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Credit Value Adjustment, Bermudan option andWrong Way RiskQian Feng1joint work with Cornelis W. Oosterlee1,21 CentrumWiskunde & Informatica, 2 Delft University of Technologyfunded by STWcollaboration with ING Bank(CWI)CVA and Bermudan option22/05/20161 / 16

Outline1Counterparty Credit Risk2Credit Value Adjustment3Bermudan option4Wrong Way Risk(CWI)CVA and Bermudan option22/05/20162 / 16

Counterparty Credit RiskCounterparty Credit RiskCounterparty Credit Risk(CCR)CCR is the risk to each party of a contract that the counterparty may fail tofulfill its obligations, causing losses to the other party.ExampleCompany A agrees to lend Company B a certain amount of money. It isexpected that company A will provide the money on time and company B willpay the money back. There is counterparty risk for both that company B maynot be able to pay the loan while company A may stop providing the agreedupon funds.OTC (over-the-counter) derivative contractInterest rate swap,FX forwards.(CWI)CVA and Bermudan option22/05/20163 / 16

Counterparty Credit RiskBasel and CCRBasel Committee on Banking SupervisionThe Basel Committee on Banking Supervision (BCBS) is a committee ofbanking supervisory authorities aiming to improve the quality of bankingsupervision worldwide.Credit exposurethe replacement cost of the contract, which is equal to the greater of the fairmarket value of the contract and zero.Exposure measurementsBasel II: Expected Exposure(EE), Potential future exposure (PFE)Basel III: Credit value adjustment (CVA), after the financial crisis2008-2009.(CWI)CVA and Bermudan option22/05/20164 / 16

Credit Value AdjustmentCredit Value AdjustmentCredit Value Adjustment (CVA)CVA is the market value of counterparty credit risk:"ZTQCVA ELGD#D(0, t)Et dPD(t) ,(1)0whereLGD: the percentage of loss given default;E: exposure Et max(Vt , 0), where Vt represents the mark-to-marketvalue of the portfolio;PD: the default probability, dPD(t) PS(t dt) PS(t) the probabilitythat default happens during time [t, t dt];Q: risk-neutral probability measure.D: the discounting factor D(0, t) 1/B(t), where banking accountRtB(t) exp( 0 rs ds) with the risk-free short rate rs ;(CWI)CVA and Bermudan option22/05/20165 / 16

Credit Value AdjustmentIntensity approachSurvival probabilityZG(s, t) exp !thu du(2),swhere hu represents the positive intensity at time u:hu du defines of probability that default time occurs during period[u, u du].hu can be a constant, a deterministic function, a stochastic process.Marginal default/survival probability model(CWI)Survival: PS(t) EQ [G(0, t)] .(3)Default: PD(t) 1 PS(t),(4)CVA and Bermudan option22/05/20166 / 16

Bermudan optionBermudan option IBermudan optionA Bermudan option is an option that be exercised on a number of dates.PayoffLet St be the stock price at time t(max(0, K St ) put,g(St ) max(0, St K ) call,(5)where K the fixed strike.Exercise datesLet T {0 t1 . . . tN T } be the collection of early-exercise dates.(CWI)CVA and Bermudan option22/05/20167 / 16

Bermudan optionBermudan option IIDefault-free value V0 max E D(0, τ )g(Sτ ) ,Q(6)τ Twhere τ is the exercise time, and let τ be the solution which represents theoptimal exercise time when maximizing the default-free value.Default-adjusted value A0 max EQ D(0, β)G(0, β)g(Sβ ) .(7)β Twhere β is the exercise time, and let β be the solution which represents theoptimal exercise time when maximizing the default-adjusted value.(CWI)CVA and Bermudan option22/05/20168 / 16

Bermudan optionOptimal exercise boundary at time tm TOptimal default-free exercise value x (tm ):Vt (x (tm )) g(x (tm )) 0,(8)where the option value at time tm conditioned on Stm x is given by QVt (x) maxE D(t, τ )g(Sτ ) Stm x .τ {tm 1 ,.,tM }(9)Optimal default-adjusted exercise value y (tm ):At (y (tm )) g(y (tm )) 0,(10)where the option value at time tm conditioned on Stm x is given by QAt (x) maxE D(t, β)G(t, β)g(Sβ ) Stm x .β {tm 1 ,.,tM }(CWI)CVA and Bermudan option22/05/2016(11)9 / 16

Bermudan optionNumerical SchemesCOS methodFourier-cosine expansion and FFTFang and Oosterlee , Ruijter and Oosterlee .Stochastic Grid Bundling Method (SGBM)Simulation, regression, bundling, and the relation of the (discounted)characteristic function and the (discounted) moments.Jain and Oosterlee .(CWI)CVA and Bermudan option22/05/201610 / 16

