On the risk management of demand deposits: quadratic hedging of interest rate margins

Abstract

This paper examines the problem of hedging banks interest rate margins. We assume that the demand’s deposits follow an exponential Lévy process with potential jumps. The forward market rates are assumed to follow the standard market model introduced by Brace et al. (Math Finance 7(2):127–155, 1997). As Adam et al. (Hedging interest rate margins on demand deposits, Université Paris 1 Panthéon-Sorbonne working paper, 2012), we consider that deposit rates depend linearly on market rates. Face to incompleteness, the liability manager must hedge both interest rate and demand deposit risks. For this purpose, we introduce various quadratic hedging criteria, allowing us to provide explicit hedging strategies that we further analyze. We illustrate in particular the impact of both the trends and the volatilities of interest rates and demand deposits.

Appendix

1.1 Proof of Corollary 2

Recall that we assume

$$\begin{aligned} dL_{t}= & {} L_{t}\left( \mu _{L}+\sigma _{L}dW_{L,t}\right) , \\ dK_{t}= & {} K_{t}\left( \mu _{K}+\sigma _{K}dW_{K,t}+\delta _{K,t}dN_{K,t}\right) , \end{aligned}$$

where both $\left( W_{L,t}\right) _{t}$ and $\left( W_{K,t}\right) _{t}$ are standard Brownian motions under the same filtration $\left( \mathcal {F} _{t}\right) _{t}$ and that the pure jump process $\left( N_{K,t}\right) _{t}$ is a Poisson process with intensity $\lambda$ with a i.i.d. sequence $\left( \delta _{K,T_{n}}\right) _{n}$ having expectation $\overline{\delta }$.

• 1. Since $L_{t}$ has a Lognormal distribution, we have:

  $$\begin{aligned} \mathbb {E}\left[ L_{t}\right]= & {} L_{0}\exp \left[ \mu _{L}t\right] , \\ Var\left[ L_{t}\right]= & {} L_{0}^{2}\exp \left[ 2\mu _{L}t\right] \left( \exp \left[ \sigma _{L}^{2}t\right] -1\right) . \end{aligned}$$

• 2. We have also:

  $$\begin{aligned} \mathbb {E}\left[ K_{t}\right] =K_{0}\exp \left[ \left( \mu _{K}+\lambda \overline{\delta }\right) t\right] , \end{aligned}$$

• 3. Notation: $\mathcal {E}\left[ .\right]$ denotes the stochastic exponential of Doléans-Dade. We have:

  $$\begin{aligned} L_{t}= & {} L_{0}\mathcal {E}\left[ \mu _{L}t+\sigma _{L}W_{L,t}\right] , \\ K_{t}= & {} K_{0}\mathcal {E}\left[ \mu _{K}+\sigma _{K}W_{K,t}+\int _{0}^{t}\int _{-\infty }^{+\infty }\delta _{K,s}dN_{K,s} \right] \end{aligned}$$

  Using Yor’s formula,¹³ we deduce:

  $$\begin{aligned} L_{t}K_{t}= & {} L_{0}K_{0} \\&\times \mathcal {E}\left[ \left( \mu _{L}+\mu _{K}+\sigma _{L}\sigma _{K}\rho \right) t+\sigma _{L}W_{L,t}+\sigma _{K}W_{K,t}+\int _{0}^{t}\int _{-\infty }^{+\infty }\delta _{K,s}dN_{K,s}\right] \\= & {} \exp \left[ \left( \mu _{L}+\mu _{K}-\frac{1}{2}\left[ \sigma _{L}^{2}+\sigma _{K}^{2}\right] \right) t+\sigma _{L}W_{L,t}+\sigma _{K} W_{K,t}\right] \prod _{T_{n}\le t}(1+\delta _{K,T_{n}}). \end{aligned}$$

Thus, we get:

• (i) Expectation of the product $L_{t}K_{t}$:

  $$\begin{aligned} \mathbb {E}\left[ L_{t}K_{t}\right] =L_{0}K_{0}\exp \left[ \left( \mu _{L}+\mu _{K}+\sigma _{L}\sigma _{K}\rho \right) t\right] \exp \left[ \lambda \overline{\delta }t\right] . \end{aligned}$$

• (ii) Covariance$\left[ L_{t};K_{t}\left( \alpha +\left( \beta -1\right) L_{t}\right) \right]$:

  $$\begin{aligned}&Cov\left[ L_{t};K_{t}\left( \alpha +\left( \beta -1\right) L_{t}\right) \right] \\&\quad =\mathbb {E}\left[ L_{t}K_{t}\left( \alpha +\left( \beta -1\right) L_{t}\right) \right] -\mathbb {E}\left[ L_{t}\right] \mathbb {E}\left[ K_{t}\left( \alpha +\left( \beta -1\right) L_{t}\right) \right] . \end{aligned}$$

  But first we have:

  $$\begin{aligned} \mathbb {E}\left[ K_{t}\left( \alpha +\left( \beta -1\right) L_{t}\right) \right] =\alpha \mathbb {E}\left[ K_{t}\right] +\left( \beta -1\right) \mathbb {E}\left[ K_{t}L_{t}\right] , \end{aligned}$$

  which implies:

  $$\begin{aligned}&\mathbb {E}\left[ K_{t}\left( \alpha +\left( \beta -1\right) L_{t}\right) \right] \\&\quad =\alpha K_{0}\exp \left[ \left( \mu _{K}+\lambda \overline{\delta }\right) t \right] +\left( \beta -1\right) L_{0}K_{0}\exp \left[ \left( \mu _{L}+\mu _{K}+\sigma _{L}\sigma _{K}\rho \right) t\right] \exp \left[ \lambda \overline{\delta }t\right] . \end{aligned}$$

  Second, we get:

  $$\begin{aligned}&\mathbb {E}\left[ L_{t}K_{t}\left( \alpha +\left( \beta -1\right) L_{t}\right) \right] \\&\quad =\alpha \mathbb {E}\left[ L_{t}K_{t}\right] +\left( \beta -1\right) \mathbb {E} \left[ L_{t}^{2}K_{t}\right] . \end{aligned}$$

  We note that:

  $$\begin{aligned} L_{t}^{2}=L_{0}^{2}\exp \left[ \left( 2\mu _{L}-\sigma _{L}^{2}\right) t+2\sigma _{L}W_{L,t}\right] . \end{aligned}$$

  Therefore, we deduce:

  $$\begin{aligned} \mathbb {E}\left[ L_{t}^{2}K_{t}\right]= & {} L_{0}^{2}K_{0}\times \mathbb {E}\left[ \exp \left[ \left( \left( 2\mu _{L}+\sigma _{L}^{2}\right) +\mu _{K}-\frac{1}{2}\left[ 4\sigma _{L}^{2}+\sigma _{K}^{2}\right] \right) t\right. \right. \\&\left. \left. +\,2\sigma _{L}W_{L,t}+\sigma _{K}W_{K,t}\right] \prod \limits _{T_{n}\le t}(1+\delta _{K,T_{n}})\right] \\= & {} L_{0}^{2}K_{0}\exp \left[ \left( \left( 2\mu _{L}+\sigma _{L}^{2}\right) +\mu _{K}+2\sigma _{L}\sigma _{K}\rho \right) t\right] \exp \left[ \lambda \overline{\delta }t\right] . \end{aligned}$$

