vectors in IR^{d}. Assume that Ee^{(u,X_{1})} is finite, and denote ρ = − log Ee^{(u,X_{1})} = −ψ(u). In this conditions, introduce the probability measure P̃ by its restrictions P̃_{t} to each F_{t} by

\frac{dP̃_{t}}{dP_{t}} = exp[(u, X_{t}) + ρt].

Then

(a) the law of the unidimensional Lévy process {(v, X_{t})}_{t≥0} under P̃ is given by the triplet

\{ \begin{matrix}ã = (a, v) + \frac{1}{2}[(v, Σu) + (u, Σv)] + ∫_{IR^{d}} e^{(u,y)}(v, y)1_{\{|(v,y)|≤1,|x|>1\}}Π(dx) \\ σ̃^{2} = (v, Σv) \\ π̃(A) = ∫_{IR^{d}} 1_{\{(v,y)∈A\}}e^{(u,y)}Π(dy). \end{matrix}(6)

(b) In the particular case when X is a diffusion with jumps which characteristic exponent given in (2) the law of the unidimensional Lévy process {(v, X_{t})}_{t≥0} under P̃ is given by the triplet

\{ \begin{matrix}ã = (a, v) + \frac{1}{2}[(v, Σu) + (u, Σv)] \\ σ̃^{2} = (v, Σv) \\ π̃(A) = ∫_{IR^{d}} 1_{\{(v,y)∈A\}}e^{(u,y)}Π(dy). \end{matrix}  (7)

Furthermore, the intensity of the Poisson process under P̃ is given by

α̃ = ∫_{IR^{d}} e^{(u,y)}Π(dy) = α ∫_{IR^{d}} e^{(u,y)}F (dy)

(c) Assume (b), and let Π(dy) = αF (dy) where F is the common distribution of the random variables {Y_{k}}_{k≥1}, and has characteristic function (under P) given by

ϕ(z) = ∫_{IR^{d}} e^{(z,y)}F (dy).

Then, the characteristic function of the same random variables under P̃ is given by

ϕ̃(θ) = \frac{ϕ(θv + u)}{ϕ(u)}.  (8)

Remark: Consider a diffusion with gaussian jumps, in what can be considered as a multidimensional extension of the jump-diffusion model proposed by Merton (1976). Then, the characteristic function corresponding to the distribution of the jumps is given by

ϕ(z) = exp[(z, μ) + \frac{1}{2}(z, Δz)],

where the d-dimensional vector μ is the drift of the jumps, and the nonnegative definite matrix Δ is the covariance. According to (8), the characteristic exponent of the jumps of the process {(v, X_{t})}_{t≥0} under the probability measure P̃ in the Lemma 3.1 is given by

ϕ̃(θ) = \frac{ϕ(θv + u)}{ϕ(u)} = exp \{θ((v, μ) + \frac{1}{2}[(v, Δu) + (u, Δv)]) + \frac{1}{2}θ^{2}(v, Δv)\}. (9)

In conclusion, jumps under P̃ are also gaussian, with mean and variance obtained in (9)

Proof. First compute the expectation under P̃ as an expectation under P.

Ẽe^{θ(v,X_{t})} = Ee^{(u+θv,X_{t})+ρt} = exp{t[ψ(u + θv, X_{t}) − ψ(u)]}.

Now, compute the characteristic exponent of (v, X),

ψ(u + θv) − ψ(u) = (a, u + θv) − (a, u) + \frac{1}{2}[(u + θv, Σu + θv) − (u, Σu)

+ ∫_{IR^{d}} (e^{(u+θv,y)} − 1 − (u + θv, y)1_{{|y|≤1}})Π(dy)

− ∫_{IR^{d}} (e^{(u,y)} − 1 − (u, y)1_{{|y|≤1}})Π(dy)

= θ\{(a, v) + \frac{1}{2}[(v, Σu) + (u, Σv)]\} + \frac{1}{2}(v, Σv)

