**2.1. Description of the Liquidity Coverage Ratio Model**

Before the 2007-2009 financial crisis, banks were prosperous with high liquidity provisioning rates, low interest rates and soaring cash outflows. This was followed by the collapse of the housing market, exploding default rates and the effects thereafter. The LCR was developed to promote short-term resilience of a bank's liquidity risk profile. This standard aims to ensure that a bank has an adequate stock of unencumbered HQLAs that consist of cash or assets that can be converted into cash at little or no loss of value in private markets to meet its liquidity needs for a 30 calendar day liquidity stress scenario (see, for instance, [1], [3] and [4]). In order to make our analysis tractable, we make the following assumption about our LCR model.

**Assumption 2.1. (Filtered Probability Space and Time Index)** *Assume that we have a filtered probability space* (Ω, F, **P**) *with filtration* {F*t*}*t*≥<sup>0</sup> *on a time index set T* = [*t*0, *t*1].

Subsequently, we study a system of stochastic differential equations (SDEs) that value HQLAs at time *<sup>t</sup>* as *<sup>x</sup>*<sup>1</sup> : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> (compare with [10]). Here, HQLAs, *<sup>x</sup>*<sup>1</sup> *<sup>t</sup>* , are stochastic because they are dependent on the stochastic rates of return on L1As and L2As (see [12] for more details). Also, NCOs at time *t*, *x*<sup>2</sup> *<sup>t</sup>* , with *<sup>x</sup>*<sup>2</sup> : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> are stochastic because their value has a reliance on random cash in- and outflows as well as liquidity provisioning. Furthermore, for *<sup>x</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup><sup>2</sup> we use the notation *xt* to denote

$$\mathbf{x}\_{t} = \begin{bmatrix} \mathbf{x}\_{t}^{1} \\ \mathbf{x}\_{t}^{2} \end{bmatrix}$$

and present the LCR, *<sup>l</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup>+, by

$$\mathbf{x}\_t = \mathbf{x}\_t^1 / \mathbf{x}\_t^2 = \mathbf{x}\_t^1 . (\mathbf{x}\_t^2)^{-1}. \tag{2}$$

It is important for banks that *lt* in (2) has to be sufficiently high to ensure high LCRs. In fact, as was mentioned before, Basel III sets the minimum value of the LCR at 1. Obviously, low values of *lt* indicate that the bank has decreased liquidity and is at high risk of causing a credit crunch (see, for instance, [13]).

does not move below an acceptable level. The control theoretic liquidity problem is to meet LCR targets with as little additional liquidity provisioning (essentially corresponding to cash injections in our chapter) as possible and optimal HQLA allocation. Section 5 provides conclusions and future directions. As was mentioned before, the additional provisioning may arise from an inflow of cash injections. We choose examples to illustrate that Basel III liquidity regulation resulted from both problematic liquidity structures and unexpected cash

In this section, we model HQLAs, NCOs and LCR in a stochastic framework by following [10] very closely. This is important for solving the optimal LCR control problem outlined

Before the 2007-2009 financial crisis, banks were prosperous with high liquidity provisioning rates, low interest rates and soaring cash outflows. This was followed by the collapse of the housing market, exploding default rates and the effects thereafter. The LCR was developed to promote short-term resilience of a bank's liquidity risk profile. This standard aims to ensure that a bank has an adequate stock of unencumbered HQLAs that consist of cash or assets that can be converted into cash at little or no loss of value in private markets to meet its liquidity needs for a 30 calendar day liquidity stress scenario (see, for instance, [1], [3] and [4]). In order to make our analysis tractable, we make the following assumption about our

**Assumption 2.1. (Filtered Probability Space and Time Index)** *Assume that we have a filtered*

Subsequently, we study a system of stochastic differential equations (SDEs) that value

because they are dependent on the stochastic rates of return on L1As and L2As (see [12]

their value has a reliance on random cash in- and outflows as well as liquidity provisioning.

*xt* = *x*1 *t x*2 *t* 

*lt* = *x*<sup>1</sup> *<sup>t</sup>* /*x*<sup>2</sup> *<sup>t</sup>* = *<sup>x</sup>*<sup>1</sup> *<sup>t</sup>* .(*x*<sup>2</sup> *<sup>t</sup>* , are stochastic

*<sup>t</sup>* , with *<sup>x</sup>*<sup>2</sup> : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> are stochastic because

*<sup>t</sup>*)<sup>−</sup>1. (2)

*probability space* (Ω, F, **P**) *with filtration* {F*t*}*t*≥<sup>0</sup> *on a time index set T* = [*t*0, *t*1].

HQLAs at time *<sup>t</sup>* as *<sup>x</sup>*<sup>1</sup> : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> (compare with [10]). Here, HQLAs, *<sup>x</sup>*<sup>1</sup>

Furthermore, for *<sup>x</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup><sup>2</sup> we use the notation *xt* to denote

outflows (see Section 5 for more details).

subsequently in Section 3.

