Pricing contingent claims in complete markets has garnered significant attention since the seminal work of Black & Scholes in 1973. Efficient hedging of contingent claims is well-established in complete markets characterized by the same interest rate for the credit and deposit accounts. (refer to Karatzas and Shreve [6] for detailed insights). However, our focus shifts to a more realistic financial market scenario, introducing a two-interest rate model where the credit rate surpasses the deposit rate, aligning more closely with real-world financial markets (as discussed by Kane and Melnikov [5]). In this paper, we consider a multi-dimensional model featuring $m+2$ securities, encompassing two risk-free assets, d stocks driven by a d-dimensional Brownian motion, and $m-d$ stocks influenced by an $(m-d)$-dimensional Poisson process.

Given the incompleteness of the market with two interest rates, we transform it into a suitable auxiliary market using a multi-dimensional jump-diffusion model incorporating two interest rates (see Korn [7] for details).

The structure of this paper unfolds as follows: Section 2 provides an overview of the market model. Section 3 delves into contingent claim valuation within complete markets, accompanied by a theorem presenting a comprehensive solution to the contingent claim problem in such markets. In Section 4, we establish a martingale measure for the new auxiliary market characterized by a higher interest rate for the credit account. Additionally, in section 5, we explore the concept of shortfall risk, acknowledging situations where achieving a perfect hedge might be infeasible, yet it remains possible to minimize the expected shortfall risk, as demonstrated towards the end of this section. The final section will explore pricing contingent claims via market completion in $(B_1,B_2,S_m)$-market, where we study no-arbitrage price bounds in incomplete markets.

Let $(\Omega ,{\mathcal{F}}_t,P,\mathbb{F})$ be a filtered probability space with a complete and right-continuous filtration $\mathbb{F}:={\left\{{\mathcal{F}}_t\right\}}_{0\le t\le T}$. Assume there are $m+2$ continuously traded securities, including two risk-free assets, d stocks driven by an ${\mathbb{R}}^d$-valued Brownian motion $W(t)={(W_1(t),\cdots ,W_d(t))}^{⊺}$, and a $(m-d)$-dimensional multivariate Poisson process $N(t)={(N_1(t),\cdots ,N_{m-d}(t))}^{⊺}$ with a positive intensity $\lambda$. This intensity is independent of W and is denoted by ${\lambda }^{(k)}(t)$, representing the rate of the jump process at time t. The process ${\lambda }^{(k)}(t)$ is $\left\{{\mathcal{F}}_t\right\}$-predictable, positive, and uniformly bounded over $[0,T]$.

The price of the $i^{th}$ stock, $S_i(t)$, is determined by the following equation where ${\tilde{N}}_k(t)=N_k(t)-{\int }_0^t{\lambda }^{(k)}(s)ds$, ${\nu }_{ik}(t)>-1$ for all $i,k$, and $t\in [0,T]$, ${\sigma }_{ij}>0$, and ${\mu }_i\in \mathbb{R}$. $\sigma$ and $\nu$ are matrix-valued processes such that $i^{th}$ row is given by ${\sigma }_{ij}=({\sigma }_{i1},\cdots ,{\sigma }_{id})$, and ${\nu }_{ik}=({\nu }_{i1},\cdots ,{\nu }_{i(m-d)})$ for $i=1,\cdots ,m$, respectively. We assume $\mu$, $\sigma$, and $\nu$ are uniformly bounded in $(t,\omega )\in [0,T]\times \Omega$. Henceforth, the dynamics of the price in 1 possess a unique solution under these assumptions. We also define the volatility coefficients $\tilde{\sigma }(t)=\left[\sigma (t)\nu (t)\right]$, forming an $m\times m$ full-rank matrix, ensuring that $\det (\tilde{\sigma }(t){\sigma }^{⊺}(t))\ne 0$ a.s. for all $t\in [0,T]$.

In our model, the market incompleteness arises from denoting two different interest rates, as described below.

Let us consider one deposit account $B_1$ with the interest rate $r_1$ and one credit account $B_2$ with the interest rate $r_2$ satisfying Given that, in reality, the credit rate is always higher than the deposit rate, we assume constant values for $r_1$ and $r_2$ such that and investors are not allowed to borrow and lend money simultaneously.

The market described above is denoted as the $(B_1,B_2,S_m)$-market.

