Efficient Lifetime Portfolio Sensitivities: AAD Versus Longstaff-Schwartz Compression Chris Kenyon 26.03.2014 Contact: Chris.Kenyon@lloydsbanking.com
Acknowledgments & Disclaimers Joint work with Andrew Green and Chris Dennis. The views expressed in this presentation are the personal views of the speaker and do not necessarily reflect the views or policies of current or previous employers. Not guaranteed fit for any purpose. Use at your own risk., 2/47
Contents Introduction Methods Implementation Example Conclusions Appendix Bibliography
Introduction Methods Implementation Example Conclusions Appendix Bibliography
Introduction Capital and funding are legal and existential requirements for financial institutions holding derivatives Requirements have expanded in the past few years (Dodd and Frank 2010; Treasury 2013; BCBS-189 2011) Pricing derivatives means including, i.e. hedging, all cashflows associated with the trade over its lifetime Calculations involve portfolios because capital and funding are netting-set and institution-level issues, 5/47
Objective Lifetime portfolio Value-at-Risk (VAR) Lifetime portfolio Expected Shortfall (ES) Motivation Key elements to price funding and capital, 6/47
Problem Scale First Look Total trades: 100,000 to 1,000,000 (or more) Trades by netting set: 1 to 100,000 (or more) Netting sets: 1,000 to 10,000 Material portfolio maturity: few months (FX) to decades (Rates) Notional size: billions to multiple trillions (USD equivalent) New aspects of problem are costs of capital and funding which tie trades together non-linearly., 7/47
Desk View Capital is part of the pricing picture which requires lifetime costs, not just costs on trade date Excess Profit Excess Profit xva ~ Resource Management KVA desk costs CVA/FVA desk costs Capital Credit Funding Market Risk (Basel 2) CCR RWA (Basel 2) CVA RWA (Basel III) CCP (BCBS 227,253) AVA (Prudent Valuation) CVA DVA FVA / Collateral cost +/- IM/IA (BCBS 261, PruVal) Institutional Costs Institutional Costs Base price Capital Modelling Under Basel III RWA-Calculation- & -Price-driven Behavior 8, 8/47
Trading Trading Choices actively compares different parts of the pricing picture, e.g. capital vs funding vs credit vs IM Trading Venue CSA with Cost Source no CSA CSA segregated bilateral IM CCP Balance Sheet Resource Cost MR Market Risk capital CCR highly reduced B2 default risk, B3 default fund Capital highly CVA reduced NA B3 but CRD IV exceptions AVA EBA term funding capitalization and VA Credit CVA reduced highly reduced NA CVA, DVA FVA highly reduced highly reduced NA liquidity gap from unsecured exposure Funding Collateral VM NA NA cheapest of cash, repo, and including optionality IM NA NA cost of funding IM Capital Modelling Under Basel III RWA-Calculation- & -Price-driven Behavior 9, 9/47
CSA Back-to-Back Corporate Interest Rate Swap A Bank Interest Rate Swap B Market Case I Trade A ve Value to Corp Trade A +ve Value to Bank Trade B -ve Value to Bank Trade B +ve Value to Mkt Variation Margin Corporate Gives Collateral to Bank Bank Rehypothecates Collateral to Market Case II Trade A +ve Value to Corp Trade A -ve Value to Bank Trade B +ve Value to Bank Trade B -ve Value to Mkt Variation Margin Bank Rehypothecates Collateral to Corporate Market gives Collateral to Bank, 10/47
CSA+CCP/IM Back-to-Back Corporate Interest Rate Swap A Bank Interest Rate Swap B CCP Case I Trade A ve Value to Corp Trade A +ve Value to Bank Trade B -ve Value to Bank Trade B +ve Value to CCP Variation Margin Corporate Gives Collateral to Bank Bank Rehypothecates Collateral to Market Initial Margin No Collateral Source Bank Borrows Unsecured Case II Trade A +ve Value to Corp Trade A -ve Value to Bank Trade B +ve Value to Bank Trade B -ve Value to CCP Variation Margin Bank Rehypothecates Collateral to Corporate Market gives Collateral to Bank Initial No Collateral Source Bank Margin Borrows Unsecured, 11/47
Calculation Methods Classification Alternatives Calculation Type Counterparty EAD Calculation Credit Risk CEM Function of Netting set Value Standardized Function of Netting set Value Internal Model Method Exposure profile Weight Calculation Standardized External Ratings FIRB Internal & External Ratings AIRB Internal & External Ratings, Internal LGDs CVA Capital Standardized Function of EAD Advanced VAR / SVAR on Regulatory CVA or CS01 Market Risk Standardized Deterministic formulae Internal Model Method VAR + SVAR CCP initial margin (IM) typically depends on VAR or ES calculation, 12/47
