No-arbitrage, change of measure and conditional Esscher transforms

Hans Bühlmann  Freddy Delbaen

Paul Embrechts Departement of Mathematics, ETH Zürich

Albert N. Shiryaev Steklov Mathematical Institute, Moscow

1 Introduction

This paper grew out of a seminar at the Department of Mathematics at the ETH, Zürich during the Summer Semester of 1995 on the subject of mathematical finance and insurance mathematics. It should be viewed as a contribution towards bridging the existing methodological gap between both fields, especially in the area of pricing derivative instruments. Both insurance and finance are interested in the fair pricing of financial products. For instance, in the case of car insurance, depending on the various characteristics of the driver, a so-called net premium is calculated which should cover the ecpected losses over the period of the contract. To this net premium, various loading factors (for costs, fluctuations, …) are added. The resulting gross premium is also subject to market forces which imply that a market-conform premium is finally charged. The more an insurance market is liquid (many potential offers of insurance, deregulated markets), the more a “correct, fair” price may be expected to emerge. Very important in the process of determining the above premium is the attitude of both parties involved towards risk. Within the more economic literature this attitude towards risk can be described through the notion of utility. Utility theory enters as a tool to provide insight into decision making in the face of uncertainty. For a very readable introduction within the context of insurance, see Bowers et al. (1989). An alternative economic tool is equilibrium theory. Depending on the economic theory used, a multitude of possible premiums may result, one of which is the time-honoured Esscher principle. Rather than being based on the expected loss itself, the Esscher principle starts from the expectation of the loss under an exponentially transformed distribution, properly normalised. In Bühlmann (1980, 1983), the Esscher principle is discussed within the utility and equilibrium framework. Besides the pricing of individual risks (claims, say), more complicated insurance products involve time and hence are based on specific stochastic processes. The classical insurance risk processes are of the compound Poisson type or their generalisations like mixed and doubly stochastic compound Poisson processes. The main feature of such processes, making them distinct from the typical diffusion type models in finance, is their jump structure. Indeed, when we turn to fair pricing in finance, the standard reasoning uses the socalled no-arbitrage (or no free lunch) approach which says that there is no such thing as a riskless gain. The precise mathematical formulation of this economic principle brings in the by now fundamental notion of risk neutral martingale measure. In the case where the underlying stochastic process is “nice” (geometric Brownian motion say), exactly one such measure exists and the fair price of a contingent claim is the expectation with respect to this measure, properly discounted. The latter, so-called complete case is rare in insurance. Due to the jump structure of standard risk processes, we are in the so-called incomplete case. As a consequence, risk cannot fully be hedged away and in most cases, there will be infinitely many such equivalent martingale measures so that pricing is directly linked to an attitude towards risk. Whereas in classical insurance, the question becomes “which premium principle to use,” within the (incomplete) finance context it becomes “which equivalent martingale measure to use.” This is exactly the point where the Esscher transform enters as one of the possible pricing candidates. Going back to a fundamental paper of Esscher (1932), the Esscher transform is by now standard methodology in insurance, gradually however its appearance within mathematical finance is becoming more and more prominent: see for instance the beautiful paper by Gerber and Shiu (1994) and the references and discussions therein. An interesting paper, coming more from the realm of mathematical finance is Grandits (1996). The present paper should be looked at in conjunction with Bühlmann et. al. (1996) where special attention is given to discrete models. As explained above, typical insurance processes involve a jump component besides a possible diffusion term. It is therefore natural to present the necessary mathematical methodology needed for discussing pricing within both insurance and finance within the wider theory of semi-martingales. This is exactly what is done in the present paper. The classical notion of Esscher transform for distribution functions is generalised to stochastic processes. For a discussion of Esscher transform in a distributional context, see Jensen (1995). In Embrechts et. al. (1985) an application to the approximation of the total claim amount distribution in the compound Poisson and negative binomial case is given.

1.1 Some notation

Suppose that a financial process (stock returns, spot rates, zero coupon bonds, value of a derivative instrument, …) S = (S_{t})_{t≥0} is given on a filtered probability space (Ω, F, (F_{t})_{t≥0}, P) where 𝔽 = (F_{t})_{t≥0} denotes the “flow of information.” Mathematically the latter means that 𝔽 consists of an increasing family of sub σ- algebras, i.e. for all s ≤ t, F_{s} ⊂ F_{t} ⊂ F. Assume further that S is of “exponential form,”

S_{t} = S_{0} e^{H_{t}} , H_{0} = 0, t ≥ 0,  (1)

where H = (H_{t})_{t≥0} is a semimartingale with respect to 𝔽 and P. The latter will be denoted by H ∈ Sem (𝔽, P) or H ∈ Sem (P).We remark that the notion of semimartingale does not depend on the measure P. More precisely, if Q ∼ P are two equivalent probability measures, then Sem (P) = Sem (Q). For a precise definition see for instance Jacod and Shiryaev (1987) and Rogers and Williams (1987). Using Itô’s formula for f ∈ C^{2}, one obtains:

f(H_{t}) = f(H_{0}) + ∫_{0}^{t} f′(H_{s−})dH_{s} + \frac{1}{2} ∫_{0}^{t} f′′(H_{s−})d⟨H^{c}⟩_{s}

+ ∑_{0 < s ≤ t} [f(H_{s}) − f(H_{s−}) − f′(H_{s−})ΔH_{s}]  (2)

where ΔH_{s} = H_{s} − H_{s−} and ⟨H^{c}⟩ is a quadratic characteristic of the continuous martingale part H^{c} of H. Hence for the case (1) above:

dS_{t} = S_{t−}dĤ_{t}  (3)

with

Ĥ_{t} = H_{t} + \frac{1}{2}⟨H^{c}⟩_{t} + ∑_{0<s≤t}(e^{ΔH_{s}} − 1 − ΔH_{s}).  (4)

In the class of semimartingales the linear equation (3) has a unique solution:

S_{t} = S_{0} E(Ĥ)_{t}  (5)

where E(Ĥ) is called the Doléans stochastic exponential

E(Ĥ)_{t} = exp \{Ĥ_{t} − \frac{1}{2}⟨Ĥ^{c}⟩_{t}\} ∏_{0<s≤t}(1 + ΔĤ_{s})e^{−ΔĤ_{s}}.  (6)

It should be remarked that for every semimartingale H = (H_{t}), with probability one,

∑_{0 < s ≤ t} |ΔH_{s}|^{2} < ∞, ∀t > 0.  (7)

From (7) it immediately follows that for each t > 0, there are only finitely many time points s ≤ t such that |ΔH_{s}| > \frac{1}{2}. Consequently, the infinite sums and products in (4) and (6) are absolutely convergent and hence Ĥ and E(Ĥ) are well defined.

1.2 Discrete time

Consider the set-up (1) but now in discrete time,

S_{n} = S_{0} e^{H_{n}}, H_{0} = 0, n = 0, 1, 2, ⋯  (8)

where H = (H_{n})_{n ≥ 0} is a stochastic sequence defined on a filtered probability space (Ω, F, (F_{n})_{n ≥ 0}, P). Clearly, (8) can formally be considered as a special case of (1) by defining

F_{t} = F_{n} , H_{t} = H_{n} , n ≤ t < n + 1.

Put

Ĥ_{n} = ∑_{0 < k ≤ n}(e^{Δ H_{k}} − 1)  (9)

(to be compared with (4)), then we obtain

S_{n} = S_{0} ∏_{0 < k ≤ n}(1 + ΔĤ_{k}) = S_{0}E(Ĥ)_{n}.  (10)

The latter should be compared with (5) and (6). In the sequel we denote

h_{k} = ΔH_{k}(= H_{k} − H_{k−1})

and

ĥ_{k} = ΔĤ_{k}(= Ĥ_{k} − Ĥ_{k−1}).

