A Simple CVA Example

In my blog post two weeks ago I posted some figures with paths and exposures. Today I want to give a short example how Monte Carlo simulation can be used for the calculation of the CVA of an option. To keep things simple we will assume that only the counterparty can default (we will calculate the unilateral CVA).

The formula for unilateral CVA is given by

where LGD is the loss given default, i.e. one minus the recovery rate, DF is the discount factor (we assume a deterministic yield) and PD is the default probability of our counterparty. The term EE(t) stands for the expected exposure at time t. Exposure describes the amount of money that the counterparty owes us if it defaults at time t. As we only want to calculate a single instrument CVA we do not need to take into account netting. Furthermore we assume that o collateral agreements are in place. We can then calculate the positive exposure

where V(t) is the mark to market value of the

instrument a time t. As we model the underlying risk factor of the instrument by a stochastic process, V(t) and therefore also E(t) are random variables.

Equity paths: Black Scholes model with constant volatility.

We can calculate the expected exposure EE(t) by calculating the expectation value over all realisations of our random variable E(t). For an option the value of a trade at time t is simply given by

where K is the strike price. 

E(t) for the call option with strike K=100

EE(t)

Having calculated the exposure we put everything together with a discretised version of formula one

.