Advanced Risk Management I

1. Advanced Risk Management I Lecture XVAs, toxic assets and prudent valuation

2. Derivatives, CVA, DVA, …

3. Plain vanilla swap (fixed-floating) • In a fixed-floating swap • the long party pays a flow of fixedsumsequal to a percentage c, defined on a yearbasis • the short party pays a flow of floatingpaymentsindexed to a market rate • The NPV is

4. Interest rate derivatives • Interest rate options are used to set a limit above (cap) or below (floor) to the value of a floating coupons. • A cap/floor is a portfolio of call/put options on interest rates, defined on the floating coupon schedule • Each option is called caplet/floorlet min(i(ti,ti+1), rcap) = i(ti,ti+1) – max(i(ti,ti+1) – rcap, 0) max(i(ti,ti+1), rfloor) = i(ti,ti+1) + max(rfloor – i(ti,ti+1), 0)

5. Call – Put = v(t,)(F – Strike) • Reminding the put-call parity applied to cap/floor we have Caplet(strike) – Floorlet(strike) =v(t,)[expected coupon – strike] =v(t,)[f(t,,T) – strike] • This suggests that the underlying of caplet and floorlet are forward rates, instead of spot rates. • This is for beginners. Graduate students know that forward rates are unbiased predictor of the future interest ratesunder the forward martingale measure (FMM)

6. Cap/Floor: Black formula • Using Black formula, we have Caplet = (v(t,tj) – v(t,tj+1))N(d1) – v(t,tj+1) KN(d2) Floorlet = (v(t,tj+1) – v(t,tj))N(– d1) + v(t,tj+1) KN(– d2) • The formula immediately suggests a replicating strategy or a hedging strategy, based on long (short) positions on maturity tj and short (long) on maturity tj+i for caplets (floorlets)

8. Swaption • Swaptions are options to enter a payer or receiver swap, for a swap rate at a given strike, at a future date. • A payer-swaption provides the right, but not the obligation, to enter a payer swap, and corresponds to call option, while the receiver swaption gives the right, but not the obligation, to enter a receiver swap, and corresponds to a put option.

9. Payer swaption • Assume one is paying floating and wants to have protection against an increase in interest rate, but does not want to buy a cap

10. Receiver swaption • Assume one is receiving floating and wants to have protection against a decrease in interest rate, but does not want to buy a floor

11. Swaption • A swaption gives the right, but not the obligation, to enter a swap contract at a future date tn for swap rate k. • Reset dates {tn ,tn+1,……tN} for the swap, with payments due at dates {tn +1 ,tn +2,……tN + 1} • Define i = ti +1–ti the daycount factors and the unit discount factor

12. Swaption valuation • The value of a swaption is computed using Swaption = A(t;n,N) EA{max[R(tn;n,N) - k ,0]} • Assuming the swap rate to be log-normally distributed (Swap Market Model), we have Black formula Swaption = A(t;n,N)Black[S(t;n,N),k,tn,(n,N),ω] with ω = 1,–1 for call (payer) and put (receiver), or explicitly: Swaption = ω[(v(t,tn) – v(t,tN))N(d1) – iv(t,ti) kN(d2)]

14. Counterparty risk in swap contracts • 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.

15. CVA and DVA • In recentyears new riskfactorshavebeenadded to the replicating portfolio, takinginto account severalvaluationadjustment. • CVA and DVA refer to adjustment for credit risk • The price of a swap isthen Swap – CVA + DVA

16. 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 tn-1 and tn. In this case, the loss for the party paying fixed is given by where R(tn,tN) is the swap rate at time tn 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).

17. 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 R(tn,tN) is the swap rate at time tn 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).

18. Credit risk: long party

19. Swap – CVA + DVA

20. CRM: theory • 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.

21. From CVA to CSA • According to the so-called ISDA Agreement the credit mitigating techniques used apply netting and the Credit Support Annex (CSA) 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.

22. CSA pricing • The frontier of pricing is about the impact of collateral. • If collateral is posted every day, then the evaluation must be made using daily discounting (EONIA) and it is risk-free (apart from loss occurring overnight). • Money for collateral must be collected by issuing debt. The cost is then the spread between the funding interest rate and EONIA. • It is more so if “rehypothecation” of collateral is not allowed.

23. FVA, MVA,… • Under both EMIR (EU) and Dodd-Frank (US) the trading in derivatives is moving from the OTC market (bilateral relationship) to «central clearing» (multilateral relationship with central clearing, i.e. futures style market). • Today the pricing of derivatives with central clearing includes the cost of funding the initial margin (MVA) and the cost of funding of the collateral posted, and the return on investment of the collateral received (FVA).