Recall that

h_{k} = ln\frac{S_{k}}{S_{k−1}}

and hence can be viewed as a compound return, whereas

ĥ_{k} = \frac{S_{k}}{S_{k−1}} − 1 = \frac{ΔS_{k}}{S_{k−1}} = e^{h_{k}} − 1

stands for simple return. Using this terminology and the correspondances stated above, (1) can be viewed as a continuous model for compound return, whereas (5) is the continuous analogon of simple return. It is useful to remark that the representation (1) lends itself naturally for statistical data analysis. However, with respect to probabilistic analysis, the representation (5) turns out to be more advantageous. An example of the latter is the following: E(Ĥ) is a local martingale if Ĥ is a local martingale.

1.3 No-arbitrage and equivalent martingale measures.

The “equivalence,” of the notions no-arbitrage, no free lunch and the existence of equivalent martingale measures belongs to the folklore of mathematical finance. The key underlying idea is the local equivalence of martingale measures, i.e. P̃ ∼^{loc} P on (Ω, F) meaning that for each t > 0, P̃_{t} ∼ P_{t} (equivalence of probability measures) where P_{t} = P |F_{t}, P̃_{t} = P̃|F_{t} and such that S = (S_{t}) is a martingale or local martingale with respect to P̃.

In discrete time, n = 0, 1, …, N, the precise formulation of the above is as follows. Equivalent are

(a) no-arbitrage, and

(b) there exists a probability measure P̃ on (Ω, F) so that P̃_{N} ∼ P_{N} and S = (S_{n})_{n ≤ N} is a P̃_{N} -martingale.

In the continuous time case, the situation is much more delicate. A solution is to be found in Delbaen and Schachermayer (1994) and (1996) and the references therein. Independent of the precise equivalence statements, the construction of all equivalent martingale measures in a particular situation is important. A slightly less ambitious goal would be the construction of certain subclasses. The main aim of our paper is exactly the solution of this technical problem. We shall also discover the so-called conditional Esscher transform as a special case of the change of measure paradigm in stochastic calculus.

2 Some facts about semimartingales

2.1 Definition

Below we summarise the basic definitions and results concerning semimartingale theory of relevance in insurance and finance. The càdlàg (right-contin-uous with left limits) stochastic process H = (H_{t})_{t ≥ 0} defined on a filtered probability space (Ω, F, (F_{t})_{t ≥ 0}, P) is a semimartingale if H admits a canonical decomposition

H_{t} = H_{0} + A_{t} + M_{t}, t ≥ 0,  (11)

where A = (A_{t}) ∈ V (a process of bounded variation), M = (M_{t}) ∈ M_{loc} (a local martingale). Furthermore, we have that for each t ≥ 0, A_{t} and M_{t} are F_{t}-measurable.

We recall that M ∈ M_{loc} if and only if there exists a sequence of (F_{t})_{t ≥ 0}-stopping times (τ_{n})_{n ≥ 1} such that τ_{n} ↑ ∞ (P -a.s.) for n → ∞ and for each n ≥ 1, the stopped process

M^{τ_{n}} = (M_{t}^{τ_{n}}) with M_{t}^{τ_{n}} = M_{t ∧ τ _{n}}, n ≥ 1,

is a martingale:

E|M_{t}^{τ_{n}}| < ∞, E(M_{t}^{τ_{n}}|F_{s}) = M_{s}^{τ_{n}} (P − a.s.), s ≤ t.

We would like to stress that local martingales are more than just martingale modulo boundedness conditions. Indeed, there exist local martingales possessing strong integrability properties which nonetheless are not martingales. See for instance Revuz and Yor (1994), Chapter V, Exercise (2.13) where a local martingale is given, bounded in L^{2}, but which is not a martingale. In the case of discrete time, we have the following nice characterisation of local martingales; see for instance Jacod and Shiryaev (1987), Chapter 1, 1.64 or Liptser and Shiryaev (1986), Chapter VII, §1. Let X = (X_{n})_{n ≥ 0} be a stochastic sequence defined on a filtered probability space (Ω, F, (F_{n})_{n ≥ 0}, P). X is assumed adapted, i.e. X_{n} is F_{n}-measurable for all n ≥ 0 and E|X_{0}| < ∞. Then the following conditions are equivalent:

(1) X is a local martingale,

(2) X is a martingale transformation, i.e. there exists a martingale Y = (Y_{n}) and a predictable sequence V = (V_{n}) (meaning that for each n ≥ 1, V_{n} is F_{n−1}-measurable) such that for n ≥ 1:

X_{n} = X_{0} + ∑_{0 < k ≤ n} V_{k}Δ Y_{k}, ΔY_{k} = Y_{k} − Y_{k−1},

(3) X is a generalised martingale, i.e.

E(|X_{n}| |F_{n−1}) < ∞, n ≥ 1,

and

E(X_{n}| F_{n−1}) = X_{n−1}.

(The key point in the latter conditions is that we do not assume integrability of X_{n}, n ≥ 1)

Remark: The condition (2) above can be interpreted as X_{n} is the value of a trading strategy V on an underlying asset Y. This shows that the notion of local martingales lies at the heart of stochastic processes in finance and insurance. Unfortunately, the continuous time analogue of the above result is false.

2.2 Semimartingale representations

Denote by μ = μ(ω; ds, dx)(or dμ) the measure describing the jump structure of H:

μ(ω; (0, t,] × A) = ∑_{0 < s ≤ t} I(ΔH_{s}(ω) ∈ A), t > 0.

where A ∈ B(ℝ \ {0}), ΔH_{s} = H_{s} − H_{s−} and I(·) stands for the indicator function. By ν = ν(ω; ds, dx) (or dv) we denote a compensator of μ, i.e. a predictable measure (see Jacod and Shiryaev (1987), Chapter II, 1.8) with the property that μ − ν is a local martingale measure. This means that for each A ∈ B(ℝ \ {0}):

(μ(ω; (0, t] × A) − ν(ω; (0, t] × A))_{t > 0}

is a local martingale with value 0 for t = 0. The latter property is almost equivalent to the local martingale property of the signed measure μ − ν. We shall not enter into the subtle difference here.

A semimartingale H = (H_{t})_{t ≥ 0} is called special if there exists a decomposition (11) with a precticable process A = (A_{t})_{t ≥ 0}. See Jacod and Shiryaev (1987) where it is also shown that every semimartingale with bounded jumps (|ΔH_{t}(ω)| ≤ b < ∞, ω ∈ Ω, t > 0) is special.

Let φ be a truncation function, e.g. φ(x) = xI(|x| ≤ 1). Then ΔH_{s} − φ(ΔH_{s}) ≠ 0 if and only if |ΔH_{s}| > b for some b > 0. Hence

H^{∨}(φ)_{t} = ∑_{0 < s ≤ t} (ΔH_{s} − φ(ΔH_{s}))

denotes the jump part of H corresponding to big jumps. The number of the latter is still finite on [0, t], for all t > 0, because for all semimartingales

∑_{0 < s ≤ t}(ΔH_{s})^{2} < ∞, P − a.s.