In the $(B_1,B_2,S_m)$-market, we denote ${\beta }_1(t)$ and ${\beta }_2(t)$ as the number of units invested in the $B_1$ and $B_2$ accounts, respectively, and $\gamma (t)=({\gamma }_1(t),\cdots ,{\gamma }_m(t))$, where ${\gamma }_i$ represents the number of units invested in the $i^{\mathrm{th}}$ stock. The portfolio process is then denoted as follows The value of the portfolio $\pi$ is given by with ${\beta }_1\ge 0$, ${\beta }_2\le 0$, and where x is the initial value (initial capital) of the portfolio. This portfolio is self-financing (SF) if The portfolio is admissible if Denote the class of admissible portfolio strategies with initial capital x by

Any non-negative ${\mathcal{F}}_t$-measurable random variable $f_T$ is called a contingent claim with maturity time T. A market is complete if and only if any contingent claim $f_T$ can be replicated. Namely, there exists an initial capital x and $\pi \in \mathrm{SF}$ such that:

Let us consider $X(t)$ (or $Y(t)$) as the investor’s wealth (or debt) at time t and call it the wealth process (or debt process) if $X(t)$ (or $-Y(t)$) is generated by a self-financing and admissible strategy.

Since the $(B_1,B_2,S_m)$-market is not a complete market, standard methods for pricing and investing do not work. To address this, we transform the market into an auxiliary market ${(B_z,S_m)}_{z\in [0,r_2-r_1]}$. In this market, $B_z$ is the bank account with the interest rate Note that the $(B_z,S_m)$-market is complete for every $r_z$ satisfying $r_z\in [0,r_2-r_1]$ for any $t\in [0,T]$.

Now, we derive the dynamics of the wealth and debt processes in the $(B_1,B_2,S_m)$-market.

By the self-financing wealth process $X(T)\ge 0$, where ${\beta }_1>0$, ${\beta }_2<0$. Then,

Denoting we obtain

Taking the same steps, one can observe that the stochastic differential equation (SDE) of the seller is as follows

A hedging strategy against f in the $(B,S_m)$-market is not necessarily a hedging strategy against f in the $(B_1,B_2,S_m)$-market. In this regard, we first pay attention to contingent claim valuation in the complete markets and then in the $(B_1,B_2,S_m)$-market.

As mentioned in the previous section, any non-negative ${\mathcal{F}}_T$-measurable random variable $f_T$ is called a contingent claim with maturity T. The $(B,S_m)$-market is complete if and only if any contingent claim $f_T$ can be replicated. This means that there exists an initial wealth x and a strategy $\pi \in \mathrm{SF}$ such that $X_T^{\pi }(x)=f_T$. We show that this is the only price for a contingent claim, preventing any arbitrage opportunities. To do that, we define a unique equivalent martingale measure. Let us consider where ${\theta }_W(t)$ is an ${\mathbb{R}}^d$-valued process, ${\theta }_N(t)$ is an ${\mathbb{R}}^{m-d}$-valued process, and $\tilde{\sigma }(t):=\left[\sigma (t)\nu (t)\right]$ is the $m\times m$ volatility matrix process. Let us define the following processes and where $t_n^{(k)}$ is the time of the n-th jump, and $N_k(t)=sup\{n:t_n^{(k)}\le t\}$ is the number of type k random jumps to the market by time t.

Here, E means expectation with respect to P.

The discount process $\gamma (t)$ is defined as

Now, we transform the problem of contingent claim valuation in the $(B_1,B_2,S_m)$-market to a suitable complete market $(B_z,S_m)$. By substituting $r(t)$ with $r_z(t)$ and defining $\widehat{\theta }(t)$, $\widehat{Z}(t)$, $\widehat{W}(t)$, $\widehat{P}$, and $\widehat{\gamma }(t)$ as in Section 3, one can obtain the same results. Then, the fair price $\widehat{p}$ of the contingent claim f in the $(B_z,S_m)$-market is given by where $\widehat{E}$ is the expectation with respect to the probability measure $\widehat{P}$.

The following lemma relates the $(B_z,S_m)$-market to the $(B_1,B_2,S_m)$-market. In other words, we obtain a condition under which the wealth processes corresponding to a portfolio process $\pi$ coincide in the $(B_1,B_2,S_m)$-market and $(B_z,S_m)$-market, respectively.

Proof:$\widehat{X}(t)$ follows the stochastic differential equation

By comparing equation 25 to the stochastic differential equation for $X(t)$ as and by the assumption then is equivalent to

By recalling the relation $a=a^{+}-a^{-}$ for $a\in \mathbb{R}$, one can find the equivalence of equation 27 and equation 24.

Following the above statement and Lemma 2, the wealth process in the $(B_z,S_m)$-market, denoted as ${\widehat{X}}_t(C_{r_z})$, coincides with the wealth process in the $(B_1,B_2,S_m)$-market, denoted as $X_t(C_{r_z})$, and

Therefore, we assert that the minimal hedge $\zeta$ in the $(B_z,S_m)$-market against $f_T$ is also a hedge in the $(B_1,B_2,S_m)$-market if the relation 24 holds.

