On a Heath–Jarrow–Morton approach for stock options

Abstract

This paper aims at transferring the philosophy behind Heath–Jarrow–Morton to the modelling of call options with all strikes and maturities. Contrary to the approach by Carmona and Nadtochiy (Finance Stoch. 13:1–48, 2009) and related to the recent contribution (Finance Stoch. 16:63–104, 2012) by the same authors, the key parameterisation of our approach involves time-inhomogeneous Lévy processes instead of local volatility models. We provide necessary and sufficient conditions for absence of arbitrage. Moreover, we discuss the construction of arbitrage-free models. Specifically, we prove their existence and uniqueness given basic building blocks.

Appendix

Proofs and further properties are to be found in the electronic supplement.

1.1 A.1 Local characteristics

Let $X$ be an $\mathbb{R}^{d}$-valued semimartingale with integral characteristics $(B,C,\nu)$ in the sense of [20, Definition II.4.1] with respect to some fixed truncation function $h:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$. By [20, I.2.9] there exist a predictable $\mathbb{R}^{d}$-valued process $b$, a predictable $\mathbb{R}^{d\times d}$-valued process $c$, a kernel $K$ from $(\varOmega\times\mathbb{R},\mathscr{P})$ to $(\mathbb{R}^{d},\mathscr{B})$, and a predictable increasing process $A$ such that

$$dB_{t}=b_{t}dA_{t},\quad dC_{t}=c_{t}dA_{t},\quad\nu(dt,dx)=K_{t}(dx)dA_{t}. $$

If $A_{t}=t$, then we call the triplet $(b,c,K)$ local or differential characteristics of $X$ relative to the truncation function $h$. Most processes in applications as, for example, diffusions, Lévy processes etc. allow local characteristics. In this case, $b$ stands for a drift rate, $c$ for a diffusion coefficient, and $K$ for a local Lévy measure representing the jump activity. If they exist, then the local characteristics are unique up to a $dP\otimes dt$-nullset on $\varOmega\times\mathbb{R}_{+}$.

1.2 A.2 Local exponents

Definition A.1

Let $(b,c,K)$ be a Lévy–Khintchine triplet on $\mathbb{R}^{d}$ relative to some truncation function $h:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$. We call the mapping $\psi:\mathbb{R}^{d}\rightarrow\mathbb{C}$,

$$ \psi(u):=iub-\frac{1}{2}u^{\top}cu+\int\big(e^{iux}-1-iuh(x)\big)K(dx), $$

(A.1)

the Lévy exponent corresponding to $(b,c,K)$. By [20, II.2.44] the Lévy exponent determines the triplet $(b,c,K)$ uniquely. If $X$ is a Lévy process with Lévy–Khintchine triplet $(b,c,K)$, then we call $\psi$ the characteristic or Lévy exponent of $X$. If (A.1) exists for all $u\in U\supset \mathbb{R}^{d}$, then we call $\psi:U\to\mathbb{C}$ defined by (A.1) the extended Lévy exponent of $X$ on $U$.

In the same vein, local characteristics naturally lead to local exponents.

Definition A.2

If $X$ is an $\mathbb{R}^{d}$-valued semimartingale with local characteristics $(b,c,K)$, then we write

$$\begin{aligned} \psi^{X}_{t}(u):=iub_{t}-\frac{1}{2}u^{\top}c_{t}u+\int\big(e^{iux}-1-iuh(x)\big)K_{t}(dx),\quad u\in\mathbb{R}^{d}, \end{aligned}$$

(A.2)

for the Lévy exponent corresponding to $(b_{t},c_{t},K_{t})$. We call the family of predictable processes $\psi^{X}(u):=(\psi^{X}_{t}(u))_{t\in\mathbb{R}_{+}},u\in\mathbb{R}^{d}$, the local exponent of $X$. Equality (A.2) implies that $u\mapsto\psi ^{X}_{t}(u)$ is the characteristic exponent of a Lévy process.

The name exponent is of course motivated by the following fact.

Remark A.3

If $X$ is a semimartingale with deterministic local characteristics $(b,c,K)$, then it is a PII, and we have

$$E[e^{iu(X_{T}-X_{t})}\vert \mathscr{F}_{t}]=E[e^{iu(X_{T}-X_{t})}]=\exp\left(\int_{t}^{T}\psi _{s}^{X}(u)ds\right) $$

for any $T\in\mathbb{R}_{+}$, $t\in[0,T]$, $u\in\mathbb{R}^{d}$; see [20, II.4.15].

