This case asks you to use a contagion model to assess the risk faced by someone exposed to the default of a clearing house (or central counterparty, CCP). As background, you may want to read the articles about the risk of CCP default.

The model will make the following simplifying assumptions.

- The clearing house has 25 members. (You are not a member. You are a client of one or more members.)
- The clearing house defaults if 5 or more members default within a month.
- Defaulting members are instantly replaced by new non-defaulting members so that there are always 25 members.
- If the clearing house defaults, the recovery rate on its obligations is R = 0.10.
- There will be no government bailout if the CCP defaults.
- The riskless rate is zero.

Default by member firm i is governed by a Poisson process J (i) t which jumps into default with intensity λt. The intensity process is common to all member firms, but, given λt, the Poisson jumps are independent across firms.

The intensity obeys the Hawkes process, dλt = κ(λ̄ − λt) dt + ∑ i β dJ (i) t , where the sum runs over the 25 active members. Thus a default by the jth member means that dJ (j) t = +1, which raises the risk of all the others defaulting.

We will discretize the model to intervals ∆t = 1/52, or one week, so that each member’s probability of default each week is λ1 /52 where λ1 is the intensity at the start of the week.

Our model for the mechanics of default is as follows. All clients’ positions are marked-to-market continually until CCP default, and thus they have zero value. Once default occurs, the CCP stops paying clients with winning positions.

The value of these positions continues to evolve after default. We will assume that the clients’ claims on the CCP are frozen at the end of one week after a default occurs. The CCP then cancels all client positions and pays the recovery rate on the positive claims.1 Continued on next page.

1Note that your initial margin is assumed to be NOT at risk. The CCP cannot seize it to pay other clients.

**Question 1.**

Default clustering (or contagion) in this model is controlled by the parameter β. So accurately estimating it will have important implications.

Assume that under the physical measure we have parameters κ = 4, λ̄ == 0.01. Assume the system is currently healthy, so that λ0 = λ̄. Simulate 10 5 histories of the system for five years with β = 0.05. How many clearing house defaults do you observe? How much does the answer change if β = 0.15.

Can you think of any ways that β could be estimated from historical data?

**Question 2 (a).**

CVA is computed under the risk neutral measure. Assume that under this measure we have κQ = 2, λ̄Q = 0.02 and βQ = β = 0.10. Re-run your simulation for 1 year with these values, assuming λ

Q 0 = λ̄

- Keep track of the date of the first CCP default (if any) along each simulated path.

Now suppose you have a 1-year futures position with the CCP in a commodity that is unrelated to the financial system (for example, sugar). Under the risk neutral measure, its futures price, f, obeys df/f = σ dW and σ = .20.

Your position has a notional value of $1 million. Use your simulations of the CCP default times and the assumptions on default loss given above to compute the CVA (in dollars) for your futures position. Explain exactly what your algorithm is doing.

**Question 2 (b).**

Now suppose you have a cleared trade with the CCP which is a long position in a 5-year CDS referencing one of the member banks. So if the CCP defaults there is a probability of 1 in 5 that your CDS is triggered. Suppose the position has a notional value of $1 million and the recovery rate on the bank’s debt is R = 0.5. What is the CVA on this position?

For this calculation, you may ignore the component that comes from the mark- to-market in states where the CCP defaults and the member bank does not. Again, explain clearly how you are computing the CVA.

**Question 3.**

This question asks you to think about the value of a “too-big-to-fail” guaran- tee that tax-payers may be providing to clearing houses. Even though the true probability (under our model) of a CCP default is very small, the guarantee’s value is determined by the risk-neutral measure.

Mathematically, our model of CCP default exposure is very similar to the default exposure of a senior tranche of a CDO. As we have seen, the 2007-2009 financial crisis was kindled by losses in senior tranches of CDOs made up of assets whose default processes were very highly correlated. So we don’t want to make the mistake of over estimating their safety again.

What is the value of a 5-year CDS on the clearing house (with notional value of $1 million). Express your answer as a price today, not a fee payment.