1 Introduction

We propose to modify the notion of a contingent claim price in the setting where the source of uncertainty is a strict local martingale rather than a martingale. More precisely, we propose to use as a pricing operator for contingent claims the minimal cost for superreplicating a given contingent claim simultaneously under two probability measures. In the case of foreign exchange markets with X modelling the exchange rate (for example, the price of one Euro in dollars), the two measures can be thought of as a “dollar measure” and a “Euro measure” corresponding to the choice of dollars or Euros as numéraires. The two measures are not equivalent if X is a strict local martingale (that is, a local martingale that is not a martingale). In this case, the cost for joint superreplication is higher than the expected value under the local martingale measure.

Our main result is Theorem 3.4, which provides a formula for the minimum joint superreplication cost in a complete market. This approach restores put-call parity and international put-call equivalence for model prices, and gives the price X 0 for the contingent claim that pays X T at time T. Our pricing formula agrees with the proposals of other authors (Lewis [25], Madan and Yor [26], Andersen [1]); the novelty is the rigorous justification of this formula as a hedging cost.

The mathematical contribution of this paper is mainly contained in Sect. 2 and the Appendix. We show how to construct the measure corresponding to a numéraire that is allowed to vanish. Towards this end, we construct the Föllmer measure for nonnegative local martingales, extending the corresponding results for strictly positive local martingales. We also develop a stochastic calculus for the suggested change of measure, in which neither measure dominates the other one.

Section 3 contains the main financial results of the paper. Our approach uses two numéraires simultaneously, which requires us to re-introduce the notions of market completeness and superreplication. We introduce a model of the market and define trading strategies and contingent claim replication. After proving our main result on the minimal replicating price, we give numerous corollaries and examples.

In Sect. 4, we consider a physical measure under which both currencies might completely devalue against the other. In such a situation, an equivalent probability measure under which the exchange rate follows local martingale dynamics cannot exist. Instead, each risk-neutral measure is only absolutely continuous with respect to the physical measure. However, as one may use both currencies as hedging instruments, superreplication of contingent claims might still be possible. We provide a set of conditions under which replicating strategies exist, and we show in Proposition 4.2 how the minimal cost for such a strategy is exactly given by the pricing operator of this paper.

This point of view gives us an interpretation of the lack of the martingale property of an exchange rate under a risk-neutral probability measure, as the positive probability of complete devaluations of currencies (corresponding to explosions of the exchange rate) occurring under some dominating probability measure. We remark that this dominating probability measure usually does not correspond to the Föllmer measure, which we shall discuss below, but is equivalent to the sum of the Föllmer measure and the original measure.

1.1 Related literature

We now link our financial results to relevant literature:

Strict local martingales, that is, local martingales that are not martingales, have recently been introduced in the financial industry to model exchange rates under the risk-neutral measure. This is due to the fact that they are able to well capture observed features of the market such as implied volatility surfaces and that they are easily analytically tractable. An important example is the class of “quadratic normal volatility” models, a family of local martingales which are studied for example in Andersen [1] and in our companion paper Carr et al. [2]. There is a vast literature on pricing options in strict local martingale models, often coined “bubbles”. For an overview of this literature, we refer the interested reader to the recent survey by Protter [31]. Heston et al. [18] were among the first to point out that put-call parity usually does not hold in strict local martingale models. For a discussion of these models specifically in the context of foreign exchange, we refer the reader to Jarrow and Protter [21]. Further models in which strict local martingales appear can be found among the class of stochastic volatility models; Sin [40] was among the first to point this out. For example, in the log-normal SABR model, if the asset price process is positively correlated with the stochastic volatility process, then it follows strict local martingale dynamics; see Example 6.1 in Henry-Labordère [17].

Several papers suggest adjustments to the pricing of contingent claims by expectations in strict local martingale models in order to address the lack of put-call parity:

Lewis [25] proposes to add a correction term to the price of a call. However, this approach lacks a clear economic motivation. As his starting point is exactly put-call parity for model prices, it is not clear how other contingent claims should be priced. Cox and Hobson [4] suggest to consider collateral requirements when pricing contingent claims; such collateral requirements correspond to a constraint on the class of admissible trading strategies. This leads to a higher contingent claim price, but usually does not restore put-call parity for model prices.

Madan and Yor [26] propose to take the limit of a sequence of prices obtained from approximating the asset price by true martingales as the price for a contingent claim. This approach also restores put-call parity for model prices. However, one might criticize that the limit of the approximating prices usually does not agree with the classical price in the case that the underlying is a true martingale. For instance, consider an arbitrage-free, complete market with corresponding risk-neutral measure \(\mathbb {Q}\), a standard \(\mathbb {Q}\)-geometric Brownian motion X=(X t ) t∈[0,T] as underlying, and a contingent claim that pays \(\mathbf {1}_{\{X_{T} \in \mathbb {N}\}} / q_{X_{T}}\) at maturity T, where \(q_{y} = \mathbb {Q}(\max_{t \in[0,T]} X_{t} \geq y)\) for all \(y \in \mathbb {R}\). This claim should have price zero as \(X_{T} \notin \mathbb {N}\) almost surely. However, if one approximates X with versions that are stopped at hitting times of integers, as in Madan and Yor’s approach, then one obtains a price of one for that claim.

We here suggest to take an economic point of view based on a replicating argument and derive a pricing operator that restores put-call parity, and therefore assigns model prices that correspond to observed market prices. We thus not only justify Lewis’ pricing operator by an economic argument, but also generalize it to a wider class of models and contingent claims.

The approach taken here can be interpreted as a link between classical pricing and pricing under Knightian uncertainty. Pricing in the classical sense corresponds to the choice of one probability measure under which a contingent claim is superreplicated. This choice implies a strong assumption on the chosen nullsets, that is, by assumption a set of events is determined to be not relevant for computing a replicating trading strategy. If the modeler were to consider another probability measure, other events would be selected, leading to a different replicating price and strategy. Indeed, one would like that the choice of probability measure should not have a large impact on the price (or, more importantly, on the hedging strategy), or at least should be quantifiable.

We remark that Delbaen and Schachermayer [7] work out the connection of strict local martingales and changes of numéraires. While we understand a change of numéraire as a combination of a change of currency and the corresponding change of measure, they start by looking at the change of currency only. Their results imply that in an arbitrage-free model, a change of currency leads to the existence of arbitrage if the corresponding exchange rate is a strict local martingale under a unique risk-neutral measure. We circumvent the appearance of arbitrage here by associating the change of numéraire with the introduction of a new probability measure that is not equivalent to the old one. It is exactly this lack of equivalence which avoids the arbitrage after the change of currency.

2 Change of measure with a nonnegative local martingale

In this section, given a probability measure \(\mathbb {Q}\), we construct and discuss a new probability measure \(\widehat{\mathbb {Q}}\) corresponding to a density process that follows local martingale dynamics only and is allowed to hit zero. It is helpful to interpret the notation of this section in a financial context. Towards this end, we interpret X as an exchange rate, for example, the price of one Euro in dollars. Then \(\mathbb {Q}\) represents the risk-neutral measure corresponding to the dollar-numéraire and \(\widehat{\mathbb {Q}}\) the risk-neutral measure corresponding to the Euro-numéraire.

The mathematical finance literature has utilized the techniques developed by Föllmer [12] and Meyer [27] to construct probability measures with a strict local martingale as density process mostly in the context of arbitrage and bubbles; see for example Delbaen and Schachermayer [6], Pal and Protter [28], Fernholz and Karatzas [11], Ruf [34], and, parallel to the present work, Kardaras et al. [23]. Here, we slightly extend this literature by allowing the local martingale to hit zero. On the other hand, true martingales possibly hitting zero as density processes have been studied by Schönbucher [39] within the area of credit risk. Schönbucher [39] terms the corresponding measure a “survival measure”. We extend this direction of research by allowing the change of measure to be determined by a local martingale only.

Throughout this section, we fix a time horizon T∈(0,∞], a stochastic basis \((\varOmega, \mathcal {F}_{T}, (\mathcal {F}_{t})_{t \in[0, T]}, \mathbb {Q})\), and a nonnegative \(\mathbb {Q}\)-local martingale X=(X t ) t∈[0,T]. We assume that x 0:=X 0>0 is deterministic, that \((\mathcal {F}_{t})_{t \in[0, T]}\) is right-continuous, and that X(ω) has right-continuous paths for all ω∈Ω; see also Lemma 1.1 in Föllmer [12] for the construction of a right-continuous version if this assumption does not hold.