Before proving this statement, we state the following lemma.

Now let us return to the proof of Statement 2.

Now, let us provide an approximation of the arbitrage-free prices for the claim $f_T={(S_T^1-K)}^{+}$. In this scenario, we calculate the supremum and infimum over auxiliary markets to find approximations for the upper and lower hedging prices of the claim. Therefore, the arbitrage-free interval of prices can be approximated as follows

Here, $C_{BS}$ represents the price of a call option driven by the Black-Scholes formula where

Here, $\Phi (.)$ is the standard normal distribution function. In Example 1, $\tilde{\lambda }(t)$ represents the total jump intensity:

One can approximate the upper and lower hedging prices for Example 1 within the interval:

By the call-put parity, a similar method can be applied to

This completes Example 1.

In this section, we study the case where the initial wealth x is less than the required expected value of $e^{-r_{z}T}{f}_T$ denoted by $\widehat{E}\left[e^{-r_{z}T}{f}_T\right]$. In this case, it is unlikely to apply a perfect hedge; however, it is possible to minimize the risk of shortfall corresponding to the initial cost constraint by considering the following optimization problem:

Here, $l^p(x)=\frac{x^p}{p}$ is the loss function with $p>1$, and $\mathcal{A}=\left\{\xi \mathrm{s}.\mathrm{t}E\right[{sup}_{0\le t\le T}|X_T^{\xi }(.)\left|\right]<\infty \}$, i.e., the set of all admissible portfolios with initial capital x. $f_T\in \left[L^{p+ϵ}(\Omega ,{\mathcal{F}}_T,P)\right]$ is the contingent claim with the maturity time T for some $ϵ$. $X_T^{\pi ,d}$ is the wealth process.

In this problem set, if x is greater than the replication cost $f_T$, the completeness of the market allows the investor to hedge the contingent claim $f_T$ without taking risks. On the other hand, if x is strictly less than the replication cost of $f_T$, there is a potential for a shortfall. We have the option to divide this problem into a perfect hedging problem of $f_T$ and a utility minimization problem.

Denote ${\mathcal{A}}_0(\alpha )$ as the set of portfolio processes $\pi (t)$ for $0\le t\le T$ and $\alpha >0$, satisfying and

Now, we introduce another optimization problem with the goal of

Now, we present a solution to the problem 54 in a two-interest rate market.

In this section, our aim is to study no-arbitrage price bounds in incomplete markets. To initiate our analysis, we examine the market $(B,S_m)$, which is characterized by multi-dimensional risky assets and one non-risky asset and results in a single interest rate. Our objective is to price contingent claims in incomplete markets, prompting a transition to a market with two different interests later on, resulting in market incompleteness.

Assuming the dynamics of the risky assets follow the equation 1, with parameters and assumptions identical, we introduce a non-risky asset governed by

Let $\pi =(\beta (t),{\gamma }_1(t),\cdots ,{\gamma }_m(t))$ be a ${\mathbb{R}}^{(m+1)}$-valued process for $t\in [0,T]$, representing a portfolio. We assume that ${\int }_0^T{\parallel \pi (t)\parallel }^{2}dt<\infty$ almost surely under the probability measure P.

The value of the portfolio, denoted by $X^{\pi }(t)$, is given by

It suffices to assume that our market is arbitrage-free if there exists an equivalent martingale measure, i.e., a measure equivalent to P under which the value of any self-financing strategy is a local martingale. The existence of this measure can be inferred by assuming at least one predictable process $\kappa ={({\kappa }_W,{\kappa }_N)}^{⊺}$, where ${\kappa }_W$ and ${\kappa }_N$ are defined on ${\mathbb{R}}^m$-valued Brownian motion and a $(m-d)$-dimensional Poisson process, respectively, such that the process $\kappa$ satisfies where $\mathbf{1}$ represents a vector of ones. Henceforth, we assume the existence of at least one process $\kappa$ as described above. Let us define the probability measure such that

Define a non-negative local martingale (See [8]) with $E\left[Z_{\kappa }(t)\right]=1$ for all $t\in [0,T]$. The sufficient condition for market completeness is the uniqueness of the equivalent martingale measure. Therefore, our market is complete if $Z_{\kappa }$ is a martingale and 62 has only one solution such that ${\kappa }_N^{(k)}>0$ for $k\in \{1,\cdots ,m-d\}$.

Assume $\Xi$ represents the set of all possible equivalent martingale measures in this market, i.e., $\Xi$ is the set of all $\kappa$ which solve relation 62 with ${\kappa }_N^{(k)}>0$ for $k\in \{1,\cdots ,m-d\}$, and $Z_{\kappa }$ is a martingale in this set. Therefore, the unique parameters of this are given by