LCR model.

**2. A Liquidity Coverage Ratio Model**

70 Dynamic Programming and Bayesian Inference, Concepts and Applications

for more details). Also, NCOs at time *t*, *x*<sup>2</sup>

and present the LCR, *<sup>l</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup>+, by

**2.1. Description of the Liquidity Coverage Ratio Model**

Bank liquidity has a heavy reliance on liquidity provisioning rates. This rate should be reduced for high LCRs and increased beyond the normal rate when bank LCRs are low. In the sequel, the stochastic process *<sup>u</sup>*<sup>1</sup> : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> is the *normal rate of liquidity provisioning per monetary unit of the bank's NCOs* whose value at time *t* is denoted by *u*<sup>1</sup> *<sup>t</sup>* . In this case, *<sup>u</sup>*<sup>1</sup> *<sup>t</sup> dt* is the normal liquidity provisioning rate per unit of the bank's NCOs over the time period (*t*, *t* + *dt*). A related concept is the *adjustment to the rate of liquidity provisioning per monetary unit of the bank's NCOs for surplus or deficit*, *<sup>u</sup>*<sup>2</sup> : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup>+, that depends on the LCR. In the case of liquidity deficit, during stress scenarios, this adjustment rate can correspond to a cash injection rate. Here the amount of surplus or deficit is reliant on the excess of HQLAs over NCOs. We denote the sum of *<sup>u</sup>*<sup>1</sup> and *<sup>u</sup>*<sup>2</sup> by the *liquidity provisioning rate u*<sup>3</sup> : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup>+, i.e.,

$$
\mu\_t^3 = \mu\_t^1 + \mu\_{t'}^2 \text{ for all } t. \tag{3}
$$

The following assumption is made in order to model the LCR in a stochastic framework (compare with [10]).

**Assumption 2.2. (Liquidity Provisioning Rate)** *The liquidity provisioning rate, u*3, *is predictable with respect to* {F*t*}*t*≥<sup>0</sup> *and provides us with a means of controlling bank LCR dynamics (see (3) for more details).*

The closed loop system will be defined such that Assumption 2.2 is met, as we shall see in the sequel. In times of deficit, for (3), we should choose the cash injection rate, *u*2, sufficiently large in order to guarantee bank liquidity. In reality, cash injections are subject to more stringent conditions (see, also, [13]).

Before and during the financial crisis, the LCR decreased significantly as extensive cash outflows took place with a consequent rising of NCOs. By contrast, banks predicted continued growth in the financial markets (see, for instance, [14]). The dynamics of the *outflows per monetary unit of the bank's NCOs*, *e* : Ω × *T* → R, is given by

$$
\delta\_t de\_t = r\_t^\varepsilon dt + \sigma\_t^\varepsilon dW\_t^\varepsilon, \ e(t\_0) = e\_0. \tag{4}
$$

where *et* is the outflows per NCO monetary unit, *<sup>r</sup><sup>e</sup>* : *<sup>T</sup>* <sup>→</sup> R is the rate of outflows per monetary unit of the bank's NCOs, the scalar *<sup>σ</sup><sup>e</sup>* : *<sup>T</sup>* <sup>→</sup> R, is the volatility in the outflows per NCO unit and *<sup>W</sup><sup>e</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> R is standard Brownian motion (compare with [10]). Moreover, we consider

$$dh\_t = r\_t^h dt + \sigma\_t^h dW\_t^h \; \; h(t\_0) = h\_{0\prime} \tag{5}$$

where the stochastic processes *<sup>h</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> is the *investment return on bank HQLAs per monetary unit of HQLAs*, *<sup>r</sup><sup>h</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> is the rate of HQLA return per HQLA unit, the scalar *<sup>σ</sup><sup>h</sup>* : *<sup>T</sup>* <sup>→</sup> R, is the volatility in the rate of HQLA returns and *<sup>W</sup><sup>h</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> R is standard Brownian motion. Before the 2007-2009 financial crisis, riskier HQLA returns were much higher than those of riskless reserves, making the former a more attractive but much riskier investment (see, also, [13]). During and after the crisis, this tendency reversed.