Any nonnegative random variable, such as X T , is explicitly allowed to take values in [0,∞]. For a nonnegative random variable Z and some set \(A \in~\mathcal {F}_{T}\), we write Z 1 A to denote the random variable that equals Z whenever ω∈ A, and otherwise is zero. For any stopping time τ, we denote the stochastic process that arises from stopping a process N=(N t ) t∈[0,T] at time τ by \(N^{\tau}= (N_{t}^{\tau})_{t \in [0,T]}\); that is, \(N^{\tau}_{t} := N_{t \wedge\tau}\) for all t∈[0,T]. For any measure \(\mathbb {P}\) on \((\varOmega , \mathcal {F}_{T})\), we denote the corresponding expectation operator by \(\mathbb {E}^{{\mathbb {P}}}\).

We define the stopping times

$$\begin{aligned} R_i &:= \inf \{t \in[0,T]: X_t > i \}, \\ S_i &:= \inf \{t \in[0,T]: X_t < 1/i \} \end{aligned}$$

for all \(i \in \mathbb {N}\), set R:=lim i↑∞ R i and S:=lim i↑∞ S i , with the convention that \(\inf\emptyset:= \mathfrak{T}\) for some transfinite time \(\mathfrak{T} > \infty\); see Appendix A for details. Define the process Y=(Y t ) t∈[0,T] by Y t :=1/X t 1 {R>t} for t∈[0,T] and the stopping times

$$\begin{aligned} S_i^Y &:= \inf \{t \in[0,T]: Y_t > i \}, \end{aligned}$$

again with \(\inf\emptyset:= \mathfrak{T}\). We observe that \(S_{i} = S_{i}^{Y}\) for all \(i \in \mathbb {N}\) if X (R+t)∧T =∞ for all t≥0. We assume that \((\mathcal {F}_{t} \cap \mathcal {F}_{R-})_{t \in[0, T]}\) is the right-continuous modification of a standard system; see Appendix B for notation and a discussion of this assumption.

The next theorem states the main result of this section: for the nonnegative \(\mathbb {Q}\)-local martingale X, there exists a probability measure under which X serves as the numéraire. We remark that we only specify the new measure on \((\varOmega, \mathcal {F}_{R-})\) and not on \((\varOmega, \mathcal {F}_{T})\). This is due to the fact that the original measure \(\mathbb {Q}\), by assumption, does not “see” any events after the stopping time R. However, a measure on \((\varOmega, \mathcal {F}_{T})\) satisfying the properties of the next theorem could be easily constructed by arbitrarily, but consistently, extending the measure \(\widehat{\mathbb {Q}}\) from \(\mathcal {F}_{R-}\) to \(\mathcal {F}_{T}\). Observe that Z 1 {R>τ∧T} is \(\mathcal {F}_{R^{-}}\)-measurable if Z is an \(\mathcal {F}_{\tau\wedge T}\)-measurable random variable for some stopping time τ.

Theorem 2.1

(Change of measure with a nonnegative local martingale)

There exists a unique probability measure \(\widehat{\mathbb {Q}}\) on \((\varOmega, \mathcal {F}_{R-})\) such that

$$\begin{aligned} \widehat{\mathbb {Q}} (A \cap \{R > \tau\wedge T \} ) = \frac{\mathbb {E}^{\mathbb {Q}} [\mathbf {1}_A X^\tau_T ]}{x_0} \end{aligned}$$
(2.1)

holds for all stopping times τ and for all \(A \in \mathcal {F}_{\tau\wedge T}\). This measure also satisfies \(\widehat{\mathbb {Q}}(R_{i} \wedge T<R) = 1\) for all \(i \in \mathbb {N}\). Moreover, we have

$$\begin{aligned} \mathbb {E}^{\widehat{\mathbb {Q}}} \bigl[Z \mathbf {1}_{\{R > \tau\wedge T\}} \bigr] &= \frac{\mathbb {E}^{\mathbb {Q}} [ Z \mathbf {1}_{\{S>\tau\wedge T\}} X^\tau_T ]}{x_0} \end{aligned}$$
(2.2)

and

$$\begin{aligned} \mathbb {E}^{{\mathbb {Q}}} \bigl[Z \mathbf {1}_{\{S > \tau\wedge T\}} \bigr] &= x_0 \mathbb {E}^{\widehat {\mathbb {Q}}} [ Z \mathbf {1}_{\{R>\tau\wedge T\}} Y^\tau_T ] \end{aligned}$$
(2.3)

for all stopping times τ and \(\mathcal {F}_{\tau\wedge T}\)-measurable random variables Z∈[0,∞].

The process Y satisfies

$$\begin{aligned} \widehat{\mathbb {Q}} \Bigl(\inf_{t \in[0,T]} Y_t \geq0 \Bigr) = 1; \end{aligned}$$

furthermore, we have

$$\begin{aligned} \widehat{\mathbb {Q}} \Bigl(\inf_{t \in[0,T]} Y_t > 0 \Bigr) = 1 \end{aligned}$$

if and only if the process X is a (uniformly integrable) \(\mathbb {Q}\)-martingale. Moreover:

  1. 1.

    Y is a \(\widehat{\mathbb {Q}}\)-supermartingale;

  2. 2.

    Y is a local \(\widehat{\mathbb {Q}}\)-martingale if and only if \({\mathbb {Q}}(S>S_{i} \wedge T) = 1\) for all \(i \in \mathbb {N}\); then \((S_{i}^{Y})_{i \in \mathbb {N}}\) is, under \(\widehat{\mathbb {Q}}\), a localizing sequence for Y;

  3. 3.

    Y is a \(\widehat{\mathbb {Q}}\)-martingale if and only if \({\mathbb {Q}}(S > T) = 1\).

Proof

Without loss of generality, assume throughout the proof that x 0=1. Observe that \(X^{R_{i}}\) is a nonnegative martingale by Lemma A.3 in the Appendix; thus, it generates a measure \(\mathbb {Q}_{i}\) on \((\varOmega, {\mathcal {F}}_{R_{i}-})\) by \(\mathrm {d}\mathbb {Q}_{i} := X^{R_{i}}_{T} \,\mathrm {d}\mathbb {Q}\) for all \(i \in \mathbb {N}\). Observe that the family of probability measures \((\mathbb {Q}_{i})_{i \in \mathbb {N}}\) is consistent, that is, \(\mathbb {Q}_{i+j}|_{{\mathcal {F}}_{R_{i}-}} = \mathbb {Q}_{i}\) for all \(i,j \in \mathbb {N}\), and that \({\mathcal {F}}_{R-} = \bigvee_{i \in \mathbb {N}} {\mathcal {F}}_{R_{i}-}\). Now, the extension Theorem V.4.1 in Parthasarathy [29] yields the existence of a probability measure \(\widehat{\mathbb {Q}}\) on \((\varOmega, {\mathcal {F}}_{R-})\) such that \(\widehat{\mathbb {Q}}|_{{\mathcal {F}}_{R_{i}-}} = \mathbb {Q}_{i}\); see Appendix B for an argument that the assumptions of that theorem hold.

