Multi-level Stochastic Valuations

Multi-level Stochastic Valuations 14 March 2016 High Performance Computing in Finance Conference 2016 Grigorios Papamanousakis Quantitative Strategist, Investment Solutions Aberdeen Asset Management 0

Contents PFE Modelling explained Multi-Curve Modelling and stochastic basis for collateral management Cash vs. No-Cash Collateral The impact of haircut on the valuation High Performance Computing Q & A 1

Sketching the problem Consider the problem of calculating the potential Future Collateral Requirements of a large derivative book for a global asset manager Purpose: Provide a probabilistic approach of what may be the collateral requirements, on various granularity levels (fund level, counterparty level, division level, company level, etc.) and different time horizons, of clients portfolio Worse Case Scenario Time Horizon Collateral Requirement Fund level Division level Company level 90% 1w 1m 1y 60mln 45mln 40mln 70% 1w 1m 1y 30mln 23mln 20mln Average case 1w 1m 1y 18mln 12mln 10mln Additional information could be extracted as well. Collateral decomposition (Cash Collateral, Gov. Bonds, HY bonds, etc.), counterparty exposure, etc. Typically the portfolio will include interest rate swaps, swaptions, inflation swaps, cross currency swaps, equity options, CDSs, etc. 2

Basic Definitions Potential Future Exposure (PFE) is defined as the maximum expected credit exposure over a specified period of time calculated at some level of confidence. PFE is a measure of counterparty credit risk. Expected Exposure (EE) is defined as the average exposure on a future date Credit Valuation Adjustment (CVA) is an adjustment to the price of a derivative to take into account counterparty credit risk. Collateral can be considered any type of valuable liquid property that is pledged by the recipient as security against credit risk. 3

PFE Modelling Explained What do we need to built a standard market PFE Model? A consistent simulation framework to project forward in time the interest rate curves. This will typically have 100-120 time-steps and around 100k -1mln scenarios Multiple stochastic processes to simulate the FX exposure for every different currency in the portfolio, aligned with the time steps and the number of simulations of the interest rate scenarios. The same idea applies for the inflation rates but with less time steps Correlate all the Brownian motions calibrating on the market data and use the Cholesky decomposition to form multivariate normal random variables and project forward in time on every scenario Re-evaluate every position, on every time step, for every scenario and aggregate per netting set to calculate the Expected Exposure - EE Calculate the collateral changes based on the CSA agreement with every counterparty and available collateral on our pool 4

Example Portfolio Value / Collateral Exposure over time Total MTM Portfolio Value for All Scenarios 100 Collateral Exposure for All Scenarios 120 80 60 100 Portfolio Value ( mln) 40 20 0-20 -40 Exposure ( mln) 80 60 40-60 20-80 -100 Jan14 Jan16 Jan18 Jan20 Jan22 Simulation Dates 0 Jan14 Jan16 Jan18 Jan20 Jan22 Simulation Dates 5

Example Cash Collateral Exposure Distribution 95% & 80% Confidence level 45 40 35 Portfolio Collateral Exposure Profiles Max Exp Collateral Exp (95%) Collateral Exp (80%) Exp Exposure (EE) Exposure ( mln) 30 25 20 15 10 5 0 Jan14 Jan16 Jan18 Jan20 Jan22 Simulation Dates 6

Dimensionality of the problem The need for HPC For every interest rate curve we will typically have 120+ simulation steps (50 year horizon) To construct the full term structure on every simulation step we need 20-25 tenor points We simulate different forward and discount curves in order to capture the stochastic credit spreads between the two curves (e.g. Sonia and Libor) The same applies for every currency in our portfolio along with a stochastic process to simulate the FX exposure Depending the degree of convergence we want to achieve we simulate 100k-1mln scenarios So for the simulation the dimensionality of the problem is: nsim x nsteps x ntenors x ncurves x ncurrencies 1mln x 120 x 25 x 15 For valuation purposes we have to take under consideration on top of that the size of the portfolio as well as the NO-Cash collateral 7

Multi-Curve Modelling Before 2007 Before the 2007 credit crunch the interest rates showed typical textbook behaviour F Depo = F FRA. The forward rate implied by two deposits coincides with the corresponding FRA rate Compounding two consecutive 3m Libor rates yields to the corresponding 6m forward Libor rate Differences were still observable on the market but generally regarded as negligible Then 2007 arrived and the crisis widened the basis between OIS and LIBOR rates 9

Multi-Curve Modelling After 2007 The LIBOR-OIS basis spread is neither deterministic nor negligible GBP 3month Libor OIS spread: 10

Multi-Curve Modelling After 2007 Exactly the same for the forward OIS LIBOR basis spread: GBP 1Yx2Y FRAs vs 1Yx2Y forward OIS spread: 11

Multi-curve modelling What do we get when we model the basis spread stochastically: Accurate pricing. Swap rates depend on the OIS discount factors Better fitting since we are able to calibrate each model separately Better correlation structures We can quantify the risk sources as we are able to decompose on different risk factors More accurate estimation of the portfolio future collateral requirements and consequently more efficient collateral management More realistic modelling Is it easy? No. 12

