Independent Random Time Default Model

• Independent Random Time Default Model is a reduced-form credit risk specification where default time is modeled as an exogenous random variable independent of market factors.
• This independence yields a deterministic survival curve and ensures the immersion property, allowing defaultable prices to factorize into survival weights and default-free valuations.
• The framework underpins valuation in options, counterparty risk, and CDS, while also extending to density-based and stochastic hazard models when independence is relaxed.

Independent Random Time Default Model denotes a reduced-form credit-risk specification in which the default time $\tau$ is modeled as an exogenous random time independent of the reference filtration generated by market factors. The market filtration is progressively enlarged to $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$; under independence the immersion property holds, the Azéma supermartingale becomes a deterministic survival curve, and defaultable prices often factorize into survival weighting times default-free valuations. This structure is used directly in random-time option pricing and in Szimayer-style counterparty-default models for equity derivatives and variable annuities, and it also appears as a simplifying case inside broader density-based, information-based, and term-structure models (Antonelli et al., 2015, Rutkowski et al., 25 May 2026).

1. Filtration enlargement, independence, and hazard structure

Let $H_t:=1_{\{\tau\le t\}}$ be the default indicator. In the standard setup, $$(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb Q)$$ carries the market information generated by tradable assets, while the enlarged filtration is

$$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)=\mathcal F_t\vee H_t.$$

A central structural assumption is the immersion, or $H$-hypothesis: every $\mathcal F_t$-martingale remains a $\mathcal G_t$-martingale. In the independent random time case, independence between $\tau$ and the market filtration implies immersion automatically, so discounted asset prices that are $\mathcal F$-martingales remain martingales after enlargement, preserving arbitrage-free pricing in the enlarged market (Antonelli et al., 2015).

The survival process, or Azéma supermartingale, is

$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$0

Under independence, $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$1 is deterministic. If $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$2 has deterministic intensity $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$3, then

$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$4

and the hazard process $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$5 is deterministic as well (Antonelli et al., 2015). In the Black–Scholes and Merton jump-diffusion specifications used for equity-linked applications, this deterministic hazard is written as $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$6, with survival

$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$7

when $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$8 is deterministic (Rutkowski et al., 25 May 2026).

The conditional-density approach makes precise what independence contributes and what it does not. If

$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$9

then the intensity, when it exists, is $H_t:=1_{\{\tau\le t\}}$0 on $H_t:=1_{\{\tau\le t\}}$1. Under independence one has $H_t:=1_{\{\tau\le t\}}$2, hence $H_t:=1_{\{\tau\le t\}}$3 is deterministic and immersion holds. Outside immersion, however, the intensity only characterizes the conditional law before default; after-default valuation depends on the full conditional density term structure, not only on $H_t:=1_{\{\tau\le t\}}$4 (0905.0559).

2. Valuation principle and survival factorization

For a payoff depending on the asset path and the random trigger time, $H_t:=1_{\{\tau\le t\}}$5, independence permits conditioning directly on $H_t:=1_{\{\tau\le t\}}$6: $H_t:=1_{\{\tau\le t\}}$7 The same mechanism gives the standard decomposition of a defaultable claim with survival payoff $H_t:=1_{\{\tau\le t\}}$8 and recovery-at-default payoff $H_t:=1_{\{\tau\le t\}}$9,

$$(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb Q)$$0

Under independence and deterministic $$(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb Q)$$1, this simplifies to

$$(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb Q)$$2

with $$(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb Q)$$3. The key simplification is the factorization of market and default components (Antonelli et al., 2015).

In the Szimayer-style independent random time model used for European-style claims, if $$(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb Q)$$4 is paid only on survival, then

$$(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb Q)$$5

For constant $$(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb Q)$$6 and $$(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb Q)$$7, the defaultable value is the default-free value multiplied by $$(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb Q)$$8 (Rutkowski et al., 25 May 2026).