**Assumption 2.3. (HQLA Classes)** *Suppose from the outset that bank HQLAs can be classified into n* + 1 *asset classes. One of these HQLAs is risk free (like Central Bank reserves) while the HQLAs* 1, 2, . . . , *n have some risk associated with them.*

Riskier HQLAs evolve continuously in time and are modeled using a *n*-dimensional Brownian motion. In this multidimensional context, the *investment returns on bank HQLAs in the k-th HQLA per monetary unit of the k-th HQLA* is denoted by *y<sup>k</sup> <sup>t</sup>* , *k* ∈ N*<sup>n</sup>* = {0, 1, 2, . . . , *n*} where *<sup>y</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup>*n*<sup>+</sup>1. Thus, the return per HQLA unit may be given by

$$y = (\mathbf{R}(t), y\_{t'}^1, \dots, y\_t^n)\_{\prime}$$

where R(*t*) represents the return on reserves and *y*<sup>1</sup> *<sup>t</sup>* , ..., *<sup>y</sup><sup>n</sup> <sup>t</sup>* represent riskier HQLA returns. Furthermore, we can model *y* as

$$dy\_t = r\_t^y dt + \Sigma\_t^y dW\_t^y \quad y(t\_0) = y\_{0\prime} \tag{6}$$

*r*

<sup>R</sup>(*t*) = *ry*<sup>0</sup>

*π<sup>t</sup>* = (*π*<sup>0</sup>

*<sup>C</sup><sup>t</sup>* <sup>=</sup> <sup>Σ</sup>*<sup>y</sup> t* Σ *y t T*

*<sup>t</sup>* <sup>=</sup> *<sup>π</sup><sup>T</sup> t* Σ*y <sup>t</sup> dW<sup>y</sup> t* ,

*dht* = [*r*

*r y <sup>t</sup>* = (*r*

Σ*y <sup>t</sup>* =

*πT t r y <sup>t</sup>* <sup>=</sup> *<sup>π</sup>*<sup>0</sup> *t r*

*πT <sup>t</sup>* <sup>Σ</sup>*<sup>y</sup> <sup>t</sup> dW<sup>y</sup>*

Then, we set

(*t*), *r*

<sup>R</sup>(*t*), *r y t T* + *r*

*<sup>t</sup>* , *<sup>π</sup><sup>T</sup>*

 0 ... 0 Σ*y t*

*<sup>t</sup>* )*<sup>T</sup>* = (*π*<sup>0</sup>

 , <sup>Σ</sup>*<sup>y</sup>*

<sup>R</sup>(*t*) + *<sup>π</sup>jT*

<sup>R</sup>(*t*) + *<sup>π</sup><sup>T</sup>*

*t r y <sup>t</sup>* <sup>+</sup> *<sup>π</sup>jT t r*

*t r y <sup>t</sup>* ]*dt* <sup>+</sup> *<sup>π</sup><sup>T</sup>*

*dit* = *r<sup>i</sup>*

*<sup>t</sup> dht* + *<sup>x</sup>*<sup>2</sup>

<sup>R</sup>(*t*)*x*<sup>1</sup>

*<sup>t</sup> dit* <sup>−</sup> *<sup>x</sup>*<sup>2</sup>

+[*x*<sup>1</sup> *t πT t* Σ*y <sup>t</sup> dW<sup>y</sup> <sup>t</sup>* <sup>−</sup> *<sup>x</sup>*<sup>2</sup> *t σe dW<sup>e</sup> t* ],

*t u*3 *<sup>t</sup> dt* <sup>−</sup> *<sup>x</sup>*<sup>2</sup>

*<sup>t</sup> det*

*dW<sup>i</sup> <sup>t</sup>* ] <sup>−</sup> *<sup>x</sup>*<sup>2</sup> *<sup>t</sup>* [*r<sup>e</sup>*

*<sup>t</sup>* ]*dt* + *<sup>x</sup>*<sup>2</sup>

*<sup>t</sup>* [*σ<sup>i</sup> dW<sup>i</sup> <sup>t</sup>* <sup>−</sup> *<sup>σ</sup><sup>e</sup>*

The SDEs (9) and (10) may be rewritten into matrix-vector form in the following way

*<sup>t</sup>* + *<sup>x</sup>*<sup>1</sup> *t πT t r y <sup>t</sup>* <sup>+</sup> *<sup>x</sup>*<sup>2</sup> *t u*1 *<sup>t</sup>* + *<sup>x</sup>*<sup>2</sup> *t u*2 *<sup>t</sup>* <sup>−</sup> *<sup>x</sup>*<sup>2</sup> *tre*

place or instability in the value of pre-existing NCOs.

*dx*<sup>1</sup> *<sup>t</sup>* = *<sup>x</sup>*<sup>1</sup>

*dx*<sup>2</sup> *<sup>t</sup>* = *<sup>x</sup>*<sup>2</sup>

(compare with [10]).

we derive models for HQLAs, *x*1, and NCOs, *x*2, given by

= [*r*

= *x*<sup>2</sup> *t* [*ri tdt* + *<sup>σ</sup><sup>i</sup>*

= *x*<sup>2</sup> *t* [*ri <sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>e</sup>*

*tdt* + *<sup>σ</sup><sup>i</sup>*

*<sup>t</sup>* , *<sup>π</sup>*<sup>1</sup>

*<sup>t</sup>* <sup>∈</sup> <sup>R</sup>*n*×*n*,

<sup>R</sup> : *<sup>T</sup>* <sup>→</sup> <sup>R</sup>+, the rate of return on riskless assets,

*<sup>t</sup>*)*T*, *<sup>π</sup>* : *<sup>T</sup>* <sup>→</sup> <sup>R</sup>*k*,

. Then, we have that (7)

<sup>R</sup>(*t*) + *<sup>π</sup><sup>T</sup>*

*<sup>t</sup>* , *h*(*t*0) = *h*0.