We now generalise the notion of local exponents to complex-valued semimartingales and more general arguments.

Definition A.4

Let $X$ be a $\mathbb{C}^{d}$-valued semimartingale, and $\beta$ a $\mathbb{C}^{d}$-valued $X$-integrable process. We call a predictable ℂ-valued process $\psi^{X}(\beta)=(\psi _{t}^{X}(\beta))_{t\in\mathbb{R}_{+}}$ the local exponent of $X$ at $\beta$ if $\psi^{X}(\beta)$ is in $L(I)$, meaning that it is Lebesgue-integrable on finite intervals, and $(\exp(i\beta \stackrel {\mbox{\tiny $\bullet $}}{}X_{t}-\int_{0}^{t}\psi_{s}^{X}(\beta)ds))_{t\in\mathbb{R}_{+}}$ is a complex-valued local martingale. We denote by $\mathscr{U}^{X}$ the set of processes $\beta$ such that the local exponent $\psi^{X}(\beta)$ exists.

The following lemma implies that $\psi^{X}(\beta)$ is unique up to a $dP\otimes dt$-nullset.

Lemma A.5

Let $X$ be a complex-valued semimartingale, and $A,B$ complex-valued predictable processes of finite variation with $A_{0}=0=B_{0}$ and such that $\exp(X-A)$ and $\exp(X-B)$ are local martingales. Then $A=B$ up to indistinguishability.

The following result shows that Definition A.4 generalises Definition A.2.

Proposition A.6

Let $X$ be an $\mathbb{R}^{d}$-valued semimartingale with local characteristics $(b,c,K)$. Suppose that $\beta$ is a $\mathbb{C}^{d}$-valued predictable and $X$-integrable process. If $\beta$ is $\mathbb{R}^{d}$-valued for any $t\in\mathbb{R}_{+}$, then $\beta\in\mathscr{U}^{X}$. Moreover, there is equivalence between

• 1. $\beta\in\mathscr{U}^{X}$;

• 2. $\int_{0}^{t}\int1_{\{-\mathrm{Im}(\beta_{s}x)>1\}}e^{-\mathrm {Im}(\beta_{s}x)}K_{s}(dx)ds<\infty$ almost surely for any $t\in \mathbb{R} _{+}$.

In this case, we have

$$\begin{aligned} \psi_{t}^{X}(\beta)=i\beta_{t}b_{t}-\frac{1}{2}\beta_{t}^{\top}c_{t}\beta_{t}+\int \big(e^{i\beta_{t}x}-1-i\beta_{t}h(x)\big) K_{t}(dx) \end{aligned}$$

(A.3)

outside some $dP\otimes dt$-nullset.

Equality (A.3) implies that the local exponent of $X$ at any $\beta\in\mathscr{U}^{X}$ is determined by the triplet $(b,c,K)$ and hence by the local exponent of $X$ in the sense of Definition A.2.

Definition A.7

Let $X^{(1)},\dots,X^{(n)}$ be semimartingales which allow local characteristics. We call them locally independent if

$$\mathscr{U}^{(X^{(1)},\dots,X^{(n)})}\cap\big(L(X^{(1)})\times\cdots \times L(X^{(n)})\big) =\mathscr{U}^{X^{(1)}}\times\cdots\times\mathscr{U}^{X^{(n)}} $$

for

$$\begin{aligned} L(X^{(1)})\times\cdots\times L(X^{(n)}) :=&\big\{ \beta=(\beta^{(1)},\dots,\beta^{(n)})\mbox{ complex-valued}:\\ &\phantom{\{} \beta^{(i)} \mbox{ $X^{(i)}$-integrable for }i=1,\dots,n\big\} \end{aligned}$$

and

$$\psi^{(X^{(1)},\dots,X^{(n)})}(\beta)=\sum_{j=1}^{n}\psi^{X^{(j)}}(\beta^{(j)}) $$

outside some $dP\otimes dt$-nullset for any $\beta=(\beta^{(1)},\dots,\beta^{(n)})\in\mathscr{U}^{(X^{(1)},\dots,X^{(n)})}$.