The same survival-ratio logic yields pre-default valuation formulas for basic credit instruments. A risky zero-coupon without recovery prices on $$(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb Q)$$9 as the default-free discount factor times $$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)=\mathcal F_t\vee H_t.$$0, while the default leg of a CDS integrates loss-given-default against $$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)=\mathcal F_t\vee H_t.$$1. These are the standard immersion-based pre-default formulas under deterministic survival (0905.0559).

3. Explicit pricing in option models

A prominent application is the Random Time Forward Starting (RTFS) option, whose payoff at maturity $$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)=\mathcal F_t\vee H_t.$$2 is

$$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)=\mathcal F_t\vee H_t.$$3

Its arbitrage-free value is

$$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)=\mathcal F_t\vee H_t.$$4

Under independence, the price decomposes into a realized-trigger part and a pre-trigger part: $$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)=\mathcal F_t\vee H_t.$$5 Here $$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)=\mathcal F_t\vee H_t.$$6 is the forward-start call with deterministic determination time $$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)=\mathcal F_t\vee H_t.$$7. In scale-invariant models,

$$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)=\mathcal F_t\vee H_t.$$8

so RTFS valuation reduces to integrating standard forward-start prices against the default-time density (Antonelli et al., 2015).

In Black–Scholes, the deterministic-time forward-start price is

$$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)=\mathcal F_t\vee H_t.$$9

For $H$0 and $H$1, the RTFS price becomes

$H$2

with closed-form cases for $H$3 and $H$4, including an explicit expression when $H$5. The same framework yields closed- or semi-closed-form formulas in Heston and in Lévy/Variance-Gamma models via Fourier methods (Antonelli et al., 2015).

For standard European options, the independent random time default model produces particularly simple multiplicative adjustments. Under Black–Scholes with dividend yield $H$6,

$H$7

and defaultable put–call parity is

$H$8

Under Merton’s Gaussian jump-diffusion, the default-free call is a Poisson mixture of Black–Scholes formulas, and independence again implies

$H$9

(Rutkowski et al., 25 May 2026).

4. Counterparty risk, CVA, and equity-protection applications

For OTC derivatives, unilateral counterparty risk enters through the standard CVA expression

$\mathcal F_t$0

Under deterministic counterparty intensity $\mathcal F_t$1 and independence between $\mathcal F_t$2 and market factors,

$\mathcal F_t$3

For RTFS options with an independent counterparty default time, the adjustment simplifies further to

$\mathcal F_t$4

If $\mathcal F_t$5 almost surely, then $\mathcal F_t$6 and the defaultable RTFS price is $\mathcal F_t$7 (Antonelli et al., 2015).

The same survival-factor structure appears in variable annuities with equity protection swaps (EPS). If the EPS cash flow is owed only if the counterparty survives to $\mathcal F_t$8, then

$\mathcal F_t$9

where $\mathcal G_t$0 is the default-free value of the static replicating portfolio. In the no-default case the static hedge replicates exactly, but counterparty default creates residual losses that cannot be fully hedged. Under the paper’s assumptions, wrong-way risk is excluded by independence and recovery is set to $\mathcal G_t$1. The resulting default adjustment under Black–Scholes is

$\mathcal G_t$2

and the default-adjusted premium is

$\mathcal G_t$3

Under Merton jump-diffusion, the corresponding default adjustment is a Poisson mixture (Rutkowski et al., 25 May 2026).

5. Embeddings in density-based and term-structure frameworks

In the conditional-density literature, independence is the case $\mathcal G_t$4, so the survival process $\mathcal G_t$5 is deterministic and immersion holds (0905.0559). Song’s framework of random times with differentiable conditional distribution makes this especially transparent: if $\mathcal G_t$6 is independent, then

$\mathcal G_t$7

so the increasing family of martingales is differentiable with respect to $\mathcal G_t$8 and density $\mathcal G_t$9. This permits an isomorphic implantation into an auxiliary model absolutely continuous with respect to a Cox model, together with conditional expectation, optional splitting, and enlargement formulas on $\tau$0 (Song, 2013).

