The impact of non-cash collateralization on the over-the-counter derivatives markets

Abstract

In this study, we propose a microeconomics model to verify effects of the non-cash collateralization on the liquidity of the over-the-counter (OTC) derivatives markets accepting both cash and non-cash assets. Liquidity is measured as an equilibrium volume of the derivatives contract. The equilibrium volume is obtained by solving the utility maximization problem of a risk-averse collateral payer who wants to optimize her/his capital. The collateral payer’s capital depends on the non-cash asset used as collateral. We consider both option and forward contracts as examples. Our sensitivity analysis shows that the optimal combination of cash and non-cash collaterals can maximize the liquidity of derivatives. Especially for option contracts, the market requires both cash and non-cash collaterals for liquidity. The empirical result related to this finding is provided. Overall, the introduction of non-cash collateralization boosts the liquidity of derivatives contracts. We also show how the arrangements of collateralization can boost the liquidity of the OTC derivatives markets. Moreover, we demonstrate that the combination of cash and non-cash collaterals to maximize liquidity differs from that to maximize the participant’s utility. This indicates that the optimal combination is not efficient in terms of Pareto criteria.

Appendices

Motivation from technical viewpoint

In this section, we provide a brief discussion of the motivation for this study from the perspective of the participant with a negative exposure (i.e., the bank).

At the contract date $t_n$, the bank enters $k_{t_n}$ units of the derivatives contract with a negative exposure. Then, the bank has to post collateral with cash or bond or both and receives the derivatives value $k_{t_n}|V_{t_n}|$ from the dealer. The cash received from the dealer is used as cash collateral. Then, the amount of the cash collateral is $\eta k_{t_n}|V_{t_n}|$, and that of the non-cash collateral is $k_{t_n}\frac{1-\eta }{1-h}\frac{|V_{t_n}|}{B_{t_n}}B_{t_n}$ considering the haircut. The dealer, who is the collateral receiver, should pay interest (at the collateral rate) on the posted cash collateral whether a default occurs or not. We denote the cash account and value of the bond held by the bank at time t as $A^C_t$ and $A^B_t$, respectively. Then, the cash account and value of the bond held by the collateral payer at the contract date $t_n$ are, respectively,

$$\begin{aligned} A^C_{t_{n}}&= k_{t_n}|V_{t_n}|-\eta k_{t_n}|V_{t_n}|=(1-\eta )k_{t_n}|V_{t_n}|, \\ A_{t_{n}}^B&= MB_{t_n}-k_{t_n}\frac{1-\eta }{1-h}\frac{|V_{t_n}|}{B_{t_n}}B_{t_n}= \left( M-k_{t_n}\frac{1-\eta }{1-h}\frac{|V_{t_n}|}{B_{t_n}} \right) B_{t_n}. \end{aligned}$$

At the next MtM date $t_{n+1}$, in case of no default, the bank recovers the posted collateral. The bank also earns interest on the cash collateral. In case of default, the bank does not recover the posted collateral at all. Then, the cash account and value of the bond held by the bank at the next MtM date $t_{n+1}$ are, respectively,

$$\begin{aligned} \begin{aligned} A^C_{t_{n+1}}&=(1+r_{t_n}\Delta t)A^C_{t_n}+(1+r_{t_n}^c\Delta t)\eta k_{t_n} |V_{t_n}|(1-1_{\tau<t_{n+1}}) \\&=k_{t_n}|V_{t_n}|+\{(1-\eta )r_{t_n}+\eta r_{t_n}^c\}k_{t_n}|V_{t_n}|\Delta t-(1+r_{t_n}^c\Delta t)\eta k_{t_n}|V_{t_n}|1_{\tau<t_{n+1}}, \\ A_{t_{n+1}}^B&=\left( M-k_{t_n}\frac{1-\eta }{1-h}\frac{|V_{t_n}|}{B_{t_n}}\right) B_{t_{n+1}}+k_{t_n}\frac{1-\eta }{1-h}\frac{|V_{t_n}|}{B_{t_n}} B_{t_{n+1}}(1-1_{\tau<t_{n+1}})\\&=MB_{t_{n+1}}-k_{t_n}\frac{1-\eta }{1-h}\frac{|V_{t_n}|}{B_{t_n}} B_{t_{n+1}}1_{\tau <t_{n+1}} \end{aligned} \end{aligned}$$

where $\tau$ is the default time of the bank. Therefore, the change in the asset amount of the collateral payer is

$$\begin{aligned} \begin{aligned} \Delta A&=A_{t_{n+1}}^C+A_{t_{n+1}}^B-(k_{t_n}|V_{t_n}|+MB_{t_n})\\&=\{(1-\eta )r_{t_n}+\eta r_{t_n}^c\}k_{t_n}|V_{t_n}|\Delta t-(1+r_{t_n}^c\Delta t)\eta k_{t_n}|V_{t_n}|1_{\tau<t_{n+1}}\\&\quad +\,M(B_{t_{n+1}}-B_{t_n})-k_{t_n}\frac{1-\eta }{1-h}\frac{|V_{t_n}|}{B_{t_n}} B_{t_{n+1}}1_{\tau <t_{n+1}}. \end{aligned} \end{aligned}$$

(A.1)

If the bank does not default (i.e., $1_{\tau <t_{n+1}}=0$), then (A.1) is

$$\begin{aligned} \begin{aligned} \Delta A&=\{(1-\eta )r_{t_n}+\eta r_{t_n}^c\}k_{t_n}|V_{t_n}|\Delta t+M(B_{t_{n+1}}-B_{t_n}). \end{aligned} \end{aligned}$$

(A.2)

From (A.2), we have

$$\begin{aligned} \frac{\partial \Delta A}{\partial \eta }=(r_{t_n}^c-r_{t_n})k_{t_n}|V_{t_n}|\Delta t \end{aligned}$$

(A.3)

if the effect of $\eta$ on the value of $V_{t_n}$ is negligibly small.