  Consequently, we get:

  $$\begin{aligned}&Cov\left[ L_{t};K_{t}\left( \alpha +\left( \beta -1\right) L_{t}\right) \right] \\&\quad =\alpha L_{0}K_{0}\exp \left[ \left( \mu _{L}+\mu _{K}+\sigma _{L}\sigma _{K}\rho \right) t\right] \exp \left[ \lambda \overline{\delta }t\right] \\&\qquad +\left( \beta -1\right) L_{0}^{2}K_{0}\exp \left[ \left( \left( 2\mu _{L}+\sigma _{L}^{2}\right) +\mu _{K}+2\sigma _{L}\sigma _{K}\rho \right) t \right] \exp \left[ \lambda \overline{\delta }t\right] \\&\qquad -L_{0}K_{0}\left( \alpha \exp \left[ \left( \mu _{L}+\mu _{K}\right) t \right] \exp \left[ \lambda \overline{\delta }t\right] \right. \\&\qquad \left. -\left( \beta -1\right) L_{0}^{2}K_{0}\exp \left[ \left( 2\mu _{L}+\mu _{K}+\sigma _{L}\sigma _{K}\rho \right) t\right] \exp \left[ \lambda \overline{\delta }t \right] \right) , \end{aligned}$$

  from which we get:

  $$\begin{aligned}&Cov\left[ L_{t};K_{t}\left( \alpha +\left( \beta -1\right) L_{t}\right) \right] =\exp \left[ \lambda \overline{\delta }t\right] \times L_{0}K_{0} \\&\quad \times \left( \begin{array}{l} \alpha \exp \left[ \left( \mu _{L}+\mu _{K}\right) t\right] \left[ \exp \left[ \sigma _{L}\sigma _{K}\rho t\right] -1\right] \\ +\left( \beta -1\right) L_{0}\exp \left[ \left( 2\mu _{L}+\mu _{K}+\sigma _{L}\sigma _{K}\rho \right) t\right] \left( \exp \left[ \left( \sigma _{L}^{2}+\sigma _{L}\sigma _{K}\rho \right) t\right] -1\right) \end{array} \right) \end{aligned}$$

1.2 Proof of Proposition 2

We have to solve

$$\begin{aligned} Min_{S\in \mathcal {H}_{2}}\,Variance\left[ S-IRM\left( K_{T},L_{T}\right) \right] ,\text { under }\mathbb {E}_{\mathbb {Q}}\left[ S \right] =0, \end{aligned}$$

which is equivalent here to

$$\begin{aligned} Min_{s}\,Var\left[ s\left( L_{T}\right) -\frac{1}{4}K_{T}\left( L_{T}-\alpha -\beta L_{T}\right) \right] , \end{aligned}$$

under

$$\begin{aligned} \mathbb {E}_{\mathbb {Q}}\left[ s\left( L_{T}\right) \right] =0. \end{aligned}$$

– First, we consider the solution under the additional assumption:

$$\begin{aligned} \mathbb {E}_{\mathbb {P}}\left[ IRM\left( K_{T},L_{T}\right) -s\left( L_{T}\right) \right] =m. \end{aligned}$$

Due to its convexity, we can apply results based on Lagrangian conditions.¹⁴ Let us introduce:

$$\begin{aligned} \mathcal {L}\left( h,\zeta ,\xi \right)= & {} Var\left[ s\left( L_{T}\right) -IRM\left( K_{T},L_{T}\right) \right] +2\zeta \left( \mathbb {E}_{\mathbb {P}}\left[ s\left( L_{T}\right) \eta _{T}\right] \right) \\&+2\xi \left( \mathbb {E}_{\mathbb {P}}\left[ IRM\left( K_{T},L_{T}\right) -s\left( L_{T}\right) \right] -m\right) , \end{aligned}$$

where $\eta _{T}$ corresponds to the Radon–Nikodym derivative $\dfrac{d \mathbb {Q}}{d\mathbb {P}}$ of the minimal martingale measure $\mathbb {Q}$ with respect to the objective probability $\mathbb {P}$.

Note that, under model assumptions, we have:

$$\begin{aligned} \eta _{T}=\exp \left[ -\frac{1}{2}\left( \frac{\mu _{L}}{\sigma _{L}}\right) ^{2}T-\left( \frac{\mu _{L}}{\sigma _{L}}\right) W_{L,T}\right] , \end{aligned}$$

from which we deduce that $\eta _{T}$ is a deterministic function of the market rate at maturity $L_{T}$ given by:

$$\begin{aligned} \eta _{T}=\chi _{T}L_{T}^{-\left( \frac{\mu _{L}}{\sigma _{L^{2}}}\right) }, \text { with }\chi _{T}=\exp \left[ -\frac{1}{2}\left[ \mu _{L}-\left( \frac{ \mu _{L}}{\sigma _{L}}\right) ^{2}\right] T\right] L_{0}^{\left( \frac{\mu _{L}}{\sigma _{L^{2}}}\right) }. \end{aligned}$$

Then, the optimization problem with the previous additional condition is equivalent to $Min\mathcal {L}\left( h,\zeta ,\xi \right)$. The first order conditions correspond to:

$$\begin{aligned}&\frac{\partial }{\partial h}Var\left[ s\left( L_{T}\right) -IRM\left( K_{T},L_{T}\right) \right] +2\zeta \left( \mathbb {E}_{\mathbb {P}}\left[ s\left( L_{T}\right) \eta _{T}\right] \right) \\&\quad +2\xi \left( \mathbb {E}_{ \mathbb {P}}\left[ IRM\left( K_{T},L_{T}\right) -s\left( L_{T}\right) \right] -m\right) =0. \end{aligned}$$

We deduce that:

$$\begin{aligned} h_{m}^{*}\left( L_{T}\right) =\mathbb {E}_{\mathbb {P}}\left[ IRM\left( K_{T},L_{T}\right) \left| L_{T}\right. \right] -\zeta \eta _{T}+\xi , \end{aligned}$$

with

$$\begin{aligned} \left\{ \begin{array}{l} \zeta -\xi =m \\ \zeta \mathbb {E}_{\mathbb {P}}\left[ \eta _{T}^{2}\right] -\xi =\mathbb {E}_{ \mathbb {Q}}\left[ IRM\left( K_{T},L_{T}\right) \right] \end{array} \right. . \end{aligned}$$

We deduce:

$$\begin{aligned} \zeta= & {} \frac{\mathbb {E}_{\mathbb {Q}}\left[ IRM\left( K_{T},L_{T}\right) \right] -m}{\mathbb {E}_{\mathbb {P}}\left[ \eta _{T}^{2}\right] -1}, \\ \xi= & {} \frac{\mathbb {E}_{\mathbb {Q}}\left[ IRM\left( K_{T},L_{T}\right) \right] -m\mathbb {E}_{\mathbb {P}}\left[ \eta _{T}^{2}\right] }{\mathbb {E}_{ \mathbb {P}}\left[ \eta _{T}^{2}\right] -1}. \end{aligned}$$

1.3 Proof of Remark 1

From definition of the interest rate margin, we deduce:

$$\begin{aligned} \mathbb {E}_{\mathbb {P}}\left[ IRM\left( K_{T},L_{T}\right) \left| L_{T}\right. \right]= & {} -\frac{1}{4}\mathbb {E}_{\mathbb {P}}\left[ K_{T}\left( \alpha +\left( \beta -1\right) L_{T}\right) \left| L_{T}\right. \right] \\= & {} -\frac{1}{4}\mathbb {E}_{\mathbb {P}}\left[ K_{T}\left| L_{T}\right. \right] \left( \alpha +\left( \beta -1\right) L_{T}\right) \end{aligned}$$