In the natural-model with jumps, the independent case is obtained by taking the Azéma supermartingale $\tau$1 deterministic, which implies $\tau$2 and $\tau$3. If $\tau$4 admits a density $\tau$5, then $\tau$6. If $\tau$7 has atoms at deterministic times $\tau$8, the hazard process acquires predictable jumps

$\tau$9

and the compensator of $\mathcal F$0 is deterministic. This embedding shows that an independent random time default model can accommodate mixed continuous–discrete default laws, including predictable calendar-time default masses (Song, 2013).

In the defaultable HJM framework with risky times, independence relative to a reference filtration implies that no new default-relevant news arrives after time $\mathcal F$1. A consistent specification is then a deterministic random measure

$\mathcal F$2

concentrated on deterministic risky dates $\mathcal F$3. The no-arbitrage conditions simplify to

$\mathcal F$4

Thus the independent random time model can be represented either as a purely absolutely continuous deterministic-hazard model or as a term-structure model with deterministic risky dates and maturity jumps (Fontana et al., 2016).

6. Limitations, non-intensity variants, and broader reinterpretations

The main limitation of the basic independent model is the exclusion of dependence between market risk and default risk. Once $\mathcal F$5 is no longer independent of $\mathcal F$6, expectation factorization breaks, the hazard process becomes stochastic, and the simple survival-times-price formulas disappear. One explicit extension proposed in the RTFS setting is an affine hazard coupling

$\mathcal F$7

where $\mathcal F$8 and $\mathcal F$9 are deterministic bounded functions and $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$00 is a positive adapted process independent of $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$01. Pricing then requires optional projection and evaluation of non-factorized expectations, typically by Monte Carlo or transform methods (Antonelli et al., 2015).

A common misconception is that an independent random time default model must always be an intensity model with compensator absolutely continuous in $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$02. Information-based constructions show otherwise. In the Brownian-bridge model

$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$03

with $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$04 independent of $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$05, the compensator of $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$06 is

$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$07

continuous and singular with respect to Lebesgue time, so no $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$08-intensity exists (Bedini et al., 2016). An analogous random-length bridge model for $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$09 yields a compensator driven by the local time of the information process at levels $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$10, again producing a totally inaccessible default time without a classical intensity (Louriki, 2022).

At the portfolio level, independence can also be scale-dependent rather than absolute. In the OU–Binomial coarse-graining framework, defaults are conditionally independent across obligors at the monthly scale given a latent path $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$11, but aggregation over $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$12 months produces a random block probability $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$13. The resulting effective same-period correlation is

$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$14

so temporal persistence alone generates overdispersion, autocorrelation, and effective default correlation at long horizons. This is explicitly interpreted as an Independent Random Time Default Model mechanism: independence at fine scales, dependence at coarse scales through a random time-aggregated hazard (Mori, 30 May 2026).

A different reinterpretation randomizes the operational clock rather than the default time directly. In the SubCIR construction, a base diffusion intensity model is time-changed by an independent Lévy subordinator $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$15, producing

$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$16

and default intensity

$\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$17

Subordination preserves eigenfunctions and maps eigenvalues $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$18 to $\mathcal G_t=\mathcal F_t\vee \sigma(\tau\wedge t)$19, so analytical tractability survives even though the new intensity becomes a nonnegative jump-diffusion or pure-jump process with two-sided mean-reverting jumps (Mendoza-Arriaga et al., 2014).

In this broader sense, the Independent Random Time Default Model is best viewed as a family of enlargement-based default specifications built around an exogenous random time. Its defining strength is tractable filtration enlargement and valuation factorization under independence; its defining boundary is that once independence, immersion, or absolute continuity in physical time is relaxed, the model passes into density-based, local-time, stochastic-hazard, or coarse-grained variants whose mathematics is materially different.