VAR and Expected Shortfall (ES) are Risk Measures VAR, Value-At-Risk, = percentile of loss distribution VAR(α) = inf{loss R : F(loss) α} ES (aka CVAR) = conditional expectation of loss given extreme event ES(β) = 1 β β 0 VAR(α)dα Both depend on calculating a distribution Both depend on (relatively) extreme events, e.g. 99th percentiles or more are commonly used, 13/47
How extreme are VAR-relevant and ES-relevant events? Percent 5 4 3 2 1 2004 2006 2008 2010 2012 2014 Basis Points 50 0 50 2004 2006 2008 2010 2012 2014 Year But are differences from 2006 relevant to 2014?, 14/47
Conditional-shock distribution Percent 30 20 10 0 10 20 30 2004 2006 2008 2010 2012 2014 Year SD percent 10 8 6 4 2006 2008 2010 2012 2014 4 40 Ratio 2 0 Percent 20 0 2 20 4 2006 2008 2010 2012 2014 2006 2008 2010 2012 2014, Year 15/47
VAR-relevant and ES-relevant events are large for historical 5Y USD Swap rates 0.10 0.08 0.06 0.04 0.02 0.00 VAR(99%, 10-day) of absolute values = 29% ES(99%, 10-day) of absolute values = 33% Relevant events are significant relative curve moves so sensitivity-based pricing methods may not provide accuracy, 16/47
Problem Scale Second Look Scenarios Original Problem Portfolio Lifetime, 17/47
Introduction Methods Implementation Example Conclusions Appendix Bibliography
Pricing by Replication Price any derivative (portfolio) by constructing another portfolio with the same cashflows, and dynamics: ˆV = Π d ˆV = dπ Price ˆV, is unadjusted price V, plus adjustments U for credit, funding, collateral, capital, etc. For details see (Green, Kenyon, and Dennis 2014). ˆV = V + U Formally we set up a PDE and then use the Feynman-Kac theorem to transform it to an expectation. We arrive at U = CVA + DVA + FCA + COLVA + KVA, 19/47
where CVA = DVA = FCA = COLVA = KVA = U = CVA + DVA + FCA + COLVA + KVA, T t λ C (u)e u t (r(s)+λ B (s)+λ C (s))ds [ E t V (u) gc (V (u), X(u)) K (u, V (u), V (u), X(u))] du S T λ t B (u)e u t (r(s)+λ B (s)+λ C (s))ds E t [V (u) g B (V (u), X(u))] du T t T t T t λ B (u)e u t (r(s)+λ B (s)+λ C (s))du E t [ɛ h (u)] du s X (u)e u t (r(s)+λ B (s)+λ C (s))ds E t [X(u)] du γ K (u)e u t (r(s)+λ B (s)+λ C (s))ds E t [K (u)] du. We need to calculate VAR and/or ES at all future times (simulation stopping dates) under all scenarios., 20/47
Calculation of VAR and ES VAR has often been calculated based on: Sensitivities and shifts: delta, S ; delta-gamma, S and SS ; delta-adjusted delta-gamma; delta-gamma-vega, adds σ. Based on very short Taylor series: f (x + a) f (x) + af (x) + a 2 1 2 f (x) + a 3 1 6 f (x) Rare to go more than two derivatives out. Essentially a local approximation with all that this implies. Cheap, usually have sensitivities already for hedging... at time zero only. Full revaluation: expensive... even at time zero Given that shocks are large, sensitivity-based methods are difficult to justify. We need a fast, but non-local, pricing method, 21/47
(Automatic) Algorithmic Differentiation, (A)AD (Mostly) automated method to provide derivatives for any computer code (Naumann 2012). Widely used in some banks to provide sensitivities (Capriotti, Lee, and Peacock 2011). Very fast: constant speed for arbitrary number of derivatives. Local method, but very general can be added to any code (memory limits may apply) However, we do not need sensitivities of prices for VAR and ES, 22/47
Longstaff-Schwartz State-space regression-based approach (Longstaff and Schwartz 2001). Of the many similar techniques appearing around 2000 this is perhaps the most widespread in actual use. Widely used for pricing high-dimensional, callable derivatives Also applied to hedging these derivatives, i.e. providing local sensitivities., 23/47
L-S Algorithm 1. Choose a series of explanatory variables for the portfolio O 1 (ω, t k ),..., O n O(ω, t k ) 2. Choose a set of functions of the explanatory variables: the basis for the regression for the value function, f 1 (O i (ω, t k )),..., f n f (O i (ω, t k )). 3. Run a simulation in the risk-neutral measure including all significant dates 4. Using backward induction from the maturity of the portfolio t K = T, obtain the functions, F(α m, O i (ω, t k ), t k ) = α m f m (O i (ω, t k )), (1) n f m=1 that provide a fast approximation to the value of the portfolio at each observation time, t k. 5. Run a second simulation to calculate metrics, using F., 24/47
