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Counterparty Risk Advanced Methods of Risk Management Umberto Cherubini

Counterparty Risk Advanced Methods of Risk Management Umberto Cherubini. In this lecture you will learn The difference between futures-style and Over-the-Counter markets. The Credit Valuation Adjustment (CVA) of derivative transactions (linear/non linear)

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Counterparty Risk Advanced Methods of Risk Management Umberto Cherubini

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  1. Counterparty Risk Advanced Methods of Risk Management Umberto Cherubini

  2. In this lecture you will learn The difference between futures-style and Over-the-Counter markets. The Credit Valuation Adjustment (CVA) of derivative transactions (linear/non linear) The impact of dependence between risk of the underlying asset and default of the counterparty. Learning Objectives

  3. Over-the-Counter Bilateral relationship Customized products Low basis risk Low liquidity Relevant risk Market Counterparty Futures-style Organized market Standardized products High basis risk High liquidity Relevant risks Market Basis risk OTC vs Futures-style

  4. Most of financial derivative contracts, and particularly those with retail counterparties are traded on what is called Over-the-Counter (OTC) market The OTC market allows the construction of customized positions for hedging or investment purposes The cost is illiquidity and credit risk (counterparty risk) Derivatives on OTC markets

  5. Consider a lineare OTC contract, i.e. forward, determined at time 0. Remember that if we only focus on the risk of price changes in the underlying asset we have CF(t) = v(t,T)EQ[S(T) –F(0)] = S(t) – P(t,T)F(0) where F(0) is the forward price at time 0. Notice that the product is linear, meaning delta = 1 and the replicating portfolio is static. The simplest example

  6. We make the assumption that: default occurs at time T, which is the maturity of the contract: this is a simplifying assumption that will be relaxed later one. We assume that if at maturity the marked-to- market value of the derivative contract is positive for the counterparty which goes in default, the other party is compelled to abide by the contract and pay its obligation. On the other side, if the value of the contract is negative for the counterparty in default the other party has an exposure equal to that value, with the same degree of seniority of the other liabilities. This assumption corresponds to the reality of legal provisions of counter party risk. Counterparty risk

  7. The value of the impact of counter party risk requires to distinguish the sign, long or short of the position. This is because counterparty risk is triggered by two events: Default of the counterparty The contract is “out-of-the-money” for the party in dafault, that it the contract has negative value for the counterparty in default. So, in case on the delivery date we have S(T) > F(0) the contract is in-the-money for the party long in the contract. If instead it is S(T) < F(0) the contract is in-the-money for the short party. In the former case the long party in the contract will be exposed to default risk, in the latter the short one will. Long and short positions

  8. Denote A the long party of the contract and B the short one. Let us introduce characteristic functions 1A and 1B assuming value 1 if the party A or B is in a default state and zero otherwise. Definiamo RRA e RRB i tassi di recupero delle due controparti. Nello stesso modo definiamo le loss-given-default LgdA = 1 – RRA e LgdB = 1 – RRB. Default and loss

  9. The pay-off value of the forward contract must take into account both its sign and its value in caso of default of the relevant counterparty. From the viewpoint of the long end of the contract we have CFA(T) = max[S(T) – F(0),0](1 –1B) + max[S(T) – F(0),0]RRB1B – – max[F(0) –S(T),0] = CF(T) – LgdB1Bmax[S(T) – F(0),0] Risk of the long party

  10. For the short end of the contract, the default event is relevant only in the hypothesis that the contract ends in-the-money. From the viewpoint of the counterparty CFB(T) = max[F(0) – S(T),0](1 –1A) + max[F(0) – S(T),0]RRA1A – – max[S(T) – F(0),0] = – CF(T) – LgdA1Amax[F(0) – S(T),0] Risk of the short party

  11. Counterparty risk corresponds to a short position is options. The option is of the call type for the long endo of the contract and of the put type for the other end of the contract. Exercise of the option is contingent on two events The value of the underlying asset at time T Default event of the relevant counterparty Counterparty risk