Lemma A.8

If $\varphi\in\varPi$, then $\mathrm{Re}(\varphi(u))\leq0$ for any $u\in \mathbb{R}$, where $\varPi$ is defined in Sect. 3.1.

The following four lemmas follow immediately from the definition of local exponents.

Lemma A.9

Let $X$ be a ℂ-valued semimartingale that allows local characteristics. Then there is equivalence between

• 1. $\exp(X)$ is a local martingale;

• 2. $-i\in\mathscr{U}^{X}$ and $\psi^{X}(-i)=0$ outside some $dP\otimes dt$-nullset.

Remark A.10

If $X$ in Lemma A.9 is real-valued and if $e^{X}$ is a local martingale, then Proposition A.6 yields that the mapping $\mathbb{R}\to\mathbb{C}$, $u\mapsto\psi^{X}_{t}(u)$ is in $\varPi$ outside some $dP\otimes dt$-nullset.

Lemma A.11

Let $X$ be a $\mathbb{C}^{d}$-valued semimartingale, $\beta$ a $\mathbb{C}^{d}$-valued and $X$-integrable process, and $u\in\mathbb{C}$. Then $u\beta\in\mathscr{U}^{X}$ if and only if $u\in\mathscr{U}^{\beta\stackrel{.}{} X}$. In that case, we have

$$\psi^{X}(u\beta)=\psi^{\beta\stackrel{.}{} X}(u). $$

Lemma A.12

Let $X,Y$ be $\mathbb{C}^{d}$-valued semimartingales, and $u\in\mathbb{C}^{d}$. Then $u\in\mathscr{U}^{X+Y}$ if and only if $(u,u)\in\mathscr{U}^{(X,Y)}$. In that case, we have

$$\psi^{X+Y}(u)=\psi^{(X,Y)}(u,u) $$

outside some $dP\otimes dt$-nullset.

Lemma A.13

Let $X,Z$ be $\mathbb{C}^{d}$-valued semimartingales, and $\beta,\gamma$ predictable $\mathbb{C}^{d}$-valued processes such that

• 1. the components of $\gamma$ are Lebesgue-integrable on finite intervals;

• 2. $\beta\gamma$ is Lebesgue-integrable on finite intervals;

• 3. $Z_{t}=Z_{0}+\int_{0}^{t}\gamma_{s} ds+X_{t}$.

Then $\beta\in\mathscr{U}^{Z}$ if and only if $\beta\in\mathscr{U}^{X}$. In that case, $\psi^{Z}(\beta)=\psi^{X}(\beta)+i\beta\gamma$ outside some $dP\otimes dt$-nullset.

1.3 A.3 Semimartingale decomposition relative to a semimartingale

Let $(X,Y)$ be an $\mathbb{R}^{1+d}$-valued semimartingale with local characteristics $(b,c,K)$, written here in the form

$$ b=\left( \textstyle\begin{array}{c}b^{X}\\ b^{Y} \end{array}\displaystyle \right), \quad c:=\left( \textstyle\begin{array}{cc}c^{X} & c^{X,Y}\\ c^{Y,X}& c^{Y} \end{array}\displaystyle \right). $$

(A.4)

Suppose that $\int_{0}^{t}\int_{(1,\infty)}e^{x}K_{s}(dx)ds<\infty$ for any $t\in\mathbb{R}_{+}$ or, equivalently, $e^{X}$ is a special semimartingale. We set

$$\begin{aligned} X^{\parallel}_{t} :=&\log \mathscr{E}\left(\big(c^{X,Y}(c^{Y})^{-1}\big)\stackrel {\mbox{\tiny $\bullet $}}{}Y^{c}_{t}+f*\big(\mu ^{(X,Y)}-\nu^{(X,Y)}\big)_{t}\right) \end{aligned}$$

for any $t\in\mathbb{R}_{+}$, where $c^{-1}$ denotes the pseudoinverse of a matrix $c$ in the sense of [1], $Y^{c}$ is the continuous local martingale part of $Y$, $\mu^{(X,Y)}$ resp. $\nu^{(X,Y)}$ denote the random measure of jumps of $(X,Y)$ and its compensator, and $f:\mathbb{R}^{1+d}\rightarrow\mathbb{R}$, $(x,y)\mapsto(e^{x}-1)1_{\{y\neq0\}}$. We call $X^{\parallel}$ and $X^{\perp}:=X-X^{\parallel}$ the dependent and independent parts of $X$ relative to $Y$.