But we have:

$$\begin{aligned}&\mathbb {E}_{\mathbb {P}}\left[ K_{T}\left| L_{T}\right. \right] \\&\quad =\mathbb {E}_{\mathbb {P}}\left[ K_{0}\exp \left[ \left( \mu _{K}-\frac{1}{2} \sigma _{K}^{2}\right) T+\sigma _{K}\rho W_{L,T}\right. \right. \\&\qquad \left. \left. +\sigma _{K}\sqrt{1-\rho ^{2} }\widetilde{W}_{K,T}\right] \prod \limits _{T_{n}\le T}(1+\delta _{K,T_{n}})\big | L_{T}\right] \\&\quad =\mathbb {E}_{\mathbb {P}}\left[ K_{0}\exp \left[ \left( \mu _{K}-\frac{1}{2} \sigma _{K}^{2}\right) T+\sigma _{K}\rho W_{L,T}\right. \right. \\&\qquad \left. \left. +\sigma _{K}\sqrt{1-\rho ^{2} }\widetilde{W}_{K,T}\right] \big | W_{L,T}\right] \exp \left[ \lambda \overline{\delta }T\right] \\&\quad =K_{0}\exp \left[ \left( \mu _{K}-\frac{1}{2}\sigma _{K}^{2}\rho ^{2}\right) T \right] \exp \left[ \sigma _{K}\rho W_{L,T}\right] \exp \left[ \lambda \overline{\delta }T\right] = \end{aligned}$$

Since we have:

$$\begin{aligned} \exp \left[ \sigma _{K}\rho W_{L,T}\right] =\phi _{T}L_{T}^{\left( \frac{ \sigma _{K}\rho }{\sigma _{L}}\right) }, \end{aligned}$$

with

$$\begin{aligned} \phi _{T}=\exp \left[ -\left( \frac{\sigma _{K}\rho }{\sigma _{L}}\right) \left( \mu _{L}-\frac{1}{2}\sigma _{L}^{2}\right) T\right] L_{0}^{\left( - \frac{\sigma _{K}\rho }{\sigma _{L}}\right) }. \end{aligned}$$

Therefore, we deduce:

$$\begin{aligned} \mathbb {E}_{\mathbb {P}}\left[ K_{T}\left| L_{T}\right. \right] =\varphi _{T}L_{T}^{\left( \frac{\sigma _{K}\rho }{\sigma _{L}}\right) } \end{aligned}$$

with

$$\begin{aligned} \varphi _{T}= & {} K_{0}\exp \left[ \left( \mu _{K}-\frac{1}{2}\sigma _{K}^{2}\rho ^{2}-\left( \frac{\sigma _{K}\rho }{\sigma _{L}}\right) \left( \mu _{L}-\frac{1}{2} \sigma _{L}^{2}\right) \right. \right. \\&\left. \left. +\lambda \overline{\delta }\right) T\right] L_{0}^{\left( -\frac{\sigma _{K}\rho }{\sigma _{L}}\right) } \end{aligned}$$

Finally, we get:

$$\begin{aligned} \mathbb {E}_{\mathbb {P}}\left[ IRM\left( K_{T},L_{T}\right) \left| L_{T}\right. \right] =-\frac{1}{4}\varphi _{T}L_{T}^{\left( \frac{\sigma _{K}\rho }{\sigma _{L}}\right) }\left( \alpha +\left( \beta -1\right) L_{T}\right) . \end{aligned}$$

Now, to compute $\mathbb {E}_{\mathbb {Q}}\left[ IRM\left( K_{T},L_{T}\right) \right]$, note that, under the risk neutral probability $\mathbb {Q}$, the process defined by:

$$\begin{aligned} \widehat{W}_{L,t}=W_{L,t}+\left( \frac{\mu _{L}}{\sigma _{L}}\right) t \end{aligned}$$

is a standard Brownian motion.

Therefore, the dynamics of process L is given by:

$$\begin{aligned} dL_{t}=L_{t}\left( \sigma _{L}d\widehat{W}_{L,t}\right) , \end{aligned}$$

and the dynamics of process K is given by:

$$\begin{aligned} dK_{t}=K_{t}\left( \left[ \mu _{K}-\sigma _{K}\rho \left( \frac{\mu _{L}}{ \sigma _{L}}\right) \right] dt+\sigma _{K}\rho d\widehat{W}_{L,t}+\sigma _{K} \sqrt{1-\rho ^{2}}d\widetilde{W}_{K,t}+\delta _{K,t}dN_{K,t}\right) . \end{aligned}$$

Thus, we deduce:

$$\begin{aligned}&\mathbb {E}_{\mathbb {Q}}\left[ K_{T}\left( \alpha +\left( \beta -1\right) L_{T}\right) \right] \\&\quad =\alpha \exp \left[ \left( \mu _{K}-\sigma _{K}\rho \left( \frac{\mu _{L}}{ \sigma _{L}}\right) +\lambda \overline{\delta }\right) T\right] \\&\qquad +\left( \beta -1\right) \exp \left[ \left( \mu _{K}-\sigma _{K}\rho \left( \frac{\mu _{L}}{\sigma _{L}}\right) +\sigma _{L}\sigma _{K}\rho \right) T\right] \exp \left[ \lambda \overline{\delta }T\right] . \end{aligned}$$

Finally we get:

$$\begin{aligned}&\mathbb {E}_{\mathbb {Q}}\left[ IRM\left( K_{T},L_{T}\right) \right] \\&\quad =-\frac{1}{4}\alpha \exp \left[ \left( \mu _{K}-\sigma _{K}\rho \left( \frac{ \mu _{L}}{\sigma _{L}}\right) +\lambda \overline{\delta }\right) T\right] \\&\qquad - \frac{1}{4}\left( \beta -1\right) \exp \left[ \left( \mu _{K}-\sigma _{K}\rho \left( \frac{\mu _{L}}{\sigma _{L}}\right) +\sigma _{L}\sigma _{K}\rho \right) T\right] \exp \left[ \lambda \overline{\delta }T\right] . \end{aligned}$$

1.4 Proof of Proposition 4

We have to solve:

$$\begin{aligned} Min_{S\in \mathcal {H}_{3}}\,Variance\left[ S-IRM\left( K_{T},L_{T}\right) \right] ,\text { under }\mathbb {E}_{\mathbb {Q}}\left[ S \right] =0, \end{aligned}$$

which is equivalent here to:

$$\begin{aligned} Min_{s}\,Var\left[ s\left( K_{T},L_{T}\right) -\frac{1}{4}K_{T}\left( L_{T}-\alpha -\beta L_{T}\right) \right] , \end{aligned}$$

under

$$\begin{aligned} \mathbb {E}_{\mathbb {Q}}\left[ s\left( K_{T},L_{T}\right) \right] =0. \end{aligned}$$

– First, we consider the solution under the additional assumption:

$$\begin{aligned} \mathbb {E}_{\mathbb {P}}\left[ IRM\left( K_{T},L_{T}\right) -s\left( L_{T},K_{T}\right) \right] =m. \end{aligned}$$