+ ∫_{IR^{d}} (e^{(u+θv,y)} − e^{(u,y)} − (θv, y)1_{{|y|≤1}})Π(dy)

= θ\{(a, v) + \frac{1}{2}[(v, Σu) + (u, Σv)] + ∫_{IR^{d}} e^{(u,y)}(v, y)1_{\{|(v,y)|≤1,|x|>1\}}Π(dx)\}

+ \frac{1}{2}(v, Σv) + ∫_{IR^{d}} (e^{(θv,y)} − 1 − (θv, y)1_{\{|y|≤1\}})e^{(u,y)}Π(dy)

giving (6). In what concerns (7), similar calculations give the result. Let us see (c). As the distribution of the jumps under P̃ is given by \frac{1}{α̃} π̃(dy),

ϕ̃(θ) = \frac{1}{α̃} ∫_{R} e^{θx}π̃(dx)

= \frac{α}{α̃} ∫_{IR} e^{(θv+u,y)}F (dy) = \frac{ϕ(θv + u)}{ϕ(u)}.

4 Examples

European derivative

Let X_{t}^{1} = rt and (X_{t}^{2}, X_{t}^{3}) be a bidimensional Lévy Process. We show how to obtain a formula for the value of an option to exchange one risky asset for another at the end of a determined period, as was considered by Margrabe (1978). Let S_{T}^{2} and S_{T}^{3} be two risky assets, a contract with payoff (S_{T}^{2} −S_{T}^{3} )^{+} can be priced using The Dual Market Method:

D = E [e^{−rT} (S_{T}^{2} − S_{T}^{3} )^{+}] .

= ∫_{A} e^{−rT} (S_{0}^{2}e^{X_{T}^{2}} − S_{0}^{3}e^{X_{T}^{3}} )dP

Assuming for simplicity S_{0}^{2} = S_{0}^{3} = 1, Then A = {ω ∈ Ω : X_{T}^{2}(ω) > X_{T}^{3}(ω)}, we proceed to applied the method:

D = ∫_{A} e^{−rT} (e^{X_{T}^{2}} − e^{X_{T}^{3}} )dP

= ∫_{{S_{T} >1}} e^{−rT} e^{X_{T}^{3}} (S_{T} − 1)dP

where S_{T} = e^{X_{T}} and X = X^{2} − X^{3}. Now, to use the dual measure, observe that ρ = − log Ee^{−r+X_{1}^{3}} = r − log Ee^{X_{1}^{3}}, then:

dP̃ = \frac{e^{X_{T}^{3}}}{Ee^{X_{T}^{3}}} dP

With all this:

D = e^{−ρT} ∫_{{S_{T} >1}}(S_{T} − 1)dP̃

D = e^{−ρT} ∫_{{S_{T} >1}} S_{T} dP̃ − e^{−ρT} ∫_{{S_{T} >1}} dP̃

Now to reduce this expression we need to assume a distribution for X and then apply the Lemma 3.1 to obtain the density of S_{T} under P̃. For example, assume a multidimensional version of the model proposed by Merton (1976), by part (c) of lemma 3.1 and taking u = (−1, 0, 1), v = (0, 1, −1),μ = (μ_{1}, μ_{2}, μ_{3}) and Δ = [σ_{ij}], we have that jumps under P̃ are also gaussian with mean (μ_{2} − μ_{3}) + (σ_{13} + σ_{23} − σ_{12} − σ_{3}^{2}) and variance σ_{2}^{2} + σ_{3}^{2} − 2σ_{23}. In this way an analogous formula to the one obtained by Merton (1976) can be obtained. It is worth noting that instead of assuming a distribution for X, we can make the following change of measure :

dP̂ = \frac{e^{X_{T}^{2}}}{Ee^{X_{T}^{2}}} dP

we obtain:

