The stochastic logarithm of

semimartingales and market

prices of risk

Frank Oertel

http://euclid.ucc.ie/pages/staff/oertel

AMaMeF Workshop on Stochastic Analysis and Computational Finance

Institute for Mathematical Sciences Imperial College London

12th November 2005

Contents

1. The Standard Black-Merton-Scholes Model Revisited

2. Semimartingales with Lévy Processes in View 3. The Stochastic Logarithm: The Germ of

Financial Mathematics?

4. Conclusion The Standard Black-Merton-Scholes

Model Revisited

Let (Ω, ℱ , ℙ) be a given probability space, T < ∞ a finite time horizon and W = (W_{t})_{0≤t≤T} be a standard Brownian motion in ℝ (SBM). Information is modelled as the augmented natural filtration 𝔽^{W} = {ℱ_{t}^{W} | 0 ≤ t ≤ T } of the SBM W, where ℱ_{T}^{W} = ℱ.

Consider two tradable assets:

• A riskless bond with (non-stochastic) price process B = (B_{t})_{0≤t≤T} ;

• A stock with stochastic price process S = (S_{t})_{0≤t≤T} .

B is modelled as B_{t} := e^{rt}, where r > 0 (riskfree rate of interest), and S follows a geometric Brownian motion:

dS_{t} = S_{t} dR_{t},

where R_{t} := μt + σW_{t} denotes the cumulated return process (μ ≥ 0 and σ > 0). Thus:

S_{t} {(Itô) \atop =} S_{0} exp (μt + σW_{t} − \frac{1}{2}σ^{2}t)
= S_{0} exp (R_{t} − \frac{1}{2}[R, R]_{t})
= S_{0} ℰ (R)_{t} . [⟵ stochastic exponential]

Let f ∈ L^{0}(Ω, ℱ , ℙ) be an arbitrary contingent claim such as a European call option, a credit swap or a reverse convertible bond.

Problem: Does there exists a consistent price system for f, and is it possible to eliminate (large parts of) the intrinsic risk of f by replicating f through a hedging portfolio?

Well-known Idea: We move to a ’risk-neutral universe’, where no ’money-machines’ exist and the stock has no ’drift’ !

Theorem 1 Assume the standard BMS-model and let ℚ ∼ ℙ on ℱ_{T}^{W}. TFAE:

(i) The discounted price process S^{B} := \frac{S}{B} is a ℚ-martingale;

(ii) \frac{dℚ}{dℙ} = Z_{T}, where Z_{T} = ℰ ( − \frac{μ − r}{σ}W )_{T} .

\frac{μ − r}{σ} is known as the Girsanov kernel resp. ’market price of risk’ or ’Sharpe ratio’.

Since ℚ ∼ ℙ, ℚ is an equivalent martingale measure (EMM).

Hence again, in the standard BMS-model there exists a unique EMM, consequently a unique risk-neutral price. The BMS-model is complete.

Imperfections of the BMS-model:

• Log returns do not behave according to a Normal distribution;

• Volatilities change stochastically over time and are clustered.

Therefore, we turn our attention to...

Semimartingales with Lévy Processes in View

From now on, we assume that a complete probability space (Ω, ℱ , ℙ) is given which in addition is endowed with a filtration 𝔽 = (ℱ_{t})_{t≥0} satisfying the usual conditions (such as 𝔽^{W}).

Definition 1 (Semimartingale) A process X = (X_{t}) in ℝ^{d} whose path is right continuous and has left limits (’càdlàg’ resp. ’RCLL’) is called a semimartingale (w.r.t. ℙ and 𝔽) if there exist an adapted local martingale M and an adapted process A of finite variation such that

X = M + A .

If in addition the process A is predictable and A_{0} = 0, the decomposition of X even is unique. In this case, X is called a special semimartingale and X̃:= A the compensator of X.

Examples: Standard Brownian motion, geometric Brownian motion, the Poisson process, and the compound Poisson process are semimartingales.

