--- title: Independent Random Time Default Model url: https://www.emergentmind.com/topics/independent-random-time-default-model type: topic --- # Independent Random Time Default Model 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 [1504.03552] [2605.25450]. ## 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 [1504.03552]. The survival process, or Azéma supermartingale, is \[ Z_t:=\mathbb Q(\tau>t\mid \mathcal F_t). \] Under independence, \(Z_t=\mathbb Q(\tau>t)\) is deterministic. If \(\tau\) has deterministic intensity \(\lambda(t)\), then \[ Z_t=\exp\!\Big(-\int_0^t \lambda(u)\,du\Big),\qquad dPD(t)=\lambda(t)Z_t\,dt, \] and the hazard process \(\Gamma_t:=-\log Z_t\) is deterministic as well [1504.03552]. In the Black–Scholes and Merton jump-diffusion specifications used for equity-linked applications, this deterministic hazard is written as \(\lambda_D(t)\), with survival \[ S(t)=\mathbb Q(\tau>t)=\exp\!\Big(-\int_0^t \lambda_D(u)\,du\Big) \] when \(\lambda_D\) is deterministic [2605.25450]. The conditional-density approach makes precise what independence contributes and what it does not. If \[ \mathbb P(\tau\in du\mid \mathcal F_t)=\alpha_t(u)\,\eta(du), \] then the intensity, when it exists, is \(\lambda_t=\alpha_t(t)/Z_t\) on \(\{t<\tau\}\). Under independence one has \(\alpha_t(u)\equiv \alpha(u)\), hence \(Z_t\) 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 \(\lambda_t\) [0905.0559]. ## 2. Valuation principle and survival factorization For a payoff depending on the asset path and the random trigger time, \(X=f(S_\cdot,\tau)\), independence permits conditioning directly on \(\tau\): \[ V_0 = \int_0^T \mathbb E^\mathbb Q\!\big[D(0,T)f(S_\cdot,u)\big]\,dF_\tau(u) + \mathbb Q(\tau>T)\,\mathbb E^\mathbb Q\!\big[D(0,T)f(S_\cdot,T^+)\big]. \] The same mechanism gives the standard decomposition of a defaultable claim with survival payoff \(\Phi(S_T)\) and recovery-at-default payoff \(R\Psi(S_\tau)\), \[ V_0^{\mathrm{def}} = \mathbb E^\mathbb Q\!\big[D(0,T)\Phi(S_T)1_{\{\tau>T\}}\big] + \mathbb E^\mathbb Q\!\big[D(0,\tau)R\Psi(S_\tau)1_{\{\tau\le T\}}\big]. \] Under independence and deterministic \(\lambda(t)\), this simplifies to \[ V_0^{\mathrm{def}} = G_T\,\mathbb E^\mathbb Q\!\big[D(0,T)\Phi(S_T)\big] + \int_0^T D(0,t)\,R\,\mathbb E^\mathbb Q[\Psi(S_t)]\,\lambda(t)G_t\,dt, \] with \(G_t=\mathbb Q(\tau>t)\). The key simplification is the factorization of market and default components [1504.03552]. In the Szimayer-style independent random time model used for European-style claims, if \(\Phi(S_T)\) is paid only on survival, then \[ V_t = \mathbb E^\mathbb Q\!\Big[e^{-\int_t^T r(u)\,du}\,\Phi(S_T)\,1_{\{\tau>T\}}\mid \mathcal F_t\Big] = e^{-\int_t^T \lambda_D(u)\,du}\, \mathbb E^\mathbb Q\!\Big[e^{-\int_t^T r(u)\,du}\Phi(S_T)\mid \mathcal F_t\Big]. \] For constant \(r\) and \(\lambda_D\), the defaultable value is the default-free value multiplied by \(e^{-\lambda_D(T-t)}\) [2605.25450]. The same survival-ratio logic yields pre-default valuation formulas for basic credit instruments. A risky zero-coupon without recovery prices on \(\{t<\tau\}\) as the default-free discount factor times \(S(T)/S(t)\), while the default leg of a CDS integrates loss-given-default against \(\lambda(u)S(u)/S(t)\). 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 \(T\) is \[ \Pi_T=\big(S_T-\kappa S_{\tau\wedge T}\big)^+. \] Its arbitrage-free value is \[ c(t,T)=\mathbb E^{\mathbb Q}\!\big[B(t,T)(S_T-\kappa S_{\tau\wedge T})^+\mid \mathcal G_t\big]. \] Under independence, the price decomposes into a realized-trigger part and a pre-trigger part: \[ c(t,T) = \mathbb E^{\mathbb Q}\!