We interpret the relationship (A.3) as follows. The bank is willing to post the cash collateral ($\eta \nearrow 1$) if $r^c>r$. However, when $r>r^c$, the bank is willing to post the non-cash asset collateral ($\eta \searrow 0$). This means that the bank must choose either a cash or non-cash asset as collateral. Moreover, her/his decision depends on the levels of interest rates at the contract date $t_n$ only. However, if the bank can default, we have

$$\begin{aligned} \frac{\partial \Delta A}{\partial \eta }= & {} (r_{t_n}^c-r_{t_n})k_{t_n}|V_{t_n}|\Delta t-(1+r_{t_n}^c\Delta t) k_{t_n}|V_{t_n}|1_{\tau<t_{n+1}}\nonumber \\&\quad +k_{t_n}\frac{1}{1-h}\frac{|V_{t_n}|}{B_{t_n}}B_{t_{n+1}}1_{\tau <t_{n+1}} \end{aligned}$$

(A.4)

from (A.1). Therefore, we deduce that the bank’s collateral asset selection is more complicated when a default occurs because she/he should not only consider the levels of interest rates at the contract date but also should be concerned about the (stochastic) change in the bond price during $[t_n,t_{n+1}]$ unlike the no default case. Moreover, since the bank is risk-averse, it might avoid choosing only either cash or non-cash asset as collateral. In this study, we show how the collateral payer combines the collateral assets and then verify the validity of the recent clearinghouse actions.

Proof of Proposition 3.1

The dealer deposits the posted cash collateral $\eta V_t$ with risk-free rate r and the returns with collateral rate $r^c$. The dealer also sources cash $\frac{1-h_p}{1-h}(1-\eta )V_t=(1-\eta )V_t$ (under the assumption $h=h_p$) by exchanging it for the posted bond worth $\frac{1}{1-h}(1-\eta )V_t$ in the repo market. The money sourced via the repo market is further deposited with risk-free rate r, and the returns are deposited with the repo rate $r^p$. Thus, the instantaneous change in collateral for the collateral receiver is

$$\begin{aligned} y_tV_tdt:=\left\{ r_t-(\eta r_t^c+(1-\eta )r_t^p) \right\} V_tdt. \end{aligned}$$

Therefore, when we denote the total collateral value at time t by $C_t$, the derivatives’ time t value is given by

$$\begin{aligned} \begin{aligned} V_t&=E_t^Q\left[ \left\{ e^{-\int _t^Tr_sds}V_T+\int _t^Te^{-\int _t^sr_udu}y_sC_sds \right\} 1_{\tau >T}\right] \\&\quad +E_t^Q\left[ \left\{ e^{-\int _t^\tau r_sds}C_\tau +\int _t^\tau e^{-\int _t^sr_udu}y_sC_sds\right\} 1_{\tau \le T} \right] . \end{aligned} \end{aligned}$$

(B.1)

The first expectation shows that the derivatives payment included the net return from investing the posted collateral without defaults, and the second expectation shows that the collateral value included the net return from investing the posted collateral at default. Equation (B.1) agrees with (A.1) of Johannes and Sundaresan (2007). The assumption of continuous and perfect collateralization yields

$$\begin{aligned} C_t=V_t \end{aligned}$$

(B.2)

for $0\le t \le T$. Then, Eq. (B.1) reduces to

$$\begin{aligned} \begin{aligned} V_t=&E_t^Q\left[ e^{-\int _t^Tr_sds}V_T+\int _t^Te^{-\int _t^sr_udu}y_sV_sds\right] . \end{aligned} \end{aligned}$$

(B.3)

Several studies have derived the pricing equation for the contingent claim $V_T$ with maturity T from (B.3) (Fujii et al. 2010; Fujii and Takahashi 2016; Johannes and Sundaresan 2007). We follow their derivations. Set

$$\begin{aligned} Z_t=e^{-\int _0^tr_sds}V_t+\int _0^te^{-\int _0^sr_udu}y_sV_sds, \end{aligned}$$

(B.4)

then

$$\begin{aligned} \begin{aligned} E_t^Q[Z_T]&=E_t^Q\left[ e^{-\int _0^Tr_sds}V_T+\int _0^Te^{-\int _0^sr_udu}y_sV_sds \right] \\&=e^{-\int _0^tr_sds}E^Q_t\left[ e^{-\int _t^Tr_sds}V_T+\int _t^Te^{-\int _t^sr_udu}y_sV_sds \right] +\int _0^te^{-\int _0^sr_udu}y_sV_sds\\&=e^{-\int _0^tr_sds}V_t+\int _0^te^{-\int _0^sr_udu}y_sV_sds\\&=Z_t. \end{aligned} \end{aligned}$$

Hence, $Z_t$ is a Q-martingale. From (B.4), it holds

$$\begin{aligned} \begin{aligned} dZ_t&=-r_te^{-\int _0^tr_sds}V_tdt+e^{-\int _0^tr_sds}dV_t+e^{-\int _0^tr_udu}y_tV_tdt. \\ \end{aligned} \end{aligned}$$