Fractional Brownian motion (if H ≠ \frac{1}{2}) and processes of type |W |^{α}, where 0 < α < 1 and W is a standard Brownian motion, do not belong to the class of semimartingales.

Stability properties: The class 𝒮 of semimartingales has a vector space structure. 𝒮 can be equipped with a topology such that it even turns into a Fréchet space. Due to Itô’s formula, 𝒮 is an algebra and stable under C^{2}- functions on open sets. Moreover, 𝒮 is stable under convex functions. In particular, it is a lattice (e. g., X ∧ Y = \frac{1}{2}(X + Y − |X − Y |)).

Definition 2 (Lévy Process) A process X = (X_{t})_{t≥0} in ℝ^{d} is a 𝔽-Lévy process if

(i) X is càdlàg; (ii) X_{0} = 0 a.s.; (iii) X is adapted and has independent incre-

ments w.r.t. 𝔽 (independent from the past); (iv) If 0 ≤ s ≤ t, then X_{t} −X_{s} =^{d} X_{t−s} (stationary

increments). In the following, we only consider the case d = 1 (to simplify notation).

Examples: Brownian motion, the Poisson process, Cauchy and α-stable processes belong to the class of Lévy processes.

Processes of type (at + σW_{t} + Y_{t})_{t≥0}, where a, σ ∈ ℝ, Y is a compound Poisson process and W is a SBM (in ℝ), are Lévy processes as well.

Further examples: Variance Gamma (VG), Normal Inverse Gaussian (NIG), CGMY, the Hyperbolic model, and the Meixner process.

Let X = (X_{t}) be an arbitrary Lévy process in ℝ. Let u ∈ ℝ and t ≥ 0. Put

φ_{Xt}(u) = ℙ̂_{Xt}(u) := 𝔼[e^{iuX_{t}}] = _{ℝ}^{∫} e^{iux} ℙ_{Xt}(dx) .

Theorem 2 (Lévy-Khintchine Formula) Let X be a Lévy process. Then there exists a Radon measure ν on ℬ(ℝ) such that ν({0}) = 0, _{ℝ}^{∫} x^{2} ∧ 1 ν(dx) < ∞, and parameters α ∈ ℝ,

σ ≥ 0 such that

φ_{Xt}(u) = 𝔼[e^{iuX_{t}}] = e^{−tψ(u)} = φ t_{X1}(u), (∗)

where

ψ(u) := \frac{σ^{2}u^{2}}{2}−iαu+_{ℝ}^{∫} (1−e^{iux}+iux11_{(−1,1)} (x))ν(dx).

Conversely, given any such triplet (α, σ^{2}, ν), there exists a probability space (Ω, ℱ , ℙ) and a corresponding Lévy process X with characteristic function (∗) - unique in distribution.

The (non-random) triple (α, σ^{2}, ν) is called the local characteristics of X. ν is known as the Lévy measure. Note that the local characteristics depends on X. Consequently, we also make use of the notation (α^{X}, (σ^{X})^{2}, ν^{X}).

Remark and Definition 1 Let X be an arbitrary adapted and càdlàg process.

ΔX_{s} := X_{s} − X_{s−} = X_{s} − lim_{u↑s} X_{u}

denotes the jump of X at time s ≥ 0, where we put X_{0−} := X_{0}. All paths of X are continuous except for the jumps. Moreover, X_{−} is leftcontinuous, locally bounded and predictable, hence adapted. ΔX denotes the corresponding jump process. If X is continuous, then of course, ΔX = 0.

For every ω ∈ Ω, the random set

M(ω) := {(s, ΔX_{s}(ω))|s > 0, ΔX_{s}(ω) ≠ 0}

is at most countable.

The jump measure of X is defined as the mapping

j_{X} : \underbrace{Ω}_{samples} × ℬ( \underbrace{ℝ^{+}}_{time} × \underbrace{ℝ}_{range of X} ) ⟶ ℕ_{0} ∪ \{∞\},

j_{X}(ω, B) := card (M(ω) ∩ B),

where ω ∈ Ω and B is an arbitrary Borel set in ℝ^{+} × ℝ. j_{X} counts, ω-by-ω, the number of all s > 0 such that ΔX_{s}(ω) ≠ 0 and (s, ΔX_{s}(ω)) ∈ B.