Observe that for all \(A \in \mathcal {F}_{\tau\wedge T}\) and stopping times τ,

$$\begin{aligned} \widehat{\mathbb {Q}} \bigl(A \cap \{R > \tau\wedge T \} \bigr) &= \lim _{i \uparrow\infty} \widehat{\mathbb {Q}} (A \cap \{ R_i > \tau \wedge T \} ) = \lim_{i \uparrow\infty} \mathbb {Q}_i (A \cap \{ R_i > \tau\wedge T \} ) \\ &= \lim_{i \uparrow\infty} \mathbb {E}^{\mathbb {Q}} \bigl[ \mathbf {1}_{A \cap \{ R_i > \tau \wedge T \}} X^{R_i}_T \bigr] = \lim_{i \uparrow\infty} \mathbb {E}^{\mathbb {Q}} \bigl[\mathbf {1}_{A \cap \{ R_i > \tau \wedge T \}} X^{\tau}_T \bigr] \\ &= \mathbb {E}^{\mathbb {Q}} \bigl[\mathbf {1}_A X^\tau_T \bigr]. \end{aligned}$$

This yields (2.1). Now, with τ=R i and A=Ω we obtain that \(\widehat{\mathbb {Q}} (R > R_{i} \wedge T ) = 1\) for all \(i \in \mathbb {N}\). This identity implies that \(\widehat{\mathbb {Q}}(A) = \widehat{\mathbb {Q}} (A \cap \{R > R_{i} \wedge T \} ) = \mathbb {E}^{\mathbb {Q}}[X^{R_{i}}_{T} \mathbf {1}_{A} ]\) for all \(A \in \mathcal {F}_{R_{i}-}\) and \(i \in \mathbb {N}\). Since \({\mathcal {F}}_{R-} = \bigvee_{i \in \mathbb {N}} \mathcal {F}_{R_{i}-}\) and \(\bigcup_{i \in \mathbb {N}} \mathcal {F}_{R_{i}-} \) is a π-system, this yields uniqueness of \(\widehat {\mathbb {Q}}\) on \((\varOmega, \mathcal {F}_{R-})\). Then (2.2) follows from (2.1) by applying the monotone convergence theorem, and (2.3) follows from (2.2) by using the fact that \(\mathbb {Q}(R>T)=1\) and formally applying (2.2) to \(Z \mathbf {1}_{\{R > \tau\wedge T\}} Y^{\tau}_{T}\) instead of Z, where τ and Z are as in the theorem.

Observe that \(\{\inf_{t \in[0,T]} Y_{t} < 0\} = \bigcup_{i \in \mathbb {N}} \{\inf_{t \in[0,R_{i})}Y_{t} < 0\}\) \(\widehat{\mathbb {Q}}\)-almost surely and conclude that Y is \(\widehat{\mathbb {Q}}\)-almost surely nonnegative by dominated convergence and applying (2.1) with τ=R i and \(A = {\{ \inf_{t \in[0,R_{i})} Y_{t} < 0\} }\) for all \(i \in \mathbb {N}\). Next, (2.1) with τ=T and A={inf t∈[0,T] Y t >0} yields that \(\widehat{\mathbb {Q}}(\{\inf_{t \in[0,T]} Y_{t} > 0\} ) = \mathbb {E}^{\mathbb {Q}}[X_{T}]\), which shows the equivalence of the positivity of Y under \(\widehat{\mathbb {Q}}\) and the martingale property of X under \(\mathbb {Q}\).

Observe that

$$\begin{aligned} \mathbb {E}^{\widehat{\mathbb {Q}}} [Y_t \mathbf {1}_A ] = \mathbb {E}^{\widehat{\mathbb {Q}}} \bigl[Y_{t} \mathbf {1}_A \mathbf {1}_{\{R > t\}} \bigr] = \mathbb {Q}(A \cap \{S >t \} ) \leq \mathbb {Q}(A \cap \{S > s \} ) =\mathbb {E}^{\widehat{\mathbb {Q}}} [Y_s \mathbf {1}_A ] \end{aligned}$$

by applying (2.3) twice (with Z=1 A and τ=t the first time and τ=s the second time), for all t∈[0,T], s∈[0,t] and \(A \in \mathcal {F}_{s}\). Thus, the process Y is a \(\widehat{\mathbb {Q}}\)-supermartingale. This implies for all stopping times τ that Y τ is a \(\widehat {\mathbb {Q}}\)-martingale if and only if \(\mathbb {Q}(S > \tau\wedge T ) = 1\), since \(\mathbb {Q}(S > \tau\wedge T ) = \mathbb {E}^{\widehat{\mathbb {Q}}} [Y_{T}^{\tau}]\) again by (2.3) with Z=1. Using τ=T yields that Y is a \(\widehat{\mathbb {Q}}\)-martingale if and only if \(\mathbb {Q}(S >T ) = 1\), and using \(\tau= S^{Y}_{i}\) yields that for all \(i \in \mathbb {N}\), \(Y^{S_{i}^{Y}} \) is a \(\widehat{\mathbb {Q}}\)-martingale if and only if \(\mathbb {Q}(S >S_{i} \wedge T ) = 1\), since \(\mathbb {Q}(S_{i}^{Y} = S_{i})=1\). We conclude by applying (i) and the equivalence of (a.1) and (a.2) in Lemma A.3 in the Appendix. □

It is important to note that the measures \(\mathbb {Q}\) and \(\widehat{\mathbb {Q}}\) are usually not absolutely continuous with respect to each other; in particular, we can have \(\widehat{\mathbb {Q}}(R \leq T) > 0 = \mathbb {Q}(R \leq T)\). Furthermore, note that the indicators in (2.2) and (2.3) can be omitted if Z is finite or \(\mathbb {Q}(S \leq T) = 0\) or \(\widehat{\mathbb {Q}}(R \leq T) = 0\), respectively. In general, however, the indicators are necessary as the example \(Z=1/X^{S}_{T}\) illustrates.

Remark 2.2

(Duality of martingale property and positivity of density processes)

Observe that we have proved the equivalence of the following statements in Theorem 2.1:

  1. 1.

    Y is a (uniformly integrable) \(\widehat{\mathbb {Q}}\)-martingale on [0,T];

  2. 2.

    \(\mathbb {Q}(S>T) = 1\).

We have also proved the equivalence of the following statements:

  1. 1.

    X is a (uniformly integrable) \({\mathbb {Q}}\)-martingale on [0,T];

  2. 2.

    \(\widehat{\mathbb {Q}}(R>T) = 1\).

We emphasize the symmetry of these two equivalences.

This duality of the martingale property of a nonnegative local martingale under one measure and its non-explosiveness under another measure has been utilized to provide conditions for the martingale property of local martingales; see in particular Cheridito et al. [3] and Ruf [35, 37].

We next derive properties of the change of measure in Theorem 2.1. In particular, we focus on understanding which of the martingale properties of stochastic processes survive the change of measure, possibly after modifying the processes. The discussion here involves local martingales on stochastic intervals. This notion generalizes the definition of local martingales; its precise definition can be found in Appendix A.

Proposition 2.3

(Equivalence of (local) martingales)

Assume the notation of Theorem 2.1 and let τ denote a stopping time and N=(N t ) t∈[0,T] a progressively measurable stochastic process taking values in [0,∞] and such that N t =N t 1 {R>t} for all t∈[0,T]. The following statements then hold:

  1. (i)

    The process \((N^{\tau}_{t} \mathbf {1}_{\{S > \tau\wedge t\}})_{t \in [0,T]}\) is a \(\mathbb {Q}\)-martingale if and only if N τ Y τ is a \(\widehat{\mathbb {Q}}\)-martingale.

  2. (ii)

    The process (N t 1 {S>t}) t∈[0,T] is a \(\mathbb {Q}\)-local martingale on [0,S) (equivalently, on [0,R∧S)) if and only if NY is a \(\widehat{\mathbb {Q}}\)-local martingale on [0,R) (equivalently, on [0,R∧S)).

  3. (iii)

    If \((N_{t}^{S_{i}} \mathbf {1}_{\{S>S_{i} \wedge t\}})_{t \in [0,T]}\) is a \(\mathbb {Q}\)-martingale for all \(i \in \mathbb {N}\), then NY is a  \(\widehat{\mathbb {Q}}\)-local martingale.

The proof of Proposition 2.3 is based on an extended version of Bayes’ formula and can be found in Appendix C. Applying (i) to N≡1 and to N=X with τ=T yields exactly the equivalences of Remark 2.2. Applying (ii) to N≡1 yields that Y is a \(\widehat{\mathbb {Q}}\)-local martingale on [0,R∧S) if and only if S is announced under \(\mathbb {Q}\) (for example, if X does not jump to zero under \(\mathbb {Q}\)). The example X=Y≡1 shows that the reverse direction in (iii) in Proposition 2.3 usually does not hold.