*t r y t* ,

Optimizing Basel III Liquidity Coverage Ratios

http://dx.doi.org/10.5772/58395

73

*<sup>t</sup>*, *i*(*t*0) = *i*0. (8)

*<sup>t</sup>* ]*dt* (9)

*<sup>t</sup>* ] (10)

<sup>R</sup>(*t*)1*n*)*T*, *<sup>r</sup><sup>y</sup>* : *<sup>T</sup>* <sup>→</sup> <sup>R</sup>*n*,

<sup>R</sup>(*t*)1*<sup>n</sup>* = *r*

*<sup>t</sup>* ,..., *<sup>π</sup><sup>k</sup>*

*t* Σ*y <sup>t</sup> dW<sup>y</sup>*

Next, we take *<sup>i</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> as the *increase of NCOs before outflows per monetary unit of NCOs*, *<sup>r</sup><sup>i</sup>* : *<sup>T</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> is the rate of NCO increase before outflows per monetary unit of NCOs, the scalar *<sup>σ</sup><sup>i</sup>* <sup>∈</sup> R is the volatility in the NCO increase before outflows per monetary unit of NCOs and *<sup>W</sup><sup>i</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> R represents standard Brownian motion (compare with [10]).

*dW<sup>i</sup>*

The stochastic process *it* in (8) may typically originate from NCOs that have recently taken

Next, we develop a simple stochastic model that replaces a more realistic system that emphasizes features that are specific to our particular study (see, also, [13]). In our situation,

*<sup>t</sup> det*

*<sup>t</sup> dt* + *<sup>σ</sup><sup>e</sup>*

*dW<sup>e</sup>*

*dW<sup>e</sup> t* ].

where *<sup>r</sup><sup>y</sup>* : *<sup>T</sup>* <sup>→</sup> <sup>R</sup>*n*+<sup>1</sup> denotes the rate of asset returns, <sup>Σ</sup>*<sup>y</sup> <sup>t</sup>* <sup>∈</sup> <sup>R</sup>(*n*+1)×*<sup>n</sup>* is a covariance matrix of HQLA returns and *<sup>W</sup><sup>y</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup>* is standard Brownian. We assume that the investment strategy *<sup>π</sup>* : *<sup>T</sup>* <sup>→</sup> <sup>R</sup>*n*+<sup>1</sup> is outside the simplex

$$S = \{ \pi \in \mathbb{R}^{n+1} : \pi = (\pi^0, \dots, \pi^n)^T, \pi^0 + \dots + \pi^n = 1, \ \pi^0 \ge 0, \dots \pi^n \ge 0 \}.$$

In this case, short selling is possible. The *investment return on bank HQLAs* is then *h* : Ω × *R* → R+, where the dynamics of *h* can be written as

$$dh\_t = \pi\_t^T dy\_t = \pi\_t^T r\_t^y dt + \pi\_t^T \Sigma\_t^y dW\_t^y \dots$$

This notation can be simplified as follows. We denote

*r* <sup>R</sup>(*t*) = *ry*<sup>0</sup> (*t*), *r* <sup>R</sup> : *<sup>T</sup>* <sup>→</sup> <sup>R</sup>+, the rate of return on riskless assets, *r y <sup>t</sup>* = (*r* <sup>R</sup>(*t*), *r y t T* + *r* <sup>R</sup>(*t*)1*n*)*T*, *<sup>r</sup><sup>y</sup>* : *<sup>T</sup>* <sup>→</sup> <sup>R</sup>*n*, *π<sup>t</sup>* = (*π*<sup>0</sup> *<sup>t</sup>* , *<sup>π</sup><sup>T</sup> <sup>t</sup>* )*<sup>T</sup>* = (*π*<sup>0</sup> *<sup>t</sup>* , *<sup>π</sup>*<sup>1</sup> *<sup>t</sup>* ,..., *<sup>π</sup><sup>k</sup> <sup>t</sup>*)*T*, *<sup>π</sup>* : *<sup>T</sup>* <sup>→</sup> <sup>R</sup>*k*, Σ*y <sup>t</sup>* = 0 ... 0 Σ*y t* , <sup>Σ</sup>*<sup>y</sup> <sup>t</sup>* <sup>∈</sup> <sup>R</sup>*n*×*n*, *<sup>C</sup><sup>t</sup>* <sup>=</sup> <sup>Σ</sup>*<sup>y</sup> t* Σ *y t T* . Then, we have that (7) *πT t r y <sup>t</sup>* <sup>=</sup> *<sup>π</sup>*<sup>0</sup> *t r* <sup>R</sup>(*t*) + *<sup>π</sup>jT t r y <sup>t</sup>* <sup>+</sup> *<sup>π</sup>jT t r* <sup>R</sup>(*t*)1*<sup>n</sup>* = *r* <sup>R</sup>(*t*) + *<sup>π</sup><sup>T</sup> t r y t* , *πT <sup>t</sup>* <sup>Σ</sup>*<sup>y</sup> <sup>t</sup> dW<sup>y</sup> <sup>t</sup>* <sup>=</sup> *<sup>π</sup><sup>T</sup> t* Σ*y <sup>t</sup> dW<sup>y</sup> t* , *dht* = [*r* <sup>R</sup>(*t*) + *<sup>π</sup><sup>T</sup> t r y <sup>t</sup>* ]*dt* <sup>+</sup> *<sup>π</sup><sup>T</sup> t* Σ*y <sup>t</sup> dW<sup>y</sup> <sup>t</sup>* , *h*(*t*0) = *h*0.