In particular, if X is a Lévy process, the set B := [0, t] × Λ (where t is fixed and Λ is a Borel set in ℝ) induces the random measure

N_{t}^{Λ} (ω) := N_{t}(ω, Λ) := j_{X}(ω, [0, t] × Λ )

which - in the Lévy case (!) - even leads to a Poisson process N^{Λ} = (N_{t}^{Λ} )_{t≥0} of intensity ν(Λ):

𝔼[N_{t}^{Λ} ] = 𝔼[N_{t}(·, Λ)] = tν(Λ) for all t ≥ 0.

Theorem 3 (Lévy Decomposition) Let X be a Lévy process with local characteristics (α, σ^{2}, ν). Then

X = A + M^{c} + M^{d}

is the sum of three independent Lévy processes, where

(i) A_{t} := αt + N_{t}^{Λ} ; ∑ ; i=1 Y_{i} is a finite variation compound Poisson process with drift having only jumps of size at least 1 (where Λ := ℝ \ (−1, 1) and Y_{i} := ΔX_{T_{i}^{Λ}} ),

(ii) M_{t}^{c} := σW_{t} =^{d} W_{σ^{2}t} is a Brownian motion,

(iii) M_{t}^{d} := _{(−1,1)}^{∫} x(N_{t}(·, dx) − tν(dx)) is a pure-

jump martingale having only jumps of size less than 1.

Corrolary 1 A Lévy process is a semimartingale.

The Stochastic Logarithm: The Germ of Financial Mathematics?

Theorem 4 (Stochastic Exponential) Let X be an ℝ-valued semimartingale such that X_{0} =

0. Put

V_{t} := _{0<s≤t}^{∏}(1 + ΔX_{s}) · e^{−ΔX_{s}} .

Then for almost all ω ∈ Ω the infinite product is absolutely convergent, and V = (V_{t}) is an adapted purely discontinuous process which is of finite variation. Put

Z_{t} := exp (X_{t} − \frac{1}{2}[X, X]_{t}^{c}) · V_{t} .

Then Z_{0} = 1, and Z = (Z_{t}) is the unique semimartingale which is a solution of the following stochastic integral equation

Z = 1 + ∫ Z_{−} dX .

Moreover, Z = Z_{−}(1 + ΔX).

The semimartingale Z is called the stochastic exponential of the semimartingale X and is denoted by ℰ (X). If X is a local martingale, so is ℰ (X).

Examples: If W is a standard Brownian motion, ℰ (W)_{t} = exp (W_{t} − \frac{1}{2}t), and if N is a Poisson process, then ℰ (N)_{t} = 2^{N_{t}}.

Let X be a Lévy process with local characteristics (α, σ^{2}, ν). Assume that the Laplace transform 𝔼[e^{−zX_{1}}] < ∞ for all z in some B_{r}(0). Then X_{t} = αt + σW_{t} + M_{t}, where α := 𝔼[X_{1}] < ∞ and M_{t} := _{ℝ}^{∫} x(N_{t}(·, dx) − tν(dx)). Let dS_{t} =

S_{t−}(μ_{t}dt+ρ_{t}dX_{t}), where μ and ρ are continuous and deterministic functions. Then

S_{t} = S_{0} ℰ ( ∫_{0}^{•} μ_{s} ds + ∫ ρ dX)_{t}
= S_{0} exp ( ∫_{0}^{t} σρ_{s} dW_{s} + ∫_{0}^{t} ρ_{s} dM_{s}
+ ∫_{0}^{t} (αρ_{s} + μ_{s} − \frac{σ^{2}ρ_{s}^{2}}{2} ) ds)
× _{0<s≤t}^{∏}(1 + ρ_{s}ΔM_{s}) · exp (−ρ_{s}ΔM_{s}).

