Let’s calculate CVA (credit value adjustment) analytically. We will see that analytical CVA calculation is quite complex even for a fairly simple transaction (a vanilla swap). A few shortcuts will help us simplify the calculation.

### Problem

Consider a five-year semi-annual vanilla payer swap in Euro: every six months (*coupon period*) we pay interest over six months at a fixed rate *K *and receive interest over the same period at Euribor 6-month rate fixed at the start of the period (*fixing date*). The number of days in the coupon period is calculated using Actual/360 convention, the notional amount is 1. The swap starts today (9 April 2010) and the fixed rate *K *equals today’s par swap rate, so mark-to-market value of the swap is zero now. There is no netting or collateral agreement for this trade. For simplicity, we ignore all other details (for instance, the fact that Euribor actually fixes 2 days before the start of the coupon period, or that the coupon payment is rolled over if it falls on a weekend). We want to calculate the credit value adjustment (CVA) for this swap. By definition, CVA is the difference between the risk-free value of the trade and its value that takes into account the possibility of the counterparty’s default [1].

### Inputs

We have:

- Discount curve
*P(t)*for Euro.*P(t)*is today’s price of a zero-coupon riskless bond that pays 1 euro at time*t*. - Implied volatility cubes
*C(t, τ, k)*and*S(**t, Ï„, k**)*for Euro.*S(**t, τ, k**)**Ï„*, strike*k*and expiry date*t*.*C(t, Ï„, k)*is the interest rate volatility implied by a caplet with tenor*τ*, strike*k*and expiry date*t*. - Recovery rate
*R*. If the counterparty defaults, we expect to recover*E*R*, where*E*is our exposure to the counterparty at the moment of default. In other words, our losses will be*(1-R)*E*. - Probability
*PD(t)*of the counterparty’s default.*PD(t)*is the probability that the counterparty defaults between now and time*t,*as viewed by the market (for instance, derived from CDS prices as shown in [2]).

### Calculation

Assuming independence between exposure and counterparty’s credit quality, CVA is given (see [1]) by the formula:

where is the end date of the trade and is the discounted expected exposure computed under an equivalent martingale measure (e.g. risk-neutral measure).

Recovery rate R and default probability PD are given, so we only need to calculate the expected exposure .

We recall that the exposure at time *t* is the maximum of the trade’s mark-to-market value (*MtM*) and zero:

Consider a contract that gives its holder the right, but no obligation, to enter the swap at time *t*. Essentially it is an option to acquire the swap at no cost at time *t*. This option is equivalent to receiving max(*MtM(t),0*) at time *t*. Therefore, its value today is the discounted expectation of max(*MtM(t),0*) calculated under an equivalent martingale measure, which is the definition of . Thus,

** the discounted expected exposure on a trade at time t equals the value of the option to acquire the trade at no cost at time t**.

Now let us fix time in the future and calculate today’s value of the option to acquire the swap at *. The MtM of the swap is defined by the value of the outstanding cashflows. If the swap’s coupon dates are and ** *for some i, then at one already knows the cashflow that is due at . Indeed, the floating rate was fixed at , so it is known at *. Remaining cashflows are not known yet, but they can be priced as a swap starting at and ending at . We see that the payoff of the option in question depends on:*

- Euribor fixing at
- Forward swap rate at

To avoid having to price a path-dependent derivative, we replace the option to enter the swap at with the sum of two options: a cap starting at and ending at and a swaption starting at and ending at . It is easy to see that the sum of the payoffs of the cap and the swaption is greater or equal to the payoff of the original option, so the sum of the values of the cap and the swaption give us a conservative estimate of the expected exposure.

We start with pricing a cap starting atÂ and ending atÂ . Its underlying rate is Euribor 6-month and strike is *K*. As shown in [3], we can use Black’s formula to price the cap. We take implied volatility

,

calculate the forward rate

and use Black’s formula to get the cap value

Next, we price the swaption part in a similar way, again taken from [3]. The implied volatility is

the forward swap rate is

and the swaption price is

The total upper boundary for discounted expected exposure at is .

Defining the quantity

we get the upper boundary for CVA as a sum

We take CVA approximation

### Conclusion

Although we were able to compute an analytical approximation for CVA, we had to make numerous assumptions to simplify the task. For derivatives more complex than swaps (or even for a portfolio of swaps) analytical computation of CVA would not be feasible. Instead, we will have to compute CVA by Monte Carlo simulation, as outlined in [1]. An example of CVA calculation with Monte Carlo is shown here.

**References**

1. Zhu, Steven H. and Pykhtin, Michael, AÂ Guide to Modeling Counterparty Credit Risk. GARP Risk Review, July/August 2007. Available at SSRN: http://ssrn.com/abstract=1032522

2. Beck, Ronald. The CDS market: a primer. Available online: http://www.dbresearch.com/PROD/DBR_INTERNET_EN-PROD/PROD0000000000185396.pdf

3. Brigo, Damiano and Mercurio, Fabio, Interest Rate Models – Theory and Practice: With Smile, Inflation and Credit. Springer; 2nd edition (August 2, 2006).

Hello Cyril, you asserts ‘To avoid having to price a path-dependent derivative, we replace the option to enter the swap at t’ with the sum of two options: a cap starting at t(i-1) and ending at t(i) and a swaption starting at t(i) and ending at t(n) . It is easy to see that the sum of the payoffs of the cap and the swaption is greater or equal to the payoff of the original option…’ May you please explain more how the sum of the 2 replacement options’ payoffs is greater or equal to the original swaption’s payoff?

Let be the value of a fixed cashflow at and be the expected value of the floating cashflow at , where the expectation is computed at time . The expected payoff of the original option is . The expected payoff of the cap is , and that of the swaption is . Denote , , then the payoff of the original option is , of the cap is , and of the swaption is . Obviously, , which means that the cap and the swaption are better or equal to the original option.