The process H(φ) = H −H^{∨}(φ) is a semimartingale with bounded jumps and hence it is special:

H(φ)_{t} = H_{0} + B(φ)_{t} + M(φ)_{t}  (12)

where B(φ) is a predictable process and M(φ) is a local martingale. Every local martingale M(φ) can be decomposed as:

M(φ) = M^{c}(φ) + M^{d}(φ)  (13)

where M^{c}(φ) is a continuous (martingale) part and M^{d}(φ) is a purely discontinuous (martingale) part,

M^{d}(φ)_{t} = ∫_{0}^{t} ∫ φ(x)d(μ − ν).  (14)

More details, including a proof of (14), are to be found in Jacod and Shiryaev (1987), Chapter II, 2.34. It is clear that

H^{∨}(φ)_{t} = ∫_{0}^{t} ∫ (x − φ(x))dμ.  (15)

Consequently H has the following canonical representation:

H_{t} = H_{0} + B(φ)_{t} + M^{c}(φ)_{t} +

∫_{0}^{t} ∫ φ(x)d(μ − ν) + ∫_{0}^{t} ∫ (x − φ(x))dμ, (16)

a formula going back to Lévy and Khintchin. The continuous martingale part M^{c}(φ) does not depend on φ and will be denoted by H^{c} (the continuous martingale part of H). Consequently,

H_{t} = H_{0} + B(φ)_{t} + H_{t}^{c} + ∫_{0}^{t} ∫ φ(x)d(μ − ν)
+ ∫_{0}^{t} ∫ (x − φ(x))dμ.  (17)

Denote by ⟨H^{c}⟩ a predictable quadratic characteristic of H^{c}, i.e. (H^{c})^{2} − ⟨H^{c}⟩ is a local martingale.

We finally arrive at the triplet of predictable characteristics of the semimartingale H:

T (φ) = (B(φ), ⟨H^{c}⟩, ν).

In the case φ(x) = xI(|x| ≤ 1) we denote B = B(φ). Then (17) takes on the form:

H_{t} = H_{0} + B_{t} + H_{t}^{c} + ∫_{0}^{t} ∫_{|x| ≤ 1} xd(μ − ν)
+ ∫_{0}^{t} ∫_{|x| > 1} xdμ.  (18)

In Jacod and Shiryaev (1987), Chapter II, 2 it is shown that if H is a semimartingale, then

ΔB(φ)_{t}(ω) = ∫ φ(x)ν(ω; {t} × dx),

where

ν(ω; {t} × dx) = ν(ω; (0, t] × dx) − ν(ω; (0, t) × dx)

and

(x^{2} ∧ 1) ∗ ν ∈ A_{loc},

i.e. the process (∫_{0}^{t} ∫ (x^{2} ∧ 1)dν)_{t ≥ 0} is locally integrable in so far that there exist stopping times τ_{n} ↑ ∞ as n → ∞, such that for n ≥ 1

E( ∫_{0}^{τ_{n}} ∫ (x^{2} ∧ 1)dν) < ∞.

Using this notation, H turns out to be a special semimartingale if and only if

(x^{2} ∧ |x|) ∗ ν ∈ A_{loc}.

Further, H is a square integrable semimartingale if and only if

x^{2} ∗ ν ∈ A_{loc}.

If H is a special semimartingale, then the canonical representation (17) is valid with φ(x) = x, i.e.

H_{t} = H_{0} + B_{t} + H_{t}^{c} + ∫_{0}^{t} ∫ xd(μ − ν), t ≥ 0,  (19)

with B = B(φ).

There are various reasons why semimartingales play a fundamental role in insurance and finance (and indeed in many more applications):

(i) They form a wide class of processes including stochastic sequences in discrete time, martingales, super- and sub-martingales, diffusion processes, diffusions with jumps, processes with independent increments (if for every λ ∈ ℝ, (Ee^{iλH_{t}})_{t ≥ 0} has bounded variation). This is especially important in the intersection of insurance and finance where models involving both a diffusion component as well as a jump component are relevant.

(ii) They form the most general class of stochastic processes for which a stochastic integration theory can be worked out, the latter is a consequence of the famous Bichteler, Dellacherie, Kussmaul, Métivier and Pellaumail theorem (see Rogers and Willliams (1987), Section IV. 16). A full stochastic calculus, including Itô’s lemma for semimartingales exists.

(iii) The knowledge that a stochastic process is not a semimartingale may have important implications in finance in so far that then often explicit arbitrage strategies can be worked out. A typical example concerning so-called fractional Brownian motion is to be found in Rogers (1995). See also Delbaen and Schachermayer (1994) where it is shown that a very weak form of the no-arbitrage property implies that the price process is already a semimartingale.

2.3 Examples

2.3.1 Discrete time

In this case we don’t really need the heavy semimartingale machinery, we only include this case for illustrative purposes. Consider the stochastic sequence H = (H_{n})_{n ≥ 0} with h_{n} = ΔH_{n} = H_{n} − H_{n−1}. Hence

H_{n} = H_{0} + ∑_{0 < k ≤ n} h_{k}
= H_{0} + ∑_{0 < k ≤ n} φ(h_{k}) + ∑_{0 < k ≤ n}(h_{k} − φ(h_{k}))  (20)
= H_{0} + ∑_{0 < k ≤ n} E[φ(h_{k})|F_{k−1}]
+ ∑_{0 < k ≤ n}(φ(h_{k}) − E[φ(h_{k})|F_{k−1}]) + ∑_{0 < k ≤ n}(h_{k} − φ(h_{k})).

Define for all A ∈ B(ℝ\ {0}), k ≥ 0 :

μ_{k}(A) = I(h_{k} ∈ A) = I(ΔH_{k} ∈ A),
ν_{k}(A) = E[I(h_{k} ∈ A)|F_{k−1}] = P (h_{k} ∈ A|F_{k−1}),

where conditional expectations are always taken as regular versions. Then

μ(ω; (0, n] × A) = ∑_{0 < k ≤ n} μ_{k}(A),
ν(ω; (0, n] × A) = ∑_{0 < k ≤ n} ν_{k}(A),

yielding the canonical representation (see (17))

H_{n} = H_{0} + B(φ)_{n} + ∑_{0 < k ≤ n} ∫ φ(x)d(μ_{k} − ν_{k})

+ ∑_{0 < k ≤ n} ∫ (x − φ(x))dμ_{k},  (21)

where

B(φ)_{n} = ∑_{0 < k ≤ n} ∫ φ(x)dν_{k}.  (22)

(We could have written ν_{k}(dx) for dν_{k} etc.) Because there is no continuous part, the characteristic triplet reduces to

T (φ) = (B(φ), 0, ν)  (23)

where

B(φ) = (B(φ)_{n})_{n ≥ 0}, ν = (ν_{n})_{n ≥ 1}.

2.3.2 Processes with independent increments (I.I.)

A process H = (H_{t})_{t ≥ 0} with I.I. is a semimartingale if and only if for each λ ∈ ℝ, (Ee^{i λ H_{t}})_{t ≥ 0} is a function of bounded variation. For a proof, see Jacod and Shiryaev (1987), Chapter II, 4.14. A remarkable fact for such processes is that their triplet of predictable characteristics only has deterministic components. If H = (H_{t}) is continuous in probability, then B(φ)_{t}, < H^{c} >_{t} and ν((0, t] × dx) are continuous in t and the Lévy–Khintchin formula yields

E exp\{iλ(H_{t} − H_{0})\} = exp \{iλB(φ)_{t} − \frac{λ^{2}}{2} C_{t}

+ ∫_{0}^{t} ∫ (e^{iλx} − 1 − iλφ(x))ν(ds × dx)} (24)

where C_{t} =< H^{c} >_{t} is the variance of the continuous Gaussian part of H, and B(φ) and ν are the first and third component in the triplet T (φ) = (B(φ), < H^{c} >, ν) of H written in semimartingale form. If the I.I. process is moreover homogeneous (stationary), also referred to as a Lévy process, then

B(φ)_{t} = tb(φ)
C_{t} = tC  (25)
ν(dt × dx) = dt × F (dx)

where F is a distribution function on ℝ. For a textbook treatment of Lévy processes, see Bertoin (1996). Hence in this case the triplet T (φ) is reduced to (b(φ), C, F (dx)).