The mapping X ↦ ℰ (X) = Z can be inverted if almost all paths of Z and Z_{−} do not go through 0 (cf. Jacod (1979), Foldes (1990), Choulli et al (1998), Kallsen/Shiryaev (2001), Jacod/Shiryaev (2nd ed. 2002)).

Theorem 5 (Stochastic Logarithm) Let

𝒮_{0}^{∼} := {X ∈ 𝒮 | { ΔX = −1} is evanescent , X_{0} = 0}

and

𝒮_{1}^{∗} := {Z ∈ 𝒮 | {ZZ_{−} = 0} is evanescent , Z_{0} = 1}.

Then the mapping

ℰ : 𝒮_{0}^{∼} ⟶^{≃} 𝒮_{1}^{∗}
X ↦ ℰ (X)

is bijective, and its inverse is given by

ℒ : 𝒮_{1}^{∗}  ⟶^{≃} 𝒮_{0}^{∼}
Z ↦ ∫ \frac{1}{Z_{−}}dZ − 1 .

In analogy to real analysis, ℒ (Z) := ∫ \frac{1}{Z_{−}} dZ −1 is called the stochastic logarithm of Z. Note that Δℒ (Z) = \frac{1}{Z_{−}}ΔZ.

Corrolary 2 Let Z ∈ 𝒮_{1}^{∗}. TFAE:

(i) Z is a local martingale;

(ii) There exists a unique local martingale M ∈ 𝒮_{0}^{∼} such that Z = ℰ (M) (outside some evanescent set);

(iii) ℒ (Z) is a local martingale.

Proposition 1 Let U, V ∈ 𝒮_{1}^{∗}. Then UV ∈ 𝒮_{1}^{∗} and

ℒ (UV) = ℒ (U) + ℒ (V) + [ℒ (U), ℒ (V)].

Moreover, \frac{1}{V} ∈ 𝒮_{1}^{∗} and

ℒ (\frac{1}{V} ) = 1 − ℒ (V) − [V, \frac{1}{V} ].

In particular, (𝒮_{1}^{∗}, ·) is a commutative group.

Theorem 6 (Modelling of Asset Prices) Let S ∈ 𝒮 such that S_{0} ≠ 0 a.s. and \frac{S}{S_{0}} ∈ 𝒮_{1}^{∗} (asset price process). Then there exists a unique semimartingale R ∈ 𝒮_{0}^{∼} (the cumulated return process) such that S = S_{0} ℰ (R), namely R = ℒ (\frac{S}{S_{0}}).

Let N ∈ 𝒮_{1}^{∗} (numeraire – e. g., a price process of Euros) and put S^{N} := \frac{S}{N} (e. g., the Euro

price process of S). Then \frac{S^{N}}{S_{0}} ∈ 𝒮_{1}^{∗}, and

S^{N} = S_{0}ℰ (Y_{N}),

where

Y_{N} := ℒ (\frac{1}{N})+R+[ ∫ N_{−}dR, \frac{1}{N} ] = ℒ (\frac{S^{N}}{S_{0}} ) ∈ 𝒮_{0}^{∼} .

Lemma 1 Let X ∈ 𝒮 and Z ∈ 𝒮_{1}^{∗} . Then \frac{1}{Z} ∈ 𝒮_{1}^{∗}, and

∫ \frac{1}{Z} d[Z, X] = X_{0} + [ℒ (Z), X]^{c} + _{0<s≤•}^{∑} \frac{ΔZ_{s}}{Z_{s}} ΔX_{s}
= X_{0} − [ℒ (\frac{1}{Z} ), X].

Proof. WLOG, let X_{0} = 0. Then

∫ \frac{1}{Z} d[Z, X] = ∫ \frac{1}{Z_{−}} d[Z, X]+ ∫ Δ(\frac{1}{Z} ) d[Z, X]
= [ℒ (Z), X] + _{0<s≤•}^{∑} Δ(\frac{1}{Z} )_{s}Δ[Z, X]_{s}
= [ℒ (Z), X] + _{0<s≤•}^{∑} Δ(\frac{1}{Z} )_{s}ΔZ_{s}ΔX_{s} .