Backwards Induction Step from t K = T 1. For each path ω at observation date t k, calculate the value of the portfolio, F (ω; t k ), as a discounted sum of cash flows C(ω, t j ) for t j > t k including future exercise decisions, giving F (ω; t k ) = K j=k+1 e t j t k r(ω,s)ds C(ω, t j ). (2) 2. For all t k perform a regression of the continuation values, F(ω; t k ) to get α m such that, n f min α m f m (O i (ω, t k )) F (ω; t k ) 2, α m R. (3) ω 3. The functions, m=1 F(α m, O i (ω, t k ), t k ) = α m f m (O i (ω, t k )), (4) provide a fast approximation to the value of the portfolio., 25/47 n f m=1
Longstaff-Schwartz-Compression for VAR and ES We make three steps: generalize LS application to all derivatives, vanilla non-callable as well as callable use the resulting regressions as a compression technique use the resulting single regression for non-local pricing for VAR and ES Main benefits: Non-local pricing available Regression provides portfolio compression into single equation Portfolio prices are constant cost independent of portfolio size and composition, 26/47
Adapting Longstaff-Schwartz-Compression to VAR/ES Main issues: t = 0 portfolio NPV has exactly 1 value, so regression impossible t > 0 state factor dynamics region << VAR shocks region If payoff does not contain a risk factor how get regression? Solutions: Focus of this talk Expand the state factor dynamics, including at t = 0. Automatically solves first two issues. Longstaff-Schwartz Augmented Compression (LSAC) Forthcoming Function composition plus regression. (f (g))) = f (g)g Regression links local derivative estimates for global derivative estimate. Often get g analytically. Longstaff-Schwartz Augmented Compression Composition (LSACC)., 27/47
Expanded State Space Two methods depending on whether, or not, there are any Bermudan-callable or American-callable options in the portfolio. With AB-Callable Use an Early Start for the simulation, i.e. start as far back as required to span the state space (calibrate and start at t s where s is the shift. A = Early-Start. Required to preserve path-continuity for comparison with continuation values from regression. Potential (mild to very-mild) loss of accuracy due to non-stationarity of processes leading to different regression functions. No AB-Callable At each stopping date and on each path shock the state space of the t = 0-calibrated simulation. A = Shocked-State No continuation value comparison required so we do not need path-continuity. No loss of accuracy. Many choices as to how to apply the shocks. We used one shock, randomly chosen, per path. Focus here is on swap portfolios with central counterparties, these are all non-american-callable and non-bermudan-callable (currently). So use Shocked-State approach., 28/47
Introduction Methods Implementation Example Conclusions Appendix Bibliography
Implementation: Preliminaries Even with LSAC, large computational problem: GPU natural because XVA (mostly) massively parallel Can we test/run same code on CPU and GPU? Approaches: Single source Mostly single-source Will our code go faster on GPU? Tested exactly the same code using single source approach and obtained x50 to x100 for interest rate swaps (K20 versus single core on Xeon) Mostly single source available from CUDA using host device functions Conceptually, use flowgraph approach, 30/47
Compute Flow Random Numbers State Variables Shocked State Variables Shocked Regression Variables & Shocked Portfolio Values Portfolio Regressions Shocked Regression Variables ES & VAR Funding Spreads and Prices Scenarios, 31/47
Introduction Methods Implementation Example Conclusions Appendix Bibliography
Example: USD Interest Rate Swap Portfolio Banks have large, to very large, portfolios of swaps with Central Counterparties that require Initial Margin. Initial Margins are typical based on VAR or ES Standard Interest Rate Swap (IRS) prices as: NPV = fixed leg floating leg (or vice versa) NPV = K τ i d(t i )n i i j L(t j 1, t j ) = 1 ( d τ ) (t j 1 ) τ j d τ 1 (t j ) τ j d(t j )n j L(t j 1, t j ) d(t) discount factor, d τ (t) tenor τ projection factor, L index rate Mildly non-linear due to ratio of projection factors Low ratio of compute to memory operations, not obviously optimal for GPU. However, did work well in our preliminary tests., 33/47
Exposure (USD) Portfolio Details Exposure profile (BLUE) when n = 100 1,400,000,000 1,200,000,000 1,000,000,000 800,000,000 600,000,000 400,000,000 200,000,000 - Portfolio Exposure (n=100 swaps) 0 5 10 15 20 25 30 Years Portfolio maturity: 30Y. Swaps: vanilla USD, 3M A360/6M Fixed n swaps with maturity i 30 where i = 1,..., n n notional = USD100M (0.5 + x) where x U(0,1) strike = K (0.5 + x) where x U(0,1), K = 2.5% gearing = (0.5 + x) where x U(0,1) P[payer] = 90%, 34/47
