The challenge in managing new financial risks: adopting an heuristic or theoretical approach

Abstract

The financial crisis began with the collapse of Lehman Brothers and the subprime asset backed securities debacle. Credit risk was turned into liquidity risk, resulting in a lack of confidence among financial institutions. In this article, we will propose a way to model liquidity risk and the credit risk in best practices. We will show that liquidity risk is a new type of risk and the current way to deal with it is based solely on observed variables without any theoretical link. We propose an heuristic approach to combine the numerous liquidity risk indicators with a logistic regression for the first time. In regards to credit risk, several articles prove that the best practice is to use an option model to appreciate this risk. We will present our methodology using stochastic diffusion for the interest rate because currently the yield curves aren’t liquid. This approach is more relevant because the basis model in prior publications has a constant interest rate or a forward rate. Both models allow a better understanding of liquidity and credit risks and the further development of research deals with the link between these two financial risks.

Appendices

Appendix 1: Interest rate stochastic models

Cf “Asset and Risk Management : Finance orientée Risque”, Louis Esch, Robert Kieffer, Thierry Lopez with the collaboration of Christian Berbé, Pascal Damel, Miche Debay and J.-F. Hannosset. De Boeck and soon in English at Wiley’s.

Appendix 2

$$\begin{aligned} dVA_t=VA_t(rdt + \sigma 'dB_t) \end{aligned}$$

We researched a suitable process so that the integrals $\int _0^t VA_tdt$ and $\int _0^t VA_tdB_t$ would have the meaning

$$\begin{aligned} VA_t=VA_0 + \int _{t=0}^T rVA_tdt + \int _{t=0}^T \sigma VA_tdB_t \end{aligned}$$

Itô’s formula as the function of g with two variables (t, B) twice continuously differentiable over $\mathbb {R}$

$$\begin{aligned} g(t,B_t)=g(0,0)+\int _{s=t}^T\frac{\delta g}{\delta B}(s, B_s)dB_s + \int _{s=t}^T(\frac{\delta g}{\delta t} + \frac{1}{2} \frac{\delta ^2g}{\delta B^2})(s,B_s)ds \end{aligned}$$

where $\frac{\delta g}{\delta t}, \frac{\delta g}{\delta B}, \frac{\delta ^2g}{\delta B^2}$ designate the partial derivatives of g. This formula can be written in the shape of a differential increment

$$\begin{aligned} dg(t,B_t)=\frac{\delta g}{\delta B}(t, B_t)dB_t + (\frac{\delta g}{\delta t} + \frac{1}{2} \frac{\delta ^2g}{\delta B^2})(t,B_t)dt \end{aligned}$$

Considering the function g(t, B), a calculation with partial derivatives, combined with Itô’s formula demonstrate

$$\begin{aligned}&\displaystyle g(t,B) = VA_0e^{((r-\frac{\sigma ^2}{2})t+\sigma B)}\\&\displaystyle \frac{\delta g}{\delta t} = r - \frac{\sigma ^2}{2}, \quad \frac{\delta g}{\delta B} = \sigma , \quad \frac{\delta ^2 g}{\delta B^2} = \sigma ^2\\&\displaystyle g(t,B_t)=g(0,0) + \int _{s=t}^T((r-\frac{\sigma ^2}{2})+\frac{1}{2}\sigma ^2)(s,B_s)ds+\sigma \int _{s=t}^Tg(s,B_s) dB_s\\&\displaystyle \Leftrightarrow g(t,B_t)=g(0,0) + \int _{s=t}^T rg(s,B_s)ds + \sigma \int _{s=t}^T g(s,B_s) dB_s\\&\displaystyle \Leftrightarrow g(t,B_t)=VA_0 + \int _{s=t}^T g(s,B_s)dVA_s \end{aligned}$$

Appendix 3

Calculation of the Call value

• $PE_D$: Strike current debt value

• $P_{CALL}$= CALL Price

• $P_{CALL}=e^{-rT}E(VA_T- PE_{DT})_+$

• $P_{CALL}=e^{-rT} E(VA_{T\{VA_T \ge PE_{DT}\}})- e^{-rT} PE_{DT} Proba\{VA_T \ge PE_{DT}\}$

with: $VA_t = VA_0e^{((r-\frac{\sigma ^2}{2})T+\sigma B_t)}$ and $B_T=N(0,\sqrt{T})$ with N a regular density law

$$\begin{aligned} N(x) = \int _{-\infty }^x e^{-\frac{\mu ^2}{2}}\frac{d\mu }{\sqrt{2\Pi }} \quad \text {and}\quad d_r=[(d_{z1}+\Pi )\times \sigma _r^\theta +\overline{r}^\theta ]dt\times P(\lambda ). \end{aligned}$$

The following quantities must be calculated

$$\begin{aligned} Proba\{VA_T \ge PE_{DT}\}= & {} \int \limits _R\left\{ VA_0e^{((r-\frac{\sigma ^2}{2})T+\sigma \mu )}\ge PE_D\right\} e^{-\frac{\mu ^2}{2}}\frac{dv}{\sqrt{2\Pi }} \Leftrightarrow \\ Proba\{VA_T \ge PE_{DT}\}= & {} \int \limits _R\left\{ VA_0e^{((r-\frac{\sigma ^2}{2})T+\sigma \sqrt{T}v)}\ge PE_D\right\} e^{-\frac{v^2}{2}}\frac{dv}{\sqrt{2\Pi }} \Leftrightarrow \end{aligned}$$