In order to better understand the change of measure suggested in this section, it is instructive to study an extreme case where the measures \(\mathbb {Q}\) and \(\widehat{\mathbb {Q}}\) are not only not absolutely continuous with respect to each other, but even singular:

Example 2.4

(Singular measures)

Assume that T∈(0,∞) and let X be defined by

$$\begin{aligned} X_t = 1+\int_0^{t\wedge\widetilde{S}} \frac{1}{\sqrt{T-u}} \,\mathrm {d}W_u\quad\hbox{for all $t \in[0,T)$,} \end{aligned}$$
(2.4)

where W=(W t ) t∈[0,T] denotes a \(\mathbb {Q}\)-Brownian motion and \(\widetilde {S} = S\) the first hitting time of −1 by \(( \int_{0}^{t} 1/\sqrt{T-u}\, \,\mathrm {d}W_{u})_{t \in[0,T)}\). Since X corresponds to a deterministically time-changed Brownian motion, we have \(\mathbb {Q}(S < T) = 1\) and thus \(\mathbb {Q}(X_{T} = 0) =1\).

Under the measure \(\widehat{\mathbb {Q}}\) of Theorem 2.1, note that Y=(Y t ) t∈[0,T], defined by Y t :=1/X t 1 {R>t} for all t∈[0,T], has the dynamics

$$\begin{aligned} \mathrm {d}Y_t = -Y_t^2 \frac{1}{\sqrt{T-t}} \,\mathrm {d}W_t^{\widehat{\mathbb {Q}}} \end{aligned}$$
(2.5)

for all t∈[0,R) and some \(\widehat{\mathbb {Q}}\)-Brownian motion \(W^{\widehat{\mathbb {Q}}} :=(W_{t}^{\widehat{\mathbb {Q}}})_{ t \in[0,R) }\). Thus, Y is just a time-change of the reciprocal of a three-dimensional \(\widehat{\mathbb {Q}}\)-Bessel process Z starting in one. To see this, define the processes Z=(Z u ) u≥0 and B=(B u ) u≥0 by Z u :=Y T(1−exp(−u)) and

$$\begin{aligned} B_u := \int_0^{T(1-\exp(-u))} \frac{1}{\sqrt{T-v}} \,\mathrm {d}W_v^{\widehat{\mathbb {Q}}} \quad\hbox{ for all $u \geq0$.} \end{aligned}$$

Then observe that \(\mathrm {d}Z_{u} = -Z_{u}^{2} \,\mathrm {d}B_{u}\) and 〈B〉 u =u for all u≥0 and conclude by applying Lévy’s theorem; see Theorem 3.3.16 in Karatzas and Shreve [22]. We then get Y t =Z log(T/(T−t)) and \(\widehat{\mathbb {Q}}(Y_{t} > 0 \mbox{ for all } t \in [0,T)) = 1 = \widehat{\mathbb {Q}}(Y_{T} = 0) \); see Sect. 3.3 of Karatzas and Shreve [22]. Indeed, note that

$$\widehat{\mathbb {Q}}(R=T) = \lim_{t \uparrow T; t <T} \widehat{\mathbb {Q}}(R>t) - \widehat{ \mathbb {Q}}(R>T)= 1 . $$

Thus, the two measures are singular with respect to each other on \(\mathcal {F}_{T}\) since we have \(\mathbb {Q}(R=T) = 0 < 1 = \widehat{\mathbb {Q}}(R=T)\); however, \(\widehat{\mathbb {Q}}\) is absolutely continuous with respect to \(\mathbb {Q}\) on \(\mathcal {F}_{t}\) for all t∈[0,T) since Y is a strictly positive, strict \(\widehat{\mathbb {Q}}\)-local martingale; see Remark 2.2. We also note that X t is a true \(\mathbb {Q}\)-martingale for all t∈[0,T), but X=X T is a strict \(\mathbb {Q}\)-local martingale.

The next example is a slight modification of the example in Delbaen and Schachermayer [9]. It here illustrates that the equivalence of two probability measures \(\mathbb {Q}\) and \(\mathbb {Q}^{Z}\) on \((\varOmega, \mathcal {F}_{T})\), under which X is a nonnegative right-continuous local martingale, does not necessarily imply the equivalence of the corresponding probability measures \(\widehat{\mathbb {Q}}\) and \(\widehat{\mathbb {Q}}^{Z}\), constructed as in Theorem 2.1. This observation is one reason why we shall assume complete markets later on.

Example 2.5

(Lack of equivalence)

Fix T∈(0,∞) and let \(\mathcal{X} = (\mathcal{X}_{t}) _{t \in [0,T]}\) and \(\mathcal{Z} = (\mathcal{Z}_{t})_{t \in [0,T]}\) denote two independent processes with the same distribution as the process X in Example 2.4. Define the stopping time

$$\begin{aligned} \tau^{\mathcal{X} } := \inf \{t \in[0,T]: \mathcal{X}_t < 1/2 \} \end{aligned}$$

with \(\inf\emptyset:= \mathfrak{T}\), and similarly \(\tau^{\mathcal{Z} }\). Define now the processes \(X := \mathcal{X}^{\tau^{\mathcal{X} } \wedge\tau^{\mathcal{Z} } }\) and \(Z := \mathcal{Z}^{\tau ^{\mathcal{X} } \wedge\tau^{\mathcal{Z}}}\). Since the stopping time \(\tau^{\mathcal{Z} }\) is independent from \(\mathcal{X}\) and satisfies \(\mathbb {Q}(\tau^{\mathcal{Z} } < T) = 1\), the process X is a strictly positive true \(\mathbb {Q}\)-martingale by a conditioning argument; and similarly, so is Z.

Define now a new probability measure \(\mathbb {Q}^{Z}\) by \(\mathrm {d}\mathbb {Q}^{Z} = Z_{T} \,\mathrm {d}\mathbb {Q}\) and observe that \(\mathbb {Q}\) and \(\mathbb {Q}^{Z}\) are equivalent and that the process X is a strict \(\mathbb {Q}^{Z}\)-local martingale since

$$\begin{aligned} \mathbb {E}^{\mathbb {Q}^{Z}}[X_T] &= \mathbb {E}^{\mathbb {Q}}[X_T Z_T] = \frac{1}{2} \mathbb {E}^{\mathbb {Q}} \bigl[X_T \mathbf {1} _{\{\tau^{\mathcal{Z} }< \tau^{\mathcal{X} }\}} + Z_T \mathbf {1}_{\{\tau ^{\mathcal{X} }< \tau^{\mathcal{Z} }\}}\bigr] \\ & =\mathbb {E}^{\mathbb {Q}} \bigl[X_T \mathbf {1}_{\{\tau^{\mathcal{Z} }< \tau^{\mathcal{X} }\}} \bigr] =1-\mathbb {E}^{\mathbb {Q}} \bigl[X_T \mathbf {1}_{\{\tau^{\mathcal{X} }< \tau^{\mathcal {Z} }\}} \bigr] = 1 - \frac{1}{2}\mathbb {Q}( \tau^{\mathcal{X} }< \tau^{\mathcal{Z} } ) \\ &= \frac{3}{4} < 1, \end{aligned}$$

where we have used the definitions of \(\mathbb {Q}^{Z}, \tau^{\mathcal{X}}\) and \(\tau^{\mathcal{Z} }\), the facts that \(\mathbb {Q}( \tau^{\mathcal{X} }=\tau ^{\mathcal{Z} }) =0\) and that X and Z have the same distribution under \(\mathbb {Q}\), and the martingale property of X under \(\mathbb {Q}\).

Let \(\widehat{\mathbb {Q}}\) and \(\widehat{\mathbb {Q}}^{Z}\) now denote the probability measures of Theorem 2.1 with X as density process. These two measures cannot be equivalent since X is a strictly positive true \(\mathbb {Q}\)-martingale, but only a strict \(\mathbb {Q}^{Z}\)-local martingale. Thus, the measures \(\mathbb {Q}\), \(\mathbb {Q}^{Z}\) and \(\widehat{\mathbb {Q}}\) are all equivalent, but only absolutely continuous with respect to \(\widehat{\mathbb {Q}}^{Z}\).