where the stochastic processes *<sup>h</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> is the *investment return on bank HQLAs per monetary unit of HQLAs*, *<sup>r</sup><sup>h</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> is the rate of HQLA return per HQLA unit, the scalar *<sup>σ</sup><sup>h</sup>* : *<sup>T</sup>* <sup>→</sup> R, is the volatility in the rate of HQLA returns and *<sup>W</sup><sup>h</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> R is standard Brownian motion. Before the 2007-2009 financial crisis, riskier HQLA returns were much higher than those of riskless reserves, making the former a more attractive but much riskier

**Assumption 2.3. (HQLA Classes)** *Suppose from the outset that bank HQLAs can be classified into n* + 1 *asset classes. One of these HQLAs is risk free (like Central Bank reserves) while the HQLAs*

Riskier HQLAs evolve continuously in time and are modeled using a *n*-dimensional Brownian motion. In this multidimensional context, the *investment returns on bank HQLAs in*

*<sup>t</sup> dW<sup>y</sup>*

matrix of HQLA returns and *<sup>W</sup><sup>y</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup>* is standard Brownian. We assume that the

*<sup>S</sup>* <sup>=</sup> {*<sup>π</sup>* <sup>∈</sup> <sup>R</sup>*n*+<sup>1</sup> : *<sup>π</sup>* = (*π*0, ..., *<sup>π</sup>n*)*T*, *<sup>π</sup>*<sup>0</sup> <sup>+</sup> ... <sup>+</sup> *<sup>π</sup><sup>n</sup>* <sup>=</sup> 1, *<sup>π</sup>*<sup>0</sup> <sup>≥</sup> 0, . . . *<sup>π</sup><sup>n</sup>* <sup>≥</sup> <sup>0</sup>}.

In this case, short selling is possible. The *investment return on bank HQLAs* is then *h* : Ω × *R* →

*t r y <sup>t</sup> dt* <sup>+</sup> *<sup>π</sup><sup>T</sup>*

*<sup>t</sup>* <sup>Σ</sup>*<sup>y</sup> <sup>t</sup> dW<sup>y</sup> t* .

*<sup>t</sup> dyt* = *<sup>π</sup><sup>T</sup>*

*<sup>t</sup>* , ..., *<sup>y</sup><sup>n</sup>*

*t* ),

*<sup>t</sup>* , ..., *<sup>y</sup><sup>n</sup>*

*<sup>t</sup>* , *k* ∈ N*<sup>n</sup>* = {0, 1, 2, . . . , *n*}

*<sup>t</sup>* represent riskier HQLA returns.

*<sup>t</sup>* <sup>∈</sup> <sup>R</sup>(*n*+1)×*<sup>n</sup>* is a covariance

*<sup>t</sup>* , *y*(*t*0) = *y*0, (6)

investment (see, also, [13]). During and after the crisis, this tendency reversed.

*the k-th HQLA per monetary unit of the k-th HQLA* is denoted by *y<sup>k</sup>*

*dyt* = *r y <sup>t</sup> dt* <sup>+</sup> <sup>Σ</sup>*<sup>y</sup>*

where *<sup>r</sup><sup>y</sup>* : *<sup>T</sup>* <sup>→</sup> <sup>R</sup>*n*+<sup>1</sup> denotes the rate of asset returns, <sup>Σ</sup>*<sup>y</sup>*

*dht* = *π<sup>T</sup>*

investment strategy *<sup>π</sup>* : *<sup>T</sup>* <sup>→</sup> <sup>R</sup>*n*+<sup>1</sup> is outside the simplex

R+, where the dynamics of *h* can be written as

This notation can be simplified as follows. We denote

where *<sup>y</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup>*n*<sup>+</sup>1. Thus, the return per HQLA unit may be given by