Due to its convexity, we can apply results based on Lagrangian conditions.¹⁵ Let us introduce:

$$\begin{aligned} \mathcal {L}\left( h,\zeta ,\xi \right)= & {} Var\left[ s\left( K_{T},L_{T}\right) -IRM\left( K_{T},L_{T}\right) \right] +2\zeta \left( \mathbb {E}_{\mathbb {P}}\left[ s\left( K_{T},L_{T}\right) \eta _{T}\right] \right) \\&+2\xi \left( \mathbb {E}_{\mathbb {P}}\left[ IRM\left( K_{T},L_{T}\right) -s\left( K_{T},L_{T}\right) \right] -m\right) , \end{aligned}$$

where $\widehat{\eta }_{T}$ corresponds to the Radon–Nikodym derivative $\dfrac{d\widehat{\mathbb {Q}}}{d\mathbb {P}}$ of the minimal martingale measure $\widehat{\mathbb {Q}}$ with respect to the objective probability $\mathbb {P}$ and under the filtration generated by $W_{L}$, $W_{K}$ and the jump component. Note that, under model assumptions, we have:

$$\begin{aligned} \widehat{\eta }_{T}=\exp \left[ -\frac{1}{2}\left( \frac{\mu _{L}}{\sigma _{L}}\right) ^{2}T-\left( \frac{\mu _{L}}{\sigma _{L}}\right) W_{L,T}\right] , \end{aligned}$$

from which we deduce that $\eta _{T}$ is a deterministic function of the market rate at maturity $L_{T}$ given by:

$$\begin{aligned} \widehat{\eta }_{T}=\chi _{T}L_{T}^{-\left( \frac{\mu _{L}}{\sigma _{L^{2}}} \right) },\text { with }\chi _{T}=\exp \left[ -\frac{1}{2}\left[ \mu _{L}-\left( \frac{\mu _{L}}{\sigma _{L}}\right) ^{2}\right] T\right] L_{0}^{\left( \frac{\mu _{L}}{\sigma _{L^{2}}}\right) }. \end{aligned}$$

Then, the optimization problem with the previous additional condition is equivalent to $Min\mathcal {L}\left( h,\zeta ,\xi \right)$. The first order conditions correspond to:

$$\begin{aligned}&\frac{\partial }{\partial h}Var\left[ s\left( L_{T}\right) -IRM\left( K_{T},L_{T}\right) \right] +2\zeta \left( \mathbb {E}_{\mathbb {P}}\left[ s\left( L_{T}\right) \widehat{\eta }_{T}\right] \right) \\&\quad +2\xi \left( \mathbb { E}_{\mathbb {P}}\left[ IRM\left( K_{T},L_{T}\right) -s\left( L_{T}\right) \right] -m\right) =0. \end{aligned}$$

We deduce that:

$$\begin{aligned} s_{m}^{**}\left( L_{T}\right) =IRM\left( K_{T},L_{T}\right) -\zeta \eta _{T}+\xi , \end{aligned}$$

with:

$$\begin{aligned} \left\{ \begin{array}{l} \zeta -\xi =m \\ \zeta \mathbb {E}_{\mathbb {P}}\left[ \widehat{\eta }_{T}^{2}\right] -\xi = \mathbb {E}_{\mathbb {Q}}\left[ IRM\left( K_{T},L_{T}\right) \right] \end{array} \right. . \end{aligned}$$

We deduce:

$$\begin{aligned} \zeta= & {} \frac{\mathbb {E}_{\mathbb {Q}}\left[ IRM\left( K_{T},L_{T}\right) \right] -m}{\mathbb {E}_{\mathbb {P}}\left[ \widehat{\eta }_{T}^{2}\right] -1}, \\ \xi= & {} \frac{\mathbb {E}_{\mathbb {Q}}\left[ IRM\left( K_{T},L_{T}\right) \right] -m\mathbb {E}_{\mathbb {P}}\left[ \widehat{\eta }_{T}^{2}\right] }{ \mathbb {E}_{\mathbb {P}}\left[ \widehat{\eta }_{T}^{2}\right] -1}. \end{aligned}$$

1.5 The locally risk-minimizing strategy

The notations and definitions in this paper are those of Jacod and Shiryaev (1987). All processes are defined on filtered probability spaces $\left( \Omega ,\mathcal {F},\mathcal {(F}_{t}\mathcal {)},P\right)$ indexed on [0, T] and satisfying the usual conditions.

Let us recall that a semimartingale $S=(S_{t})_{t\in \left[ 0,1\right] }$ has the following canonical decomposition:

$$\begin{aligned} S=S_{0}+M+A, \end{aligned}$$

where $S_{0}$ is finite-valued and $\mathcal {F}_{0}$-measurable, A is a process with finite variation and M is a local martingale starting at zero. When A is predictable, this decomposition is unique. We obtain similar results in continuous time (see Ansel and Stricker 1993), as shown in what follows.

1.5.1 Continuous time model

The price S has the following canonical decomposition:

$$\begin{aligned} S=M+\lambda .\left\langle M,M\right\rangle , \end{aligned}$$

where $\lambda$ is a predictable process satisfying:

$$\begin{aligned} \lambda ^{2}.\left\langle M,M\right\rangle <\infty . \end{aligned}$$

Ansel and Stricker (1993) established the following existence result:

• If $1-\lambda \Delta M>0$ a.s. then $\mathcal {E}(-\,\lambda .M)$ is a positive local martingale, the minimal martingale measure is a probability and its density process is given by $\mathcal {E}(-\,\lambda .M)$.

• If $[1-\lambda \Delta M\le 0]>0$, then the minimal martingale measure is not a probability.

We consider a financial market with two traded assets: a riskless money market account and a risky asset, referred to as the bond and the stock, respectively. The value of the money market account at time t is given by $B_{t}=B_{0}e^{rt}$, where r is the riskfree interest rate. Without loss of generality, we take $r=0$ and $B_{0}=1$ in the following. The evolution of the stock price is described by a stochastic process $(S_{t})_{t\in \{0,1,\ldots ,T\}}$ on a probability space $(\Omega ,\mathcal {F}_{t},\mathbb {P})$, with some filtration $(\mathcal {F}_{t})_{t\in \{0,1,\ldots ,T\}}$ on $(\Omega , \mathcal {F})$ ($\mathbb {P}$ is usually called the historical probability). The price $S_{t}$ is taken $\mathcal {F}_{t}$-measurable and square-integrable.

General Case Let $(\Omega ,\mathcal {F},\mathbb {P})$ be a probability space with a filtration $(\mathcal {F}_{t})_{t\in [0,T]}$ satisfying the usual conditions of right-continuity and completeness. Assume that $\mathcal {F} _{0}$ is trivial and that $\mathcal {F}_{T}=\mathcal {F}$. The price $S_{t}$ is again supposed to be $\mathcal {F}_{t}$-measurable and square-integrable. It is a semimartingale with a decomposition:

$$\begin{aligned} S_{t}=S_{0}+A_{t}+M_{t}, \end{aligned}$$

such that:

• 1. M is a square-integrable martingale with $M_{0}=0$.

• 2. A is a predictable process of finite variation |A| with $A_{0}=0$.

We denote the predictable quadratic variation of the martingale M by $\left\langle M,M\right\rangle$. A trading strategy will be of the form $(q_{S},q_{B})$ satisfying the following conditions:

• 1. $(q_{S,t})_{t\in [0,T]}$ is a predictable process.