(B.5)

By arranging (B.5), we have the (linear) backward stochastic differential equation,

$$\begin{aligned} \begin{aligned} -dV_t&=-(r_t-y_t)V_tdt-e^{\int _0^tr_sds}dZ_t,\quad V_T=Z_T. \\ \end{aligned} \end{aligned}$$

(B.6)

By solving (B.6), we have the derivatives pricing formula under full collateralization,

$$\begin{aligned} V_t=E_t^Q\left[ e^{-\int _t^T(r_s-y_s)ds}V_T\right] . \end{aligned}$$

Calculation of statistics

Each term of (3.5) is calculated as follows.

$$\begin{aligned} \begin{aligned} E_{t_n}[g(t_{n+1};\eta )]&=E_{t_n}[g_0(t_{n+1};\eta )(1-1_{\tau<t_{n+1}})+g_1(t_{n+1};\eta )1_{\tau<t_{n+1}}]\\&=E_{t_n}[g_0(t_{n+1};\eta )(1-1_{\tau<t_{n+1}})]+E_{t_n}[g_1(t_{n+1};\eta )1_{\tau<t_{n+1}}],\\ Cov_{t_n}[B_{t_{n+1}},g(T;\eta )]&=Cov_{t_n}[B_{t_{n+1}},g_0(t_{n+1};\eta )(1-1_{\tau<t_{n+1}})]\\&\quad +Cov_{t_n}[B_{t_{n+1}},g_1(t_{n+1};\eta )1_{\tau<t_{n+1}}]\\&=E_t[B_{t_{n+1}}g_0(t_{n+1};\eta )(1-1_{\tau<t_{n+1}})]\\&\quad -E_{t_n}[B_{t_{n+1}}]E_t[g_0(t_{n+1};\eta )(1-1_{\tau<t_{n+1}})]\\&\quad +E_{t_n}[B_{t_{n+1}}g_1(t_{n+1};\eta )1_{\tau<t_{n+1}}]\\&\quad -E_{t_n}[B_{t_{n+1}}]E_{t_n}[g_1(t_{n+1};\eta )1_{\tau<t_{n+1}}],\\ Var_{t_n}[g(t_{n+1};\eta )]&=Var_{t_n}[g_0(t_{n+1};\eta )(1-1_{\tau<t_{n+1}})]+Var_{t_n}[g_1(t_{n+1};\eta )1_{\tau<t_{n+1}}]\\&\quad +2Cov_t[g_0(t_{n+1};\eta )(1-1_{\tau<t_{n+1}}),g_1(t_{n+1};\eta )1_{\tau<t_{n+1}}]\\&=E_{t_n}[(g_0(t_{n+1};\eta )(1-1_{\tau<t_{n+1}}))^2]-E_{t_n}[g_0(t_{n+1};\eta )(1-1_{\tau<t_{n+1}})]^2\\&\quad +E_{t_n}[(g_1(t_{n+1};\eta )1_{\tau<t_{n+1}})^2]-E_{t_n}[g_1(t_{n+1};\eta )1_{\tau<t_{n+1}}]^2\\&\quad +2(E_{t_n}[g_0(t_{n+1};\eta )(1-1_{\tau<t_{n+1}})g_1(t_{n+1};\eta )1_{\tau<t_{n+1}}]\\&\quad -E_{t_n}[g_0(t_{n+1};\eta )(1-1_{\tau<t_{n+1}})]E_{t_n}[g_1(t_{n+1};\eta )1_{\tau<t_{n+1}}])\\&=E_{t_n}[g_0(t_{n+1};\eta )^2(1-1_{\tau<t_{n+1}})]-E_{t_n}[g_0(t_{n+1};\eta )(1-1_{\tau<t_{n+1}})]^2\\&\quad +E_{t_n}[g_1(t_{n+1};\eta )^21_{\tau<t_{n+1}}]-E_{t_n}[g_1(t_{n+1};\eta )1_{\tau<t_{n+1}}]^2\\&\quad -2E_{t_n}[g_0(t_{n+1};\eta )(1-1_{\tau<t_{n+1}})]E_{t_n}[g_1(t_{n+1};\eta )1_{\tau <t_{n+1}}], \end{aligned} \end{aligned}$$

since $(1-1_{\tau<t_{n+1}})^2=1-1_{\tau <t_{n+1}}$, $(1_{\tau<t_{n+1}})^2=1_{\tau <t_{n+1}}$ and $(1-1_{\tau<t_{n+1}})\times 1_{\tau <t_{n+1}}=0$.