*y* = (R(*t*), *y*<sup>1</sup>

1, 2, . . . , *n have some risk associated with them.*

72 Dynamic Programming and Bayesian Inference, Concepts and Applications

where R(*t*) represents the return on reserves and *y*<sup>1</sup>

Furthermore, we can model *y* as

Next, we take *<sup>i</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> as the *increase of NCOs before outflows per monetary unit of NCOs*, *<sup>r</sup><sup>i</sup>* : *<sup>T</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> is the rate of NCO increase before outflows per monetary unit of NCOs, the scalar *<sup>σ</sup><sup>i</sup>* <sup>∈</sup> R is the volatility in the NCO increase before outflows per monetary unit of NCOs and *<sup>W</sup><sup>i</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> R represents standard Brownian motion (compare with [10]). Then, we set

$$d\dot{\mathbf{u}}\_{t} = \mathbf{r}\_{t}^{\dot{\mathbf{t}}} \mathbf{d}t + \sigma^{\dot{\mathbf{t}}} d\mathcal{W}\_{t\prime}^{\dot{\mathbf{t}}} \quad \mathbf{i}(t\_{0}) = \mathbf{i}\_{0}. \tag{8}$$

The stochastic process *it* in (8) may typically originate from NCOs that have recently taken place or instability in the value of pre-existing NCOs.

Next, we develop a simple stochastic model that replaces a more realistic system that emphasizes features that are specific to our particular study (see, also, [13]). In our situation, we derive models for HQLAs, *x*1, and NCOs, *x*2, given by

$$\begin{split}d\mathbf{x}\_{t}^{1} &= \mathbf{x}\_{t}^{1}dh\_{t} + \mathbf{x}\_{t}^{2}\boldsymbol{u}\_{t}^{3}dt - \mathbf{x}\_{t}^{2}de\_{t} \\ &= [r^{\mathbf{R}}(t)\mathbf{x}\_{t}^{1} + \mathbf{x}\_{t}^{1}\tilde{\boldsymbol{\pi}}\_{t}^{T}\tilde{\boldsymbol{\tau}}\_{t}^{\vartheta} + \mathbf{x}\_{t}^{2}\boldsymbol{u}\_{t}^{1} + \mathbf{x}\_{t}^{2}\boldsymbol{u}\_{t}^{2} - \mathbf{x}\_{t}^{2}\boldsymbol{r}\_{t}^{\varepsilon}]dt \\ &+ [\mathbf{x}\_{t}^{1}\tilde{\boldsymbol{\pi}}\_{t}^{T}\tilde{\boldsymbol{\Sigma}}\_{t}^{\vartheta}d\boldsymbol{W}\_{t}^{\vartheta} - \mathbf{x}\_{t}^{2}\boldsymbol{\sigma}^{\varepsilon}d\boldsymbol{W}\_{t}^{\varepsilon}], \\ d\mathbf{x}\_{t}^{2} &= \mathbf{x}\_{t}^{2}d\boldsymbol{i}\_{t} - \mathbf{x}\_{t}^{2}d\boldsymbol{e}\_{t} \\ &= \mathbf{x}\_{t}^{2}[r\_{t}^{i}dt + \boldsymbol{\sigma}^{i}d\boldsymbol{W}\_{t}^{i}] - \mathbf{x}\_{t}^{2}[r\_{t}^{\varepsilon}dt + \boldsymbol{\sigma}^{\varepsilon}d\boldsymbol{W}\_{t}^{\varepsilon}] \\ &= \mathbf{x}\_{t}^{2}[r\_{t}^{i} - r\_{t}^{\varepsilon}]dt + \mathbf{x}\_{t}^{2}[\boldsymbol{\sigma}^{i}d\boldsymbol{W}\_{t}^{i} - \boldsymbol{\sigma}^{\varepsilon}d\boldsymbol{W}\_{t}^{\varepsilon}]. \end{split} \tag{10}$$

The SDEs (9) and (10) may be rewritten into matrix-vector form in the following way (compare with [10]).

**Definition 2.4. (Stochastic System for the LCR Model)** *Define the* stochastic system for the LCR model *as*

$$d\mathbf{x}\_{l} = A\_{l}\mathbf{x}\_{l}dt + \mathbf{N}(\mathbf{x}\_{l})u\_{l}dt + a\_{l}dt + \mathbf{S}(\mathbf{x}\_{l}, \mathbf{u}\_{l})dW\_{l\prime} \tag{11}$$

*S*(*xt*, *ut*)*dWt* =

Furthermore, *W*1, *W*2, and *W*<sup>3</sup> represent *Wy*, *We*, and *W<sup>i</sup>*

