Why do we need to split market and default information into 2 separate filtrations?

Question

The reduced-form approach to modelling derivatives with credit risk normally assumes the existence of two filtrations:

• A market filtration $(\mathscr{F}_t)_{t\geq0}$ carrying market and economic information (such as stock prices or interest rates); and
• A default filtration $(\mathscr{H}_t)_{t\geq0}$ carrying information about the default time of the counterparty in scope.

Pricing is then performed under a full filtration $(\mathscr{G}_t)_{t\geq0}$ defined as: $$\forall t\geq 0, \quad\mathscr{G}_t:=\mathscr{F}_t\vee\mathscr{H}_t$$ Why do we need to split the information into two separate filtrations? Alternatively, under which modelling assumptions is this framework necessary? Most papers on pricing claims with credit risk readily make the (H)-hypothesis: any $\mathscr{F}_t$-martingale remains a $\mathscr{G}_t$-martingale. I wonder, what is the point of this convoluted setting? There must be a specific technical reason but I haven't yet found any paper which clearly spells it out.

Being the devil's advocate⁽¹⁾, let us consider a market with the following characteristics:

• The market includes a traded asset $S$ driven by a Brownian Motion $(W_t)_{t\geq0}$.
• There exists a default process $H_t:=\pmb{1}_{\{\tau\geq t\}}$, where $\tau$ is the default time.
• There exists a deterministic hazard rate $\gamma_t$ which specifies the distribution of the default process.
• The asset price and the default time are independent.
• The market is endowed with a single filtration $(\mathscr{F}_t)_{t\geq0}$ generated by $W_t$ and $H_t$.

We want to price a $\mathscr{F}_T$-measurable contingent claim $\xi$ written on the asset $S$ and subject to credit risk, $T>t$. Then: $$\begin{align} V_t&=E\left(\left.\xi\pmb{1}_{\{\tau>T\}}\right|\mathscr{F}_t\right) \\ &=E\left(\left.\xi\right|\mathscr{F}_t\right) E\left(\left.\pmb{1}_{\{\tau>T\}}\right|\mathscr{F}_t\right) \\ &=\upsilon_t \left(1-P\left(\left.\tau\leq T\right|\mathscr{F}_t\right)\right) \\[2.5pt] &=\upsilon_te^{\Gamma_t-\Gamma_T} \end{align}$$ where $V$ (resp. $\upsilon$) is the defaultable (resp. risk-free) value of the claim and $\Gamma_t$ is defined as: $$\Gamma_t:=\int_0^t\gamma_s\text{d}s$$ I don’t see any issue with this setting.

_{(1) Edit: my initial example was wrong, I used incorrectly the tower law for nested conditional expectations given $\mathscr{F}_t$ contains information on both $W_t$ and $H_t$ hence $\mathscr{F}_t\notin\sigma(W_s, s\leq T)$. I've modified the example to include an independence assumption, but the question remains: when and why do we need the setting with two filtrations?}

Answer 1

I think you are absolutely correct if the hazard rate is deterministic, although I think you are forgetting a discounting factor in your example. But sometimes the hazard rate cannot be assumed to be deterministic (e.g. when pricing CVA and DVA). Here the hazard rate is instead assumed to follow a stochastic process itself, such that $\mathbf{1}_{\{\tau>t\}}$ is the first jump in a Cox process.

It is common to assume that the hazard rate follows a Cox–Ingersoll–Ross process (or extensions to this), which is a mean-reverting square-root diffusion process with SDE $$d \gamma_t=\kappa(\theta-\gamma_t)dt+\sigma\sqrt{\gamma_t}dW_t$$ with Feller-constraint $2\kappa\theta\geq \sigma^2$ to make sure that the origin is inaccessible forcing $\gamma_t>0$ for all $t$.

In general it is assumed that the hazard rate is adapted to the filtration generated by the (default-free) market variables: $\mathscr{F}_t$. Conditional on this information the number of jumps between times $s<t$ is Poisson and the probability of $n$ jumps is thus given by $$\frac{\left(\Gamma_{t}-\Gamma_{s}\right)^{n}}{n!}e^{-\left(\Gamma_{t}-\Gamma_{s}\right)}$$ The probability of zero jumps (no default) is consequently given by $e^{-\left(\Gamma_{t}-\Gamma_{s}\right)}$.