The multi-curve world During the last 5 years or so there were significant efforts to define a consistent multi-curve framework that would be broadly acceptable. We are still trying We have two main ways to model the basis spread and on each of them sub categories Implicit modelling, where we model the joint evolution of OIS rates and curve rates for different market tenors (Ref. [4], [7], [8], [10]). Explicit modelling, where we have three main categories Additive spreads, as in [1], [8], [10] Multiplicative spreads, as in [5], [6] Instantaneous spreads, as in [2] 13

Portfolio Impact of Multi Curve Modelling The impact of the multi curve modelling is highly dependent on the portfolio Although for the valuation of an IRS portfolio we discount using SONIA (for both legs), for the forward curve we use LIBOR. In that case the impact on the value of the portfolio is subject to: The discount curve we use for discounting both legs of the swaps (SONIA) The forward curve we use for the floating leg of the swap (LIBOR) The discount curve we use for discount the liabilities (LIBOR) In case we are using IRSs within a LDI portfolio the LIBOR-OIS curve difference has a significant impact on the hedge effectiveness. 14

Portfolio Impact of Multi Curve Modelling Discounted Exposure ( mln) 45 40 35 30 25 20 15 10 5 Portfolio level Discounted Collateral Requirements Discounted 95% Exp Col Single Discounted 95% Exp Col Dual Discounted 80% Exp Col Single Discounted 80% Exp Col Dual 0 Jan14 Jan15 Jan16 Jan17 Jan18 Jan19 Jan20 Jan21 Jan22 Jan23 Simulation Dates 15

Cash vs Bond Collateral Introducing the collateral pool on a multi-curve framework We assume that the fund on the previous example has a collateral pool with 15mln Cash 10mln of Gilts with 10% haircut 10mln of AAA Bonds with 20% haircut 20mln of HY Bonds with 30% haircut We now model a dynamic collateral portfolio that changes over time We re-evaluate the collateral on every time step, assuming there is no credit migration on the collateral posted Reconstruct the new collateral portfolio and simulate forward (additional computational complexity) 17

Bond Collateral Exposure Profiles - Results Exposure ( mln) 60 50 40 30 20 Bond Collateral Exposure Profiles Max Bond Collateral Exp (95%) Bond Collateral Exp (95%) Max Cash Collateral Exp (95%) Cash Collateral Exp (95%) Bond Collateral Exp (80%) Cash Collateral Exp (80%) Exp Exposure (EE) 10 0 Jan14 Jan16 Jan18 Jan20 Jan22 Simulation Dates 18

Computational Impact When we include No-Cash collateral instruments in our collateral pool the complexity of problem increases exponentially We now have a dynamic portfolio that changes continuously, adding an optimization depending collateral decision on every step we re-evaluate the portfolio The moment that the first No-Cash Collateral is been posted either from us or our counterparty, we need to start simulating the credit migration risk associated with that bond This is clearly an computational intensive exercise, ideal for parallelization and acceleration using HPC 19

21 NVIDIA GPU Computing: Graphical Processing Units

22 Data-Flow Computation on FPGA

Decomposing the problem Preprocessing the portfolio Fetch all the interest rate curve / FX data live from Bloomberg / Thomson Reuters Read the portfolio and construct all the asset classes and projected cash-flows Preprocess the portfolio for different calendars, payment/receive/reset frequencies Mainly a task we use only CPUs for portfolios with less than 10000 derivatives. Banks with larger portfolios may consider parallelize this step as well. Monte Carlo simulation Perform the simulation for the various stochastic processes and construct the full term structure for every simulation step Post process the portfolio Re-Evaluate the portfolio on every scenario and simulation step adjusting for the collateral posted from either us or our counterparties Acceleration using HPC is essential! 23

Term structure k Tenor Points GPU Implementation Each thread calculates the forward interest rate for the i-th period Each thread block calculates a Monte Carlo scenario with different tenor dates Monte Carlo simulation evolution N time steps T 1 T 2 T 3 T N-2 T N-1 T N r 1 (T 1 ) r 2 (T 1 ) r 2 (T 2 ) r 3 (T 1 ) r 3 (T 2 )... r k-1 (T 1 ) r k-1 (T 2 ) r k-1 (T 3 ) r k-1 (T N-2 ) r k-1 (T N-1 ) r k-1 (T N ) r k (T 1 ) r k (T 2 ) r k (T 3 ) r k (T N-2 ) r k (T N-1 ) r k (T N ) 24

Acceleration Results CPU execution times & Speedup Test Environment: CPU: an Intel Xeon X5650 CPU with 6 cores 12 threads running at 2.67GHz an NVIDIA Tesla K40 (2880 cores) with the CUDA version 6.5 a Xilinx Vertex-6 V-SX4757 FPGA card with Maxeler Max-Compiler Number of paths CPU GPU GPU/CPU FPGA FPGA/CPU 50K 21.5s 20.2s 1x 1.1s 19.5x 100K 43.3s 22.8s 2x 1.1s 39x 500K 251.8s 20.2s 12.5x 2.4s 102x 1M 503.6s 22.2s 25x 4.1s 120x 25

