Stochastic calculus is used to model financial derivatives such as credit default swaps (CDS) by considering the dynamic changes in the credit risk associated with the underlying asset, such as a bond or a loan. This typically involves using stochastic differential equations to represent the evolution of certain financial quantities over time. In the case of credit risk modeling, we can use stochastic calculus to track the credit spread – the premium required by an investor to take on the credit risk – as well as the hazard rate, which represents the risk of default.
To begin, let’s introduce the main building blocks of credit risk modeling. We consider a credit risky asset issued by a reference entity, which has a hazard rate (or default intensity) λ(t). We assume that the time until default is a random variable τ that has a conditional probability density function f(t|ℱt), given the filtration ℱt which represents the available information until time t. The survival probability until time t is given by:
S(t) = Pr(τ > t) = ∫t∞f(u|ℱt)du
The hazard rate λ(t) is defined as the default probability per unit of time, conditional on the reference entity’s not having defaulted until time t. It can be expressed as:
$$\lambda(t) = -\frac{\partial \ln S(t)}{\partial t}$$
One widely used approach to pricing credit risky securities, and in particular CDS, is the intensity-based approach, which is based on modeling the hazard rate (or the default intensity) λ(t). This can be done by considering a stochastic process λ(t)0 ≤ t ≤ T, which could be mean-reverting, such as the Cox-Ingersoll-Ross (CIR) process, or follow a more general affine process.
A popular stochastic model for the hazard rate is the Cox process, given by:
$$d\lambda(t) = \kappa(\theta - \lambda(t)) dt + \sigma \sqrt{\lambda(t)} dW(t)$$
where κ denotes the speed of mean reversion, θ is the long-term mean hazard rate, σ is the volatility term, and dW(t) is a Brownian motion term.
To price a CDS contract, the protection seller receives a periodic premium from the protection buyer until the reference entity defaults or the CDS matures. The premium leg (PL) of the CDS is given by the present value of expected premium payments, while the default leg (DL) represents the expected payout by the protection seller upon default. The CDS spread is determined such that the premium leg and default leg are equal. Let R represent the recovery rate of the asset upon default, and T be the maturity of the CDS contract; then, the CDS spread (s) is given by:
$$\begin{aligned}
\text{PL}(s) & = \text{DL}(s) \sum_{i=1}^{n} s E\left[ e^{-\int_{0}^{t_i} r(u)du} \cdot (S(t_{i-1}) - S(t_i)) \right] \\
& = (1-R) E\left[ \int_{0}^{T} e^{-\int_{0}^{t} r(u)du} dS(t) \right]\end{aligned}$$
Here, r(u) denotes the risk-free rate at time u, and the expectation is taken with respect to the risk-neutral measure.
In this setup, we have combined the dynamics of credit spreads and stochastic interest rates in a framework that allows us to price and analyze credit derivatives like CDS contracts. By solving the above equation numerically, we can quickly reprice the contracts as market conditions change, such as changes in credit spreads or the interest rate environment. The use of stochastic calculus in credit risk modeling allows us to better understand the dynamics of credit spreads, assess the risks associated with credit events, and develop effective risk management strategies for credit portfolios.