For the constant intensity process $\lambda$ defined above, each term in (3.5) is calculated as follows.

$$\begin{aligned} E_{t_n}[g(t_{n+1};\eta )]= & {} E_{t_n}[g_0(t_{n+1};\eta )1_{\tau> t_{n+1}}]+E_{t_n}[g_1(t_{n+1};\eta )(1-1_{\tau> t_{n+1}})]\\= & {} E_{t_n}[E[g_0(t_{n+1};\eta )1_{\tau> t_{n+1}}|{\mathcal {F}}^W_{t_{n+1}}]]\\&\quad +E_{t_n}[E[g_1(t_{n+1};\eta )(1-1_{\tau> t_{n+1}})|{\mathcal {F}}^W_{t_{n+1}}]]\\= & {} E_{t_n}[g_0(t_{n+1};\eta )E[1_{\tau> t_{n+1}}|{\mathcal {F}}^W_{t_{n+1}}]]\\&\quad +E_{t_n}[g_1(t_{n+1};\eta )E[(1-1_{\tau> t_{n+1}})|{\mathcal {F}}^W_{t_{n+1}}]]\\= & {} E_{t_n}[e^{-\lambda \Delta t}g_0(t_{n+1};\eta )]+E_{t_n}[(1-e^{-\lambda \Delta t})g_1(t_{n+1};\eta )]\\= & {} e^{-\lambda \Delta t}E_{t_n}[g_0(t_{n+1};\eta )]+(1-e^{-\lambda \Delta t})E_{t_n}[g_1(t_{n+1};\eta )],\\ Cov_{t_n}[B_{t_{n+1}},g(t_{n+1};\eta )]= & {} E_{t_n}[B_Tg_0(t_{n+1};\eta )1_{\tau>t_{n+1}}]-E_{t_n}[B_{t_{n+1}}]E_{t_n}[g_0(t_{n+1};\eta )1_{\tau>t_{n+1}}]\\&\quad +E_{t_n}[B_{t_{n+1}}g_1(t_{n+1};\eta )(1-1_{\tau>t_{n+1}})]\\&\quad -E_{t_n}[B_{t_{n+1}}]E_{t_n}[g_1(t_{n+1};\eta )(1-1_{\tau>t_{n+1}})]\\= & {} E_{t_n}[e^{-\lambda \Delta t}B_{t_{n+1}}g_0(t_{n+1};\eta )]-E_{t_n}[B_{t_{n+1}}]E_{t_n}[e^{-\lambda \Delta t}g_0(t_{n+1};\eta )]\\&\quad +E_{t_n}[(1-e^{-\lambda \Delta t})B_{t_{n+1}}g_1(t_{n+1};\eta )]\\&\quad -E_{t_n}[B_{t_{n+1}}]E_{t_n}[(1-e^{-\lambda \Delta t})g_1(t_{n+1};\eta )]\\= & {} e^{-\lambda \Delta t}E_{t_n}[B_{t_{n+1}}g_0(t_{n+1};\eta )]-e^{-\lambda \Delta t}E_{t_n}[B_{t_{n+1}}]E_{t_n}[g_0(t_{n+1};\eta )]\\&\quad +(1-e^{-\lambda \Delta t})E_{t_n}[B_{t_{n+1}}g_1(t_{n+1};\eta )]\\&\quad -(1-e^{-\lambda \Delta t})E_{t_n}[B_{t_{n+1}}]E_{t_n}[g_1(t_{n+1};\eta )]\\= & {} e^{-\lambda \Delta t}\{E_{t_n}[B_{t_{n+1}}g_0(t_{n+1};\eta )]-E_{t_n}[B_{t_{n+1}}]E_{t_n}[g_0(t_{n+1};\eta )]\}\\&\quad +(1-e^{-\lambda \Delta t})\{E_{t_n}[B_{t_{n+1}}g_1(t_{n+1};\eta )]\\&\quad -E_{t_n}[B_{t_{n+1}}]E_{t_n}[g_1(t_{n+1};\eta )]\},\\ Var_{t_n}[g(t_{n+1};\eta )]= & {} E_{t_n}[g_0(t_{n+1};\eta )^21_{\tau>t_{n+1}}]-E_{t_n}[g_0(t_{n+1};\eta )1_{\tau>t_{n+1}}]^2\\&\quad +E_{t_n}[g_1(t_{n+1};\eta )^2(1-1_{\tau>t_{n+1}})]\\&\quad -E_{t_n}[g_1(t_{n+1};\eta )(1-1_{\tau>t_{n+1}})]^2\\&\quad -2E_{t_n}[g_0(t_{n+1};\eta )1_{\tau>t_{n+1}}]E_{t_n}[g_1(t_{n+1};\eta )(1-1_{\tau >t_{n+1}})]\\= & {} e^{-\lambda \Delta t}E_{t_n}[g_0(t_{n+1};\eta )^2]-e^{-\lambda \Delta t}E_{t_n}[g_0(t_{n+1};\eta )]^2\\&\quad +(1-e^{-\lambda \Delta t})E_{t_n}[g_1(t_{n+1};\eta )^2]-(1-e^{-\lambda \Delta t})E_{t_n}[g_1(t_{n+1};\eta )]^2\\&\quad -2e^{-\lambda \Delta t}(1-e^{-\lambda \Delta t})E_{t_n}[g_0(t_{n+1};\eta )]E_{t_n}[g_1(t_{n+1};\eta )]\\= & {} e^{-\lambda \Delta t}\{E_{t_n}[g_0(t_{n+1};\eta )^2]-E_{t_n}[g_0(t_{n+1};\eta )]^2\}\\&\quad +(1-e^{-\lambda \Delta t})\{E_{t_n}[g_1(t_{n+1};\eta )^2]-E_{t_n}[g_1(t_{n+1};\eta )]^2\}\\&\quad -2e^{-\lambda \Delta t}(1-e^{-\lambda \Delta t})E_{t_n}[g_0(t_{n+1};\eta )]E_{t_n}[g_1(t_{n+1};\eta )]. \end{aligned}$$

Closed formula for collateralized option price

We derive a closed formula for the collateralized option price by following Kim (2002).