2.3.3 Brownian motion with drift and Poisson jumps

Suppose that

H_{t} = bt + σW_{t} + ∑_{k=1}^{N_{t}} ξ_{k}  (26)

where ξ, ξ_{1}, ξ_{2}, … are i.i.d. random variables with F (x) = P (ξ ≤ x), N = (N_{t})_{t ≥ 0} is a homogeneous Poisson process with intensity λ > 0, and W = (W_{t})_{t ≥ 0} is standard Brownian motion. Suppose furthermore that the processes W, N and (ξ_{i}) are jointly independent. In this formulation, H in (26) in the recent literature either occurs as a classical risk process perturbed by Browian motion (see Gerber (1970)) or as a model for catastrophic insurance futures (see for instance Cummins and Geman (1995)) Then

H_{t} = bt + σW_{t} + ∑_{k=1}^{N_{t}} ξ_{k}  (27)

= bt + σW_{t} + ∫_{0}^{t} ∫ xdμ
= (bt + ∫_{0}^{t} ∫ φ(x)dν) + (σW_{t} + ∫_{0}^{t} ∫ φ(x)d(μ − ν))
+(∫_{0}^{t} ∫ (x − φ(x))dμ)
= t(b + λ ∫ φ(x)F (dx)) + (σW_{t} + ∫_{0}^{t} ∫ φ(x)d(μ − ν))
+(∫_{0}^{t} ∫ (x − φ(x))dμ).

Consequently,

T (φ) = (B(φ), ⟨H^{c}⟩, ν)

where

B(φ)_{t} = t(b + λ ∫ φ(x)F (dx)),
< H^{c} >_{t} = σ^{2}t,  (28)
dν = λdtF (dx).

2.3.4 Diffusion processes with jumps.

These processes can be viewed as semimartingales with predictable characteristic triplet T (φ) = (B(φ), C, ν) where

B(φ)_{t} = ∫_{0}^{t} b(s, H_{s})ds, (b = B_{φ}),
C_{t} = ∫_{0}^{t} C(s, H_{s})ds  (29)
ν(ω; dt × dx) = dt × K_{t}(H_{s}(ω), dx),

where K_{t}(x, dy) is a Borel transition kernel from ℝ_{+} × ℝ in ℝ; see Jacod and Shiryaev (1987), Chapter III, 2.

2.4 Conditional Esscher transforms

Consider a semimartingale H = (H_{t})_{t ≥ 0} with triplet T = (B, C, ν) where we dropped for notational convenience the dependence on φ. Also for simplicity, we take φ(x) = xI(|x| ≤ 1). We first introduce the cummulant process A(u) = (A(u)_{t})_{t ≥ 0} associated with H:

A(u)_{t} = iuB_{t} − \frac{1}{2}u^{2}C_{t} + ∫ (e^{i u x} − 1 − iuφ(x))ν((0, t] × dx).  (30)

Suppose that ΔA(u) ≠ −1, then the stochastic exponential G(u) = E(A(u)) defined in (6) cannot take zero values. Now define the process

X_{t}(u) = \frac{e^{i u H_{t}}}{E(A(u))_{t}} , t ≥ 0.  (31)

An important property of semimartingales is the following characterisation:

H is a semimartingale with triplet (B, C, ν)

if and only if  (32)

X = (X_{t}(u))_{t≥0} is a local martingale for every u ∈ ℝ;

see Jacod and Shiryaev (1987), Chapter II, 2.49. For discrete time processes,

(30) reduces to

ΔA(u)_{n} = ∫ (e^{i u x} − 1)ν_{n}(dx) = E(e^{iuh^{n}} − 1|F_{n−1}).  (33)

However, in this case (6) implies that

E(A(u))_{n} = ∏_{0 < k ≤ n}(1 + ΔA(u)_{k}),

= ∏_{0 < k ≤ n} E(e^{iuh^{n}}|F_{n−1}).

Hence in discrete time for a stochastic sequence H = (H_{n}) with ΔH_{n} = h_{n} and so that E(e^{iuh^{n}}|F_{n−1}) ≠ 0, n ≥ 1, the sequence

(\frac{e^{i u H_{n}}}{∏_{0 < k ≤ n} E(e^{i u h_{k}}|F_{k−1})})_{n ≥ 1}  (34)

is a local martingale. Of course we don’t need the deep characterisation result (32) in order to prove (34), a more direct argument can be given in this case. Similarly, suppose Ee^{a_{k}}^{h_{k}} < ∞, k ≥ 1, for some constants a_{1}, a_{2}, …, then the sequence Z = (Z_{n})_{n ≥ 1} with Z_{0} = 1 and

Z_{n} = ∏_{k ≤ n} \frac{e^{a_{k}}^{h_{k}}}{E(e^{a_{k}}^{h_{k}}|F_{k−1})} , n ≥ 1 ,  (35)

is a martingale. The latter follows immediately from the adaptiveness of H and elementary properties of conditional expectation. Property (35) allows us to construct a family of measures {P̃_{N}} such that dP̃_{N} = Z_{N}dP_{N} and P̃_{N} = P̃_{N+1}|F_{N}. The conditional distribution

P̃_{N}(h_{N} ∈ A|F_{N−1}) = E[I_{A}(h_{N})\frac{e^{a_{N}} ^{h_{N}}}{E(e^{a_{N}} ^{h_{N}} |F_{N−1})}|F_{N−1}]  (36)

is called the conditional Esscher transform. In the traditional actuarial context, the h_{i}’s are independent and hence (36) reduces to an unconditional expectation, the Esscher transform:

P̃_{N}(h_{N} ∈ A) = E[I_{A}(h_{N})\frac{e^{a_{N}} ^{h_{N}}}{Ee^{a_{N}} ^{h_{N}}} ].  (37)

3 Predictable Conditions for S ∈ M_{loc} (P), S ∈ M (P)

3.1 One asset.

In order to investigate whether S ∈ M(P) (i.e. S is a P -martingale) it may be more convenient to first look for conditions so that S ∈ M_{loc}(P) (i.e. S is a local P -martingale) and then use the result in Jacod and Shiryaev (1987), Chapter I, 1.47 that a local martingale S is a uniformly integrable martingale if and only if S belongs to the class (D), that is the set of random variables {S_{T} : T finite stopping time} is uniformly integrable. We hence start with the representation (5), i.e.

S_{t} = S_{0}E(Ĥ)_{t}

and use the property (see Section 1.2) that

S ∈ M_{loc} (P) if and only if Ĥ ∈ M_{loc} (P).  (38)

From (4) and (17) we obtain:

Ĥ_{t} = H_{t} + \frac{1}{2}⟨H^{c}⟩_{t} + ∫_{0}^{t} ∫ (e^{x} − 1 − x)dμ
= H_{0} + B(φ)_{t} + H_{t}^{c} + \frac{1}{2}⟨H^{c}⟩_{t} + ∫_{0}^{t} ∫ φ(x)d(μ − ν)  (39)
+ ∫_{0}^{t} ∫ (x − φ(x))dμ + ∫_{0}^{t} ∫ (e^{x} − 1 − x)dμ

= H_{0} + B(φ)_{t} + H_{t}^{c} + \frac{1}{2}⟨H^{c}⟩_{t} + ∫_{0}^{t} ∫ φ(x)d(μ − ν)

+ ∫_{0}^{t} ∫ (e^{x} − 1 − φ(x))dμ.