D = e^{−ρT} ∫_{\{S_{T} >1\}} e^{X_{T}^{2} −X_{T}^{3}} \frac{e^{X_{T}^{3}}}{Ee^{X_{T}^{3}}} dP − e^{−ρT} P̃(S_{T} > 1)

D = e^{−ρT} \frac{Ee^{X_{T}^{2}}}{Ee^{X_{T}^{3}}} P̂(S_{T} > 1) − e^{−ρT} P̃(S_{T} > 1)

which is analogous to the formula obtained by Gerber and Shiu (1994). The original formula obtained by Margrabe (1978) was extended by Stulz (1982) to contracts with payoffs:

(min{S_{2}(T), S_{3}(T)} − K)^{+}

And finally extended to the case of several assets by Johnson (1987). Unfortunately our method does not applied, since the payoffs are not longer homogenous of any degree, unless K = 0.

American derivative

Now consider an American perpetual swap, it is a derivative with the payoff function at any time t given by

f(S_{t}^{2}, S_{t}^{3}) = (S_{t}^{2} − S_{t}^{3})^{+}

then using the Dual market method, the pricing problem of this derivative would be :

A_{T} = sup_{τ∈M_{T}} Ẽ [e^{−ρτ}(S_{τ} − S_{0}^{3})^{+}] = Ẽ [e^{−ρτ^{∗}}(S_{τ^{∗}} − S_{0}^{3})^{+}] ,

and this problem can be solved using the following proposition

Proposition 4.1. Let M = sup_{o≤t≤τ} X_{t} with τ an independent exponential random variable with parameter ρ, then Ẽe^{M} < ∞ and

A(S_{0}^{2}, S_{0}^{3}) = \frac{Ẽ [S_{0}^{2}e^{M} − S_{0}^{3}Ẽ(e^{M})]}{Ẽ(e^{M})}

the optimal stopping time is

τ_{c}^{∗} = inf{t ≥ 0, S_{t} ≥ S_{0}^{3}Ẽ(e^{M})}

Proof. See Mordecki (2002).

5 Put-Call Duality and Symmetry

In this section we will obtain the put-call duality relationship as a particular case of our Lema 3.1 and then we will derive the put-call symmetry. Now consider a Lévy market where X_{t}^{1} = rt, X_{t}^{2} = 0 and X_{t}^{3} = X_{t}. In other words we have a riskless asset that we denote by B = {B_{t}}_{t≥0}, with

B_{t} = e^{rt}, r ≥ 0,

where we take B_{0} = 1 for simplicity, and a risky asset that we denote by S = {S_{t}}_{t≥0},

S_{t} = S_{0}e^{X_{t}},  S_{0} = e^{x} > 0.  (10)

In this section we assume that the stock pays dividends, with constant rate δ ≥ 0, and as in section 2, we assume that the probability measure P is the chosen equivalent martingale measure. In other words, prices are computed as expectations with respect to P, and the discounted and reinvested process {e^{−(r−δ)t}S_{t}} is a P –martingale. In terms of the characteristic exponent of the process this means that

ψ(1) = r − δ,  (11)

based on the fact, that Ee^{−(r−δ)t+X_{t}} = e^{−t(r−δ+ψ(1))} = 1, and condition (11) can also be formulated in terms of the characteristic triplet of the process X as

a = r − δ − σ^{2}/2 − ∫_{IR} (e^{y} − 1 − y1_{{|y|≤1}})Π(dy).  (12)

In the case, when

X_{t} = σW_{t} + at (t ≥ 0),  (13)

where W = {W_{t}}_{t≥0} is a Wiener process, we obtain the Black–Scholes– Merton (1973) model (see [4],[20]). In the market model considered we introduce some derivative assets. More precisely, we consider call and put options, of both European and American types. Let us assume that τ is a stopping time with respect to the given filtration F, that is τ: Ω → [0, ∞] belongs to F_{t} for all t ≥ 0; and introduce the notation

C(S_{0}, K, r, δ, τ, ψ) = Ee^{−rτ}(S_{τ} − K)^{+}  (14)
P(S_{0}, K, r, δ, τ, ψ) = Ee^{−rτ}(K − S_{τ})^{+}  (15)

If τ = T, where T is a fixed constant time, then formulas (14) and (15) give the price of the European call and put options respectively.