Starting $u=\sqrt{T}v$

$$\begin{aligned} Proba\{VA_T \ge PE_{DT}\}= \int \limits _R\left\{ \ln VA_0e^{\left( \left( r-\frac{\sigma '^2}{2}\right) T+\sigma '\sqrt{T}v\right) }\ge \ln PE_D\right\} e^{-\frac{v^2}{2}}\frac{dv}{\sqrt{2\Pi }} \Leftrightarrow \end{aligned}$$

Using the properties of the Nepierian logarithm

$$\begin{aligned} Proba\{VA_T \ge PE_{DT}\}= \int \limits _R\left\{ \frac{\ln \frac{VA_0}{PE_D} + (r-\frac{\sigma ^2}{2})T}{\sigma \sqrt{T}} \ge -v\right\} e^{-\frac{v^2}{2}}\frac{dv}{\sqrt{2\Pi }} \Leftrightarrow \end{aligned}$$

Specifying accurately the rate distribution model and stating $v=-w$

$$\begin{aligned}&Proba\{VA_T \ge PE_{DT}\}=\\&\quad \int \limits _R\left\{ \frac{\ln \frac{VA_0}{PE_D} + [((d_{z1}+\Pi )\times \sigma ^\theta _r+\overline{r}^\theta )\times P(\lambda ) - \frac{\sigma ^2}{2}]T}{\sigma \sqrt{T}} \ge w\right\} e^{-\frac{w^2}{2}}\frac{dw}{\sqrt{2\Pi }}\\&\quad \Leftrightarrow Proba\{VA_T \ge PE_{DT}\}=N \left[ \frac{\ln \frac{VA_0}{PE_D} + [((d_{z1}+\Pi )\times \sigma ^\theta _r+\overline{r}^\theta )\times P(\lambda ) - \frac{\sigma ^2}{2}]T}{\sigma \sqrt{T}} \right] \\ \end{aligned}$$

The following quantity must also be calculated : $E(VA_{T\{VA_T \ge PE_{DT}\}})$

$$\begin{aligned} E(VA_{T\{VA_T \ge PE_{DT}\}})= & {} \int \limits _R\left\{ VA_0e^{((r-\frac{\sigma ^2}{2})T+\sigma \mu } \ge PE_D\right\} VA_0 e^{((r-\frac{\sigma ^2}{2})T+\sigma \mu } e^{-\frac{\mu ^2}{2T}} \frac{d\mu }{\sqrt{2\Pi T}}\\ \Leftrightarrow E(VA_{T\{VA_T \ge PE_{DT}\}})= & {} VA_0 + e^{rT} \int \limits _R\left\{ \frac{\ln \frac{{ VA}_0}{PE_D} + (r-\frac{\sigma ^2}{2})T}{\sigma \sqrt{T}} \ge -v\right\} \\&\times e^{-\frac{\sigma ^2}{2}T + v\sigma \sqrt{T}-\frac{v^2}{2}}\frac{dv}{\sqrt{2\Pi }}\\ \Leftrightarrow E(VA_{T\{VA_T \ge PE_{DT}\}})= & {} VA_0 + e^{rT} \int \limits _R\left\{ \frac{\ln \frac{VA_0}{PE_D} + (r-\frac{\sigma ^2}{2})T}{\sigma \sqrt{T}} \ge -v\right\} \\&\times e^{-\frac{(v-\sigma \sqrt{T})^2}{2}}\frac{dv}{\sqrt{2\Pi }}\\ \end{aligned}$$

Stating $w=v-\sigma \sqrt{T}$ and specifying accurately the rate distribution model

$$\begin{aligned}&E(VA_{T\{VA_T \ge PE_{DT}\}})=\\&\quad VA_0e^{rT} \int \limits _R\left\{ \frac{\ln \frac{VA_0}{PE_D} + [((d_{z1}+\Pi )\times \sigma ^\theta _r+\overline{r}^\theta )\times P(\lambda ) - \frac{\sigma ^2}{2}]T}{\sigma \sqrt{T}} \ge -(w+\sigma \sqrt{T})\right\} \\&\qquad \times e^{-\frac{w^2}{2}}\frac{dw}{\sqrt{2\Pi }}\\&\qquad \Leftrightarrow Proba\{VA_T \ge PE_{DT}\}\\&\qquad =VA_0e^{rT}N \left[ \frac{\ln \frac{VA_0}{PE_D} + [((d_{z1}+\Pi )\times \sigma ^\theta _r+\overline{r}^\theta )\times P(\lambda ) - \frac{\sigma ^2}{2}]T}{\sigma \sqrt{T}} \right] . \end{aligned}$$

At last, we obtain, the valuation of the option

$$\begin{aligned} \left( \begin{array}{l} PCALL=VA_0~N(d1(VA_0,PE_D,T,r,\sigma '))-PE_De^{T[(d_{z1}+\Pi )\times \sigma _r^\theta +\overline{r}^\theta ]\times P(\lambda )} ~ N(d2(VA_0, PE, T, r,\sigma '))\\ d1=\frac{\ln \frac{VA_0}{PE_D}+\left( [(d_{z1}+\Pi )\times \sigma _r^\theta +\overline{r}^\theta ]\times P(\lambda )-\frac{\sigma '^{2}}{2}\right) T}{\sigma '\sqrt{T}}\\ d2=d1-\sigma '\sqrt{T} \end{array} \right) . \end{aligned}$$