1.4 A.4 Option pricing by Fourier transform

We denote by

$$\begin{aligned} \mathscr{F}f(u):=\lim_{C\rightarrow\infty}\int_{-C}^{\infty}f(x)e^{iux}dx \end{aligned}$$

(A.5)

the (left-)improper Fourier transform of a measurable function $f:\mathbb{R}\rightarrow\mathbb{C}$ for any $u\in\mathbb{R}$ such that the expression exists. If $f$ is Lebesgue-integrable, then the improper Fourier transform and the ordinary Fourier transform (i.e., $u\mapsto\int f(x)e^{iux}dx$) coincide. In our application in Sect. 3, the improper Fourier transform exists for any $u\in\mathbb{R}\setminus\{0\}$. Moreover, we denote by

$$\begin{aligned} \mathscr{F}^{-1} g(x):=\frac{1}{2\pi}\left(\lim_{\varepsilon\downarrow0}\int _{\varepsilon}^{\infty}e^{-iux}g(u)du +\lim_{\varepsilon\downarrow0}\int_{-\infty}^{-\varepsilon} e^{-iux}g(u)du\right) \end{aligned}$$

(A.6)

an improper inverse Fourier transform, which is suitable for our application in Sect. 3.

Proposition A.14

Let $(\varOmega, \mathscr{F},P)$ be a probability space, and $\mathscr{G}\subset \mathscr{F}$ a sub-$\sigma$-field. Let $Y$ be a random variable with $E[e^{Y}\vert \mathscr{G}]=1$ and define

$$\mathscr{O}(x):= \textstyle\begin{cases}E[(e^{Y-x}-1)^{+}\vert \mathscr{G}] & \textit{ if }x\geq0,\\ E[(1-e^{Y-x})^{+}\vert \mathscr{G}] & \textit{ if }x< 0. \end{cases} $$

Then we have

$$\begin{aligned} \mathscr{O}(x) = & \mathscr{F}^{-1}\bigg(u\mapsto\frac{1-E[e^{iuY}\vert \mathscr{G}]}{u^{2}+iu}\bigg)(x),\\ E[e^{iuY}\vert \mathscr{G}] = & 1-(u^{2}+iu)\mathscr{F}\big(x\mapsto\mathscr{O}(x)\big)(u) \end{aligned}$$

for any $u,x\in\mathbb{R}$.

1.5 A.5 Bochner integrals in Fréchet spaces

Option price surfaces are interpreted as elements of the Fréchet space $L^{1}(\mathbb{R}_{+},E)$ in Sect. 4. In order to derive existence and uniqueness results, we need to consider stochastic differential equations in such spaces; see the electronic supplement. These in turn rely on a properly defined Bochner integral. Let $F$ be a vector space, and $(\Vert\cdot\Vert_{n})_{n\in\mathbb{N}}$ an increasing sequence of separable semi-norms on $F$ such that

• 1. $\|x\|_{n}=0$ $\forall n\in\mathbb{N}$ only if $x=0$;

• 2. if $(x_{k})_{k\in\mathbb{N}}$ is a $\|\cdot\|_{n}$-Cauchy sequence for all $n\in\mathbb{N}$, then there exists $x\in F$ with $\lim_{k\to\infty}\|x_{k}-x\|_{n}=0$ for any $n\in\mathbb{N}$.

Then

$$d(x,y):=\sum_{n\in\mathbb{N}}2^{-n}(1\wedge\Vert x-y\Vert_{n}) $$

defines a complete, translation-invariant, separable metric on the Fréchet space $F$.

Example A.15

We are mainly interested in the case where $F:= L^{1}(\mathbb{R}_{+},E)$ and $\Vert\cdot\Vert_{n}:= \Vert\cdot\Vert_{n,n}$ as defined in Sect. 4 or, alternatively, $F=E$ itself.

Fix a $\sigma$-finite measure space $(\varGamma,\mathscr{G},\mu)$. The goal of this part is to define a Bochner integral $\int fd\mu$ for measurable functions $f:\varGamma\to F$ with values in the Fréchet space $F$; see Definition A.17. If $f$ is simple and integrable in the sense that it is a linear combination of indicators of sets in $\mathscr{G}$ with finite $\mu$-measure, then the integral $\int fd\mu$ is naturally defined as a sum.