• 2. The process $\int _{0}^{t}q_{S,u}dS_{u}$ is a semimartingale of class $\mathcal {S}^{2}$, i.e:

  $$\begin{aligned} {E}[\int _{0}^{T}q_{S,u}^{2}d\left\langle M,M\right\rangle _{u}+\int _{0}^{T}|q_{S,u}|d|A|_{u})^{2}]<\infty . \end{aligned}$$

• 3. $(q_{B,t})_{t\in [0,T]}$ is adapted.

  The value of the portfolio at time t corresponding to $\phi$ is then given by the process $V(\phi )$ defined by $V_{t}(\phi )=q_{S,t}S_{t}+q_{B,t}$. It is right-continuous and satisfies: $V_{t}(\phi )\in \mathbb {L}^{2}(\mathbb {P})$.

  The cumulated cost up to time t is defined by:

  $$\begin{aligned} C_{t}=V_{t}(\phi )-\int _{0}^{t}q_{S,u}dS_{u}. \end{aligned}$$

  As measure of risk, Schweizer (1991) introduces the conditional mean square error process:

  $$\begin{aligned} R_{t}(\phi )=\mathbb {E}_{\mathbb {P}}[(C_{T}(\phi )-C_{t}(\phi ))^{2}| \mathcal {F}_{t}]. \end{aligned}$$

  A strategy $\phi$ is mean-self-financing if $C(\phi )$ is a martingale. A contingent claim is attainable if and only if there exists an admissible strategy $\phi$ such that $V_{T}(\phi )=H_{T}$ and $R_{t}(\phi )=0$, $\ \forall t\;a.s.$

As mentioned in Schweizer (1991), it may happen that no strategy $\phi$ can minimize $R_{t}(\phi )$ for all t. Therefore, it is necessary to weaken this condition. This gives birth to the notion of locally risk-minimizing strategy, which is defined as follows:

A trading strategy $\Delta =(\delta ,\epsilon )$ is called a small perturbation if it satisfies the following conditions:

• 1. $\delta$ is bounded.

• 2. $\int _{0}^{t}|\delta _{u}|d|A_{u}|$ is bounded.

• 3. $\delta _{T}=\epsilon _{T}=0$.

The underlying idea is to introduce the concept of local variation of a trading strategy. To this end, consider partitions $\tau =(t_{i})_{i}$ of the interval [0, T]. Such partitions will always satisfy:

$$\begin{aligned} 0=t_{0}<t_{1}<\cdots <t_{N}=T, \end{aligned}$$

with a mesh size defined by: $|\tau |=\max _{i}(t_{i}-t_{i-1})$. A sequence $(\tau _{n})_{n\in {{\mathbb {N}}}}$ of partitions will be increasing if $\tau _{n}\subseteq \tau _{n+1}$ for all n. It will be called 0-convergent if it satisfies:

$$\begin{aligned} \lim _{n\rightarrow +\infty }|\tau _{n}|=0. \end{aligned}$$

If $\Delta$ is a small perturbation and (s, t] is a subinterval of [0, T], the small perturbation $\Delta |(s,t]=(\delta |(s,t],\epsilon |(s,t])$ is defined by setting:

$$\begin{aligned} \delta |(s,t](\omega ,u)=\delta _{u}(\omega )1_{(s,t]}(u)\; \text {and}\; \epsilon |(s,t](\omega ,u)=\epsilon _{u}(\omega )1_{[s,t)}(u). \end{aligned}$$

The asymmetry corresponds to the fact that $\delta$ is predictable and $\epsilon$ adapted.

Let $\phi$ be a trading strategy, $\Delta$ a small perturbation and $\tau$ a partition of [0, T]. Then, the R-ratio is defined by

$$\begin{aligned} r^{\tau }[\phi ,\Delta ](\omega ,t)=\sum _{t_{i}\in \tau }\frac{ R_{t_{i}}(\phi +\Delta |(t_{i},t_{i+1}])-R_{t_{i}}(\phi )}{{E}[\left\langle M,M\right\rangle _{t_{i+1}}-\left\langle M,M\right\rangle _{t_{i}}| \mathcal {F}_{t_{i}}]}(\omega )\mathbb {I}_{(t_{i},t_{i+1}]}(t). \end{aligned}$$

The strategy $\phi$ is called locally risk-minimizing if

$$\begin{aligned} \lim \inf _{n\rightarrow +\infty }r^{\tau _{n}}[\phi ,\Delta ]\ge 0,a.s. \end{aligned}$$

or every small perturbation $\Delta$ and every increasing 0-converging sequence $(\tau _{n})$ of partitions of [0, T]. The ratio $r^{\tau }[\phi ,\Delta ]$ can be interpreted as a measure for the total change of riskiness if $\phi$ is locally perturbed by $\Delta$ along the partition $\tau$.

Suppose that:

H1:

For $\mathbb {P}$-almost all $\omega$, the measure on [0, T] induced by $\left\langle M,M\right\rangle (\omega )$ has the whole interval [0, T] as support (the martingale M is not locally constant: for example, a diffusion process with a strictly positive diffusion coefficient or a point process with a strictly positive intensity). Then, Schweizer (1991) shows that if a trading strategy is locally risk-minimizing then it is mean-self-financing.

Recall a main result about the computation of the locally risk-minimizing strategy in continuous time. We denote by $\widehat{\mathbb {Q}}$ the minimal martingale measure, and by $\hat{V}$ the conditional expectation of the contingent claim payoff H under $\widehat{\mathbb {Q}}$, namely

$$\begin{aligned} \hat{V}_{t}=\mathbb {E}_{\widehat{\mathbb {Q}}}[H|\mathcal {F}_{t}]. \end{aligned}$$

Suppose furthermore that:

H2:

A is continuous.

H3:

A is absolutely continuous with respect to $\left\langle M,M\right\rangle$ with a density $\alpha$ satisfying $\mathbb {E}_{M}[|\alpha |Log^{+}|\alpha |]<\infty$.

H4:

S is continuous at time T (no fixed time of discontinuity at T).

H5:

There exists a square-integrable $\mathbb {P}$-martingale $\widetilde{M}$ such that M and $\widetilde{M}$ form a $\mathbb {P}$-basis of $\mathbb {L}^{2}(\mathbb {P})$, and S and $\widetilde{M}$ form a $\mathbb { Q}$-basis of $\mathbb {L}^{2}(\mathbb {Q})$ (see Schweizer 1991, p. 355).

Proposition 7

(See Schweizer 1991, p. 355) Under conditions H1–5:

• (i) Every square-integrable contingent claim payoff H admits a decomposition:

  $$\begin{aligned} H=E\left[ H\right] _{\widehat{\mathbb {Q}}}+\int _{0}^{T}\xi _{u}^{H,\widehat{ \mathbb {Q}}}dS_{u}+\int _{0}^{T}\zeta _{u}^{H,\widehat{\mathbb {Q}}}d \widetilde{M}_{u},\widehat{\mathbb {Q}}\,a.s., \end{aligned}$$

  (16)

  where $\xi ^{H,\widehat{\mathbb {Q}}}$ and $\zeta ^{H, \widehat{\mathbb {Q}}}$ are predictable.