Since Δℒ (Z) = \frac{ΔZ}{Z_{−}}, we have

[ℒ (Z), X] = [ℒ (Z), X]^{c} + _{0<s≤•}^{∑} \frac{ΔZ_{s}}{Z_{s−}} ΔX_{s} ,

and the first equation follows. On the other hand, due to Proposition 1,

[ℒ (\frac{1}{Z} ) , X] = [1 − ℒ (Z) − [Z, \frac{1}{Z} ] , X]
= −[ℒ (Z), X] − [[Z, \frac{1}{Z} ], X]
=^{!} −[ℒ (Z), X] − _{0<s≤•}^{∑} ΔZ_{s}Δ(\frac{1}{Z} )_{s}ΔX_{s}
= − ∫ \frac{1}{Z} d[Z, X] .  _{□}

Theorem 7 (Log-Version of Girsanov) Let ℚ be an arbitrary probability measure on ℱ such that ℚ is equivalent to ℙ and ℚ = ℙ on ℱ_{0}. Put Z_{t} := 𝔼_{ℙ}[\frac{dℚ}{dℙ}|ℱ_{t}]), where 0 ≤ t ≤ T. Then Z_{0} = 1 ℙ-a.s., and both, Z and \frac{1}{Z} have strictly positive paths ℙ-a.s.. In particular, Z and \frac{1}{Z} can be represented as a stochastic exponential respectively, ℒ (Z) exists, and Δℒ (Z) = \frac{1}{Z_{−}}ΔZ > −1 (outside some evanescent set). Moreover, Z is a uniformly integrable ℙ-martingale. If M ∈ ℳloc (ℙ), then M̃ ∈ ℳloc (ℚ), where

M̃ := M + [M, ℒ (\frac{1}{Z} ) ]
= M − [M, ℒ (Z) ]^{c} − _{0<s≤•}^{∑} \frac{ΔZ_{s}}{Z_{s}} ΔM_{s} .

(The càdlàg version of) Z is known as the density process.

Proposition 2 Let ℚ, Z be as in Theorem 7. Then ℒ (Z) is a Lévy process if and only if log (Z) is a Lévy process. Moreover,

ν^{ℒ (Z)} = ν^{log (Z)} ∘ ψ^{−1},

where ψ(y) := e^{y} − 1 (y ∈ ℝ).

Theorem 7 immediately implies a generalisation of a result of Kazamaki which had been proven for continuous local martingales only (cf. Kazamaki (1994)):

Corrolary 3 (Girsanov Transformation) Let ℚ and Z be as in Theorem 7. Then the mapping

Φ_{ℙ,ℚ} : ℳloc (ℙ)  ⟶^{≃} ℳloc (ℚ)
M ⟼ M + [M, ℒ (\frac{1}{Z} )]

is an (algebraic) isomorphism, and Φ−1_{ℙ,ℚ}(N) =

N+[N, ℒ (Z)] =^{!} Φ_{ℚ,ℙ}(N) for any N ∈ ℳloc (ℚ).

Problem: How do we choose ’good’ density processes Z, i. e., ’good’ ELMMs?

• Given an arbitrary (incomplete!) semimartingale market (S, N), any density process Z_{ℙ,ℚ} – originating from any ELMM ℚ ∼ ℙ – can be represented as a stochastic exponential:

Z_{ℙ,ℚ} = ℰ (X).

In accordance with the BMS case, we call −X = −ℒ (Z_{ℙ,ℚ}) market price of risk process.

• Interesting observation in the BMS case:

−\frac{μ − r}{σ}W = ℒ (Z_{ℙ,ℚ}) =^{!} (g ∘ ℒ)(\frac{S^{B}}{S_{0}}),

where g : 𝒮_{0}^{∼} ⟶ 𝒮_{0}^{∼} is given by

g(X)_{t} := −\frac{𝔼_{ℙ}[X_{t}]}{Var_{ℙ}[X_{t}]}(X_{t} − 𝔼_{ℙ}[X_{t}]).