\big[B(t,T)(S_T-\kappa S_\tau)^+\mid \mathcal G_t\big]1_{\{0<\tau\le t\}} + 1_{\{\tau>t\}} \Big[ \int_t^T c(t,u,T)\lambda(u)e^{-\int_t^u \lambda(s)\,ds}\,du + (1-\kappa)S_t e^{-\int_t^T \lambda(s)\,ds} \Big]. \] Here \(c(t,u,T)\) is the forward-start call with deterministic determination time \(u\). In scale-invariant models, \[ c(t,u,T) = \mathbb E^\mathbb Q[B(t,u)S_u\mid \mathcal F_t]\cdot \mathrm{call}(u,1,\kappa,T) = S_t\,\mathrm{call}(u,1,\kappa,T), \] so RTFS valuation reduces to integrating standard forward-start prices against the default-time density [1504.03552]. In Black–Scholes, the deterministic-time forward-start price is \[ c^{BS}(t,u,T) = S_t\Big[\mathcal N(c_1\sqrt{T-u})-e^{-r(T-u)}\mathcal N(c_2\sqrt{T-u})\Big]. \] For \(\kappa=1\) and \(\tau\sim \mathrm{Exp}(\lambda)\), the RTFS price becomes \[ c(t,T) = 1_{\{0<\tau\le t\}}\mathrm{call}^{BS}(t,S_t,S_\tau;\sigma,r,T) + 1_{\{\tau>t\}}S_t\big[A_1(t,T)-A_2(t,T)\big], \] with closed-form cases for \(A_1\) and \(A_2\), including an explicit expression when \(\lambda=r\). The same framework yields closed- or semi-closed-form formulas in Heston and in Lévy/Variance-Gamma models via Fourier methods [1504.03552]. For standard European options, the independent random time default model produces particularly simple multiplicative adjustments. Under Black–Scholes with dividend yield \(q\), \[ C^{\mathrm{def}}(t)=e^{-\lambda_D\tau}C_{BS}(t),\qquad P^{\mathrm{def}}(t)=e^{-\lambda_D\tau}P_{BS}(t), \] and defaultable put–call parity is \[ C^{\mathrm{def}}(t)-P^{\mathrm{def}}(t) = e^{-\lambda_D\tau}\big[S_t e^{-q\tau}-K e^{-r\tau}\big]. \] Under Merton’s Gaussian jump-diffusion, the default-free call is a Poisson mixture of Black–Scholes formulas, and independence again implies \[ C_{MJD}^{\mathrm{def}}(t)=e^{-\int_t^T \lambda_D(u)\,du}C_{MJD}(t), \qquad P_{MJD}^{\mathrm{def}}(t)=e^{-\int_t^T \lambda_D(u)\,du}P_{MJD}(t) \] [2605.25450]. ## 4. Counterparty risk, CVA, and equity-protection applications For OTC derivatives, unilateral counterparty risk enters through the standard CVA expression \[ \mathrm{CVA}(t) = (1-R)\, \mathbb E^\mathbb Q\!\big[ 1_{\{t<\tau_C\le T\}}\,B(t,\tau_C)\,\mathrm{NPV}(\tau_C)^+ \mid \mathcal F_t \big]. \] Under deterministic counterparty intensity \(\lambda_C(u)\) and independence between \(\tau_C\) and market factors, \[ \mathrm{CVA}(0) = (1-R)\int_0^T D(0,t)\, \mathbb E^\mathbb Q[E_t^+]\,\lambda_C(t)\,G_t^C\,dt. \] For RTFS options with an independent counterparty default time, the adjustment simplifies further to \[ \mathrm{CVA}(t,T)=(1-R)\,\mathbb Q(t<\tau_C\le T)\,c(t,T). \] If \(\tau=\tau_C\) almost surely, then \(\mathrm{CVA}(t,T)=(1-R)c(t,T)\) and the defaultable RTFS price is \(\bar c(t,T)=R\,c(t,T)\) [1504.03552]. 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 \(T\), then \[ V_{EPS}^{\mathrm{def}}(t)=e^{-\int_t^T \lambda_D(u)\,du}\,V_{EPS}^0(t), \] where \(V_{EPS}^0\) 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 \(R=0\). The resulting default adjustment under Black–Scholes is \[ DA_{BS} = (1-e^{-\lambda_D T})\, \widehat p\,e^{rT}\, N\!\left( \frac{\ln(1+\widehat l)-(r+\tfrac12\sigma^2)T}{\sigma\sqrt T} \right), \] and the default-adjusted premium is \[ c^D = H(0)+e^{-rT}DA. \] Under Merton jump-diffusion, the corresponding default adjustment is a Poisson mixture [2605.25450]. ## 5. Embeddings in density-based and term-structure frameworks In the conditional-density literature, independence is the case \(\alpha_t(u)\equiv \alpha(u)\), so the survival process \(Z_t=\mathbb P(\tau>t\mid \mathcal F_t)\) is deterministic and immersion holds [0905.0559]. Song’s framework of random times with differentiable conditional distribution makes this especially transparent: if \(\tau\) is independent, then \[ M_t^u=\mathbb Q(\tau