5.1 Put–Call duality

The following proposition presents a relationship that we have called Put-Call duality.

Proposition 1. Consider a Lévy market with driving process X with characteristic exponent ψ(z) given by

ψ(q) = iaq − \frac{1}{2}σ^{2}q^{2} + ∫_{IR} (e^{iqy} − 1 − iqy1_{\{|y|<1\}})Π(dy),  (16)

defined on the set

ℂ_{0} = {z = p + iq ∈ ℂ: ∫_{{|y|>1}} e^{py}Π(dy) < ∞}.  (17)

Then, for the expectations introduced in (14) and (15) we have

C(S_{0}, K, r, δ, τ, ψ) = P(K, S_{0}, δ, r, τ, ψ̃),  (18)

where

ψ̃(z) = ãz + \frac{1}{2}σ̃^{2}z^{2} + ∫_{IR} (e^{zy} − 1 − zy1_{\{|y|≤1\}})Π̃(dy)  (19)

is the characteristic exponent (of a certain Lévy process) that satisfies

ψ̃(z) = ψ(1 − z) − ψ(1), for 1 − z ∈ ℂ_{0},

and in consequence,

\begin{cases}ã & = δ − r − σ^{2}/2 − ∫_{IR} (e^{y} − 1 − y1_{\{|y|≤1\}})Π̃(dy), \\ σ̃ & = σ, \\ Π̃(dy) & = e^{−y}Π(−dy). \end{cases}  (20)

Proof. In this market the martingale Z = {Z_{t}}_{t≥0} defined by (5) is given by

Z_{t} = e^{X_{t}−(r−δ)t} (t ≥ 0).  (21)

As we have done in the latter section we introduce the dual martingale measure P̃ given by its restrictions P̃_{t} to F_{t} by

\frac{dP̃_{t}}{dP_{t}} = Z_{t},

where P_{t} is the restriction of P to F_{t}. Now

C(S_{0}, K, r, δ, τ, ψ) = Ee^{−rτ}(S_{0}e^{X_{τ}} − K)^{+} = EZ_{τ}e^{−δτ}(S_{0} − Ke^{−X_{τ}})^{+}
= Ẽe^{−δτ}(S_{0} − KeX̃_{τ})^{+}

where Ẽ denotes expectation with respect to P̃, and the process X̃ = {X̃_{t}}_{t≥0} given by X̃_{t} = −X_{t} (t ≥ 0) is the dual process (see [3]). In order to conclude the proof, that is, in order to verify that

Ẽe^{−δτ}(S_{0} − KeX̃_{τ})^{+} = P(K, S_{0}, δ, r, τ, ψ̃),

we must verify the dual process X̃ is a Lévy process with characteristic exponent defined by (19) and (20). To this end take u = (−1, 0, 1) and v = (0, 0 − 1) in Lemma 3.1 part (a). This concludes the proof.

Some remarks are in order. Our Proposition 1 is very similar to Proposition 1 in Schroder (1999). The main difference is that the particular structure of the underlying process (Lévy process are a particular case of the model considered in [25]) allows to completely characterize the distribution of the dual process X̃ under the dual martingale measure P̃, and to give a simpler proof. The proof of the proposition motivates us to introduce the following market model. Given a Lévy market with driving process characterized by ψ in (16), consider a market model with two assets, a deterministic savings account B̃ = {B̃_{t}}_{t≥0}, given by