24. Prudent valuation and AVA

25. 1: Introduction Overview Prudent valuation is a fairly new topic (first full application Q1 2016), which allows rather different applications and may lead to substantiallydifferent results, even for similar portfolios. Mark to market Fair value Mark to model IPV Exclusions Scope of application Prudent value Simplified approach 0.1% Formula CRR art. 34, 105 EBA RTS 90% confidence level Core approach 9 AVAs Expert based Diversification Systems and controls Fall back

26. 1: Introduction Prudent valuation history April 2012 FSA “Regulatory Prudent Valuation Return”, Policy Statement August 2008 FSA “Dear CEO letter” 2004 2005 2008 2009 2010 2011 2012 2006 2007 November 2010 FSA “Product Control Findings and Prudent Valuation Presentation” June 2004 BCBS “International Convergence of Capital Measurement and Capital Standards” (Basel 2)

27. 1: Introduction Prudent valuation history 8 November 2013 EBA Quantitative Impact Study 31 March 2014 EBA Final Draft RTS and first application of prudent valuation EU Commission requests changes to EBA RTS 2012 2013 2014 2015 13 November 2012 EBA Discussion Paper (EBA/DP/2012/03) 10 July 2013 EBA Consultation Paper (EBA/CP/2013/28) 1 January 2014 CRR 575/2013 Q1 or Q2 2015 Expected final approval of EBA RTS by the European Commission Prudent valuation in place Dashed = unofficial

28. 1: Introduction Prudent valuation history • The idea of prudent valuation dates back to Basel 2 regulation (see BCBS, “International Convergence of Capital Measurement and Capital Standards – A revised framework”, June 2004). • In particular, sec. VI (“Trading book issues”), ch. B (“Prudent valuation guidance”), par. 690-701 set the requirements for prudent valuation in terms of • systems and controls, • valuation methodologies, • valuation adjustments or reserves, impacting regulatory capital (not P&L). • The CRR inherited most of the contents in its art. 105.

29. 2: Theoretical background • Theoretical background • Price opacity & financial crisis: the current crisis, and the Enron case before, has introduced the problem of valuation as a mean of diffusion of losses among financial institutions and assets • Pricing beyond Black-Scholes: the problem of getting the price wrong is linked to the fact that, already after the 19th October1987 market crash, the standard Black-Scholes setting of normal distribution of assets returns and perfect replication in continuous time of all financial products proved wrong • Market incompleteness & illiquidity: after Black-Scholes, new sources of risk, not traded in the market, such as volatility and correlation (smile and skew) have surfaced as key valutation elements. The hedging problem has become more complex and perfect hedging impossible (the market incompleteness problem). Moreover, if hedging can be done (volatility swaps or correlation swaps), it has to be done in highly illiquid markets, or even with OTC transations • Credit risk: “unearned credit spreads”, that is expected loss due to default of the counter party has become the major element of a financial product of contract. This has added even more focus on hedging complexities.

30. 2:Theoretical background A history of financial crises • September-October, 1998: LTCM, the major issue of the crisis is the impossibility to replicate financial derivatives in continuous time, and in perfectly liquid markets. It is the first case of incomplete markets. • December, 2001: Enron, the issue is lack of transparency in accounting data. The impact was uncertainty of valuation of similar companies or companies with the same auditor (Arthur Andersen). It was called “financial contagion by incomplete inforrmation”. • May 2005: the sudden drop in credit correlation triggered losses in financial intermediaries absorbing equity risk in securitization deals. It was a case about correlation uncertainty and hedging risk. Equity hedging strategies based on mezzanine were turned into losses by a major decrease in correlation. • 2007-2008: subprime crisis. The crisis themes were illiquidity, lack of transparency and an increase in correlation (systemic risk). On top of that, the peculiar issue of the crisis was the role played by the accounting standards in spreading contagion across intermediaries.

31. 2:Theoretical background Accounting and the subprime crisis . • “Default losses on US subprime mortgages about 500 billion dollars. • But in a mark-to-market world, deadly losses are valuation losses • Valuation losses as high as 4 trillion • Major banks failed without single penny of default • BIS study of rescue package: 5 trillions in committed resources. “ Eli Remolona, IV Annual Risk Management Conf. Singapore, July 2010

32. 2:Theoretical background Toxic assets . • “Financial assets the value of which has fallen significantly and may fall further, especially as the market for them has frozen. This may be due to hidden risks within the assets becoming visible or due to changes in extremal market environment” FT Lexicon • Toxic assets are a matter of: • Liquidity (“market frozen”) • Opacity and ambiguity (“hidden risks becoming visible”) • “Extremal market environment”

