Default-Time Expectations Under Independent Rates and Intensity
Summary
The document addresses how to express an expectation involving default before a horizon and an accumulated interest-rate factor. Its response extends the inhomogeneous Poisson setup to a Cox process, allowing stochastic default intensity, and relates the survival probability to the expectation of the exponential of negative cumulative intensity. Under independence between rates and default, it conditions on the default time and connects the expectation to a maturity-dependent rate expectation integrated against changes in survival probability.
There is a material sign and definition caveat: the question uses positive exponentials and defines its intensity-related function with a positive exponent, while the answer switches to negative exponents and identifies the intensity function with survival probability. It also uses the rate expectation with a negative exponent, interpreting it as a zero-coupon bond price. Thus the displayed result does not directly establish the requested identity as written. The response further notes that a positive valuation date requires filtration assumptions, including a martingale invariance condition.
Key ideas
- For a Cox process, stochastic intensity can define default time, with survival probability expressed using negative cumulative intensity.
- Independence allows conditioning on default time to separate the rate expectation from default survival.
- Integrating the bond-price expectation against changes in survival probability yields a default-time expectation under the stated negative-exponent convention.
- The question’s positive exponents and definitions do not match the answer’s negative-exponent formulation.
- A nonzero valuation date requires additional assumptions about market and default information.
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Full text
# Simplifying an expectation function of default time and rates
# Simplifying an expectation function of default time and rates
I have the following expectation to calculate :
$$ \mathbf{E}\left[ e^{\int_{t_0}^{\tau} r_s ds} \mathbf{1}_{\{\tau < T\}}\right] $$
More precisely, I want to show that :
$$ \mathbf{E}\left[ e^{\int_{t_0}^{\tau} r_s ds} \mathbf{1}_{\{\tau < T\}}\right] = \int_{t_0}^T P(s) dQ(s)$$
where $P(s) \equiv \mathbf{E}\left[ e^{\int_{t_0}^{s} r_u du} \right]$ and $Q(s) \equiv \mathbf{E}\left[ e^{\int_{t_0}^{s} \lambda_u du} \right]$, $\tau$ is the first jump of a Poisson process with default intensity $(\lambda_t)_t$.
The only hypothesis I have is that rates and default are independent $-$ rates don't follow a special process.
## Answer by Gordon (score 1)
https://quant.stackexchange.com/a/36915
For an in-homogeneous Poisson process, the intensity process $\lambda_t$ is assumed to be deterministic. More generally, we can define $\tau$ to be the first jump time of a Cox process, or a conditional Poisson process (see Chapter 6 of the book Credit Risk). We assume that $t_0=0$ is the valuation date. Then the intensity process $\lambda_t$ can be stochastic, and \begin{align*} Q(t) \equiv P(\tau > t) = E\left(e^{-\int_{t_0}^t \lambda_s ds} \right). \end{align*} Moreover, given the independence assumption, \begin{align*} E\left(e^{-\int_{t_0}^{\tau} r_u du} \pmb{1}_{t_0<\tau<T} \right) &= E\left(E\left(e^{-\int_{t_0}^{\tau} r_u du} \pmb{1}_{t_0<\tau<T} \,|\, \tau \right)\right)\\ &=E\left(P(\tau) \pmb{1}_{\tau<T}\right)\\ &=-\int_{t_0}^TP(s) dQ(s), \end{align*} where \begin{align*} P(s) = E\left(e^{-\int_{t_0}^s r_u du} \right) \end{align*} is the price of a zero-coupon bond with maturity $s$ and unit face value.
> Comments: If we assume that $t_0>0$, then the whole question should be changed. For example, we need to define the filtration for the market information and the enlarged filtration generated by the process defining the default time as well as the market information. Moreover, certain martingale invariance property, or $\mathcal{H}$-Hypothesis, should also be assumed.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.