Bermudan optionOptimal early exercise boundary100no defaulth 0.03h 0.395stock value90continuation region85807570Exercise region6500.050.10.150.20.25timeOptimal early exercise values of a Bermudan put option with constant intensityh {0.03, 0.3} over period [0, T ]. Parameters S0 100, r 0.01, σ 0.4.Expiration T 0.25, early exercise steps M 10, strike K 100.(CWI)CVA and Bermudan option22/05/201611 / 16

Bermudan optionBermudan optionsβ : exercised by maximizing default-adjusted valueIntensitydefault-adjusted valuedefault-free valueCVAh 0.03h 0.37.79437.40777.84167.81620.04730.4085τ : exercised by maximizing default-free valueIntensitydefault-adjusted valuedefault-free valueCVAh 0.03h 0.37.79367.37007.84227.84220.04860.4722(CWI)CVA and Bermudan option22/05/201612 / 16

Wrong Way RiskWrong Way RiskWrong Way Risk(WWR)This type of risk occurs when exposure to a counterparty is adverselycorrelated with the credit quality of that counterparty.ExampleA put option written by bank A on equity of bank B, and bank A and bank Bhave similar portfolio:Credit quality of bank A and bank B goes worse;stock price of bank B decreases;market value of the put option increases;the likelihood of default of bank A increases;(CWI)CVA and Bermudan option22/05/201613 / 16

Wrong Way RiskStochastic intensityHull-White model dxt 1 2r σ dt σdWt1 ,2(12)dyt γ(ȳ yt )dt ηdWt2 ,(13)ht ψ(t) yt ,(14)where xt log(St ) is the log-stock value, r represents the risk-free short rate,σ 0 the constant implied volatility; γ 0 corresponds to reversion speed, ȳto long run reverting level and η 0 to the volatility of variable yt , and dWt1and dWt2 are two Wiener processes with correlation dWt1 · dWt2 ρdt, and ρthe correlation coefficient; deterministic function ψ(t) satisfies ψ(0) h0 y0 .Drawback: intensity may become negative.(CWI)CVA and Bermudan option22/05/201614 / 16

Wrong Way RiskCVA stress testing-European put ion ;CVA stress testing of w.r.t correlation. Parameters S0 100, r 0.01, σ 0.4, γ 0.2, ȳ 0.1, η 0.2; marginal survival function exp( 0.3t) andy0 0.3. An European put option with expiration T 0.25 and strike K 100.Worst WWR ratio 1.18; best RWR ratio 0.82.(CWI)CVA and Bermudan option22/05/201615 / 16

Wrong Way RiskConclusion and future workConclusionCredit risk may change the early exercise strategy for credit-risk-alertinvestors: the optimal early exercise values are increased for put optionsand decreased for call options;Credit risk and WWR cannot be eliminated by changing the exercisestrategy.FutureWrong way risk(CWI)CVA and Bermudan option22/05/201616 / 16

Wrong Way RiskBibliographyF. Fang and C. W. Oosterlee. A Fourier-Based Valuation Method for Bermudan and Barrier Options under Heston’s Model.SIAM Journal on Financial Mathematics, 2(1):439–463, 2011.M. J. Ruijter and C. W. Oosterlee. Two-Dimensional Fourier Cosine Series Expansion Method for Pricing FinancialOptions. SIAM Journal on Scientific Computing, 34(5):B642–B671, 2012. 1994.T. R. Bielecki and M. Rutkowski. Credit risk: Modeling, Valuation and Hedging. Springer Science & Business Media, 2002.S. Jain and C. W. Oosterlee. The Stochastic Grid Bundling Method: Efficient Pricing of Bermudan Options and TheirGreeks . Applied Mathematics and Computation, 269:412 – 431, 2015.Y. Shen, J. van der Weide, and J. Anderluh. A Benchmark Approach of Counterparty Credit Exposure of Bermudan Optionunder Levy Process: The Monte Carlo-COS Method. Procedia Computer Science, 18(0):1163 – 1171, 2013. 2013International Conference on Computational Science.I. Ruiz, R. Pachón, and P. Del Boca. Optimal Right and Wrong Way Risk, 2013.J. Hull and A. White. CVA and Wrong-Way Risk. Financial Analysts Journal, 68(5):58–69, 2012.J. Gregory. Counterparty Credit Risk: The New Challenge for Global Financial Markets. The Wiley Finance Series. JohnWiley & Sons, 2010.Feng, Runhuan, and Hans W. Volkmer. An identity of hitting times and its application to the valuation of guaranteedminimum withdrawal benefit. arXiv preprint arXiv:1307.7070 (2013).(CWI)CVA and Bermudan option22/05/201616 / 16

Credit Value Adjustment Credit Value Adjustment Credit Value Adjustment (CVA) CVA is the market value of counterparty credit risk: CVA EQ " LGD Z T 0 D(0;t)E tdPD(t) #; (1) where LGD: the percentage of loss given default; E: exposure E t max(V t;0), where V t represents the mark-to-market value of the portfolio;