33. Securitization deals Senior Tranche Originator Junior 1 Tranche Special Purpose Vehicle SPV Sale of Assets Junior 2 Tranche … Tranche Equity Tranche

34. Tranche • A tranche is a bond issued by a SPV, absorbing losses higher than a level La (attachment) and exausting principal when losses reach level Lb (detachment). • The nominal value of a tranche (size) is the difference between Lb and La . Size = Lb – La

35. Synthetic CDOs Senior Tranche Originator Junior 1 Tranche Protection Sale Special Purpose Vehicle SPV Junior 2 Tranche CDS Premia Interest Payments Investment … Tranche Collateral AAA Equity Tranche

36. Securitization zoology • Cash CDO vs Synthetic CDO: pools of CDS on the asset side, issuance of bonds on the liability side • Funded CDO vs unfunded CDO: CDS both on the asset and the liability side of the SPV • Bespoke CDO vs standard CDO: CDO on a customized pool of assets or exchange traded CDO on standardized terms • CDO2: securitization of pools of assets including tranches • Large CDO (ABS): very large pools of exposures, arising from leasing or mortgage deals (CMO) • Managed vs unmanaged CDO: the asset of the SPV is held with an asset manager who can substitute some of the assets in the pool.

37. Large CDO • Large CDO refer to securitization structures which are done on a large set of securities, which are mainly mortgages or retail credit. • The subprime CDOs that originated the crisis in 2007 are examples of this kind of product. • For these products it is not possible to model each and every obligor and to link them by a copula function. What can be done is instead to approximate the portfolio by assuming it to be homogeneous .

38. 2: Theoretical background A simple example • Take a very simple financial product, that is an equity linked note promising to pay a participation to the increase in some stock market index in five years. • The replicating portfolio of the product is made up by: • A zero coupon bond five years maturity • An option with five years exercise time • The credit risk spread of the issues • The main sources of valuation uncertainty are then: • The calibration of the five year zero coupon from fixed income market data: data used and bootstrapping techniques. This valuation problem is common to other fixed income products. • The main source of uncertainty is volatility over the five year time horizon. Here there may be shortage of data or a problem of techniques. On one side, exchange traded derivatives do not have a liquid market for that maturity and there is a problem to extend implied volatility beyond the traded maturities. On the other side, OTC quotes take into account CVA/DVA. • The calibration of the conditional default probabilities up to five years from CDS and defaultable bond data. The solution may differ for data used and the bootstrapping techniques applied. • Other sources of uncertainty: correlations among risk factors-

39. 2: Theoretical background Incomplete markets: definition • Complete markets are defined by all financial products being “attainable”. This means that the payoff of every financial contract or product can be exactly replicated by some trading strategy. This implies lack of frictions and continuous rebalancing of the replicating portfolio. Markets are assumed to be perfectly liquid and trading is costless. • If markets are complete, there exists a unique Equivalent Martingale Measure (EMM) such that the price of each and every asset can be computed by the expected value under such measure, and discounted with the risk-free rate. With complete markets the price of each financial product would be unique, and there would be no valuation uncertainty problem. • In real world markets are incomplete and there exists a valuation uncertainty problem. The reason is that no perfect hedge exists. More precisely, the reasons for incomlplete markets are: • There are not enough assets to hedge all possible risk factors (no enough Arrow-Debreu prices) • Replicating portfolios cannot be rebalanced in continuous time in such a way as to allow for a perfect hedge • There is not enough liquidity in the market, particularly in stress times, to allow rebalancing of the replicating portfolios.

40. 2: Theoretical background Incomplete markets: theory • From a technical point of view, selecting a price in incomplete markets amounts to choose a probability measure (pricing kernel) in a set of probability measures. This set contains the probabilities such that the price of each product is a martingale. This implies that for each product it is not possible to find a replicating strategy that attains the product for sure. Price of the contract(t) = EQ[pay-off (T) | Q  ] • The problem is then to define: • The set of probabilities includes all the risk-neutral probabilities • A strategy to select a probability in the set. • Notice that the problem of selecting a probability amounts to selecting a lottery. So, a possible strategy to select a specific probability is to use expected utility or some of its extensions. • Hedging error: every probability measure that is chosen is subjected to hedging error. Based on this, for example, one could select the probability with the lowest hedging variance, in the set with some expected hedging cost.