B̃_{t} = e^{δt}, r ≥ 0,

and a stock S̃ = {S̃_{t}}_{t≥0}, modelled by

S̃_{t} = KeX̃_{t}, S_{0} = e^{x} > 0,

where X̃ = {X̃_{t}}_{t≥0} is a Lévy process with characteristic exponent under P̃ given by ψ̃ in (19). This market is the auxiliary market in Detemple (2001), and we call it dual market; accordingly, we call Put–Call duality the relation (18). It must be noticed that Peskir and Shiryaev (2001) propose the same denomination for a different relation in [23]. Finally observe, that in the dual market (i.e. with respect to P̃), the process {e^{−(δ−r)t}S̃_{t}} is a martingale. As a consequence, we obtain the Put–Call symmetry in the Black–Scholes– Merton model: In this case Π = 0, we have no jumps, and the characteristic exponents are

ψ(z) = (r − δ − σ^{2}/2)z + σ^{2}z^{2}/2,
ψ̃(z) = (δ − r − σ^{2}/2)z + σ^{2}z^{2}/2.

and relation (18) is the result known as put–call symmetry.

5.2 Symmetric markets

It is interesting to note, that in a market with no jumps the distribution (or laws) of the discounted (and reinvested) stocks in both the given and dual Lévy markets coincide. It is then natural to define a market to be symmetric when this relation hold, i.e. when

L(e^{−(r−δ)t+X_{t}} | P ) = L(e^{−(δ−r)t−X_{t}} | P̃),  (22)

meaning equality in law. In view of (20), and to the fact that the characteristic triplet determines the law of a Lévy processes, we obtain that a necessary and sufficient condition for (22) to hold is

Π(dy) = e^{−y}Π(−dy).  (23)

This ensures Π̃ = Π, and from this follows a − (r − δ) = ã − (δ − r), giving (22), as we always have σ̃ = σ. Condition (23) answers a question raised by Carr and Chesney (1996), see [5].

5.3 Examples and applications

In this section we consider that the Lévy measure of the process has the form

Π(dy) = e^{βy}Π_{0}(dy),

where Π_{0}(dy) is a symmetric measure, i.e. Π_{0}(dy) = Π_{0}(−dy). In many cases, the Lévy measure has a Radon-Nikodym density, and we have

Π(dy) = e^{βy}p(y)dy,  (24)

where p(x) = p(−x), that is, the function p(x) is even. In this way, we want to model the asymmetry of the market through the parameter β. As a consequence of (23), we obtain that when β = −1/2 we have a symmetric market. This proposal is similar, in certain sense, to the skewness premium introduced by Bates (1997) in [2]. The idea is to describe numerically the departure from the symmetry, the main difference with Bates (1997) is that the parameter β is a property of the market, independent of the derivative asset considered. It is also interesting to note, that practically all parametric models proposed in the literature, in what concerns Lévy markets, including diffusions with jumps, can be reparametrized in the form (24) (with the exception of Kou (2000), see anyhow Kou and Wang (2001)). Let us consider some examples

5.3.1 Generalized Hyperbolic Model

This model has been proposed by Eberlein and Prause (1998) as they allow for a more realistic description of asset returns (see [8]). This model has σ = 0, and a Lévy measure given by (24), with

p(y) = \frac{1}{|y|}( ∫_{0}^{∞} \frac{exp ( − \sqrt{2z + α^{2}}|y|)}{π^{2}z(J_{|λ|}^{2}(δ\sqrt{2z}) + Y_{|λ|}^{2}(δ\sqrt{2z}))}dz + 1_{\{λ≥0\}}λe^{−α|y|}),

where α, β, λ, δ are real parameters that satisfy the conditions 0 ≤ |β| < α, and δ > 0; and J_{λ}, Y_{λ} are the Bessel functions of the first and second kind (for details see [8]). Particular cases are the hyperbolic distribution, obtained when λ = 1; and the normal inverse gaussian when λ = −1/2. The statistical estimation β = −24.91 is given in [8] for the daily returns of the DAX (German stock index) for the period 15/12/93 to 26/11/97 (The other parameters are also estimated). This indicates the absence of symmetry.