To elaborate on this, under both measures \(\widehat{\mathbb {Q}}\) and \(\widehat{\mathbb {Q}}^{Z}\), the process 1/X is a martingale and follows the same dynamics as the process Y in (2.5), stopped at time \(\tau:= {\tau}^{\mathcal{X} } \wedge{\tau}^{\mathcal{Z} } \). However, the distribution of τ is different under the two measures. More precisely, under \(\widehat{\mathbb {Q}}\), the stopping time \(\tau\leq{\tau}^{\mathcal{Z} }\) is bounded by the first time that the nonnegative \(\widehat{\mathbb {Q}}\)-local martingale \(\mathcal{Z}\) starting in 1 hits 1/2, thus \(\widehat{\mathbb {Q}}({\tau} <T) =1\); see also (ii) of Proposition C.3 in the Appendix. However, both 1/X and 1/Z are \(\widehat{\mathbb {Q}}^{Z}\)-martingales and the event that neither of these hits 2 has positive probability under \(\widehat{\mathbb {Q}}^{Z}\); thus \(\widehat{\mathbb {Q}}^{Z}(\tau> T) > 0\). This yields \(\widehat{\mathbb {Q}}(R \leq T) = 0 < \widehat{\mathbb {Q}}^{Z}(R \leq T)\), despite \({\mathbb {Q}}\) and \({\mathbb {Q}}^{Z}\) being equivalent.

3 Minimal joint replication price

In this section, we derive and discuss a representation of a contingent claim price, which we define as the minimal cost of replicating the contingent claim’s payoff under two probability measures simultaneously; specifically, the measure under which the underlying follows local martingale dynamics (\(\mathbb {Q}^{\$}\)) and the measure that corresponds to the change of numéraire (). We interpret a nonnegative \(\mathbb {Q}^{\$}\)-local martingale X as the current market value of one Euro in dollars, and the process Y:=1/X under the measure , derived from \(\mathbb {Q}^{\$}\) via the density process X (see Theorem 2.1), as the current market value of one dollar in Euros under its corresponding numéraire measure.

Remark 3.1

(Arbitrage and strict local martingales)

Modelling asset prices with strict local martingales usually leads to features of contingent claim prices that, on the first look, seem to imply simple arbitrage opportunities and do not reflect our economic understanding of financial markets. To elaborate on this issue, we remind the reader of the standard definition of a contingent claim price in a complete market framework as the minimal (super-)replicating cost of this contingent claim; here the replication occurs almost surely under the unique risk-neutral measure.

Using such contingent claim prices then usually results in the loss of standard put-call parity in strict local martingale models; see for example Cox and Hobson [4]. Even more disturbingly, the minimal replicating price for an asset modelled as a strict local martingale in a complete market is below its current value. Yet, due to an admissibility constraint on trading strategies, these models do not yield arbitrage opportunities; see also Delbaen and Schachermayer [5, 8]. For example, the strategy of shorting the asset modelled by a strict local martingale and replicating its payoff for a lower cost is not admissible, as it might lead to unbounded negative wealth before the strategy matures; more details on this argument are discussed in Ruf [36].

Throughout this section, we again assume a time horizon T∈(0,∞] and a filtered probability space \((\varOmega, \mathcal {F}_{T}, (\mathcal {F}_{t})_{t \in [0,T]}, \mathbb {Q}^{\$})\) that satisfies the technical conditions of Appendix B. We fix a nonnegative \(\mathbb {Q}^{\$}\)-local martingale X with almost surely càdlàg paths (and right-continuous for all ω∈Ω), define the stopping times \((R_{i})_{i \in \mathbb {N}}\), \((S_{i})_{i \in \mathbb {N}}\), S , and R as in Sect. 2, and assume that \(\mathbb {Q}^{\$}(S>S_{i} \wedge T) = 1\) for all \(i \in \mathbb {N}\); that is, X is assumed not to jump to zero. As above, we define a process Y=(Y t ) t∈[0,T] by Y t :=1/X t 1 {R>t} for all t∈[0,T]. As illustrated in Theorem 2.1, there exists a probability measure , which corresponds to the probability measure with X as numéraire, symbolically . We then extend the measure , currently defined on \((\varOmega, \mathcal {F}_{R-})\), to a measure on \((\varOmega, \mathcal {F}_{T})\), which we again denote, with a slightly misuse of notation, by ; see Appendix B.

For some \(d \in \mathbb {N}\), we assume the existence of d+1 tradable assets with nonnegative almost surely càdlàg price processes (and right-continuous for all ω∈ Ω), denoted by S $=(S $,(i)) i=0,1,…,d in dollars and by S €=(S €,(i)) i=0,1,…,d in Euros, with \(S^{\$, (i)} = (S^{\$, (i)}_{t})_{t \in [0,T]}\) and for all i=0,1,…,d. We assume that the processes S $,(i) have \(\mathbb {Q}^{\$}\)-local martingale dynamics and S €,(i) have -local martingale dynamics for all i=0,1,…,d. Moreover, we assume that these price processes describe the same assets and are consistent; that is, we assume that for all t∈[0,T] and i=0,1,…,d. Thus, given a dollar price process S $,(i) for the ith asset, the process S €,(i) denotes the price for the same asset in Euros for all i=0,1,…,d, as it is the dollar price multiplied by the price of one dollar in Euros. This relationship holds up to time R∧S; any event “beyond” that has probability zero under one of the two measures.

By combining Proposition 2.3 and Proposition A.4 in the Appendix, given the \(\mathbb {Q}^{\$}\)-dynamics of S $, we can always construct -local martingales S € which satisfy \(S^{\$, (i)}_{t} \mathbf {1}_{\{R \wedge S>t\}} = S^{\$, (i)}_{t} \mathbf {1}_{\{R \wedge S>t\}}Y_{t}\) for all t∈[0,T] and i=0,1,…,d. More precisely, the equivalence in (ii) of Proposition 2.3 and our standing assumption that \(\mathbb {Q}^{\$}(S>S_{i} \wedge T)=1\) for all \(i \in \mathbb {N}\) first yield that S $,(i) Y is a -local martingale on [0,R∧S). Secondly, since R∧S is foretellable under , by Theorem 2.1, any -local martingale on [0,R∧S) can be extended to a local martingale on [0,R∧S], and then of course to a local martingale on [0,T] in an arbitrary manner after R∧S since the dynamics under one measure only determine the dynamics under the other measure up to the stopping time R∧S.

We suppose that S $,(0) and S €,(1) denote the dollar and Euro money market account, each assumed to pay zero interest; that is, S $,(0)≡1≡S €,(1). Thus, S $,(1)=X denotes the price of one Euro in dollars and S €,(0)=Y the price of one dollar in Euros. More generally, part (iii) of Proposition 2.3 yields that if S $,(i) is a \(\mathbb {Q}^{\$}\)-martingale for some i=1,…,d, then for all t∈[0,T]; in other words, the martingale property of S $,(i) under \(\mathbb {Q}^{\$}\) forces S €,(i) to hit zero under at time R∧S. Vice versa, if S €,(i) is a -martingale for some i=1,…,d, then \(S^{\$, (i)}_{t} \mathbf {1}_{\{ S \leq t\}} = 0\) for all t∈[0,T].

We now are ready to define a trading strategy, relying on stochastic integrals with respect to the (d+1)-dimensional local martingales S $ and S €. We refer to Sects. I.4d and III.4a in Jacod and Shiryaev [20] for a discussion of stochastic integrals when the filtration does not satisfy the “usual assumptions’’, in the case of d=1 or all price processes being continuous, and to Jacod [19] for the general case. We denote by L(S $) and L(S €) the space of all predictable processes that are integrable with respect to S $ and S €, respectively, under the corresponding measures \(\mathbb {Q}^{\$}\) and .

Stochastic integration is used in the following definition:

Definition 3.2

(Trading strategy)

A trading strategy is an \(\mathbb {R}^{d+1}\)-valued process η∈L(S $)∩L(S €) such that

  • its corresponding dollar wealth process \(V^{{\$}, \eta} = (V^{{\$ }, \eta}_{t}) _{t \in [0,T]}\) and Euro wealth process defined by

    stay nonnegative almost surely under the corresponding measure \(\mathbb {Q}^{\$}\) and , respectively, and

  • the self-financing condition holds, that is,

    where the dynamics are computed under the corresponding measure \(\mathbb {Q}^{\$}\) and .

We say that η is a trading strategy for initial capital v∈[0,∞) expressed in dollars if

$$\begin{aligned} v = V^{{\$}, \eta}_0 = \sum_{i=0}^d \eta^{(i)}_0 S^{{\$},(i)}_0 \end{aligned}$$

holds; and similarly for initial capital v expressed in Euros.

Thus, at any time t∈[0,T], each component of η t determines the current number of shares of each asset held at that point of time. Note that

(3.1)

Thus, the nonnegativity condition on V $,η implies the one on , but only up to the stopping time R. Moreover, a simple application of Itô’s rule yields that the self-financing condition under \(\mathbb {Q}^{\$}\) implies the one under , but again only up to the stopping time R; see also Geman et al. [14].