Suppose now that |e^{x} − 1 − φ(x)| ∗ ν ∈ A_{loc} (i.e. the process (∫_{0}^{t} ∫ |e^{x} − 1 − φ(x)|dν)_{t ≥ 0} is locally integrable), then

∫_{0}^{t} ∫ (e^{x} − 1 − φ(x))dμ = ∫_{0}^{t} ∫ (e^{x} − 1 − φ(x))dν
+ ∫_{0}^{t} ∫ (e^{x} − 1 − φ(x))d(μ − ν), (40)

where the last integral is a local martingale; see Jacod and Shiryaev (1987), Chapter II, 1.28 and Liptser and Shiryaev (1986), Chapter III, § 5. Hence from (39),

Ĥ_{t} = H_{0} + B(φ)_{t} + H_{t}^{c} + \frac{1}{2}⟨H^{c}⟩_{t} + ∫_{0}^{t} ∫ (e^{x} − 1 − φ(x))dν
+ ∫_{0}^{t} ∫ (e^{x} − 1 − φ(x))d(μ − ν) + ∫_{0}^{t} ∫ φ(x)d(μ − ν)
= H_{0} + K_{t} + H_{t}^{c} + ∫_{0}^{t} ∫ (e^{x} − 1)d(μ − ν),

where (see (30))

K_{t} = A(−i)_{t} = B(φ)_{t} + \frac{1}{2}⟨H^{c}⟩_{t}

+ ∫_{0}^{t} ∫ (e^{x} − 1 − φ(x))dν. (41)

Therefore Ĥ_{t} = K_{t} + (local martingale)_{t}. Since K = (K_{t}) is a predictable process, it follows that

Ĥ ∈ M_{loc} (P) if and only if K ≡ 0.

See Jacod and Shiryaev (1987), Chapter I,3.16 and Liptser and Shiryaev (1986), Chapter I,6, Theorem 4 for more details.

3.2 Two assets

Suppose that we now have a second asset S^{0} = (S_{t}^{0})_{t ≥ 0} with

S_{t}^{0} = S_{0}^{0} e^{H_{t}^{0}} ·  (42)

Similar to the discussions above (see (4)) we introduce

Ĥ_{t}^{0} = H_{t}^{0} + \frac{1}{2}⟨H^{0c}⟩_{t} + ∑_{0 < s≤ t}(e^{Δ H_{s}^{0}} − 1 − ΔH_{s}^{0})  (43)

and obtain

S_{t}^{0} = S_{0}^{0} E(Ĥ^{0})_{t}.

Therefore

\frac{S_{t}}{S_{t}^{0}} = \frac{S_{0}}{S_{0}^{0}} \frac{E(Ĥ)_{t}}{E(Ĥ^{0})_{t}} .  (44)

It is now easy to check by Itô’s formula that

E(Ĥ^{0})_{t}^{−1} = E(−Ĥ^{∗})_{t}  (45)

where

Ĥ_{t}^{∗} = Ĥ_{t}^{0} − ⟨Ĥ^{0c}⟩_{t} − ∑_{0 < s ≤ t} \frac{(ΔĤ_{s}^{0})^{2}}{1 + ΔĤ_{s}^{0}} ·  (46)

From (44) and (45) we obtain that

\frac{S_{t}}{S_{t}^{0}} = \frac{S_{0}}{S_{0}^{0}} E(Ĥ_{t}) E (−Ĥ^{∗})_{t}.  (47)

If in general U, V ∈ Sem(P), then the so-called Yor addition formula (see for instance Rogers and Williams (1987), Section IV. 19) yields

E(U)E(V) = E(U + V + [U, V])  (48)

where the quadratic covariation process

[U, V]_{t} = ⟨U^{c}, V^{c}⟩_{t} + ∑_{0 < s ≤ t} ΔU_{s}ΔV_{s}.  (49)

So from (47), (48):

\frac{S_{t}}{S_{t}^{0}} = \frac{S_{0}}{S_{0}^{0}} E(Ĥ − Ĥ^{∗} + [Ĥ, −Ĥ^{∗}])_{t}.

It is not difficult to check that

Ĥ − Ĥ^{∗} + [Ĥ, −Ĥ^{∗}] = Ĥ − Ĥ^{0} + ⟨Ĥ^{0c} − Ĥ^{c}, Ĥ^{0c}⟩

+ ∑ \frac{ΔĤ^{0}(ΔĤ^{0} − ΔĤ)}{1 + ΔĤ^{0}} ·

If S^{0} stands for a riskless asset, i.e. H^{0} is predictable, then Ĥ^{0c} = H^{0c} = 0 and

\frac{S_{t}}{S_{t}^{0}} = \frac{S^{0}}{S_{0}^{0}} E(Ĥ − Ĥ^{0} + ∑ \frac{ΔĤ^{0}(ΔĤ^{0} − ΔĤ)}{1 + ΔĤ^{0}}).  (50)

Hence

\frac{S}{S^{0}} ∈ M_{loc}(P)

if and only if  (51)

Ĥ − Ĥ^{0} + ∑ \frac{ΔĤ^{0}(ΔĤ^{0} − ΔĤ)}{1 + ΔĤ^{0}} ∈ M_{loc}(P)

The result (51) can be very useful in finding sufficient conditions for S/S^{0} to be a local P -martingale. For instance, if ΔĤ^{0} = 0, then Ĥ_{t}^{0} = H_{t}^{0} (we suppose that H^{0} is predictable) and we obtain

K_{t} − H_{t}^{0} ≡ 0 implies \frac{S}{S^{0}} ∈ M_{loc}(P).  (52)

Also, if ΔĤ = ΔĤ^{0}, then

K_{t} − H_{t}^{0} − ∑_{0 < s ≤ t}(e^{Δ H_{t}^{0}} − 1 − Δ H_{s}^{0}) ≡ 0

implies  (53)

\frac{S}{S^{0}} ∈ M_{loc}(P).

3.3 Examples

3.3.1 Discrete time. In the case of discrete time

Ĥ − Ĥ^{0} + ∑ \frac{ΔĤ^{0}(ΔĤ^{0} − ΔĤ)}{1 + ΔĤ^{0}} = ∑ \frac{ΔĤ − ΔĤ^{0}}{1 + ΔĤ^{0}} ,

so that because of (51),

\frac{S}{S^{0}} ∈ M_{loc}(P)

if and only if

∑_{k ≤ n} \frac{ĥ_{k} − ĥ_{k}^{0}}{1 + ĥ_{k}^{0}} ∈ M_{loc}(P) ·  (54)

However ĥ_{k} = e^{h_{k}} − 1, ĥ_{k}^{0} = e^{h_{k}^{0}} − 1, so that by F_{k−1}-measurability of h_{k}^{0} we obtain the following sufficient condition

E(e^{h_{k}}|F_{k−1}) = e^{h_{k}^{0}}, k ≥ 1 implies \frac{S}{S^{0}} ∈ M_{loc}(P).

3.3.2 Processes with independent increments Suppose that H = (H_{t})_{t ≥ 0} is a process with independent increments, the triplet T (φ) given by (25) and let Ĥ_{t}^{0} = rt. Then

K_{t} = t(b(φ) + \frac{C}{2} + ∫ (e^{x} − 1 − φ(x))F (dx)),

so that

b(φ) + \frac{C}{2} + ∫ (e^{x} − 1 − φ(x))F (dx) = r implies \frac{S}{S^{0}} ∈ M_{loc}(P).

3.3.3 Brownian motion with drift and Poisson jumps For the notation, see section 2.3.3. In this case

K_{t} = t(b + \frac{σ^{2}}{2} + λ ∫ (e^{x} − 1)F (dx)),

whence

b + \frac{σ^{2}}{2} + λE(e^{ξ} − 1) = τ implies \frac{S}{S^{0}} ∈ M_{loc}(P).