5.3.2 The CGMY market model

This Lévy market model, proposed by Carr et al. (2002) in [6] is characterized by σ = 0 and Lévy measure given by (24), where the function p(y) is given by

p(y) = \frac{C}{|y|^{1+Y}} e^{−α|y|}.

The parameters satisfy C > 0, Y < 2, and G = α + β ≥ 0, M = α − β ≥ 0, where C, G, M, Y are the parameters used in [6]. For studying the presence of a pure diffusion component in the model, condition σ = 0 is relaxed, and risk neutral distribution are estimated in a five parameters model. Values of β = (G − M)/2 are given for different assets, and in the general situation, the parameter β is negative, and less than −1/2.

5.3.3 Diffusions with jumps

Consider the jump–diffusion model proposed by Merton (1976) in [21]. The driving Lévy process in this model has Lévy measure given by

Π(dy) = λ\frac{1}{δ\sqrt{2π}} e^{−(y−μ)^{2}/(2δ^{2})},

and is direct to verify that condition (23) holds if and only if 2μ + δ^{2} = 0. This result was obtained by Bates (1997) in [2]. The Lévy measure also corresponds to the form in (24), if we take β = μ/δ^{2}, and

p(y) = λ\frac{1}{δ\sqrt{2π}} e^{(−(y^{2}+μ^{2})/(2δ^{2}))}.

A recent alternative jump distribution was proposed by Kou and Wang (2001) in [17]. The Lévy measure has the form (24), where

p(y) = λe^{−α|y|}.

It can be observed that this is a particular case of the CGMY model, when Y = −1. In another model Eraker, Johansen and Polson (2000) introduce compound Poisson jumps into stochastic volatility processes, the Lévy measure is :

Π(dy) = \frac{λ}{η}e^{− \frac{y}{η}} dy, y > 0

which is also a particular case of CGMY model.

6 Conclusions

In this paper we have extended the results obtained by Gerber and Shiu (1996) for the bidimensional Geometric Brownian Motion to the case of bidimensional Geometric Lévy motion. We have shown that using the Dual market method it is possible to price many derivatives, with payoffs homogenous of any degree, written in terms of two assets driven by geometric Lévy motions, in the European case and for the American perpetual case. Another important fact in this paper is the possibility of having a stochastic discount, this allow us to consider derivatives as quanto derivatives. Also we have derived a put-call relation that we call put-call duality, different from the one obtained by Peskir and Shiryaev (2001), that allowed to obtain the put–call symmetry relation as a particular case. Then, we derive necessary and sufficient conditions for this symmetry to hold and verify if the symmetry is satisfied by some models used na literature. Many extensions of the above results are of interest, in particular the extension to derivatives on several assets.

7 Appendix

How to obtain an EMM

The procedure introduced in this section is in spirit of Gerber and Shiu (1994). Take the original probability measure P and suppose that relative prices \{\frac{S^{j}}{S^{1}} \} are not martingales under P, then define the following probability:

\frac{dP^{θ}}{dP} = \frac{e^{θ·X′}}{Ee^{θ·X′}}

Now we need that \{\frac{S_{t}^{j}}{S_{t}^{1}} \} be a martingale under P^{θ} for some θ, as S_{0}^{1} = 1, then is enough to prove that

S_{0}^{j} = E^{θ}(\frac{S_{t}^{j}}{S_{t}^{1}} ) ∀j, ∀t

1 = E^{θ}(e^{X_{t}^{j}−X_{t}^{1}})

Defining ¯1_{j} = (−1, 0, …, \underbrace{1}_{j−position}, …, 0), we have

1 = M(¯1_{j}, 1; θ),  (25)

where M(z, t; θ) = \frac{M(z+θ,t)}{M(θ,t)} and M(θ, t) = E(e^{θ·X′_{t}}). From the solution of (25) we have the EMM.

References

[1] Araujo, A., and R. Oliveira (1997): “On the Pricing of European and American Swaps Options Pricing”. IMPA Preprint B 122 (1997).