At first, under measure Q given by (4.1), each stochastic process is represented as

$$\begin{aligned} \begin{aligned}&\frac{dB_t}{B_t}=r_tdt+\sigma _Bd{\tilde{W}}_{1t}, \\&dr_t=\kappa _r(a_r-r_t)dt+b_r{(\zeta _{21}d{\tilde{W}}_{1t}+\zeta _{22}d{\tilde{W}}_{2t})}, \\&dr_t^c=\kappa _c(a_c-r_t^c)dt+b_c{(\zeta _{31}d{\tilde{W}}_{1t}+\zeta _{32}d{\tilde{W}}_{2t}+\zeta _{33}d{\tilde{W}}_{3t})}, \\&dr_t^p=\kappa _p(a_p-r_t^p)dt+b_p{(\zeta _{41}d{\tilde{W}}_{1t}+\zeta _{42}d{\tilde{W}}_{2t}+\zeta _{43}d{\tilde{W}}_{3t}+\zeta _{44}d{\tilde{W}}_{4t})}\\ \end{aligned} \end{aligned}$$

where ${\tilde{W}}=({\tilde{W}}_1,{\tilde{W}}_2,\ldots ,{\tilde{W}}_4)$ is a four-dimensional standard Brownian motion under measure Q.

We set

$$\begin{aligned} Z_{s,T}=e^{-\int _s^T(r_t-y_t)dt}(B_T-K). \end{aligned}$$

Then, for the dynamics introduced in Sect. 4.1, we have

$$\begin{aligned} \begin{aligned} Z_{s,T}&=e^{-\int _s^T(r_t-y_t)dt}\left( B_se^{\int _s^Tr_tdt-\frac{1}{2}\sigma _B^2(T-s)+\int _s^T\sigma _Bd{\tilde{W}}_{1t}}-K \right) \\&=B_se^{\int _s^Ty_tdt-\frac{1}{2}\sigma _B^2(T-s)+\int _s^T\sigma _Bd{\tilde{W}}_{1t}}-e^{-\int _s^T(r_t-y_t)dt}K. \end{aligned} \end{aligned}$$

Define

$$\begin{aligned} I_r(s,t)&=\int _s^tr_udu\\&=(t-s){\bar{r}}+\frac{1}{\kappa }(1-e^{-\kappa (t-s)})(r_s-{\bar{r}})\\&\quad +\frac{b}{\kappa }\left( \zeta _{21}\int _s^t(1-e^{-\kappa (t-u)})d{\tilde{W}}_{1u}+\zeta _{22}\int _s^t(1-e^{-\kappa (t-u)})d{\tilde{W}}_{2u} \right) ,\\ I_c(s,t)&=\int _s^tr_u^cdu\\&=(t-s){\bar{r}}^c+\frac{1}{\kappa _c}(1-e^{-\kappa _c(t-s)})(r_s^c-{\bar{r}}^c)\\&\quad +\frac{b_c}{\kappa _c}\left( \zeta _{31}\int _s^t(1-e^{-\kappa _c(t-u)})d{\tilde{W}}_{1u}+\zeta _{32}\int _s^t(1-e^{-\kappa _c(t-u)})d{\tilde{W}}_{2u}\right. \\&\quad \left. +\zeta _{33}\int _s^t(1-e^{-\kappa _c(t-u)})d{\tilde{W}}_{3u} \right) ,\\ I_p(s,t)&=\int _s^tr_u^pdu\\&=(t-s){\bar{r}}^p+\frac{1}{\kappa _p}(1-e^{-\kappa _p(t-s)})(r_s^p-{\bar{r}}^p)\\&\quad +\frac{b_p}{\kappa _p}\biggl (\zeta _{41}\int _s^t(1-e^{-\kappa _p(t-u)})d{\tilde{W}}_{1u}+\zeta _{42}\int _s^t(1-e^{-\kappa _p(t-u)})d{\tilde{W}}_{2u}\\&\quad +\zeta _{43}\int _s^t(1-e^{-\kappa _p(t-u)})d{\tilde{W}}_{3u}+\zeta _{44}\int _s^t(1-e^{-\kappa _p(t-u)})d{\tilde{W}}_{4u} \biggr ),\\ M_1(s,t)&=m(s,t)-\eta m_c(s,t)-(1-\eta ) m_p(s,t),\\ M_2(s,t)&=-\eta m_c(s,t)-(1-\eta ) m_p(s,t)\\ X_{1}(s,t)&=\frac{b}{\kappa }\left( \zeta _{21}\int _s^t(1-e^{-\kappa (t-u)})d{\tilde{W}}_{1u}+\zeta _{22}\int _s^t(1-e^{-\kappa (t-u)})d{\tilde{W}}_{2u} \right) \\&\quad -\eta \frac{b_c}{\kappa _c}\left( \zeta _{31}\int _s^t(1-e^{-\kappa _c(t-u)})d{\tilde{W}}_{1u}+\zeta _{32}\int _s^t(1-e^{-\kappa _c(t-u)})d{\tilde{W}}_{2u}\right. \\&\quad \left. +\zeta _{33}\int _s^t(1-e^{-\kappa _c(t-u)})d{\tilde{W}}_{3u} \right) \\&\quad -(1\!-\!\eta )\frac{b_p}{\kappa _p}\biggl (\zeta _{41}\int _s^t(1-e^{-\kappa _p(t-u)})d{\tilde{W}}_{1u}+\zeta _{42}\int _s^t(1-e^{-\kappa _p(t-u)})d{\tilde{W}}_{2u}\\&\quad +\zeta _{43}\int _s^t(1-e^{-\kappa _p(t-u)})d{\tilde{W}}_{3u}+\zeta _{44}\int _s^t(1-e^{-\kappa _p(t-u)})d{\tilde{W}}_{4u} \biggr )\\&\quad +\int _s^t\sigma _Bd{\tilde{W}}_{1u},\\ X_{2}(s,t)&=-\eta \frac{b_c}{\kappa _c}\left( \zeta _{31}\int _s^t(1-e^{-\kappa _c(t-u)})d{\tilde{W}}_{1u}+\zeta _{32}\int _s^t(1-e^{-\kappa _c(t-u)})d{\tilde{W}}_{2u}\right. \\&\quad \left. +\zeta _{33}\int _s^t(1-e^{-\kappa _c(t-u)})d{\tilde{W}}_{3u} \right) \\&\quad -(1-\eta )\frac{b_p}{\kappa _p}\biggl (\zeta _{41}\int _s^t(1-e^{-\kappa _p(t-u)})d{\tilde{W}}_{1u}+\zeta _{42}\int _s^t(1\!-\!e^{-\kappa _p(t-u)})d{\tilde{W}}_{2u}\\&\quad +\zeta _{43}\int _s^t(1-e^{-\kappa _p(t-u)})d{\tilde{W}}_{3u}+\zeta _{44}\int _s^t(1-e^{-\kappa _p(t-u)})d{\tilde{W}}_{4u} \biggr ),\\ \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} m(s,t):&=E_s^Q[I_r(s,t)]=(t-s){\bar{r}}+\frac{1}{\kappa }(1-e^{-\kappa (t-s)})(r_s-{\bar{r}}), \\ m_c(s,t):&=E_s^Q[I_c(s,t)]=(t-s){\bar{r}}^c+\frac{1}{\kappa _c}(1-e^{-\kappa _c(t-s)})(r_s^c-{\bar{r}}^c), \\ m_p(s,t):&=E_s^Q[I_p(s,t)]=(t-s){\bar{r}}^p+\frac{1}{\kappa _p}(1-e^{-\kappa _p(t-s)})(r_s^p-{\bar{r}}^p), \\ \end{aligned} \end{aligned}$$