For each fixed $n\in\mathbb{N}$, denote the set of measurable Bochner-integrable functions from $(\varGamma,\mathscr{G},\mu)$ to the complete, separable, semi-normed space $(F,\Vert\cdot\Vert_{n})$ by $\mathscr{L}^{1}((\varGamma,\mathscr{G},\mu), (F,\Vert\cdot\Vert_{n}))$. Recall that a measurable function $f:\varGamma \to F$ is called Bochner-integrable (relative to $\|\cdot\|_{n}$) if there is a sequence $(f^{(k)})_{k\in\mathbb{N}}$ of simple integrable functions with $\lim_{k\to\infty}\int\|f^{(k)}-f\|_{n}\,d\mu=0$. Equivalently, $f:\varGamma\to F$ is measurable and $\int\|f\|_{n}\,d\mu< \infty$; see, for example, Lemma D.7 in the electronic supplement. In this case, there is some $x\in F$ such that

$$\lim_{k\to\infty}\left\|\int f^{(k)}d\mu-x\right\|_{n}=0 $$

for any such sequence. This element $\int fd\mu:=x$ is called the $(\Vert\cdot\Vert_{n})$-Bochner integral of $f$.

Note that we do not identify functions that are $\mu$-a.e. identical. Therefore, the space $\mathscr{L}^{1}((\varGamma,\mathscr{G},\mu),(F,\Vert\cdot\Vert_{n}))$, together with the semi-norm $\|f\|:= \int\Vert f\Vert_{n} \,d\mu$, is a complete semi-normed space but in general not a Banach space. Moreover, versions of the $\Vert\cdot\Vert_{n}$-Bochner integral may differ by $\Vert\cdot\Vert_{n}$-distance zero.

We are now ready to define the desired integral for Fréchet-space-valued functions.

Proposition A.16

Let

$$\begin{aligned} &f\in\mathscr{L}^{1}\big((\varGamma,\mathscr{G},\mu),(F,d)\big):= \bigcap_{n\in\mathbb{N}} \mathscr{L}^{1}\big((\varGamma,\mathscr{G},\mu ),(F,\Vert\cdot\Vert_{n})\big)\\ &\,=\bigg\{ f:\varGamma\to F: f\textit{ measurable with } \int\Vert f\Vert_{n} \,d\mu< \infty, n\in \mathbb{N}\bigg\} . \end{aligned}$$

Then there is one and only one $x\in F$ such that $x$ is a version of the $\Vert\cdot\Vert_{n}$-Bochner integral $\int fd\mu$ for any $n\in\mathbb{N}$.

Definition A.17

(Bochner integral)

We call the set $\mathscr{L}^{1}((\varGamma,\mathscr{G},\mu),(F,d))$ the space of Bochner-integrable functions and $\int fd\mu:=x$ from Proposition A.16 the corresponding Bochner integral.

We identify a right-continuous increasing function $X:\mathbb{R} _{+}\to \mathbb{R}$ with its corresponding Lebesgue–Stieltjes measure $\mu$ on $\mathbb{R} _{+}$. For $t\in \mathbb{R} _{+}$ and $\varphi: \mathbb{R} _{+}\to F$, we write

$$\varphi\in\mathscr{L}^{1}\big(([0,t],X),(F,d)\big) $$

if $\varphi1_{[0,t]}\in\mathscr{L}^{1}((\mathbb{R} _{+},\mathscr{B}(\mathbb{R} _{+}),\mu),(F,d))$. Moreover, we write

$$\int_{0}^{t}\varphi_{s}dX_{s}:= \int\varphi1_{[0,t]}d\mu\in F $$

for the integral from Definition A.17.

For right-continuous increasing processes $X$ and for all measurable functions $\varphi:\varOmega\times \mathbb{R} _{+}\to \mathbb{R}$, both $\varphi\in\mathscr{L}^{1}(([0,t],X),(F,d))$ and $\int_{0}^{t}\varphi_{s}dX_{s}$ are to be interpreted in a pathwise sense, that is, for any fixed $\omega\in\varOmega$.