  12. The value of the product from the point of view of the long end of the contract will be given by CFA(t) = S(t) – v(t,T)F(0) – EQ[v(t,T)LgdB1Bmax[S(T) – F(0),0]] From the viewpoint of the short end of the contract we will then have CFB(T) = – S(t) + v(t,T)F(0) – EQ[v(t,T)LgdA1Amax[F(0) – S(T),0]] Contract evaluation

  13. Counterparty risk is represented by EQ[v(t,T)Lgdi1imax[(S(T) – F(0)),0]] with i = A, B and  = 1(–1) for call (put) options Counterparty risk is made by Interest rate risk Market risk of the underlying Credit risk of the counterparty Recovery risk All these factors may be correlated. Risk factors

  14. In what follows we will assume that Interest rate is independent of the other risk factors Default risk of the counterparty is not dependent on other risk factors. A simple model

  15. The value of counterparty risk is then v(t,T)EQ[Lgdi1i] EQ[max[(S(T) – F(0)),0]] Notice tha in case of a zero-coupon-bond issued by the party i we have Di = v(t,T) – v(t,T)EQ[Lgdi1i], or Di = v(t,T) – v(t,T)ELi, with ELi =EQ[Lgdi1i] the expected loss. In case of independence, then ELi v(t,T) EQ[max[(S(T) – F(0)),0] Evaluation

  16. Effect 1: ruling out counterparty risk leads to undervaluation of the overall exposure to credit risk Effect 2: if one does not consider counterparty risk, he comes out with the wrong price, and the wrong hedge. Effect 3: counterparty risk makes linear product non linear, so that changes in volatility may affect the value of the contract even though it is linear and one would not expect any effect. Effects of counterparty risk

  17. The sensitivity of the contract to small changes in the underlying is no more that of a linear contract. We get in fact A = 1 – ELBN() B = – 1 + ELAN(– ) with  =(ln(S(t)/F(0))+(r + ½ 2)(T – t))/[(T – t)1/2 ] Greek letters

  18. The second order effect of finite changes in the underlying is now given by – n()/ [S(t)(T – t)1/2] Changes in volatility affect the value of the position through a vega effect – S(t)n()/ [(T – t)1/2 ]  =(ln(S(t)/F(0))+(r – ½ 2)(T – t))/[(T – t)1/2 ] Gamma and Vega

  19. Forward contract Notional 1 million Volatility 20% Maturity 1 year Counterparty Loss given default (Lgd): 100% 1 year default probability: 5% Counterparty risk at the origin of the contract, for both the long and the short end of the contract: 3983 An example

  20. Long position

  21. Long position delta

  22. Counterparty risk (long)

  23. Short position

  24. Short position delta

  25. Counterparty risk (short)

  26. Counterparty risk and vol. (long)

  27. Counterparty risk and vol. (short)

  28. Assume now that default may occur before maturity, for example by a time . The value of exposure for the long position is now max[S() – P( ,T)F(0), 0 ] and for the short position max[P( ,T)F(0) – S(), 0 ] The value of exposure is given by a sequence of options that will be multiplied times the value of the default probability of the counterparty in the sub-periods. Default before maturity

  29. Partition the lifetime of the contract in a grid of dates {t1,t2,…tn} Denote Gj(ti) the survival probability of counterparty j = A, B beyond time ti. Compute [GB(ti -1) – GB(ti) ]Call(S(ti), ti; P(ti ,T)F(0), ti ) [GA(ti -1) – GA(ti) ] Put(S(ti), ti; P(ti ,T)F(0), ti ) respectively for long and short positions Aggregate the values obtained in this way from 1 through n. Counterparty risk

  30. In a swap cotnract both the legs are exposed to counterparty risk. In the event of default of one of the two parties the other takes a loss equal to the marked to market value of the contract, equal to the net value of the cash-flows. Remember that the net value of the swap contract is positive for the long end of the contract if the swap rate on the day of default of the contract is greater than the rate on the origin of the contract. Counterparty risk in swap contracts

  31. Swap counterparty risk exposure • Assume the set of dates at which swap payments are made be {t1, t2,…, tn} and default of the counterparty that receives fixed payments (B) took place between time tj-1 and tj. In this case, the loss for the party paying fixed is given by where sr is the swap rate at time tj and k is the swap rate at the origin of the contract. Notice that this is the payoff of a payer swaption (a call option on a swap).