then $Z_{s,T}$ is rewritten as

$$\begin{aligned} Z_{s,T}=B_se^{M_1(s,T)-\frac{1}{2}\sigma _B^2(T-s)+X_{1}(s,T)}-Ke^{M_2(s,T)+X_{2}(s,T)}. \end{aligned}$$

Since ${\tilde{W}}=({\tilde{W}}_1,{\tilde{W}}_2,{\tilde{W}}_3,{\tilde{W}}_4)$ is the four-dimensional standard Brownian motion, $X_1$ and $X_2$ respectively follow the normal distribution. Thus, $X=(X_1,X_2)$ follows the two-dimensional normal distribution;

$$\begin{aligned} (X_1(s,t),X_2(s,t))^\top \sim {\mathcal {N}}_2(\mu ^{s,t},\Sigma ^{s,t}) \end{aligned}$$

where the mean vector is $\mu ^{s,t}\equiv (0,0)^\top$ ($\forall t>s$) and the covariance matrix is

$$\begin{aligned} \Sigma ^{s,t}= & {} \left( \begin{array}{cc} \Sigma _{11}^{s,t} &{} \Sigma _{12}^{s,t} \\ \Sigma _{21}^{s,t} &{} \Sigma _{22}^{s,t} \\ \end{array} \right) , \\ \Sigma _{11}^{s,t}= & {} v(s,t)+\eta ^2v_c(s,t)+(1-\eta )^2v_p(s,t)+\sigma _B^2(t-s)\\&\quad -2\eta u_{rc}(s,t)-2(1-\eta )u_{rp}(s,t)\\&\quad +2\sigma _B\left( (t-s)-\frac{1}{\kappa }(1-e^{-\kappa (t-s)})\right) +2\eta (1-\eta )u_{cp}(s,t)\\&\quad -2\sigma _B\eta \left( (t-s)-\frac{1}{\kappa _c}(1-e^{-\kappa _c (t-s)})\right) \\&\quad -2\sigma _B(1-\eta )\left( (t-s)-\frac{1}{\kappa _p}(1-e^{-\kappa _p (t-s)})\right) ,\\ \Sigma _{12}^{s,t}= & {} \Sigma _{21}^{s,t}\\= & {} -\eta u_{rc}(s,t)-(1-\eta )u_{rp}(s,t)+\eta ^2v_c(s,t)\\&\quad +2\eta (1-\eta )u_{cp}(s,t)+(1-\eta )^2v_p(s,t)\\&\quad -\eta \sigma _B \frac{b_c}{\kappa _c}\zeta _{31}\left( (t-s)-\frac{1}{\kappa _c}(1-e^{-\kappa _c(t-s)})\right) \\&\quad -(1-\eta )\sigma _B \frac{b_p}{\kappa _p}\zeta _{41}\left( (t-s)-\frac{1}{\kappa _p}(1-e^{-\kappa _p(t-s)})\right) ,\\ \Sigma _{22}^{s,t}= & {} \eta ^2v_c(s,t)+(1-\eta )^2v_p(s,t)+2\eta (1-\eta )u_{cp}(s,t), \\ v(s,t):= & {} Var_s^Q[I_r(s,t)]=\left( \frac{b}{\kappa }\right) ^2(\zeta _{21}^2+\zeta _{22}^2)\frac{1}{2\kappa }(4e^{-\kappa (t-s)}\\&\quad -e^{-2\kappa (t-s)}+2\kappa (t-s)-3), \\ v_c(s,t):= & {} Var_s^Q[I_c(s,t)]=\left( \frac{b_c}{\kappa _c}\right) ^2(\zeta _{31}^2+\zeta _{32}^2+\zeta _{33}^2)\frac{1}{2\kappa _c}(4e^{-\kappa _c(t-s)}\\&\quad -e^{-2\kappa _c(t-s)}+2\kappa _c(t-s)-3), \\ v_p(s,t):= & {} Var_s^Q[I_p(s,t)]=\left( \frac{b_p}{\kappa _p}\right) ^2(\zeta _{41}^2+\zeta _{42}^2+\zeta _{43}^2+\zeta _{44}^2)\frac{1}{2\kappa _p}(4e^{-\kappa _p(t-s)}\\&\quad -e^{-2\kappa _p(t-s)}+2\kappa _p(t-s)-3), \\ u_{rc}(s,t):= & {} Cov_s^Q[I_r(s,t),I_c(s,t)]\\= & {} \frac{b}{\kappa }\frac{b_c}{\kappa _c}(\zeta _{21}\zeta _{31}+\zeta _{22}\zeta _{32})\left( (t-s)-\frac{1}{\kappa }-\frac{1}{\kappa _c}+\frac{1}{\kappa +\kappa _c}+\frac{1}{\kappa }e^{-\kappa (t-s)}\right. \\&\quad \left. +\frac{1}{\kappa _c}e^{-\kappa _c(t-s)}-\frac{1}{\kappa +\kappa _c}e^{-(\kappa +\kappa _c)(t-s)}\right) , \\ u_{rp}(s,t):= & {} Cov_s^Q[I_r(s,t),I_p(s,t)]\\= & {} \frac{b}{\kappa }\frac{b_p}{\kappa _p}(\zeta _{21}\zeta _{41}+\zeta _{22}\zeta _{42})\left( (t-s)-\frac{1}{\kappa }-\frac{1}{\kappa _p}+\frac{1}{\kappa +\kappa _p}+\frac{1}{\kappa }e^{-\kappa (t-s)}\right. \\&\quad \left. +\frac{1}{\kappa _p}e^{-\kappa _p(t-s)}-\frac{1}{\kappa +\kappa _p}e^{-(\kappa +\kappa _p)(t-s)}\right) ,\\ u_{cp}(s,t):= & {} Cov_s^Q[I_c(s,t),I_p(s,t)]\\= & {} \frac{b_c}{\kappa _c}\frac{b_p}{\kappa _p}(\zeta _{31}\zeta _{41}+\zeta _{32}\zeta _{42}+\zeta _{33}\zeta _{43})\\&\times \left( (t-s)-\frac{1}{\kappa _c}-\frac{1}{\kappa _p}+\frac{1}{\kappa _c+\kappa _p}+\frac{1}{\kappa _c}e^{-\kappa _c(t-s)}+\frac{1}{\kappa _p}e^{-\kappa _p(t-s)}\right. \\&\quad \left. -\frac{1}{\kappa _c+\kappa _p}e^{-(\kappa _c+\kappa _p)(t-s)}\right) .\\ \end{aligned}$$

At this point, the inequality of $Z_{s,T}\ge 0$ is equivalent to

$$\begin{aligned} X_{1}(s,T)-X_{2}(s,T)\ge C(s,T), \end{aligned}$$

where

$$\begin{aligned} C(s,t)=-\ln \frac{B_s}{K}-M_1(s,t)+M_2(s,t)+\frac{1}{2}\sigma _B^2(t-s). \end{aligned}$$

Therefore, the time-s price of collateralized option $V_s$ with maturity T is given by

$$\begin{aligned} \begin{aligned} V_s&=E^Q_s[\max (Z_{s,T},0)]\\&=E^Q_s\left[ \left\{ B_se^{M_1(s,T)-\frac{1}{2}\sigma _{B}^2(T-s)+X_{1}(s,T)}-Ke^{M_2(s,T)+X_{2}(s,T)} \right\} 1_{Z_{s,T}\ge 0} \right] \\&=E^Q_s\left[ B_se^{M_1(s,T)-\frac{1}{2}\sigma _{B}^2(T-s)+X_{1}(s,T) }1_{X_{1}(s,T)-X_{2}(s,T)\ge C(s,T)}\right] \\&\quad -E_s^Q\left[ Ke^{M_2(s,T)+X_{2}(s,T)}1_{X_{1}(s,T)-X_{2}(s,T)\ge C(s,T)} \right] \\&=:\text {(I)}-\text {(II)}. \end{aligned} \end{aligned}$$

(D.1)

Terms (I) and (II) in (D.1) are solved as follows.