Conclusions Multi-curve modelling is here to stay Impact varies, but dual curve discounting is essential for accurate pricing No-Cash Collateral management is not trivial Multiple broker/fund specific CSA agreements Haircuts do have a significant impact on collateral and fund s liquidity The dynamic portfolio grows the computational complexity exponentially Monte Carlo framework provides a sensible solution High Performance Computing using is essential Significant increases the performance of Monte-Carlo simulation Memory allocation problems have to be taken under consideration Total cost of ownership Solution depends on the business case In-house development has significant Person Risk Outsourcing? 26

Thank you / Any questions? 27

References 1. Amin, A. (2010). Calibration, simulation and hedging in a Heston Libor market model with stochastic basis 2. Andersen, L. and Piterbarg, V. (2010). Interest rate modelling, in three volumes. Atlantic Financial Press 3. Brigo, D. and Mercurio, F. (2006). Interest Rate Models, Theory and Practice. Springer Finance. 4. Crepey, S. Grbac, Z. Ngor, N. Skovmand, D. A Levy HJM multiple-curve model with application to CVA computation 5. Henrad, M. (2007) The irony in derivatives discounting. Wilmott Magazine 6. Henrad, M. (2013). Multi-curves Framework with Stochastic Spread: A Coherent Approach to STIR Futures and Their Options. OpenGamma Quant Research 7. Kenyon, C. (2010). Post-shock short-rate pricing. Risk 8. Mercurio, F. and Xie, Z. (2012). The basis goes stochastic basis. Risk, Dec 2012 9. Mercurio, F. (2010). A LIBOR market model with stochastic basis 10. Moreni, N. and Pallavicini, A. (2010). Parsimonious HJM modelling for multiple yieldcurve dynamics. Working paper 28

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Appendix 30

Interest Rate Modelling Setup of the Hull-White single factor interest rate model: This is a short rate model and is defined as: dr t = θ t αr t dt + σdz Where r, is the short rate at time t, α the mean reversion speed and σ is the volatility. The drift function θ t is defined as: Where: θ t = F t 0, t + αf 0, t + σ2 2α (1 e 2αt ) F 0, t : is the instantaneous forward rate at time t F t 0, t : is the partial derivative of F with respect to time By applying the equations above we define the short rates and the corresponding forwards on every scenario one every time steps (nsim x nsteps x ncurves) 31

Deriving the full yield curve Once we have simulated the short rate path we can generate the full curve at each simulate date using the formulas: ln A t, T R t, T = 1 T t ln A t, T + 1 T t B t, T r(t) = ln P(0,T) P(0,t) + B t, T F 0, t 1 4a 3 σ 2 e at e at 2 (e 2at 1) B t, T = 1 e a(t t) a Where R(t, T) is the zero rate on time step t and P(t, T) is the price of the zero coupon bond at t that pays one pound at time maturity T. Using the above formulas we construct the a full term structure for every simulation point (nsim x nsteps x ntenors x ncurves) and we are able to evaluate our derivative portfolio and the corresponding collateral. 32

Explicit Modelling of the basis spread Additive spreads: S k x t = L k x t F k x t Where x are the different market tenors, k the simulation time steps, S the spread, L is the forward Libor rate and F the forward OIS rate. Multiplicative spreads: 1 + τ k x S k x t = 1 + τ k x L k x t 1 + τ k x F k x t Instantaneous spreads: P L t, T = P D (t, T)e T t s u du Where P D (t, T) denotes the Libor discount factor at t with maturity T 33

Modelling the Basis Spread We follow Mercurio s & Xie s [8] additive spread approach by slightly twisting their parameterisation. We simulate the OIS curve and on top of that the credit spread that provide us with the Libor rates to simplify our calibration. x For each tenor x and time interval [T k 1 t, T x k (t)] we assume that forward Libor curve L x k t is a function of the OIS forward rate R x k t, the forward basis spread S x k t and an independent martingale X x k t Using a simple affine function we derive the Libor rates formula: L k x t = L k x 0 + 1 + α k x R k x t R k x 0 + β k x [X k x t X k x 0 ], where α k x and β k x are real constant parameters, for every k and x. 34

Modelling the Basis Spread β k x models the volatility of basis spreads and α k x the correlation between OIS rates and the corresponding spreads. So we have: a x k = Corr(R x x k T k 1, S x x k T k 1 ) Var[S k x x T k 1 ] Var[R x x k T k 1 ] β x k = [1 Corr R x x k T k 1, S x x k T k 1 ] Var[S x x k T k 1 ] Var[X x x k T k 1 ] The basis factors X k x may be different on every time step and tenor point. For the simulation of X k x we will follow a simple geometric Brownian motion. Under this framework we are able to simulate on every time step both the Libor and OIS curves and price our derivatives using dual curve discounting 35