Regression and Basis Functions Regression is linear combination of 2m + 1 basis functions Constant m swaps, and m annuities, with length i 30, i = 1,..., m m Can make any swap of the same maturity but different fixed rates, from linear combination of a swap and an annuity Can make forward-starting swaps from two swaps of different maturities, 35/47
Zero Yield (cts) Shocks 40% 20% 0% -20% -40% 10-Day Conditioned Relative USD Zero Yield Moves [02-06-2008,28-05-2009] 0 5 10 15 20 25 30 Maturity VAR(1%) VAR(99%) ES(1%) ES(99%) Regulators have defined an svar as stressed VAR, i.e. VAR where the shocks are taken from a one-year period of significant stress, n = 258 Avoids pro-cyclicality, 36/47
Errors in Regression Accuracy of Regression 8.00% 6.00% 4.00% 2.00% mean max 0.00% 9 21 41 81 Basis Size (#instruments) Very good with just a few basis functions. Fewer basis functions are used at later times (because some of them have matured), 37/47
Average Error in Use %/100 (basis points) Accuracy of VAR and ES 30 20 10 0-10 -20 20 50 100 1000 10000 Portfolio Size (#swaps) Value VAR ES Better than pricing accuracy because of averaging effects despite the extreme nature of VAR and ES., 38/47
Compute Time for VAR and ES Speed 5.E+06 4.E+06 3.E+06 2.E+06 1.E+06 0.E+00 Direct Regression Portfolio Size (#swaps) ES and VAR calculations take most of the time (more for larger problems) Regression is a constant-time algorithm Speedup increases with larger problem size, reaching x100 for medium-sized swap portfolio (10,000 swaps)., 39/47
IM FVA cost / Collateral FVA cost FVA: IM (Bilateral Initial Margin) vs Collateral 50% 40% 30% 20% 10% 0% 50 100 1000 Funding Spread (bps) IM/FVA Expensive, even assuming that get paid OIS on IM. N.B. IM FVA is a function of VAR (or ES) not exposure. Even if you have zero exposure you will still have an IM FVA charge., 40/47
Introduction Methods Implementation Example Conclusions Appendix Bibliography
Conclusions Some lifetime capital and funding costs require shocked calculations, and the critical shocks are large so sensitivity-based methods are not suitable. Longstaff-Schwartz-Augmented-Compression (LSAC) is a constant-time pricing approach independent of portfolio size. GPU (K20) = 100 Longstaff-Schwartz- Augmented-Compression = 100 GPU + Longstaff-Schwartz-Augmented-Compression = 10,000 IM FVA is expensive, easily same order as usual FVA for collateralized portfolios: and it depends on VAR (or ES) not exposure., 42/47
Introduction Methods Implementation Example Conclusions Appendix Bibliography
A good risk measure m is: Coherent Monotonic: m(x) m(y ) X Y Translation Invariant: m(x + a) = m(x) + a a R Positive Homogeneous: m(ax) = a m(x) a 0 Subadditive: m(x + Y ) m(x) + m(y ) Can also describe in terms of convexity Elicitable Capable of being elicited, i.e. observed Expectations cannot be backtested Measure Coherent Elicitable VAR Expected Shortfall BCBS 265: continue to backtesting with VAR, 44/47
Introduction Methods Implementation Example Conclusions Appendix Bibliography
BCBS-189 (2011). Basel III: A global regulatory framework for more resilient banks and banking systems. Basel Committee for Bank Supervision. Capriotti, L., J. Lee, and M. Peacock (2011). Real-time counterparty credit risk management in Monte Carlo. Risk 24(6). Available at http://www.risk.net/digital assets/2966/capriotti.pdf. Dodd, C. and B. Frank (2010). Dodd-Frank Wall Street Reform and Consumer Protection Act. H.R. 4173, http://www.sec.gov/about/laws/wallstreetreform-cpa.pdf. Green, A., C. Kenyon, and C. Dennis (2014). KVA, Capital Valuation Adjustment. SSRN abstract=2400324. Longstaff, F. and E. Schwartz (2001). Valuing american options by simulation: A simple least-squares approach. The Review of Financial Studies 14(1), 113 147. Naumann, U. (2012). The Art of Differentiating Computer Programs. SIAM. Treasury (2013). 12 CFR Parts 208, 217, and 225. Regulatory Capital Rules: Regulatory Capital Rules: Regulatory Capital, Implementation of Basel III, Capital Adequacy, Transition Provisions, Prompt Corrective Action, Standardized Approach for Risk-weighted Assets, Market Discipline and Disclosure Requirements, Advanced Approaches Risk-Based Capital Rule, and Market Risk Capital Rule; Final Rule. Federal Register, Vol. 78(198), pp62017-62291. Department of the Treasury., 46/47
Thanks for your attention questions?, 47/47