4 Predictable conditions for the existence of a locally equivalent probability measure P̃ such that S ∈ M_{loc}(P̃), S ∈ M (P̃)

4.1 General results

If we have a measure P̃ ∼^{loc} P, then the likelihood (Radon–Nikodym derivative) process Z = (Z_{t})_{t≥0} with

Z_{t} = \frac{dP̃_{t}}{dP_{t}}  (55)

is strictly positive (Z_{t} > 0, P and P̃-a.s., t ≥ 0; see for instance Rogers and Williams 1987, Theorem IV, 17.1. We therefore can define the process M = (M_{t})_{t≥0} as follows:

M_{t} = ∫_{0}^{t} \frac{dZ_{s}}{Z_{s−}} ,  (56)

which satisfies M ∈ M_{loc}(P). Since dZ_{t} = Z_{t−} dM_{t}, we have that

Z_{t} = Z_{0} E(M)_{t}  (57)

where

E(M)_{t} = exp\{M_{t} − \frac{1}{2}⟨M^{c}⟩_{t}\} ∏_{0 <s ≤ t}(1 + ΔM_{s})e^{−ΔM_{s}}.  (58)

The local martingale property of M implies that the following decomposition holds:

M_{t} = M_{0} + ∫_{0}^{t} β_{s}dH_{s}^{c} + ∫_{0}^{t} ∫ W (·, s, x)d(μ − ν) + M̃_{t},

where β and W satisfy some integrability conditions (see Jacod and Shiryaev (1987), Chapter III, 4.24) and M̃ is a residual martingale part which is orthogonal to ∫_{0}^{.} β_{s}dH_{s}^{c} and ∫_{0}^{.} ∫ W (·, s, x)d(μ − ν). Unfortunately, we do not have sufficient tools in order to control the properties of M̃. However, for many interesting cases, M̃ ≡ 0. The latter for instance holds if the triplet T (φ) = (B, C, ν) for H defines the distribution of H uniquely. The following are cases where this property holds:

(i) Processes with independent increments.

(ii) Strong solutions of stochastic differential equations with respect to Brownian motion.

(iii) In the case of Poisson random measure in discrete time where ν(ω, {n} × A) = P (ΔH_{n} ∈ A|F_{n−1}), n ≥ 1, which gives us the possibility to calculate the (unconditional) distribution of (H_{n})_{n≥0}.

A possible approach consists of considering the structure of Z under the assumption that P̃ ∼^{loc} P exists. Hence assume that Z = (Z_{t})_{t≥0} satisfies the representation (57)–(58), where

M_{t} = M_{0} + ∫_{0}^{t} β_{s}dH_{s}^{c} + ∫_{0}^{t} ∫ W (., s, x) d(μ − ν).  (59)

Can we from this representation deduce the existence of P̃? This approach may work if at least the characteristic triplet of H defines the measure P (i.e. the law of H) uniquely. We assume the finite horizon case 0 ≤ t ≤ T < ∞ and normalise E Z_{T} = 1. In this case we can simply define

dP̃_{T} = Z_{T} dP_{T} .

The difficult part in this plan de campagne is to find conditions on (β, W) and (B, C, ν) which imply that Z = (Z_{t})_{0≤t≤T} is a martingale with E Z_{T} = 1. A whole series of papers exists on this topic, see for instance Jacod and Memin (1976), Liptser and Shiryaev (1972), Novikov (1975, 1979), Lepingle and Memin (1978) and Grigelionis (1971). (See Schachermayer (1993) and Delbaen and Schachermayer (1997) for a case where M̃ cannot be taken to be zero!) So suppose that M = (M_{t})_{0≤t≤T} defined as in (59) is a positive martingale with E Z_{T} = 1. We now want to understand which conditions on (β, W) imply that S ∈ M_{loc}(P̃_{T}). First observe that

SZ ∈ M_{loc}(P_{T}) implies S ∈ M_{loc}(P̃_{T}),  (60)

(see Jacod and Shiryaev, Chapter III, 3.8) so that it suffices to find conditions implying

E(Ĥ)E(M) ∈ M_{loc}(P_{T}) ·  (61)

Also note that

ĤZ ∈ M_{loc}(P_{T}) ⇒ Ĥ ∈ M_{loc}(P̃_{T})
⇔ E(Ĥ) ∈ M_{loc}(P̃_{T})
⇔ S ∈ M_{loc}(P̃_{T}),

so that instead of checking (61), one may look for conditions implying

ĤE(M) ∈ M_{loc}(P_{T}).  (62)

One easily shows that (61) and (62) are equivalent.

From (61) and Yor’s formula ((48)) one obtains:

E(Ĥ)E(M) = E(Ĥ + M + [Ĥ, M])
= E(Ĥ + M + ⟨Ĥ^{c}, M^{c}⟩ + ∑ ΔĤΔM) ·  (63)

Moreover, (41) yields

Ĥ_{t} = K_{t} + H_{t}^{c} + ∫_{0}^{t} ∫ (e^{x} − 1)d(μ − ν) ·  (64)

From (59) and (64) , assuming that the process [Ĥ, M] is locally integrable,

we can find its compensator { \atop [Ĥ, M]}. The latter is a predictable process with

the property that [Ĥ, M] − { \atop [Ĥ, M]} ∈ M_{loc}(P). The following form results see also Jacod and Shiryaev, Chapter II, 2.17:

{ \atop [Ĥ, M]_{t}} = ∫_{0}^{t} β_{s}d⟨H^{c}⟩_{s} + ∫_{0}^{t} ∫ W (e^{x} − 1)dν

− ∑_{s ≤ t} ∫ W (s, x)ν({s} × dx) ∫ (e^{x} − 1)ν({s} × dx). (65)

It turns out to be convenient to denote W = Y − 1. The main reason for this is the following. If H is a P -semimartingale with triplet (B, C, ν) and dP̃_{T} = Z_{T} dP_{T}, then H is also a P̃_{T} -semimartingale with triplet (B̃, C̃, ν̃) where dν̃ = Y dν, Y (ω, t, x) is positive and predictable and the process W in the definition of M (see (59)) has the following representation (Jacod and Shiryaev, Chapter III, 5.19)

W = Y − 1 + \frac{Ŷ − a}{1 − a} I(a < 1),  (66)
a = (a_{t}(ω)) where a_{t}(ω) = ν(ω; {t} × ℝ)
Ŷ_{t} = ∫ Y (ω, t, x)ν(ω; {t} × dx) ·

Both in the so-called quasi-left continuous case (i.e. a_{t} ≡ 0) as well as in the discrete-time case where a_{t}(ω) = P (ΔH_{t} ∈ ℝ|F_{t−1}) = 1 we have that W = Y − 1. Therefore, as a corollary we obtain

{ \atop [Ĥ, N]_{t}} = ∫_{0}^{t} β_{s}d⟨H^{c}⟩_{s} + ∫_{0}^{t} ∫ (Y − 1)(e^{x} − 1)dν ·  (67)

Together with (59), (63) and (64) we are led to the following result. Suppose that Z = (Z_{t})_{t≤T} is a positive martingale with dZ_{t} = Z_{t} − dM_{t} where M = (M_{t})_{t≤T} is given by (59) and E | Z_{T} | = 1. Then in the cases where

ν(ω; {t} × ℝ) ∈ {0, 1},

the condition

K_{t} + ∫_{0}^{t} β_{s}d⟨H^{c}⟩_{s} + ∫_{0}^{t} ∫ (Y − 1)(e^{x} − 1)dν = 0, t ≤ T,

implies that there exists a measure P̃_{T} constructed by (56), (59), (66) such that

P̃_{T} ∼ P_{T} and S ∈ M_{loc}(P̃_{T}).