  32. Swap counterparty risk exposure • Assume the set of dates at which swap payments are made be {t1, t2,…, tn} and default of the counterparty that pays fixed payments (A) took place between time tj-1 and tj. In this case, the loss for the party receiving fixed is given by where sr is the swap rate at time tj and k is the swap rate at the origin of the contract. Notice that this is the payoff of a receiver swaption (a put option on a swap).

  33. Credit risk: long party

  34. Swap credit risk: Baa

  35. A more general approach is to account for dependence between the two main events under consideration Exercise of the option Default of the counterparty Copula functions can be used to describe the dependence structure between the two events above. Dependence structure

  36. Consider a vulnerable digital call (VDC) option paying 1 euro if S(T) > K (event A). In this case, if the counterparty defaults (event B), the option pays the recovery rate RR. The payoff of this option is VDC = v(t,T)[H(A,Bc)+RR H(A,B)] = v(t,T) [Ha – H(A,B)+RR H(A,B)] = v(t,T)Ha – (1 – RR)H(A,B) = DC – v(t,T) Lgd C(Ha, Ha) Vulnerable digital call option

  37. Consider a vulnerable digital put (VDP) option paying 1 euro if S(T) ≤ K (event Ac). In this case, if the counterparty defaults (event B), the option pays the recovery rate RR. The payoff of this option is VDP = DP – v(t,T)(1 – RR)H(Ac,B) = P(t,T)Ha – v(t,T)(1 – RR)H(Ac,B) = P(t,T)Ha – v(t,T)(1 – RR)[Hb – C(Ha, Hb)] = v(t,T)(1 – Ha)– v(t,T) Lgd [Hb – C(Ha, Hb)] = v(t,T) – VDC – v(t,T) Lgd Hb Vulnerable digital put option

  38. Define the expected loss EL = Lgd Hb. If D(t,T) is a defaultable ZCB issued by the counterparty we have D(t,T) = v(t,T)(1 – EL) Notice that copula duality implies a clear no-arbitrage relationship VDC + VDP = v(t,T) – v(t,T) EL = D(t,T) Buying a vulnerable digital call and put option from the same counterparty is the same as buying a defaultable zero-coupon bond Vulnerable digital put call parity

  39. Vulnerable call and put options

  40. Vulnerable put-call parity

  41. Several techniques are used on the market to mitigate countrerparty risk. The ispiration of these techniques is the structure of futures-style, markets, based on three key principles Margins Evaluation (marking-to-market) and settlemen t of profits and losses before maturity of the contract. Compensation of profits and losses on different positions Risk mitigation clauses make more the computation of CVA more involved. Unfortunately, there is not much literature on the subject. Credit risk mitigation

  42. In principle one can think of different techniques to mitigate counterparty risk Margin deposit at the origin of the contract Position evaluation at daily on weekly period and requirement of the payment of a collateral. Netting agreement so that in case of default the net exposure between the counterparty is liquidated. CRM: theory

  43. According to the so-called ISDA Agreement the credit mitigating techniques used apply netting and the Credit Annex requiring periodic marking-to-market of the exposures. Unfortunately, there is no evidence on the diffusion of these techniques in the market practice (for example) it seems that Goldman Sachs did not use them. CRM: practice

  44. Assume a counterparty A with CFi positions in forward contracts i = 1, 2,…,p, with delivery prices Fi and delivery dates Ti with the same counterparty B. The value of each position is CFi = [Si(t) – v(t,Ti)Fi] where  = 1 represents long positions and  = – 1 short positions. A simple example

  45. CVA with netting • Assume that the counterparty B get into default at time . The value of the exposure at that date is equal to the pay-off of a basket option

  46. As it is well known basket options can only be evaluated by Monte Carlo simulation. The idea is then to select a grid of dates {t1,t2,…tn} and for each one of these to evaluate a basket option, with strike A(ti). CVA is now computed, for each date, as [G(ti-1) – G(ti)]Basket Option(S1, …Sp, ti; A(ti), ti) where G(ti) is the survival probability of counterparty beyond time ti. Extension of the use of collateral occur according to the same lines as in the univariate exposure. Monte Carlo simulation

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