Counterparty Credit Risk and CVA

Options Pricing · Medium · Free problem

Consider a single-payment derivative that pays $X \geq 0$ at maturity $T$ from a counterparty to you. The risk-free discount factor is $e^{-rT}$.

Model the counterparty's default time $\tau$ as independent of $X$, with a constant hazard rate $\lambda$ (i.e., $\tau$ is exponentially distributed). If default occurs, you recover a fraction $R \in [0, 1]$ of the claim's risk-free value.

  1. Compute the survival probability $Q(\tau > T)$ and the default probability $Q(\tau \leq T)$.
  1. Derive the time-0 value of the claim including default risk, $V_{\text{def}}$, in terms of $E_Q[X]$, $r$, $\lambda$, and $R$.
  1. Define the Credit Valuation Adjustment (CVA) as the difference between the risk-free value and the defaultable value, and simplify.

Hints

  1. With a constant hazard rate, the default time follows an exponential distribution -- recall the survival function of an exponential random variable.
  2. Split the claim value into two scenarios (survival and default), weight by their probabilities, and use the independence of $\tau$ and $X$ to separate the expectations.
  3. Factor out $e^{-rT} E_Q[X]$ from both the risk-free and defaultable values. The CVA simplifies to a product of three intuitive terms: loss-given-default, default probability, and risk-free exposure.

Worked Solution

How to Think About It: CVA is one of the most practically important quantities in derivatives pricing -- it answers the question "how much less is my trade worth because my counterparty might not pay me?" Before any math, the structure is intuitive: you have a claim worth $e^{-rT} E_Q[X]$ if the counterparty is bulletproof. If they might default, you lose a fraction $(1 - R)$ of that value with some probability. So CVA should look like: loss-given-default $\times$ default probability $\times$ risk-free exposure. That is exactly what we will derive.

Quick Sanity Checks: If $\lambda = 0$ (no default risk), CVA should be zero and the defaultable value should equal the risk-free value. If $R = 1$ (full recovery), CVA should also be zero -- you get paid in full even in default. If $R = 0$ and $\lambda$ is large, CVA should approach the full risk-free value.

Derivation:

Part 1: Survival and default probabilities.

With constant hazard rate $\lambda$, the default time $\tau$ is exponentially distributed with rate $\lambda$. Its CDF is $Q(\tau \leq t) = 1 - e^{-\lambda t}$, so:

$Q(\tau > T) = e^{-\lambda T}$

$Q(\tau \leq T) = 1 - e^{-\lambda T}$

This is the standard result for a Poisson default process: the survival probability decays exponentially with the hazard rate.

Part 2: Defaultable claim value.

There are two scenarios at maturity $T$:

  • No default ($\tau > T$, probability $e^{-\lambda T}$): You receive the full payoff $X$.
  • Default ($\tau \leq T$, probability
    - e^{-\lambda T}$): You recover $R$ times the risk-free value $e^{-rT} E_Q[X]$.

Since $\tau$ and $X$ are independent, the time-0 value is:

$V_{\text{def}} = e^{-rT} E_Q[X] \cdot e^{-\lambda T} + R \cdot e^{-rT} E_Q[X] \cdot (1 - e^{-\lambda T})$

Factoring out $e^{-rT} E_Q[X]$:

$V_{\text{def}} = e^{-rT} E_Q[X] \left[ e^{-\lambda T} + R(1 - e^{-\lambda T}) \right]$

This simplifies to:

$V_{\text{def}} = e^{-rT} E_Q[X] \left[ R + (1 - R) e^{-\lambda T} \right]$

Sanity check: when $\lambda = 0$, the bracket becomes $R + (1-R) = 1$, so $V_{\text{def}} = e^{-rT} E_Q[X]$ (the risk-free value). When $R = 1$, the bracket is

$ regardless of $\lambda$. Both match our expectations.

Part 3: CVA.

The risk-free value is $V_{\text{rf}} = e^{-rT} E_Q[X]$. The CVA is:

$\text{CVA} = V_{\text{rf}} - V_{\text{def}} = e^{-rT} E_Q[X] - e^{-rT} E_Q[X] \left[ R + (1-R) e^{-\lambda T} \right]$

$\text{CVA} = e^{-rT} E_Q[X] \left[ 1 - R - (1-R) e^{-\lambda T} \right]$

$\text{CVA} = (1 - R)(1 - e^{-\lambda T}) \cdot e^{-rT} E_Q[X]$

This has a clean interpretation:

$\text{CVA} = \underbrace{(1 - R)}_{\text{LGD}} \times \underbrace{(1 - e^{-\lambda T})}_{\text{default prob}} \times \underbrace{e^{-rT} E_Q[X]}_{\text{risk-free exposure}}$

Practical Interpretation: A trader pricing a derivative with a risky counterparty charges the risk-free price minus the CVA. The formula decomposes neatly: LGD $(1-R)$ is your loss fraction in default, $(1 - e^{-\lambda T})$ is the probability you suffer that loss, and $e^{-rT} E_Q[X]$ is the exposure at stake. In practice, $\lambda$ is backed out from CDS spreads (the CDS spread $s \approx \lambda(1-R)$ for short maturities), and $R$ is typically assumed around 40% for senior unsecured.

Answer:

  1. $Q(\tau > T) = e^{-\lambda T}$, $\quad Q(\tau \leq T) = 1 - e^{-\lambda T}$
  1. $V_{\text{def}} = e^{-rT} E_Q[X] \left[ R + (1-R) e^{-\lambda T} \right]$
  1. $\text{CVA} = (1-R)(1 - e^{-\lambda T}) \, e^{-rT} E_Q[X]$

Intuition

CVA captures the price you pay for counterparty credit risk, and its structure is universal: it is always some version of "probability of loss times severity of loss times size of exposure." This problem strips away the complexity of real CVA calculations (netting, collateral, wrong-way risk, stochastic interest rates) to reveal that clean core. In practice, the independence assumption between default and exposure is the big simplification -- when the counterparty is more likely to default precisely when your exposure is largest (wrong-way risk, as in the 2008 crisis), CVA can be dramatically higher than this formula suggests.

The connection to CDS markets is worth internalizing. Since a CDS spread roughly equals $\lambda(1-R)$, you can read off the market-implied CVA for a trade directly from the counterparty's CDS curve. This is how desks actually compute CVA charges in practice -- not by estimating hazard rates from fundamentals, but by extracting them from traded credit instruments. The exponential survival probability is just the simplest building block; real CVA engines use term structures of hazard rates calibrated to the full CDS curve.

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