Denote $D(t,T)=e^{-\int_t^Tr_u du}$ such that the risk-neutral valuation of $\xi$ becomes $$\tag{1}V_t=\mathbb{E}\left[D(t,T)\cdot\xi\cdot \mathbf{1}_{\{\tau>T\}}\middle|\mathscr{G}_t\right]$$ Note that $$\tag{2}\mathbb{E}\left[\mathbf{1}_{\{\tau>T\}}\middle|\mathscr{F}_T\vee\mathscr{H}_t\right]=\mathbf{1}_{\{\tau>t\}}e^{-\left(\Gamma_T-\Gamma_t\right)}$$ And note that conditioning on the full filtration would just yield $\mathbb{E}\left[\mathbf{1}_{\{\tau>T\}}\middle| \mathscr{G}_T\right]=\mathbf{1}_{\{\tau>T\}}$, which does not simplify the expression. This means that we can simplify the risk-neutral valuation by utilising the tower property \begin{align*} V_t&=\mathbb{E}\left[D(t,T)\cdot\xi \cdot\mathbb{E}\left[\mathbf{1}_{\{\tau>T\}}\middle|\mathscr{F}_T\vee\mathscr{H}_t\right]\middle|\mathscr{G}_t\right]\\ &=\mathbb{E}\left[D(t,T)\cdot\xi \cdot\mathbf{1}_{\{\tau>t\}}e^{-\left(\Gamma_T-\Gamma_t\right)}\middle|\mathscr{G}_t\right]\\ &=\mathbf{1}_{\{\tau>t\}}\mathbb{E}\left[D(t,T)\cdot\xi \cdot e^{-\left(\Gamma_T-\Gamma_t\right)}\middle|\mathscr{G}_t\right]\\ \tag{3} &=\mathbf{1}_{\{\tau>t\}}\mathbb{E}\left[D(t,T)\cdot\xi \cdot e^{-\left(\Gamma_T-\Gamma_t\right)}\middle|\mathscr{F}_t\right] \end{align*} where the second last equality is due to $\mathbf{1}_{\{\tau>t\}}$ being $\mathscr{G}_{t}$-measurable and the last equality is due to the expectation not depending on default information anymore. To simplify further we can assume independence between the discounting factor, the $T$-claim and the hazard rate to obtain $$V_{t}=\mathbf{1}_{\{\tau>t\}}P(t,T)v_{t}\mathbb{E}\left[ e^{-\left(\Gamma_T-\Gamma_t\right)}\middle|\mathscr{F}_t\right]$$ where $P(t,T)$ is a Zero Coupon Bond and the expectation is the probability of not defaulting between $t$ and $T$. This probability can be stripped from spreads on relevant Credit Default Swaps for example.

If we cannot assume independence (for example when there is Wrong Way Risk) then the stochastic dynamics of the hazard rate has to be assumed. So the reason for using two different filtrations is to be able to simplify the expectation when the hazard rate is stochastic.

Answer 2

Your ${\cal F}$ is actually ${\cal G}$, that is the already enlarged filtration/probability space. So, the claim here seems to be that we do not have to consider the smaller, market filtration, ${\cal F}$.

But, before we invoke Hypothesis (H), only this is true:

$$ E\left[1_{\tau>T}|{\cal G}_t\right] = 1_{\tau>t} E\left[e^{\Gamma_t -\Gamma_T}|{\cal F}_t\right]\left(\not= e^{\Gamma_t -\Gamma_T}\right), $$

when $\gamma_t$ is a stochastic process (also note the presence of $1_{\tau>t}$ even when it is deterministic).

More generally, for $X$ ${\cal F}_T$-measurable (and integrable), we have:

$$ E\left[X1_{\tau>T}|{\cal G}_t\right] = 1_{\tau>t} E\left[Xe^{\Gamma_t -\Gamma_T}|{\cal F}_t\right]\left(\not= E\left[X|{\cal F}_t\right]e^{\Gamma_t -\Gamma_T}\right). $$

Hypothesis (H) is equivalent to $\sigma$-algebras $ {\cal F}_\infty $ and ${\cal G}_t$ being conditionally independent given ${\cal F}_t$ under $Q$, for all $t\geq 0$. It is also equivalent to:

$$ P(\tau \leq t |{\cal F}_t)=P(\tau \leq t |{\cal F}_\infty) $$

for all $t\geq 0$.

As it happens, the canonical construction of default time, based on a given ${\cal F}$-progressively measurable hazard rate process $\gamma_t$ (does not need to be deterministic) and a random variable $\zeta$ which is uniform on $[0,1]$ and independent of ${\cal F}$, supported by the enlarged space $(\Omega, {\cal G}, P)$, with $$ \tau := \inf \; \{t\geq 0| e^{-\Gamma_t} < \zeta \}, $$ automatically implies that Hypothesis (H) holds. (Cox processes do too.)

Indeed:

$$ P(\tau >t |{\cal F}_\infty)= P(e^{-\Gamma_t} \geq \zeta |{\cal F}_\infty) = e^{-\Gamma_t}$$

and

$$ P(\tau >t |{\cal F}_t)= E[P(e^{-\Gamma_t} \geq \zeta |{\cal F}_\infty)|{\cal F}_t] = e^{-\Gamma_t},$$

as $\Gamma_t$ is ${\cal F}_t$-measurable.

More on Hypothesis (H) information interpretation here.