We call any pair of \(\mathcal {F}_{T}\)-measurable random variables (D $,D €) a contingent claim if D € 1 {R∧S>T}=D $ 1 {R∧S>T} Y T . The random variable D $ (D €) corresponds to the dollar (Euro) price of a contingent claim, as seen by the dollar (Euro) investor. We remind the reader that the event {S≤T} has zero -probability, but might have positive \(\mathbb {Q}^{\$}\)-probability, and the converse statement holds for the event {R≤T}.

We represent a contingent claim as a pair of random variables so that we can exactly express its payoff both in dollars and in Euros including in the event of X hitting infinity. For example, the contingent claim (X T ,1) pays off one Euro at maturity, while the contingent claim (X T ,1 {R>T}) pays off one Euro if the price of one Euro in dollars did not explode. For K≥0, the claims \(D^{C,\$}_{K} := ((X_{T}-K)^{+}, (1-K Y_{T})^{+})\) and \(D^{P,\$}_{K} := ((K-X_{T})^{+},(K Y_{T}-1)^{+})\) are called call and put, respectively, on one Euro with strike K and maturity T. Equivalently, by exchanging the first with the second component and X T with Y T , we define calls and puts on one dollar and denote them by and . In foreign exchange markets, self-quantoed calls are traded, defined as \(D^{\mathrm{SQC},\$}_{K} := X_{T} D^{C,\$}_{K} = (X_{T}(X_{T}-K)^{+}, (X_{T}-K)^{+})\) for some K≥0.

We shall assume that the market is complete both for the dollar investor and for the Euro investor; that is, for any contingent claim (D $,D €) with D $, D € taking values in [0,∞), there exist trading strategies η $ and η € such that

and such that \(V^{\$, \eta^{\$}}\) is a \(\mathbb {Q}^{\$}\)-martingale and is a -martingale. The replicability of any contingent claim under \(\mathbb {Q}^{\$}\) does not necessarily imply that any contingent claim can be replicated under , since the two measures are not equivalent in general.

Remark 3.3

(A seeming paradox)

Let the exchange rate X be a strict \(\mathbb {Q}^{\$}\)-local martingale hitting zero with positive probability. Then Y is a strict -local martingale and we have the following paradox. Under the dollar measure, one can replicate the payoff of one Euro less than one Euro; simultaneously, under the Euro measure, one can replicate the payoff of one dollar for less than one dollar. To conclude, the exchange rate reflects an overly high price (compared to their replicating cost) both for the dollar and for the Euro; thus it is at the same time too high and too low for the dollar. This paradox can be explained by recalling that the two measures \(\mathbb {Q}^{\$}\) and are not equivalent; and therefore the investors are concerned with different events when replicating a Euro or a dollar.

The next theorem constitutes the core result of this section; we recall our standing assumption that \(\mathbb {Q}^{\$}(S>S_{i} \wedge T) = 1\) for all \(i \in \mathbb {N}\).

Theorem 3.4

(Minimal joint replicating price)

Define the dollar and Euro pricing operators as

(3.2)
(3.3)

for a contingent claim D=(D $,D €). Whenever D is nonnegative, the minimal joint Q $- and Q €-replicating price expressed in dollars (Euros) is p $(p €). More precisely, there exists a trading strategy η for initial capital p $(D) (expressed in dollars) with

(3.4)

and there exist no \(\widetilde{p} < p^{\$}(D)\) and no trading strategy \(\widetilde{\eta}\) for initial capital \(\widetilde{p}\) (expressed in dollars) such that (3.4) holds with η replaced by \(\widetilde{\eta}\).

Proof

The second equality in (3.3) follows directly from Theorem 2.1. Since the market is assumed to be complete, there exist trading strategies ν for initial capital \(p^{(1)} := \mathbb {E}^{ \mathbb {Q}^{\$}}[D^{\$}]\) (in dollars) and θ for initial capital (in Euros) such that V $,ν is a \(\mathbb {Q}^{\$}\)-martingale, is a -martingale, and

To show (3.4), we now prove that the trading strategy η:=ν+θ replicates D $ under \(\mathbb {Q}^{\$}\) and D € under ; the initial cost for η, expressed in dollars, is exactly p $=p (1)+x 0 p (2). Moreover, note the identities \(V^{{\$},\eta}_{T} = V^{{\$},\nu}_{T} + V^{{\$},\theta}_{T}\) and . In order to prove that η is a trading strategy, it suffices to prove that (a) \(\mathbb {Q}^{\$}(V^{{\$},\theta}_{T} > 0) = 0\) and (b) .

For (a), note that \((V^{\$,\theta}_{t} \mathbf {1}_{\{S > t\}})_{t \geq0}\) is a \(\mathbb {Q}^{\$}\)-local martingale by (i) in Proposition 2.3 with \(N = (V^{\$,\theta}_{t} \mathbf {1}_{\{R > t\}})_{t \geq0}\) and τ=R i for all \(i \in \mathbb {N}\). By taking differences, so is \((V^{\$,\theta}_{t} \mathbf {1}_{\{S \leq t\}})_{t \geq0}\), which implies \(\mathbb {Q}^{\$}(\{V^{{\$},\theta}_{T} > 0\} \cap\{S \leq T\}) = 0\). Observe next that

by (2.3) and (3.1) for all \(i \in \mathbb {N}\), which yields (a).

For (b), it suffices to show that is a (nonnegative) -local martingale. Since is one, it only remains to show, by (3.1), that NY with \(N := (V^{{\$},\nu}_{t} \mathbf {1}_{\{R > t\}})_{t \in [0,T]}\) is also a -local martingale. However, this follows directly from (iii) in Proposition 2.3 since N was assumed to be a \(\mathbb {Q}^{\$}\)-martingale.

Next, we show that η corresponds to the cheapest trading strategy. To that end, let \(\widetilde{p} \in [0, \infty)\) and \(\widetilde{\eta}\) be a trading strategy for initial capital \(\widetilde{p}\) (expressed in dollars) that superreplicates D $ under \(\mathbb {Q}^{\$}\) and D € under . Then \(\widetilde{p} = M_{0} + N_{0}\), where M and N are the martingale and strict local martingale part of the Riesz decomposition \(V^{{\$},\widetilde{\eta}} = M + N\) under \(\mathbb {Q}^{\$}\) with \(\mathbb {Q}^{\$}(N_{T} = 0) = 1\). More explicitly, \(M_{t} := \mathbb {E}^{ \mathbb {Q}^{\$}}[V^{{\$},\widetilde{\eta}}_{T}\, |\, \mathcal {F}^{0}_{t}]\) and \(N_{t} := V^{{\$},\widetilde{\eta}}_{t} - M_{t}\) for all t∈[0,T]; see Theorem 2.3 of Föllmer [13] for the case of a not completed filtration.

Note that \(M = V^{{\$},\widetilde{\nu}}\) and \(N = V^{{\$},\widetilde {\theta}}\) for some trading strategies \(\widetilde{\nu}\) and \(\widetilde {\theta}\) with \(\widetilde{\eta} = \widetilde{\nu} + \widetilde{\theta}\). Since \(\widetilde{\nu}\) superreplicates D $ under \(\mathbb {Q}^{\$}\), we obtain \(M_{0} \geq \mathbb {E}^{ \mathbb {Q}^{\$}}[D^{\$}]\). As in (b) in the first part of the proof, we have . Thus, \(\widetilde{\theta}\) superreplicates D € under . This implies that , which yields that \(\widetilde{p} = M_{0} + N_{0} \geq p^{\$}(D)\). □

Theorem 3.4 yields the smallest amount of dollars (Euros) needed to superreplicate a claim D under both measures \(\mathbb {Q}^{\$}\) and . The corresponding replicating strategy is, as the proof illustrates, a sum of two components. The first component is the standard strategy that replicates the claim under one of the two measures; the second component replicates the claim under the events that only the other measure can “see”.

The next few corollaries are direct implications of the last theorem. We usually formulate them only in terms of the dollar pricing operator p $, but they symmetrically also hold for the Euro pricing operator p €.