Suppose now that we are interested in the construction of probability measures P̃_{T} with

P̃_{T} ∼ P_{T} and \frac{S}{S^{0}} ∈ M_{loc}(P̃_{T}).

In this case,

\frac{S_{t}}{S_{t}^{0}} Z_{t} = \frac{S_{0}}{S_{0}^{0}} Z_{0} E(Ĥ)_{t} E^{−1}(Ĥ^{0})_{t} E(M)_{t} ·  (68)

Because of (51), E(Ĥ)E^{−1}(Ĥ^{0}) = E(H̄) where

H̄ = Ĥ − Ĥ^{0} + ∑ \frac{ΔĤ^{0}(ΔĤ^{0} − ΔĤ)}{1 + ΔĤ^{0}} ·

Moreover, by (63),

E(Ĥ)E^{−1}(Ĥ^{0})E(M) = E(H̄)E(M)
= E(H̄ + M + [H̄, M])
= E(H̄ + M + ⟨H̄^{c}, M^{c}⟩ + ∑ ΔH̄ΔM)
= E(Ĥ − Ĥ^{0} + M + ⟨Ĥ^{c} − Ĥ^{0c}, M^{c}⟩

+ ∑ \frac{ΔĤ − ΔĤ^{0}}{1 + ΔĤ^{0}} ΔM

+ ∑ \frac{ΔĤ^{0}(ΔĤ^{0} − ΔĤ)}{1 + ΔĤ^{0}})

≡ E(I), say ·  (69)

Again we assume that H^{0} is predictable so that H^{0c} = 0 and consequently

I = Ĥ − Ĥ^{0} + M + ⟨Ĥ^{c}, M^{c}⟩

+ ∑ \frac{(ΔM − ΔĤ^{0})(ΔĤ − ΔĤ^{0})}{1 + ΔĤ^{0}} · (70)

Compare this expression with (51) when M ≡ 0 and (63) when Ĥ^{0} ≡ 0. In order to find now a predictable condition ensuring that S/S^{0} ∈ M_{loc}(P̃), as before, we search for a decomposition

I = Predictable process + local martingale·

Another way of putting this is: if Ĩ is the compensator (i.e. predictable part) of I, then

Ĩ = 0 implies \frac{S}{S^{0}} ∈ M_{loc}(P̃) ·  (71)

Returning to (70), observe the following facts.

(a) Ĥ_{t} − Ĥ_{t}^{0} + M_{t} = (K_{t} − Ĥ_{t}^{0}) + Ĥ_{t}^{c} + ∫_{0}^{t} ∫ (e^{x} − 1)d(μ − ν)
+ ∫_{0}^{t} β_{s}dH_{s}^{c} + ∫_{0}^{t} ∫ W d(μ − ν)
= (K_{t} − Ĥ_{t}^{0}) + (local martingale)_{t},

(b) ⟨Ĥ^{c}, M̂^{c}⟩_{t} = ∫_{0}^{t} β_{s}d⟨Ĥ^{c}⟩_{s}·

The calculation of the compensator of the last (i.e. ∑ −) term in (70) is in general involved. However, for many interesting special cases (including those already discussed in previous sections) the compensator Ĩ can be obtained in explicit form. Rather than pursuing the general case as outlined above, in the next section we shall look at some examples.

4.2 Some examples

4.2.1 Discrete time

In this case,

ΔI = ΔM + \frac{(1 + ΔM)(ΔĤ − ΔĤ^{0})}{1 + ΔĤ^{0}}
= \frac{ΔM(1 + ΔĤ) + (ΔĤ − ΔĤ^{0})}{1 + ΔĤ^{0}} ·  (72)

Again denote h_{k} = ΔH_{k}, whence ΔĤ_{k} = e^{ΔH_{k}} − 1 = e^{h_{k}} − 1; we use the same notation for H^{0}. Hence, with

ΔM_{n} = ∫ (Y_{n}(x; ω) − 1)(μ_{n}(dx) − ν_{n}(dx))
= Y_{n}(h_{n}; ω) − E(Y_{n}(h_{n}; ω)|F_{n−1}),

we obtain from (72) that

ΔI_{n} = ΔM_{n}e^{h_{n}−h_{n}^{0}} + e^{h_{n}−h_{n}^{0}} − 1
= e^{h_{n}−h^{0}_{n}}(ΔM_{n} + 1) − 1 ·  (73)

Together with the assumed predictability of (h_{n}^{0}), we obtain from (73) the following key result:

E[e^{h_{n}}(Y_{n}(h_{n}; ω) − E(Y_{n}(h_{n}; ω)|F_{n−1}) + 1)|F_{n−1}] = e^{h_{n}^{0}}, n ≥ 1

implies  (74)

\frac{S}{S^{0}} ∈ M_{loc}(P̃)·

Therefore, the existence problem of a (local) matingale measure P̃ is reduced to finding (Y_{n}) which satisfy (74). This task may still seem to be formidable in the stated generality. It is exactly at this point that the conditional Esscher transform defined in (36) enters naturally. Indeed, we assume that

Y_{n}(h_{n}; ω) = \frac{e^{a_{n}h_{n}}}{E(e^{a_{n}h_{n}}|F_{n−1})},  (75)

where the unknown functions a_{n} are F_{n−1}-measurable. Our aim is to determine the a_{n}’s in the special case of (75). With (74) we arrive at the following equation:

E[e^{(a_{n}+1)h_{n}}|F_{n−1}] = e^{h_{n}^{0}}E[e^{a_{n}h_{n}}|F_{n−1}] ·  (76)

If the increment sequence (h_{n}) is iid and h_{n}^{0} ≡ h_{1}^{0} say, then for a ≡ a_{n} we obtain the equation:

Ee^{(a+1)h_{1}} = e^{h_{1}^{0}}Ee^{ah_{1}} ·  (77)

Hence in this case, the Esscher transform allows for a special construction of (Y_{n}) by reducing the problem to finding constants (a_{n}) or predictable functions (a_{n}(ω)) satisfying (76). In Gerber and Shiu (1994), the construction (77) is applied in a finance context. See also the references in the latter paper for further reading on the subject. Embrechts (1996) discusses the Esscher transform in the light of financial versus actuarial pricing systems.

4.2.2 Processes with stationary, independent increments (S.I.I.)

Let H = (H_{t}) be a process with S.I.I., continuous in probability, and triplet

B(φ)_{t} = tb(φ)
C_{t} = tC
ν(dt × dx) = dtF (dx) ·

Moreover, H_{t}^{0} = rt, say. Then K_{t} is defined in Section 3.3.2. In this case,

I = Ĥ − Ĥ^{0} + M + [Ĥ, M],

and from (59), (64), (67) we obtain:

K_{t} + ∫_{0}^{t} β_{s}d⟨H^{c}⟩_{s} + ∫_{0}^{t} ∫ (Y − 1)(e^{x} − 1)dν = rt  (78)

implies

\frac{S}{S^{0}} ∈ M_{loc}(P̃)·

The sufficient condition (78) can be rewritten as:

t(b(φ) + \frac{C}{2} + ∫ (e^{x} − 1 − φ(x))F (dx)) + C ∫_{0}^{t} β_{s}ds
+ ∫_{0}^{t} ∫ (Y − 1)(e^{x} − 1)dsF (dx) = rt ·  (79)

Because of the homogeneity (i.e. incremental stationarity) of the process, it seems reasonable to take β_{s}(ω) ≡ β, Y (s.x.ω) ≡ Y (x). Then for unknown β and Y (x), (79) reduces to:

C(\frac{1}{2} + β) + b(φ) + ∫ (e^{x} − 1 − φ(x))F (dx)

+ ∫ (Y (x) − 1)(e^{x} − 1)F (dx) = r. (80)

Take as particular case the standard Black–Scholes set-up in finance, i.e.