*dxt* = *Atxtdt* +

**2.2. Description of the simplified LCR model**

<sup>R</sup>(*t*) + *r<sup>e</sup>*

*<sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>i</sup>*

)2(<sup>1</sup> <sup>−</sup> *xt*)<sup>2</sup> + (*σ<sup>i</sup>*

*<sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>e</sup>*

*<sup>t</sup>* + (*σ<sup>e</sup>*

*<sup>t</sup>* <sup>−</sup> (*σ<sup>e</sup>*

earlier, denoted in this section by *lt* = *x*<sup>1</sup>

*dxt* = *xt*[*r*

+[*u*<sup>1</sup> *<sup>t</sup>* + *<sup>u</sup>*<sup>2</sup>

+[(*σ<sup>e</sup>*

that *<sup>u</sup>* = (*u*2, *<sup>π</sup>*) affects only the SDE of *<sup>x</sup>*<sup>1</sup>

that *<sup>π</sup>* affects the variance of *<sup>x</sup>*<sup>1</sup>

*u*<sup>2</sup> affects only the drift of *x*<sup>1</sup>

where *M<sup>j</sup>*

*by the SDE*

[15]) to determine

 [*πT <sup>t</sup> <sup>C</sup><sup>t</sup><sup>π</sup><sup>t</sup>*]

+ <sup>0</sup> <sup>−</sup>*σ<sup>e</sup>* <sup>0</sup> <sup>−</sup>*σ<sup>e</sup>*

*<sup>t</sup>* and the drift of *<sup>x</sup>*<sup>1</sup>

*<sup>t</sup>* . Then (11) becomes

*n* ∑ *j*=0 [*Bj xt*]*u<sup>j</sup>*

= 3 ∑ *j*=1 [*M<sup>j</sup>*

1/2 0 0 0

> *xtdW*<sup>2</sup> *<sup>t</sup>* + 0 0 0 *σ<sup>i</sup> xtdW*<sup>3</sup> *t*

(*ut*)*xt*]*dW<sup>j</sup>*

*t* ,

(*ut*) is the matrix notation used to denote matrices with entries related to *ut*.

*<sup>t</sup>* but not that of *<sup>x</sup>*<sup>2</sup>

*<sup>t</sup>dt* + *atdt* +

The model can be simplified if attention is restricted to the system with the LCR, as stated

**Definition 2.5. (Stochastic Model for a Simplified LCR)** *Define the* simplified LCR system

)<sup>2</sup> + (*σ<sup>i</sup>*

The model is derived as follows. The starting point is the two-dimensional SDE for *x* = (*x*1, *x*2)*<sup>T</sup>* as in the equations (9) and (10). Next, we use the Itô's formula (see, for instance,

)2*x*<sup>2</sup> *<sup>t</sup>* + *<sup>x</sup>*<sup>2</sup> *t πT <sup>t</sup> <sup>C</sup><sup>t</sup><sup>π</sup><sup>t</sup>*]

)<sup>2</sup> + *r y t T <sup>π</sup><sup>t</sup>*]*dt*

*<sup>t</sup>* .(*x*<sup>2</sup>

*<sup>t</sup>* via the term *<sup>x</sup>*<sup>1</sup>

3 ∑ *j*=1 [*M<sup>j</sup>*

*<sup>t</sup>*)−<sup>1</sup> (compare with [10]).

*t r y t T*

(*ut*)*xt*]*dW<sup>j</sup>*

)2]*dt* (13)

1/2*dWt*, *x*(*t*0) = *x*0.

*xtdW*<sup>1</sup> *<sup>t</sup>* +

, respectively. From (11) it is evident

Optimizing Basel III Liquidity Coverage Ratios

http://dx.doi.org/10.5772/58395

75

*<sup>t</sup>* . In particular, for (11) we have

*<sup>π</sup><sup>t</sup>*. On the other hand,

*<sup>t</sup>* . (12)

*with the various terms in this SDE being*

$$\begin{aligned} u\_t &= \begin{bmatrix} u\_t^2 \\ \tilde{\boldsymbol{\pi}}\_t \end{bmatrix}, \quad u: \Omega \times T \to \mathbb{R}^{n+1}, \\\ A\_t &= \begin{bmatrix} r^\mathbb{R}(t) & -r\_t^\varepsilon \\ 0 & r\_t^\mathbb{I} - r\_t^\varepsilon \end{bmatrix}; \\\ N(\mathbf{x}\_t) &= \begin{bmatrix} \mathbf{x}\_t^2 \, \mathbf{x}\_t^1 \tilde{\boldsymbol{\pi}}\_t^\mathbb{I} \\ 0 & 0 \end{bmatrix}; \ a\_t = \begin{bmatrix} \mathbf{x}\_t^2 \boldsymbol{u}\_t^1 \\ 0 \end{bmatrix}; \\\ S(\mathbf{x}\_t, \boldsymbol{u}\_t) &= \begin{bmatrix} \mathbf{x}\_t^1 \, \tilde{\boldsymbol{\pi}}\_t^\mathbb{I} \tilde{\boldsymbol{\Sigma}}\_t^\mathbb{I} - \mathbf{x}\_t^2 \boldsymbol{\sigma}^\varepsilon & 0 \\ 0 & -\mathbf{x}\_t^2 \boldsymbol{\sigma}^\varepsilon \, \mathbf{x}\_t^2 \boldsymbol{\sigma}^i \end{bmatrix}; \\\ W\_t &= \begin{bmatrix} W\_t^y \\ W\_t^z \\ W\_t^i \end{bmatrix} \end{aligned}$$