Corollary 3.5

(Linearity of pricing operator)

The pricing operator p $ of (3.2) is linear on its domain: for any claims and and any \(a \in \mathbb {R}\) such that D 1 and D 2 are both in the domain of p $, we have

$$\begin{aligned} p^{\$}(D_1 + a D_2) = p^{\$}(D_1) + a p^{\$}(D_2), \end{aligned}$$

where .

Proof

The statement follows directly from the linearity of expectations. □

Corollary 3.6

(Martingale property of wealth process)

The wealth process V $,η of Theorem 3.4 is a \(\mathbb {Q}^{\$}\)-local martingale and thus does not introduce an arbitrage opportunity. It is a strict \(\mathbb {Q}^{\$}\)-local martingale if and only if the \(\mathbb {Q}^{\$}\)-local martingale X is a strict \(\mathbb {Q}^{\$}\)-local martingale and

Similarly, the wealth process is a -local martingale. It is a strict -local martingale if and only if the -local martingale Y is a strict -local martingale and

$$\mathbb {Q}^{\$}\bigl( \{D^{\$} > 0 \} \cap{\{S \leq T\}} \bigr) > 0. $$

Proof

The local martingale property of the wealth processes under the corresponding measures follows directly from their definition. The lack of martingale property follows from checking when the quantities p $(D) and p €(D) in (3.2) and (3.3) satisfy \(p^{\$}(D) > E^{ \mathbb {Q}^{\$}}[D^{\$}]\) and , respectively. □

Corollary 3.7

(Price of a Euro)

The minimal joint \(\mathbb {Q}^{\$}\)- and -superreplicating price of (X T ,1) is x 0 (expressed in dollars) or 1 (expressed in Euros).

Proof

Recall (2.2), which implies the identity . □

The corresponding replicating strategy is the buy-and-hold strategy of one Euro.

Corollary 3.8

(Put-call parity)

The prices of puts and calls simplify under the pricing operator p $ to

(3.5)

moreover, the put-call parity

$$\begin{aligned} p^{\$}(D^{C,\$}_K) + K = p^{\$}(D^{P,\$}_K) + x_0 \end{aligned}$$
(3.6)

holds, where \(K \in \mathbb {R}\) denotes the strike of the call \(D^{C,\$}_{K}\) and put \(D^{P,\$}_{K}\).

Proof

The statement follows directly from (3.2) and the linearity of expectations. □

We refer to Madan and Yor [26] for alternative representations of the call price in (3.5).

Giddy [15] introduces the notion of international put-call equivalence which relates the price of a call in one currency with the price of a put in the other currency; see also Grabbe [16].

Corollary 3.9

(International put-call equivalence)

The pricing operators p $ and p € satisfy international put-call equivalence: for all K>0,

Proof

We obtain

where we have used the identities of Corollary 3.8 and (2.3). The second equivalence follows in the same way or from the put-call parity for model prices in (3.6). □

The next remark discusses how our result motivates and generalizes Lewis’ generalized pricing formulas.

Remark 3.10

(Lewis’ generalized pricing formulas)

Within Markovian stochastic volatility models, Lewis [25] derives call and put prices which exactly correspond to (3.2) when applied to the call payoff \(D^{C,\$}_{K}\) or put payoff \(D^{P,\$}_{K}\). Lewis starts from the postulate that put-call parity holds, and then shows that the correction term that is added to the expected payoff under \(\mathbb {Q}^{\$}\) corresponds to the probability of some process exploding under another measure (corresponding here to ). We start here instead from an economic argument by defining the price as the minimal superreplicating cost for a contingent claim under two possibly non-equivalent measures that arise from a change of numéraire. We then show that this directly implies put-call parity for model prices. This approach also yields a generalization of Lewis’ pricing formula to arbitrary, possibly path-dependent contingent claims.

Example 3.11

(Singular measures (continued))

We continue here our discussion of Example 2.4 with \(\mathbb {Q}^{\$}= \mathbb {Q}\) and . Although the exchange rate X is a \(\mathbb {Q}^{\$}\)-local martingale, from the classical point of view of a dollar investor, the minimal superreplicating price of one Euro at time T is zero because under \(\mathbb {Q}^{\$}\) there are only paths under which this contingent claim becomes worthless. However, by means of the correction term, (3.2) yields a price p $((X T ,1))=x 0 when considering the minimal joint \(\mathbb {Q}^{\$}\)- and -superreplicating price of one Euro. For the self-quantoed call \(D^{\mathrm{SQC},\$}_{K}\), the classical price would again be zero; however, considering also the paths that the Euro investor under can see, Theorem 3.4 suggests a price \(p^{\$}(D^{\mathrm{SQC},\$}_{K})=\infty\) since .

In many applications, however, the measures \(\mathbb {Q}^{\$}\) and do not become singular. Often, one of them is absolutely continuous with respect to the other. In this case, the formulas for computing p $(D) and p €(D) simplify:

Corollary 3.12

(Absolutely continuous measures)

If \(\mathbb {Q}^{\$}(S>T) = 1\), that is, if Y is a -martingale, then p € can be computed as

If , that is, if X is a \(\mathbb {Q}^{\$}\)-martingale, then

Proof

Assume that \(\mathbb {Q}^{\$}(S>T) = 1\). Then Remark 2.2 implies that \(\mathbb {Q}^{\$}\) is absolutely continuous with respect to . Thus, if a trading strategy superreplicates D € -almost surely for a Euro investor, then it also superreplicates D $ \(\mathbb {Q}^{\$}\)-almost surely for a dollar investor. The second statement can be shown analogously. □

Example 3.13

(Reciprocal of the three-dimensional Bessel process)

We set d=T=1 and let X denote a nonnegative \(\mathbb {Q}^{\$}\)-local martingale distributed as the reciprocal of a three-dimensional Bessel process starting in 1; in particular, there exists a Brownian motion W=(W t ) t∈[0,T] such that

$$\begin{aligned} X_t = 1+\int_0^{t} X^2_u \,\mathrm {d}W_u \quad\hbox{for all $t \in[0,T]$.} \end{aligned}$$

It is well known that X is strictly positive and that Y is a -Brownian motion stopped when it hits zero; see for example Delbaen and Schachermayer [6]. Since X is strictly positive, the discussion in Remark 2.2 yields that \(\mathbb {Q}^{\$}\) is absolutely continuous with respect to ; thus, Corollary 3.12 applies.

Let us study the self-quantoed call \(D^{\mathrm{SQC},\$}_{K}\). Since Brownian motion hits 0 in any time interval with positive probability, we obtain that X hits ∞ with positive -probability. This yields directly a minimal joint \(\mathbb {Q}^{\$}\)- and -superreplicating price of \(p^{\$ }(D^{\mathrm{SQC},\$}_{K}) =\infty\). It is interesting to note that as in Example 3.11, the classical price is finite; in fact,

for all K≥0, where we have plugged in the density of killed Brownian motion; see Exercise III.1.15 in Revuz and Yor [32].

We remark that as a corollary of Remark 2.2, our setup contains only positive “bubbles” under the corresponding measure. A bubble is usually defined as the difference of the current price and the expectation of the terminal value of an asset, that is, \(x_{0} - \mathbb {E}^{ \mathbb {Q}^{\$}}[X_{T}]\) and , respectively. It is possible that both bubbles are strictly positive; however, negative bubbles cannot occur by the supermartingale property of the asset price processes under the corresponding measure. This contrasts Jarrow and Protter [21], where negative bubbles are discussed, however only when considering the dollar measure \(\mathbb {Q}^{\$}\), which is not the risk-neutral measure of a Euro investor.

In the next section, we provide an interpretation of a bubble (lack of the martingale property of the exchange rate) as the possibility of a hyperinflation under some dominating “real-world” measure \(\mathbb {P}\). If for both currencies such hyperinflations have positive \(\mathbb {P}\)-probability, then X and Y both have a positive bubble.