$$\begin{aligned} \begin{aligned} \text {(I)}&=B_se^{-\frac{1}{2}\sigma _B^2(T-s)+M_1(s,T)}E_s^Q\left[ e^{X_{1}(s,T)}1_{X_{1}(s,T)-X_{2}(s,T)\ge C(s,T)}\right] \\&=B_se^{-\frac{1}{2}\sigma _B^2(T-s)+M_1(s,T)}\int \int _{x_1-x_2\ge C(s,T)}e^{x_1}\phi _2(x|\mu ^{s,T},\Sigma ^{s,T})dx_1dx_2\\&=B_se^{-\frac{1}{2}\sigma _B^2(T-s)+M_1(s,T)}\int \int _{(1,-1) x\ge C(s,T)}e^{(1,0) x}\phi _2(x|\mu ^{s,T},\Sigma ^{s,T})dx\\&=B_se^{-\frac{1}{2}\sigma _B^2(T-s)+M_1(s,T)}\exp \left( (1,0)\mu ^{s,T}+\frac{1}{2}(1,0)\Sigma ^{s,T} (1,0)^\top \right) \\&\times \Phi \left( \frac{(1,-1) (\mu ^{s,T}+\Sigma ^{s,T} (1,0)^\top )-C(s,T)}{\sqrt{(1,-1) \Sigma ^{s,T} (1,-1)^\top }} \right) \\&=B_se^{M_1(s,T)+\frac{1}{2}(\Sigma _{11}^{s,T}-\sigma _B^2(T-s))}\Phi \left( \frac{\Sigma _{11}^{s,T}-\Sigma _{12}^{s,T}-C(s,T)}{\sqrt{D(s,T)}} \right) \\&=B_se^{M_1(s,T)+\frac{1}{2}(\Sigma _{11}^{s,T}-\sigma _B^2(T-s))}\Phi (d_1(s,T)), \end{aligned}\nonumber \\ \end{aligned}$$

(D.2)

where x means $x=(x_1,x_2)^\top$, $\phi _2(\cdot |a,b)$ is the probability density function of the two-dimensional normal distribution with mean vector a and covariance matrix b, $\Phi (\cdot )$ is the cumulative distribution function of the standard normal distribution,

$$\begin{aligned} D(s,t)=\Sigma _{11}^{s,t}-2\Sigma _{12}^{s,t}+\Sigma _{22}^{s,t}, \end{aligned}$$

and

$$\begin{aligned} d_1(s,t)=\frac{\Sigma _{11}^{s,t}-\Sigma _{12}^{s,t}-C(s,t)}{\sqrt{D(s,t)}}. \end{aligned}$$

From the forth line to the fifth line in (D.2), we used Lemma 1 in Kim (2002).

$$\begin{aligned} \begin{aligned} \text {(II)}&=Ke^{M_2(s,T)}E_s^Q\left[ e^{X_{2}(s,T)}1_{X_{1}(s,T)-X_{2}(s,T)\ge C(s,T)} \right] \\&=Ke^{M_2(s,T)}\int \int _{(1,-1) x\ge C(s,T)}e^{(0,1) x}\phi _2(x|\mu ^{s,T},\Sigma ^{s,T})dx\\&=Ke^{M_2(s,T)}\exp \left( (0,1)\mu ^{s,T}+\frac{1}{2}(0,1)\Sigma ^{s,T} (0,1)^\top \right) \\&\times \Phi \left( \frac{(1,-1)(\mu ^{s,T}+\Sigma ^{s,T} (0,1)^\top ) -C(s,T)}{\sqrt{(1,-1)\Sigma ^{s,T} (1,-1)^\top }} \right) \\&=Ke^{M_2(s,T)+\frac{1}{2}\Sigma _{22}^{s,T}}\Phi (d_2(s,T))\\ \end{aligned} \end{aligned}$$

where

$$\begin{aligned} d_2(s,t)=d_1(s,t)-\sqrt{D(s,t)}=\frac{\Sigma _{12}^{s,t}-\Sigma _{22}^{s,t}-C(s,t)}{\sqrt{D(s,t)}}. \end{aligned}$$

Closed formula for collateralized forward valuation

The time-s value $V_s$ of the forward contract maturing at time T given the forward price $F_B$ of $B_T$ is solved as follows.

$$\begin{aligned} \begin{aligned} V_s&=E_s^Q\left[ e^{-\int _s^T(r_t-y_t)dt}B_T\right] -E_s^Q\left[ e^{-\int _s^T(r_t-y_t)dt}F_B\right] \\&=E^Q_s\left[ B_se^{M_1(s,T)-\frac{1}{2}\sigma _{B}^2(T-s)+X_{1}(s,T)}\right] -E^Q_s\left[ e^{M_2(s,T)+X_{2}(s,T)}F_B\right] \\&=B_se^{M_1(s,T)-\frac{1}{2}\sigma _{B}^2(T-s)}E^Q_s\left[ e^{X_{1}(s,T)}\right] -F_Be^{M_2(s,T)}E^Q_s\left[ e^{X_{2}(s,T)}\right] \\&=B_se^{M_1(s,T)+\frac{1}{2}(\Sigma _{11}^{s,T}-\sigma _{B}^2(T-s))}-F_Be^{M_2(s,T)+\frac{1}{2}\Sigma _{22}^{s,T}}, \end{aligned} \end{aligned}$$

where $M_j(s,T)$ and $\Sigma _{jj}^{s,t}$ ($j=1,2$) are given in Appendix D.

Since the forward price $F_B$ is given as a solution of

$$\begin{aligned} V_0=0, \end{aligned}$$

$F_B$, therefore, is given by

$$\begin{aligned} F_B=B_0e^{M_1(0,T)-M_2(0,T)+\frac{1}{2}(\Sigma _{11}^{0,T}-\Sigma _{22}^{0,T}-\sigma _{B}^2T)}. \end{aligned}$$