• (ii) If $(S_{t})_{t\in [0,T]}$ has continuous trajectories, then the locally risk-minimizing strategy is given by:

  $$\begin{aligned} \xi _{t}^{H,\widehat{\mathbb {Q}}}=\dfrac{d\left\langle \widehat{V},S\right\rangle _{t}^{\widehat{\mathbb {Q}}}}{d\left\langle S,S\right\rangle _{t}^{ \widehat{\mathbb {Q}}}}, \end{aligned}$$

  where $<.,.>^{\widehat{\mathbb {Q}}}$ is the predictable quadratic covariation with respect to $\widehat{\mathbb {Q}}$.

• (iii) The unique locally risk-minimizing strategy for the contingent claim H is given by $\theta =\xi ^{H,\widehat{\mathbb {Q}}}$.

Let us remark that the preceding decomposition leading to the characterization of the locally risk-minimizing strategy holds under two different probability measures, namely under the historical probability for the first one, and under the minimal martingale measure for the second one. Nevertheless, as mentioned in Heath et al (1999), under the assumptions H1–5, a strategy is locally risk-minimizing if and only if it is pseudo-locally risk-minimizing (i.e. its cost process C is a square integrable $\mathbb {P}$-martingale, strongly $\mathbb {P}$-orthogonal to the $\mathbb {P}$-martingale part M of S). In that case, finding a locally risk-minimizing strategy is equivalent to finding a decomposition of H:

$$\begin{aligned} H=H_{0}+\int _{0}^{T}\xi _{u}^{H}dS_{u}+R_{T}^{H}, \end{aligned}$$

so that $H_{0}$ is in $\mathbb {L}^{2}(\mathcal {F}_{0},\mathbb {P})$ and $L^{H}$ is a square integrable $\mathbb {P}$-martingale null at 0 and strongly $\mathbb {P}$-orthogonal to M. In that case, $\theta =\xi ^{H}$ and $C_{t}=H_{0}+R_{t}^{H}$.

We apply such decomposition to $H=IRM(K_{T},L_{T})$.

1.6 Proof of Proposition 6

Recall that the processes L and K satisfy:

$$\begin{aligned} dL_{t}=L_{t}\left( \mu _{L}dt+\sigma _{L}dW_{L,t}\right) , \end{aligned}$$

where $\left( W_{L,t}\right) _{t}$ is a standard Brownian motion under the historical probability measure $\mathbb {P}$ and where $\mu _{L}$ and $\sigma _{L}$ are assumed to be constant. The demand deposit amount follows:

$$\begin{aligned} dK_{t}=K_{t}\left( \mu _{K}dt+\sigma _{K}dW_{K,t}+\delta _{K,t}dN_{K,t}\right) , \end{aligned}$$

where $\left( W_{K,t}\right) _{t}$ is a standard Brownian motion and $\left( N_{K,t}\right) _{t}$ is a pure jump process. The trend $\mu _{K}$ and the volatility $\sigma _{K}$ are assumed to be constant, $\left( N_{K,t}\right) _{t}$ is a Poisson process with intensity $\lambda$ and that the sequence $\left( \delta _{K,T_{n}}\right) _{n}$ is i.i.d. with expectation $\overline{ \delta }$ and standard deviation $\sigma _{\delta }$. We have also:

$$\begin{aligned} dW_{K,t}=\rho dW_{L,t}+\sqrt{1-\rho ^{2}}d\widetilde{W}_{K,t}, \end{aligned}$$

where $\left( \widetilde{W}_{K,t}\right) _{t}$ is a Brownian motion independent of $\left( W_{L,t}\right) _{t}$, while $W_{L,t}$ and $W_{K,t}$ have a correlation parameter $\rho t$.

Therefore, we note that the martingale part M of process L is given by $M_{t}=\sigma _{L}\int _{0}^{t}L_{t}dW_{L,t}$ and that we can choose the square-integrable $\mathbb {P}$-martingale $\widetilde{M}$ as follows:

$$\begin{aligned} \widetilde{M}_{t}=\sqrt{1-\rho ^{2}}\int _{0}^{t}K_{t}d\widetilde{W} _{K,t}+\int _{\mathbb {R}}K_{t-}\delta (t,K_{t-},x)(\mu (dt,dx)-\nu (dt,dx)), \end{aligned}$$

where $\mu (dt,dx)$ is the counting measure of the compound Poisson process $\int _{0}^{t}\int _{\mathbb {R}}\delta (u,K_{u-})dN_{K,u}$ and $\nu (dt,dx)$ is its compensator measure (i.e. $\nu (dt,dx)=\lambda \overline{\delta }dt$). Under previous assumptions, the locally risk-minimizing strategy is given by:

$$\begin{aligned} \theta _{t}=\dfrac{d\left\langle \widehat{IRM},L\right\rangle _{t}^{ \widehat{\mathbb {Q}}}}{d\left\langle L,L\right\rangle _{t}^{\widehat{ \mathbb {Q}}}}. \end{aligned}$$

Thus, we have to compute both terms $d\left\langle \widehat{IRM} ,L\right\rangle _{t}^{\widehat{\mathbb {Q}}}$and $d\left\langle L,L\right\rangle _{t}^{\widehat{\mathbb {Q}}}$. For this latter one, since under $\widehat{\mathbb {Q}}$, we have:

$$\begin{aligned} dL_{t}=L_{t}\left( \sigma _{L}d\widehat{W}_{L,t}\right) , \end{aligned}$$

where $\left( \widehat{W}_{L,t}\right) _{t}$ is a Brownian motion under $\widehat{\mathbb {Q}}$, we deduce that:

$$\begin{aligned} d\left\langle L,L\right\rangle _{t}^{\widehat{\mathbb {Q}}}=\sigma _{L}^{2}dt. \end{aligned}$$

To compute $d\left\langle \widehat{IRM},L\right\rangle _{t}^{\widehat{ \mathbb {Q}}}$, we note that:

$$\begin{aligned} dK_{t}= & {} K_{t}\left( \mu _{K}dt+\sigma _{K}dW_{K,t}+\delta _{K,t}dN_{K,t}\right) , \\= & {} K_{t}\left( \mu _{K}dt+\sigma _{K}\left( \rho dW_{L,t}+\sqrt{1-\rho ^{2}}d \widetilde{W}_{K,t}\right) +\delta _{K,t}dN_{K,t}\right) , \\= & {} K_{t}\left( \mu _{K}dt+\sigma _{K}\left( \rho \left[ d\widehat{W}_{L,t}- \frac{\mu _{L}}{\sigma _{L}}dt\right] +\sqrt{1-\rho ^{2}}d\widetilde{W} _{K,t}\right) +\delta _{K,t}dN_{K,t}\right) , \\= & {} K_{t}\left( \left[ \mu _{K}-\rho \sigma _{K}\frac{\mu _{L}}{\sigma _{L}} \right] dt+\sigma _{K}\rho d\widehat{W}_{L,t}+\sigma _{K}\sqrt{1-\rho ^{2}}d \widetilde{W}_{K,t}+\delta _{K,t}dN_{K,t}\right) , \end{aligned}$$

where $\widehat{W}_{L,t}$, $\widetilde{W}_{K,t}$ and $\left( N_{K,t}-\lambda \overline{\delta }t\right)$ are orthogonal martingales under $\widehat{ \mathbb {Q}}$.