4 A physical measure

In this section, we start by specifying a physical probability measure \(\mathbb {P}\) instead of a risk-neutral probability measure \(\mathbb {Q}^{\$}\). We again assume a time horizon T∈(0,∞] and a filtered probability space \((\varOmega, \mathcal {F}_{T}, (\mathcal {F}_{t})_{t \in [0,T]}, \mathbb {P})\) that satisfies the technical conditions of Appendix B. Let X=(X t ) t∈[0,T] denote a process taking values in [0,∞] with right-continuous paths for all ω∈Ω. Define the stopping times \((R_{i})_{i \in \mathbb {N}}, (S_{i})_{i \in \mathbb {N}}, S\) and R as in Sect. 2 and assume that \(\mathbb {P}(S_{i} \wedge T< S) = 1\) for all \(i \in \mathbb {N}\), X (R+t)∧T =∞ if R≤T and X (S+t)∧T =0 if S≤T for all t≥0. In particular, this assumption implies that no oscillations can occur; that is, the events H $:={R≤T} and H €:={S≤T} are disjoint. Suppose that \(\mathbb {P}(H^{\$}) <1\) and . Define again Y=(Y t ) t∈[0,T] by Y t :=1/X t 1 {R>t} for all t∈[0,T].

Under the physical probability measure \(\mathbb {P}\), the events H $ and H € may both have positive probability. We interpret these events as the complete devaluation (hyperinflation) of the dollar or Euro currency with respect to the other. Such hyperinflations have been observed; for example, the exchange rate between the American and German currencies changed by a factor of 1010 from January 1922 to December 1923, as described in Sargent [38]. During a hyperinflation, the interest rate of the inflating currency tends to become very large, and so far, we have assumed zero interest rates. We may reinterpret X T as the forward exchange rate of dollars per Euro at time T, as opposed to a spot exchange rate. This is consistent with various interest rate assumptions, including the possibility that the dollar (respectively Euro) interest rate should explode when the dollar (respectively Euro) experiences a hyperinflation.

If \(\mathbb {P}(H^{\$}) > 0\) and , then no risk-neutral measure equivalent to \(\mathbb {P}\) can exist such that either X or Y follow local martingale dynamics. Nevertheless, pricing and hedging of contingent claims might still be possible. Towards this end, let us introduce the two artificial measures

where we have conditioned the physical measure \(\mathbb {P}\) on the events \(H^{\$^{c}}\) and that no hyperinflation occurs. Note that both measures \(\mathbb {P}^{\$}\) and are absolutely continuous with respect to \(\mathbb {P}\) and that \(\mathbb {P}\) is absolutely continuous with respect to their average .

As in Sect. 3, for some \(d \in \mathbb {N}\), we assume the existence of d+1 tradable assets with nonnegative price processes, denoted by S $=(S $,(i)) i=0,1,…d in dollars and by S €=(S €,(i)) i=0,1,…d in Euros; as before, with \(S^{\$, (i)} = (S^{\$, (i)}_{t})_{t \in [0,T]}\) and for all i=0,1,…,d. We also assume again that for all t∈[0,T] and i=0,1,…,d and that S $,(1)=X and S €,(0)=Y. Moreover, we assume that S $ and S € have càdlàg paths \(\mathbb {P}^{\$}\)- and -almost surely, respectively.

Suppose that there exists exactly one probability measure \(\mathbb {Q}^{\$}\) () that is equivalent to \(\mathbb {P}^{\$}\) () and under which the processes S $ (S €) are local martingales. Consider the condition of no obvious hyperinflations (NOH):

  1. (NOH)

    The probability measures \(\mathbb {P}^{\$}\) and are equivalent on \(\mathcal {F}_{(R_{i} \wedge S_{j})-}\) for all \(i, j \in \mathbb {N}\).

This condition corresponds to an environment in which at no time, one knows that a certain hyperinflation will occur \(\mathbb {P}\)-almost surely; that is, hyperinflations occur as a surprise. To see this, assume that the condition (NOH) does not hold. Then there exist \(i,j \in \mathbb {N}\) and a set \(A \in \mathcal {F}_{(R_{i} \wedge S_{j})-}\) such that for example \(\mathbb {P}^{\$ }(A) = 0\) and , which implies A⊂H $ (modulo \(\mathbb {P}\)). Thus, for some paths one knows that a hyperinflation will occur before it occurs. As the next lemma shows, the condition (NOH) brings us back to the framework of Theorems 2.1 and 3.4.

Lemma 4.1

((NOH) and change of numéraire)

The following two conditions are equivalent:

  1. (i)

    The condition (NOH) holds.

  2. (ii)

    The equality in (2.1) of Theorem 2.1 holds for all stopping times τ and \(A \in \mathcal {F}_{\tau\wedge T}\) with \(\widehat{\mathbb {Q}}\) replaced by and \(\mathbb {Q}\) replaced by \(\mathbb {Q}^{\$}\).

The proof of this equivalence is based on the assumption that is the unique probability measure equivalent to such that S € are -local martingales. It is contained in Appendix D.

The next proposition yields that the minimal cost (expressed in dollars) for replicating a contingent claim D=(D $,D €) \(\mathbb {P}\)-almost surely is given by (3.2) in Theorem 3.4 if condition (NOH) holds. More precisely, one can find a trading strategy η, in the sense of Definition 3.2, such that the corresponding terminal wealth satisfies . Since \(\mathbb {P}\) and are equivalent, we interpret η as a replication strategy under the physical measure \(\mathbb {P}\).

Proposition 4.2

(Minimal replication cost under \(\mathbb {P}\))

Assume that the condition (NOH) holds. Then the minimal replicating cost for a contingent claim D=(D $,D €) under \(\mathbb {P}\) is exactly the one computed in Theorem 3.4.

Proof

Theorem 3.4 yields the minimal joint replicating cost of a claim under \(\mathbb {Q}^{\$}\) and under any extension of the corresponding measure after a change of numéraire. However, Lemma 4.1 shows that is exactly such an extension if (NOH) holds. This yields the assertion since \(\mathbb {Q}^{\$}\) and are equivalent to \(\mathbb {P}^{\$ }\) and , respectively. □

We also obtain an interpretation of the lack of martingale property of X under the risk-neutral measure \(\mathbb {Q}^{\$}\) as the possibility of an explosion under the physical measure \(\mathbb {P}\):

Corollary 4.3

(Interpretation of the lack of the martingale property)

Assuming the condition (NOH), we have \(\mathbb {P}(H^{\$}) > 0\) if and only if X is a strict \(\mathbb {Q}^{\$}\)-local martingale; equivalently, we have if and only if Y is a strict -local martingale.

Proof

Note that \(\mathbb {P}(H^{\$}) > 0\) if and only if , which is equivalent to . Lemma 4.1 and Remark 2.2 then yield the first equivalence of the assertion. The second follows in the same manner. □

If the condition (NOH) does not hold, then Theorem 3.4 still provides an upper bound for the minimal replicating cost of a contingent claim. However, as the next example shows, the expression in (3.2) usually does not give the smallest minimal replicating cost under \(\mathbb {P}\).

Example 4.4

(Condition (NOH) not satisfied)

We fix d=1 and T=1. Let U denote an \(\mathcal {F}_{0}\)-measurable random variable taking values in {−1,1} with \(\mathbb {P}(U = 1) \in(0,1)\). Furthermore, define X=(X t ) t∈[0,T] by X t :=(Z t )U for all t∈[0,1], where Z=(Z t ) t∈[0,T] has the same distribution as the process in (2.4). Thus H $={U=−1} and H €={U=1}, and the condition (NOH) is not satisfied.

Consider the contingent claim D=(1,1), which pays either one dollar if the dollar does not hyperinflate or otherwise one Euro. Then (3.2) yields a price (in dollars) of p $(D)=1+1=2. However, at time zero, it is already known which of the two currencies defaults, as U is \(\mathcal {F}_{0}\)-measurable. Thus, the trading strategy η=(1 {U=1},1 {U=−1}), holding one unit of the corresponding currency, perfectly replicates the contingent claim at an initial cost of only one dollar.

5 Conclusion

Based on a replication argument, we have introduced a novel pricing operator for contingent claims that restores put-call parity and international put-call equivalence for model prices. If the underlying is a true martingale, our pricing operator is just the classical replication price. Furthermore, we have interpreted the lack of the martingale property of an underlying price process under the risk-neutral probability as the positive probability of an explosion (hyperinflation) under some dominating physical measure.

Two directions of future research arise. First, we have focused on the case of two currencies, corresponding to one exchange rate only. It would be interesting to extend the results of this paper to multiple currencies and to find a consistent way to describe devaluations of currencies with respect to several other currencies. The numéraire-free approach taken in Yan [41] might be very useful. Second, throughout this paper we have relied on the assumption that markets are complete. Again, it would be interesting to consider incomplete markets and to develop a theory of joint superreplication in such markets.