*where W<sup>y</sup> <sup>t</sup>* , *<sup>W</sup><sup>e</sup> <sup>t</sup> and W<sup>i</sup> <sup>t</sup> are mutually (stochastically) independent standard Brownian motions. It is assumed that for all t* <sup>∈</sup> *<sup>T</sup>*, *<sup>σ</sup><sup>e</sup> <sup>t</sup> >* 0, *<sup>σ</sup><sup>i</sup> <sup>t</sup> >* 0 *and C<sup>t</sup> >* 0 *(compare with [10]).*

We can rewrite (11) as follows.

$$\begin{aligned} N(\mathbf{x}\_t)u\_t &:= \begin{bmatrix} \mathbf{x}\_t^2 \\ \mathbf{0} \end{bmatrix} u\_t^2 + \begin{bmatrix} \mathbf{x}\_t^1 \overline{r}\_t^{y^T} \\ \mathbf{0} \end{bmatrix} \widetilde{\pi}\_t \\ &:= \begin{bmatrix} 0 \ 1 \\ \mathbf{0} \ 0 \end{bmatrix} \mathbf{x}\_t u\_t^3 + \sum\_{j=1}^n \begin{bmatrix} \mathbf{x}\_t^1 \overline{r}\_t^{y^j} \\ \mathbf{0} \end{bmatrix} \widetilde{\pi}\_t^j \\ &:= B\_0 \mathbf{x}\_t u\_t^0 + \sum\_{j=1}^n \begin{bmatrix} \overline{r}\_t^{y^j} \ \mathbf{0} \\ \mathbf{0} \ \mathbf{0} \end{bmatrix} \mathbf{x}\_t \widetilde{\pi}\_t^j \\ &:= \sum\_{j=0}^n [B^j \mathbf{x}\_t] u\_{t'}^j \end{aligned}$$

and

$$\begin{aligned} S(\mathbf{x}\_t, \boldsymbol{\mu}\_t)dW\_t &= \begin{bmatrix} [\tilde{\boldsymbol{\pi}}\_t^T \tilde{\boldsymbol{C}}\_t \tilde{\boldsymbol{\pi}}\_t]^{1/2} & 0\\ 0 & 0 \end{bmatrix} \mathbf{x}\_t dW\_t^1 + \\ &+ \begin{bmatrix} 0 & -\boldsymbol{\sigma}^\varepsilon \\ 0 & -\boldsymbol{\sigma}^\varepsilon \end{bmatrix} \mathbf{x}\_t dW\_t^2 + \begin{bmatrix} 0 & 0\\ 0 & \boldsymbol{\sigma}^i \end{bmatrix} \mathbf{x}\_t d\mathcal{W}\_t^3 \\ &= \sum\_{j=1}^3 [M^j(\boldsymbol{\mu}\_t)\mathbf{x}\_t] d\mathcal{W}\_{t'}^j \end{aligned}$$

where *M<sup>j</sup>* (*ut*) is the matrix notation used to denote matrices with entries related to *ut*. Furthermore, *W*1, *W*2, and *W*<sup>3</sup> represent *Wy*, *We*, and *W<sup>i</sup>* , respectively. From (11) it is evident that *<sup>u</sup>* = (*u*2, *<sup>π</sup>*) affects only the SDE of *<sup>x</sup>*<sup>1</sup> *<sup>t</sup>* but not that of *<sup>x</sup>*<sup>2</sup> *<sup>t</sup>* . In particular, for (11) we have that *<sup>π</sup>* affects the variance of *<sup>x</sup>*<sup>1</sup> *<sup>t</sup>* and the drift of *<sup>x</sup>*<sup>1</sup> *<sup>t</sup>* via the term *<sup>x</sup>*<sup>1</sup> *t r y t T <sup>π</sup><sup>t</sup>*. On the other hand, *u*<sup>2</sup> affects only the drift of *x*<sup>1</sup> *<sup>t</sup>* . Then (11) becomes

$$d\mathbf{x}\_{l} = A\_{l}\mathbf{x}\_{l}dt + \sum\_{j=0}^{n} [\mathbf{B}^{j}\mathbf{x}\_{l}]u\_{l}^{j}dt + a\_{l}dt + \sum\_{j=1}^{3} [M^{j}(u\_{l})\mathbf{x}\_{l}]dW\_{l}^{j}.\tag{12}$$