We have:

$$\begin{aligned} \widehat{IRM}_{t}=\mathbb {E}_{\widehat{\mathbb {Q}}}[IRM_{T}|\mathcal {F} _{t}]=-\frac{1}{4}\mathbb {E}_{\widehat{\mathbb {Q}}}[K_{T}\left( \alpha +\left( \beta -1\right) L_{T}\right) |\mathcal {F}_{t}]= \end{aligned}$$

Step 1: Note that:

$$\begin{aligned}&\mathbb {E}_{\widehat{\mathbb {Q}}}\left[ K_{T}\left| \mathcal {F} _{t}\right. \right] \\&\quad =\mathbb {E}_{\widehat{\mathbb {Q}}}\left[ \left. \begin{array}{l} K_{0}\exp \left[ \begin{array}{l} \left( \mu _{K}-\rho \sigma _{K}\frac{\mu _{L}}{\sigma _{L}}-\frac{1}{2} \sigma _{K}^{2}\right) T+\sigma _{K}\rho \left( \widehat{W}_{L,t}+\left[ \widehat{W}_{L,T}-\widehat{W}_{L,t}\right] \right) \\ +\sigma _{K}\sqrt{1-\rho ^{2}}\left( \widetilde{W}_{K,t}+\left[ \widetilde{W} _{K,T}-\widetilde{W}_{K,t}\right] \right) \end{array} \right] \\ \prod \limits _{T_{n}\le t}(1+\delta _{K,T_{n}})\prod \limits _{t<T_{n}\le T}(1+\delta _{K,T_{n}}) \end{array} \right| \mathcal {F}_{t}\right] \\&\quad =K_{0}\exp \left[ \left( \mu _{K}-\rho \sigma _{K}\frac{\mu _{L}}{\sigma _{L}}-\frac{1}{2}\sigma _{K}^{2}\right) t+\sigma _{K}\rho \widehat{W} _{L,t}+\sigma _{K}\sqrt{1-\rho ^{2}}\widetilde{W}_{K,t}\right] \\&\qquad \times \exp \left[ \left( \mu _{K}-\rho \sigma _{K}\frac{\mu _{L}}{\sigma _{L}} \right) \left( T-t\right) \right] \prod \limits _{T_{n}\le t}(1+\delta _{K,T_{n}})\exp \left[ \lambda \overline{\delta }\left( T-t\right) \right] . \end{aligned}$$

Thus, we get:

$$\begin{aligned} \mathbb {E}_{\widehat{\mathbb {Q}}}\left[ K_{T}\left| \mathcal {F} _{t}\right. \right] =K_{t}\times \exp \left[ \left( \mu _{K}-\rho \sigma _{K} \frac{\mu _{L}}{\sigma _{L}}+\lambda \overline{\delta }\right) \left( T-t\right) \right] . \end{aligned}$$

Recall also that we have:

$$\begin{aligned} \exp \left[ \sigma _{K}\rho W_{L,t}\right] =\widehat{\phi }_{t}\left( \frac{ L_{t}}{L_{0}}\right) ^{\left( -\frac{\sigma _{K}\rho }{\sigma _{L}}\right) }, \end{aligned}$$

with

$$\begin{aligned} \widehat{\phi }_{t}=\exp \left[ -\left( \frac{\sigma _{K}\rho }{\sigma _{L}} \right) \left( \mu _{L}-\frac{1}{2}\sigma _{L}^{2}\right) t\right] . \end{aligned}$$

But:

$$\begin{aligned} \widehat{W}_{L,t}=W_{L,t}+\left( \frac{\mu _{L}}{\sigma _{L}}\right) t. \end{aligned}$$

Thus:

$$\begin{aligned} \exp \left[ \sigma _{K}\rho \widehat{W}_{L,t}\right] =\widehat{\phi } _{t}\exp \left[ \sigma _{K}\rho \left( \frac{\mu _{L}}{\sigma _{L}}\right) t \right] \left( \frac{L_{t}}{L_{0}}\right) ^{\left( -\frac{\sigma _{K}\rho }{ \sigma _{L}}\right) }. \end{aligned}$$

Therefore, we deduce another expression of $\mathbb {E}_{\widehat{\mathbb {Q}}} \left[ K_{T}\left| \mathcal {F}_{t}\right. \right]$, namely:

$$\begin{aligned} \mathbb {E}_{\widehat{\mathbb {Q}}}\left[ K_{T}\left| \mathcal {F} _{t}\right. \right] =\varphi _{t}L_{t}^{\left( -\frac{\sigma _{K}\rho }{ \sigma _{L}}\right) }, \end{aligned}$$

with

$$\begin{aligned} \varphi _{t}= & {} \widehat{\phi }_{t}\exp \left[ \sigma _{K}\rho \left( \frac{\mu _{L}}{\sigma _{L}}\right) t\right] \\&\times K_{0}L_{0}^{\left( \frac{\sigma _{K}\rho }{\sigma _{L}}\right) }\exp \left[ \left( \mu _{K}-\frac{1}{2}\sigma _{K}^{2}\right) t+\left( \mu _{K}-\rho \sigma _{K}\frac{\mu _{L}}{\sigma _{L}}\right) \left( T-t\right) \right. \\&\left. +\sigma _{K} \sqrt{1-\rho ^{2}}\widetilde{W}_{K,t}+\lambda \overline{\delta }\left( T-t\right) \right] \prod \limits _{T_{n}\le t}(1+\delta _{K,T_{n}}). \end{aligned}$$

Step 2: Note that:

$$\begin{aligned} L_{t}= & {} L_{0}\mathcal {E}\left( \sigma _{L}\widehat{W}_{L,t}\right) ,\\ K_{t}= & {} K_{0}\mathcal {E}\left( \left[ \mu _{K}-\sigma _{K}\rho \left( \frac{ \mu _{L}}{\sigma _{L}}\right) \right] t+\sigma _{K}\rho \widehat{W} _{L,t}+\sigma _{K}\sqrt{1-\rho ^{2}}\widetilde{W}_{K,t}\right) \prod \limits _{T_{n}\le t}(1+\delta _{K,T_{n}}). \end{aligned}$$

Therefore, applying Yor’s formula, we deduce:

$$\begin{aligned} K_{t}L_{t}= & {} K_{0}L_{0}\\&\mathcal {\times }\exp \left( \left[ \mu _{K}-\sigma _{K}\rho \left( \frac{\mu _{L}}{\sigma _{L}}\right) -\frac{1}{2}\left( \sigma _{L}^{2}+\sigma _{K}^{2}\right) \right] t \right. \\&\left. +\left[ \sigma _{L}+\sigma _{K}\rho \right] \widehat{W} _{L,t}+\sigma _{K}\sqrt{1-\rho ^{2}}\widetilde{W}_{K,t}\right) \prod \limits _{T_{n}\le t}(1+\delta _{K,T_{n}}). \end{aligned}$$