[2] Bates, D.S. (1997): “ The skewness premium: Option Pricing under Asymmetric Processes. Advances in Futures and Options Research”, 9, 51–82.

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

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

[5] Carr, P. and Chesney, M. (1996): “ American Put Call Symetry”. Preprint.

[6] Carr, P., Geman, H., Madan, D.B.,and Yor, M. (2002): “ The Fine Structure of Assets Returns: An Empirical Investigation”. Journal of Business, vol 75, no.2, 305–332.

[7] Detemple, J. (2001): “American options: symmetry property”. In: Option pricing, interest rates and risk management Eds. Jouini, Cvitanic, Musiela. Cambridge University Press, 67–104.

[8] Eberlein, E., and Prause, K. (1998): “The Generalized Hyperbolic Model: Financial Derivatives and Risk Measures”. Universität Freibug i. Br. Preprint 56, November.

[9] Eraker, B., Johannes, M., and Polson, N. (2000): “The Impact of Jumps in Volatility and Returns”. Manuscript. University of Chicago.

[10] Geman, H., El Karoui, N., and Rochet, J. (1995): “Changes of numéraire, changes of probability measure and option pricing,” J. Appl. Probab, 32, no. 2, 443–458.

[11] Gerber, H. U., and E. S. W. Shiu, (1994): “Option Pricing by Esscher-transform”, Transactions of the Society of Actuaries, vol. 46, 99-191.

[12] Gerber, H. U., and E. S. W. Shiu (1996): “Martingale Approach to Pricing Perpetual American Options on Two Stocks”. Math. Finance, vol. 6, no. 3, 303–322.

[13] Jacka, S.D. (1991): “Optimal stopping and the American put,” Math. Finance, 1, No.2, 1–14.

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

[15] Johnson, H. (1987): “Options on the Maximum and Minimum of Several Assets”. J. of Financial and Quantitative Analysis, 22, 277–283.

[16] Kou, S.G., (2000): “A Jump Diffusion Model for Option Pricing with three Properties: Leptokurtic Feature, Volatility Smile, and Analytical Tractability”. Preprint.

[17] Kou, S.G. and Wang H., (2001): “ Option pricing Under a Double Exponential Jump Diffusion Model”. Preprint.

[18] Kramkov, D. O., and Mordecki, E. (1994): “An integral option”. (Russian) Teor. Veroyatnost. i Primenen. 39, no. 1, 201–211; translation in Theory Probab. Appl. 39, no. 1, 162–172 (1995)

[19] Margrabe, W. (1978): “The Value of an Option to Exchange One Asset for Another,” J. Finance, 33, 177–186.

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

[21] Merton, R.C. (1976): “Option pricing when the underlying stock returns are discontinuous,” J. of Financial Economics, 3, 125–144.

[22] Mordecki, E. (2002) “Optimal Stopping and Perpetual Options for Lévy processes,”Finance and Stochastics, 4, 473-493

[23] Peskir, G., and A. N. Shiryaev (2001): “A note on the call-put parity and a call-put duality”. Theory Probab. Appl. 46, 181–183.

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

[25] Schroder, Mark. (1999): “Change of Numeraire for Pricing Futures, Forwards, and Options”. The Review of Financial Studies, vol. 12, No. 5, 1143–1163.

[26] Shepp,L. A., and A. N. Shiryaev (1994): “A new look at the Russian option”. (Russian) Teor. Veroyatnost. i Primenen. 39, no. 1,130–149; translation in Theory Probab. Appl. 39, no. 1, 103–119 (1995)

[27] Shiryaev, A.N., Kabanov, Y. M., Kramkov, D.O., and Melnikov, A.V. (1994): “On the pricing of options of European and American types, II. Continuous time”. Theory of Probability and Applications, 39, 80–129.

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

[29] Stulz, R. (1982): “Options on the Minimum or the Maximum of two Risky Assets”. J. of Financial Economics, 10, 161–185.