Problem: Let (S, N) be as above and ℚ an arbitrary ELMM on ℱ. Then

\frac{dℚ}{dℙ} =^{?} ℰ ((g̃ ∘ ℒ)(\frac{S^{N}}{S_{0}} ))_{T} .

How do we construct g̃? If we know (all) g̃, we know (all) ℚ!

Definition 3 Let M ∈ ℳ_{loc ,0}(ℙ) w.r.t. 𝔽. M satisfies the strong property of predictable representation (SPPR) if the linear operator

T : L_{M}(ℙ) ⟶ ⟶ ℳ_{loc ,0}(ℙ)
H ⟼ ∫ HdM

is onto.

Theorem 8 (He/Wang/Yan (1992)) Let X be a Lévy process and 𝔽 = 𝔽^{X} the augmented natural filtration of X. Assume that X is a (local) martingale. Then X satisfies the SPPR if and only if X is a standard Brownian motion or a compensated Poisson process, up to a constant factor.

Theorem 9 Let M ∈ ℳ_{loc ,0}(ℙ) satisfy the SPPR, and let ℚ, Z be as in Theorem 7. Then there exists a predictable process H such that

Z = 𝔼_{ℙ}[\frac{dℚ}{dℙ}|ℱ_{•}] = ℰ ( ∫ HdM) = exp (Y^{H})

outside some evanescent set, where

Y_{t}^{H} := ∫_{0}^{t} H_{s}dM_{s}−\frac{1}{2} ∫_{0}^{t} H_{s}^{2}d[M^{c}, M^{c}]_{s}+(f∗j_{∫ HdM} )_{t}.

Proof. X̃ := ℒ (Z) ∈ ℳ_{loc ,0}(ℙ). Since Z =

ℰ (X̃), the SPPR of M (applied to X̃) implies the second equation. X := log (Z) = log (ℰ (X̃)) is the logarithmic transform of X̃ = ℒ (exp (X)). Now apply Lemma 2.6 of [J. Kallsen, A. N. Shiryaev (2002). The cumulant process and Esscher’s change of measure. FS 6, 397-428]. □

Corrolary 4 Let ℚ be as in Theorem 7. Then 𝔼_{ℙ}[\frac{dℚ}{dℙ}|ℱ W_{•} ] = ℰ (∫ HdW), where H is 𝔽^{W} -predictable, and

W̃_{t} := W_{t} − ∫_{0}^{t} H_{s}ds

defines a ℚ-Brownian motion W̃.

Corrolary 5 (Exponential Utility and Hedging) Let M ∈ ℳ_{loc ,0}(ℙ) satisfy the SPPR, and let ℚ, H be as in Theorem 9.

𝔼_{ℙ}[\frac{dℚ}{dℙ} log (\frac{dℚ}{dℙ})] = α^{H} + β^{H},

where

α^{H} := 𝔼_{ℚ}[ ∫_{0}^{T} H_{s} dM_{s} − \frac{1}{2} ∫_{0}^{T} H_{s}^{2} d[M^{c}, M^{c}]_{s}]

and

β^{H} := 𝔼_{ℚ}[ _{0<s≤T}^{∑} ( log (\frac{Z_{s}}{Z_{s−}}) − \frac{ΔZ_{s}}{Z_{s−}} )].

Consequently,

𝔼_{ℙ}[\frac{dℚ}{dℙ} log (\frac{dℚ}{dℙ})] = α^{H}

if and only if M is continuous.

If in addition log (Z) resp. ℒ (Z) is a Lévy process, we can calculate β^{H} more explicitly:

\frac{β^{H}}{T} = ∫_{ℝ} f(x) ν^{ℒ (Z)} (dx)= ∫_{ℝ}(1+y−e^{y}) ν^{log (Z)}(dy),

where f(x) := log (x + 1) − x (x > −1).

Conclusion

Stochastic logarithms appear to be the key ingredients in financial mathematics! They do not only describe relative returns and link hedging with the calculation of minimal entropy martingale measures - and hence with utilityindifference. They even determine the structure of the Girsanov transformation.

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