Consequently, we have:

$$\begin{aligned}&\mathbb {E}_{\widehat{\mathbb {Q}}}\left[ K_{T}L_{T}\left| \mathcal {F} _{t}\right. \right] =K_{0}L_{0}\mathcal {\times } \\&\qquad \mathbb {E}_{\widehat{\mathbb {Q}}}\left[ \left. \begin{array}{l} \exp \left( \begin{array}{l} \left[ \mu _{K}-\sigma _{K}\rho \left( \frac{\mu _{L}}{\sigma _{L}}\right) - \frac{1}{2}\left( \sigma _{L}^{2}+\sigma _{K}^{2}\right) \right] T+\left[ \sigma _{L}+\sigma _{K}\rho \right] \left( \widehat{W}_{L,t}+\left[ \widehat{ W}_{L,T}-\widehat{W}_{L,t}\right] \right) \\ +\sigma _{K}\sqrt{1-\rho ^{2}}\left( \widetilde{W}_{K,t}+\left[ \widetilde{W} _{K,T}-\widetilde{W}_{K,t}\right] \right) \end{array} \right) \\ \prod \limits _{T_{n}\le t}(1+\delta _{K,T_{n}})\prod \limits _{t<T_{n}\le T}(1+\delta _{K,T_{n}}) \end{array} \right| \mathcal {F}_{t}\right] \\&\quad =K_{0}L_{0}\exp \left[ \left[ \mu _{K}-\sigma _{K}\rho \left( \frac{\mu _{L} }{\sigma _{L}}\right) -\frac{1}{2}\left( \sigma _{L}^{2}+\sigma _{K}^{2}\right) \right] T\right. \\&\qquad \left. +\left[ \sigma _{L}+\sigma _{K}\rho \right] \widehat{W}_{L,t}+\sigma _{K}\sqrt{1-\rho ^{2}}\widetilde{W}_{K,t}\right] \\&\qquad \times \exp \left[ \frac{1}{2}\left( \sigma _{K}^{2}+\sigma _{L}^{2}+2\rho \sigma _{K}\sigma _{L}\right) \left( T-t\right) \right] \prod \limits _{T_{n}\le t}(1+\delta _{K,T_{n}})\exp \left[ \lambda \overline{ \delta }\left( T-t\right) \right] . \end{aligned}$$

Therefore, we get:

$$\begin{aligned}&\mathbb {E}_{\widehat{\mathbb {Q}}}\left[ K_{T}L_{T}\left| \mathcal {F} _{t}\right. \right] =K_{t}L_{t}\mathcal {\times } \\&\qquad \exp \left[ \left[ \mu _{K}-\sigma _{K}\rho \left( \frac{\mu _{L}}{\sigma _{L}}\right) -\frac{1}{2}\left( \sigma _{L}^{2}+\sigma _{K}^{2}\right) \right] \left( T-t\right) \right] \\&\quad \times \exp \left[ \frac{1}{2}\left( \sigma _{K}^{2}+\sigma _{L}^{2}+2\rho \sigma _{K}\sigma _{L}\right) \left( T-t\right) \right] \exp \left[ \lambda \overline{\delta }\left( T-t\right) \right] . \end{aligned}$$

which is equivalent to:

$$\begin{aligned} \mathbb {E}_{\widehat{\mathbb {Q}}}\left[ K_{T}L_{T}\left| \mathcal {F} _{t}\right. \right] =K_{t}L_{t}\exp \left[ \left[ \mu _{K}-\sigma _{K}\rho \left( \frac{\mu _{L}}{\sigma _{L}}\right) +\left( \rho \sigma _{K}\sigma _{L}\right) +\lambda \overline{\delta }\right] \left( T-t\right) \right] . \end{aligned}$$

Finally, we get the value of the margin interest rate computed with respect to the minimal martingale measure:

$$\begin{aligned} \mathbb {E}_{\widehat{\mathbb {Q}}}\left[ IRM\left( K_{T},L_{T}\right) \left| \mathcal {F}_{t}\right. \right] =-\frac{1}{4}\left( \alpha \mathbb {E}_{ \widehat{\mathbb {Q}}}\left[ K_{T}\left| \mathcal {F}_{t}\right. \right] +\left( \beta -1\right) \mathbb {E}_{\widehat{\mathbb {Q}}}\left[ K_{T}L_{T}\left| \mathcal {F}_{t}\right. \right] \right) , \end{aligned}$$

which implies that:

$$\begin{aligned}&\mathbb {E}_{\widehat{\mathbb {Q}}}\left[ IRM\left( K_{T},L_{T}\right) \left| \mathcal {F}_{t}\right. \right] \\&\quad = -\frac{1}{4}\left( \begin{array}{l} \alpha K_{t}\times \exp \left[ \left( \mu _{K}-\rho \sigma _{K}\frac{\mu _{L} }{\sigma _{L}}+\lambda \overline{\delta }\right) \left( T-t\right) \right] \\ + \\ \left( \beta -1\right) K_{t}L_{t}\mathcal {\times }\exp \left[ \left[ \mu _{K}-\sigma _{K}\rho \left( \frac{\mu _{L}}{\sigma _{L}}\right) +\left( \rho \sigma _{K}\sigma _{L}\right) +\lambda \overline{\delta }\right] \left( T-t\right) \right] \end{array} \right) \end{aligned}$$

Thus, the process $\mathbb {E}_{\widehat{\mathbb {Q}}}\left[ IRM\left( K_{T},L_{T}\right) \left| \mathcal {F}_{t}\right. \right]$ has the following form:

$$\begin{aligned} \mathbb {E}_{\widehat{\mathbb {Q}}}\left[ IRM\left( K_{T},L_{T}\right) \left| \mathcal {F}_{t}\right. \right] =f(t,T)K_{t}+h(t,T)K_{t}L_{t}, \end{aligned}$$

with

$$\begin{aligned} \left\{ \begin{array}{l} f(t,T)=-\frac{\alpha }{4}\exp \left[ \left( \mu _{K}-\rho \sigma _{K}\frac{ \mu _{L}}{\sigma _{L}}+\lambda \overline{\delta }\right) \left( T-t\right) \right] , \\ \text {and} \\ h(t,T)=-\frac{\left( \beta -1\right) }{4}\exp \left[ \left[ \mu _{K}-\sigma _{K}\rho \left( \frac{\mu _{L}}{\sigma _{L}}\right) +\left( \rho \sigma _{K}\sigma _{L}\right) +\lambda \overline{\delta }\right] \left( T-t\right) \right] . \end{array} \right. \end{aligned}$$

Now we can determine $d\left\langle \widehat{IRM},L\right\rangle _{t}^{ \widehat{\mathbb {Q}}}$. For this latter purpose, we must determine the martingale component of $\widehat{IRM}_{t}$ driven by the Brownian motion $\left( \widehat{W}_{L,t}\right) _{t}$with respect to the minimal martingale measure. Using Ito’s formula, its stochastic differential is given by:

$$\begin{aligned} \left( f(t,T)K_{t}\rho \sigma _{K}+h(t,T)K_{t}L_{t}\left[ \sigma _{L}+\rho \sigma _{K}\right] \right) d\widehat{W}_{L,t}. \end{aligned}$$

Finally, we deduce:

$$\begin{aligned} d\left\langle \widehat{IRM},L\right\rangle _{t}^{\widehat{\mathbb {Q}}}= \left[ \left( f(t,T)K_{t}\rho \sigma _{K}+h(t,T)K_{t}L_{t}\left[ \sigma _{L}+\rho \sigma _{K}\right] \right) \sigma _{L}L_{t}\right] dt, \end{aligned}$$

and the locally risk minimizing strategy is given by:

$$\begin{aligned} \theta _{t}=\dfrac{d\left\langle \widehat{IRM},L\right\rangle _{t}^{ \widehat{\mathbb {Q}}}}{d\left\langle L,L\right\rangle _{t}^{\widehat{ \mathbb {Q}}}}=\frac{\left( f(t,T)K_{t}\rho \sigma _{K}+h(t,T)K_{t}L_{t}\left[ \sigma _{L}+\rho \sigma _{K}\right] \right) }{\sigma _{L}L_{t}}. \end{aligned}$$