41. 2: Theoretical background Alternative theories for price bounds • There are two different approaches to address valuation uncertainty. In both cases the price bounds are obtained by assuming interval valuation. • Uncertain Volatility Model: • Volatility is assumed be included in a given interval • This leads to two conservative pricing bounds (BSB PDE functions) • Avellaneda, Levy and Paràs (1996) AMMF • Choquet pricing • Interval probabilities (MMEU, Gilboa and Schmeidler, 1989) • Conservative valuation (Choquet integral) • Cherubini (1997) AMF, Cherubini and Della Lunga (2001) AMF AMF = Applied Mathematical Finance • MMEU: assume the worst possible probability scenario and select the choice tha yields the maximum expected utility.

42. 2: Theoretical background Uncertain Volatility Model • Set the delta-neutral portfolio • Volatility choice

43. 2: Theoretical background Uncertain Volatility Model • The Black-Scholes formula becomes non linear (Black-Scholes-Baremblatt) where

44. 2: Theoretical background Choquet pricing • Long and short positions • Long and short positions are represented by Choquet integral with respect to capacities. • Given a function f(.) and a non-additive measure , the lower Choquet integral and the upper Choquet integral is

45. 2: Theoretical background Dynamic replication of illiquid derivatives • Now assume you are trading a derivative with a costumer, maybe for a large quantity of the underlying asset (concentration risk) or for an illiquid underlying. In this case, standard textbook references for the pricing of options do not apply, since the production process of the derivative has an impact on the underlying asset. • Here the only process is to start with a dynamics of the underlying asset and to try a replication strategy, allowing for the liquidity cost of rebalancing the portfolio, and the funding cost of changing the leverage position. So, the market price incorporates liquidity costs, both in the sens of market liquidity and funding liquidity. Both the sources of cost are all the more relevant the larger the size of the position. • The problem of finding an optimal trade-off between liquidity cost and liquidity risk is extremely involved. In fact, it requires to define trading strategies: how many times to rebalance, when, whether at fixed intervals or contingent on some rule.

46. 3: Regulation Overview • Articles 34 and 105 of Capital Requirements Regulation (CRR, n. 575/2013), in force since 1 January 2014, require financial institutions to apply prudent valuation to all fair value positions (included positions outside the trading book), setting a new prudential requisite for regulatory capital including valuation uncertainty. • The difference between the prudent value and the fair value, accounted in the institution’s balance sheet, is called “Additional Valuation Adjustment” (AVA), and is directly deducted from the Core Equity Tier 1 (CET1) capital. • Following the CRR, the EBA published a Discussion Paper (EBA/DP/2012/03), a Consultation Paper (EBA/CP/2013/28), and a Final Draft (EBA/RTS/2014/06), to be approved by the EU Commission, setting the Regulatory Technical Standards (RTS) for prudent valuation. • The EBA Final Draft defines the AVA calculation methodology using two alternative approaches, named Simplified Approach and Core Approach. The Final Draft sets also the requirements on systems, controls and documentation that should support the prudent valuation process.

47. 3: Regulation Fair Value Vs Prudent Value Fair Value • Regulation: IFRS13 • Application: balance sheet • Percentile: 50% (expected value) • The price that would be received to sell an asset or be paid to transfer a liability in an orderly transaction between market participants at the measurement date • Must include all the factors that a market participants would use, acting in their economic best interest. • Atoms: single trades. • Fair value adjustments • Non-entity specific Prudent value • Regulation: CRR/EBA • Application: CET1 • Percentile: 90% • Must reflect the exit price at which the institution can trade within the capital calculation time horizon. • Atoms: valuation positions subject to a specific source of priceuncertainty • Entity specific • Subject to diversification benefit (50% weight for MPU, CoCo, MoRi AVAs)

48. The AVA hierarchy 3: Regulation EBA RTS: core approach [2/3] Core approach Additional Valuation Adjustments CVA/FVA AVAs Other AVAs Main AVAs Investing & Funding Cost (FVA) Art. 13 Future Admin Costs (FAC) Art. 15 Market Price Uncertainty (MPU) Art. 9 Close Out Costs (CoCo) Art. 10 Model Risk (MoRi) Art. 11 Early Termination (EaT) Art. 16 Operational Risk (OpR) Art. 17 Unearned Credit Spread (CVA) Art. 12 Concen- trated Positions (CoPo) Art. 14 Market risk factors 50% weights for diversification Market risk factors Split onto main AVAs Non-market risk factors 100% weights, no diversification

49. 4: AVA calculation Definitions and basic assumptions

50. 4: AVA calculation Definitions and basic assumptions [10] Price distribution, fair value, fair value adjustment, prudent value, AVA What about realprice distributions...? Fair value adjustment AVA Fair value (mean) Fair value adjusted Prudent value (quantile)