S_{t} = e^{μt+σW_{t}} , S_{t}^{0} = e^{rt}·

In this case, b(φ) = μ, C = σ^{2}, ν ≡ 0 so that the condition (80) reduces to the well-known equation

μ + σ^{2}(\frac{1}{2} + β) = r  (81)

and M_{t} = βH_{t}^{c} = βσW_{t}. It should be stressed that (81) can be obtained much more easily directly, i.e. without using the general theory introduced above. Indeed

Z_{t} = exp \{βσW_{t} − \frac{(βσ)^{2}}{2} t\}

and

\frac{S_{t}}{S_{t}^{0}} Z_{t} = exp\{(μ − r)t + σW_{t}\}exp \{βσW_{t} − \frac{(βσ)^{2}}{2} t\}
= exp \{σ(1 + β)W_{t} − \frac{σ^{2}(1 + β)^{2}}{2} t\} exp \{(σ^{2}(\frac{1}{2} + β) + μ − r)t\} ·

Since

(exp \{σ(1 + β)W_{t} − \frac{σ^{2}(1 + β)^{2}}{2} t\})_{t≥0}

is a P -martingale, condition (81) immediately implies that S/S^{0} ∈ M (P) and so S/S^{0} ∈ M (P̃) where dP̃_{t} = Z_{t}dP_{t}.

4.2.3 Brownian motion with drift and Poisson jumps

Consider the model (26) with triplet representation (28). Hence from (80) we obtain the following condition for (β, Y (x)):

σ^{2}(\frac{1}{2} + β) + b + λ ∫ Y (x)(e^{x} − 1)F (dx) = r  (82)

or equivalently,

σ^{2}(\frac{1}{2} + β) + b + λE(e^{ξ} − 1)Y (ξ) = r.

Compare this condition with the condition in section 3.3.3, where β = 0, Y ≡ 1 and P̃_{t} = P_{t}. If we consider a solution

Y (x) = \frac{e^{αx}}{Ee^{αξ}}

for suitable α, then we get for (α, β):

σ^{2}(\frac{1}{2} + β) + b + λ\frac{E(e^{ξ} − 1)e^{αξ}}{Ee^{αξ}} = r,

or with ψ(α) = Ee^{αξ},

βσ^{2} + λ\frac{ψ(α + 1)}{ψ(α)} = r − b − \frac{σ^{2}}{2} − λ.

Conclusion In order to price and hedge derivative instruments in insurance and finance, a no-arbitrage approach leads to the construction of equivalent (local) martingale measures of specific semimartingales. For a general class of such processes, including discrete models, processes with stationary and independent increments and certain diffusion models with jumps, a general construction toward obtaining such measures is outlined. Though these methods are well known in the literature on general stochastic processes, we found it useful to summarise the main results and applications of this theory to the context of insurance and finance. In doing so, we hope to contribute to closing the methodological gap currently existing between both fields. The main common tool concerns the so-called Esscher transform, a time honoured tool in insurance risk theory. Its construction is generalised to the so-called conditional Esscher transform which may serve a similar purpose within more general pricing models.

Acknowledgment The authors would like to thank a referee for the careful reading of the first version of this paper.

Department of Mathematics  Steklov Mathematical Institute
ETHZ  Vavilova
CH-8092 Zürich  GSP-1 117966 Moscow
Switzerland  Russia

References

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

N.L. Bowers, Jr., H.U. Gerber, J.C. Hickman, D.A. Jones and C.J. Nesbitt (1989) Actuarial Mathematics. The Society of Actuaries, Itasca, Illinois.

H. Bühlmann (1980) An economic premium principle.

ASTIN Bulletin, 11, 52–60.

H. Bühlmann (1983) The general economic premium principle.

ASTIN Bulletin, 14, 13–21.

H. Bühlmann, F. Delbaen, P. Embrechts, A. N. Shiryaev (1996) Fundamental Theorem of Asset Pricing, Esscher Transform and Change of Measure in The Finite Horizon Discrete Multiperiod Model. Preprint, ETH Zürich.

D. Cummins, H. Geman (1995) Pricing catastrophe futures and call spreads: an arbitrage approach. Journal of Fixed Income, 4, 46–57.

F. Delbaen, W. Schachermayer (1994) A general version of the fundamental theorem of asset pricing. Mathematische Annalen 300, 463– 520.

F. Delbaen, W. Schachermayer (1996) The fundamental theory of asset pricing for unbounded processes. Manuscript ETH Zürich, Universität Wien, to appear.

F. Delbaen, W. Schachermayer (1997) A simple counter-example to several problems in the theory of asset pricing. Mathematical Finance Volume 7.

P. Embrechts (1996) Actuarial versus financial pricing of insurance. Proceedings of the Conference on Risk Management in Insurance, The Wharton School, Philadelphia.

P. Embrechts, J.L. Jensen, M. Maejima and J.L. Teugels (1985)

Approximations for compound Poisson and Pólya processes. Adv. Appl. Prob. 17, 623–637.

F. Esscher (1932) On the probability function in the collective theory of risk. Skandinavisk Aktuarietidskrift 15, 175–195.

H.J. Furrer, H. Schmidli (1994) Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion. Insurance: Math. and Econom. 15, 23–36.

H.U. Gerber (1970) An extension of the renewal equation and its application to the collective theory of risk. Skandinavisk Aktuarietidskrift, 205–210.

H.U. Gerber, E.S.W. Shiu (1994) Option pricing by Esscher transforms. Transactions of the Society of Actuaries XLVI, 99–191.

P. Grandits (1996) The p-optimal martingale measure and its asymptotic relation with the Esscher transform. Preprint, University of Vienna.

B.I. Grigelionis (1971) On absolute continuity of measures corresponding to stochastic processes. Lit. Mathem. Sb. 11, 783–794.

J. Jacod, J. Memin (1976) Caractéristiques locales et conditions de continuité absolue pour les semi-martingales. Z. Wahrsch. verw. Gebiete 35, 1–7.

J. Jacod, A. N. Shiryaev (1987) Limit Theorems for Stochastic Processes. Springer, Berlin.

J.L. Jensen (1995) Saddlepoint approximations. Clarendon Press, Oxford.

D. Lepingle, J. Memin (1978) Sur l’intégrabilité uniforme des martingales exponentielles. Z. Wahrsch. verw. Gebiete 42, 175–203.

R. Liptser, A.N. Shiryaev (1972) On absolute continuity of measures associated with diffusion processes with respect to Wiener measure. Izv. Akad. Nauk SSSR, Sec. Mat. 36, 874–889.

R. Liptser, A.N. Shiryaev (1986) Theory of Martingales. Kluwer, Amsterdam.

A.A. Novikov (1975) On discontinuous martingales. Theory Prob. Appl. 20, 11–26.

A.A. Novikov (1979) On conditions for uniform integrability of continuous non-negative martingales. Theory Prob. Appl. 24, 820–824.

D. Revuz, M. Yor (1994) Continuous Martingales and Brownian Motion. Second Edition. Springer, Berlin.

L.C.G. Rogers (1995) Arbitrage with fractional Brownian motion. Preprint, University of Bath.

L.C.G. Rogers, D. Williams (1987) Diffusions, Markov Processes, and Martingales. Volume 2: Itô Calculus. Wiley, Chichester.

W. Schachermayer (1993) A Counterexample to Several Problems in the Theory of Asset Pricing. Mathematical Finance 3, 217–230.