A new approach to multiple curve Market Models of Interest Rates. Rodney Hoskinson

Transcription

1 A new approach to multiple curve Market Models of Interest Rates Rodney Hoskinson Rodney Hoskinson This presentation has been prepared for the Actuaries Institute 2014 Financial Services Forum. The Institute Council wishes it to be understood that opinions put forward herein are not necessarily those of the Institute and the Council is not responsible for those opinions.

2 Agenda Motivation/Context current challenges in term structure modelling Volatility modelling New model with multiple regimes Model statics implied volatility surface Market calibration example

3 Motivation/Context

4 New spread dynamics

5 Long expiry smile

6 Option-implied volatility regimes

7 Modelling requirements Non-zero and non-constant spread multiple curves Smile and smile evolution Arbitrage-free Tractable - rapid pricing and calibration formulas Capable of allowing for: Counterparty credit risk Collateral and re-hypothecation Funding liquidity risk Central clearing

8 Multiple curve term structure models with stochastic volatility Stochastic evolution of 2 curves: Discount rates and forward rates; or Discount rates and spreads Price contracts linked to Libor, Euribor or equivalent Instantaneous or discrete forwards Separate stochastic evolution of rate volatility

9 Model Definition

10 Stochastic Volatility modelling in term structure models - current methods 1. Heston (mean-reverting process): dddd(tt) = κκ zz 0 zz tt dddd + ηη(tt) zz tt dddd(tt) κκ mean reversion speed zz 0 mean reversion level ηη(tt) or ηη volatility of variance 2. SABR: (non mean reverting): dddd(tt) = ααzz tt dddd(tt) αα volatility of variance

11 Regime Switching + Mean Reversion Heston: dddd(tt) = κκ zz 0 zz tt dddd + ηη(tt) zz tt dddd(tt) Markov-switching Heston: dddd tt = κκ ff αα tt, tt zz tt dddd + ηη tt zz tt dddd tt ff(, ) deterministic function of αα tt and tt αα tt - continuous time Markov chain with discrete state space

12 Volatility Simulation Sample Path Two Heston Volatilities with Markov-Switching Mean Reversion Function of Markov Chain, Stochastic variance Trading Days

13 Interest Rate Definitions Discrete Forward Rate Model Times TT 0, TT 1, TT 2 to TT NN Discrete tenor xx spacing (e.g. xx = 3mm, 6mm) At time tt < TT nn, for period TT nn to TT nn+1 : LL nn xx tt = FF nn xx tt + SS nn xx tt - FRA rate - par swap rate against Libor over (TT nn, TT nn+1 ) exchanged at TT nn+1 FF nn xx tt - forward rate on OIS (overnight indexed swap) curve (for discounting) SS nn xx tt - forward rate of FRA-OIS spread

14 Example Full Model Dynamics Full Collateralisation OIS equation (n=1...n): ddff nn xx tt = zz 1 tt λλ nn FF tt φφ nn FF tt, FF nn xx tt ddyy FF nn tt Spread equation (n=1 N): ddss nn xx tt = zz 2 tt λλ nn SS tt TT φφ nn SS tt, SS nn xx tt ddyy SS nn tt OIS volatility: ddzz 1 tt = κκ 1 ff αα tt, tt zz 1 tt dddd + ηη 1 tt zz 1 tt ddzz 1 nn tt Spread vol.: ddzz 2 tt = κκ 2 gg αα tt, tt zz 2 tt dddd + ηη 2 tt zz 2 tt ddzz 2 nn (tt) λλ nn FF tt, λλ nn SS tt - deterministic volatility vectors defining vol term structure φφ nn FF tt, xx(tt) = bb nn tt xx(tt) + 1 bb nn tt xx(0) - displaced lognormal skew YY FF nn tt, YY SS nn tt, ZZ 1 nn tt, ZZ 2 nn (tt) Independent standard Brownian motions (YY FF nn tt, YY SS nn tt vectors) under OIS TT nn -forward martingale measure

15 Model Statics

16 Model-Generated Volatility Surface One SV, mean reversion rate = 0.25, volatility of variance = 1.5, 16 state Markov chain

17 Volatility surface Markov Chain Impact

18 SABR vs Markov-Switching Heston

19 Traditionally Mean Reversion Kills the Smile Displaced lognormal Heston, volatility of variance = 1.5, OIS and Spread Skew 0.6

20 Kappa can now increase short end smile Markov-switching Heston, Volatility of variance = 0.5, OIS and Spread Skew 0.6

21 Smiles Without SV Sum of 2 Displaced Lognormal Processes

22 Empirics Term Parameter Calibration to Swaption Cube

23 Swaption cube Swaption option to enter interest rate swap at a fixed rate (strike) instead of the prevailing future par forward swap rate Swaption Cube Implied volatility (price) by 3 factors option expiry, underlying swap maturity, strike Bloomberg swaption cube implied volatilities available for Option expiries 3M, 1Y, 5Y, 10Y, 20Y and 30Y Swap maturities for each expiry 2Y, 5Y, 10Y, 20Y, 30Y 9 Strikes for each expiry/maturity pair At The Money and +/- 25,50,100,200 basis points 30 smiles each with 9 strikes Euro area money market : 6 month tenor

24 Swaption Cube 2yr/5yr Euro 1/9/11

25 Swaption Cube 10yr/20yr Euro 1/9/11

26 Swaption Cube 30yr Euro 1/9/11

27 Term Parameter Calibration Cube data as parameters of a set of models One model per smile Constant parameters Per smile parameters Global parameters Example for each swaption (expiry nn, maturity mm) : OIS : xx Spread: ddss nn,mm xx FF ddff nn,mm tt = zz nn,mm FF tt λλ nn,mm tt = λλ SS bb SS xx SS nn,mm bb FF xx FF nn,mm FF nn,mm tt + 1 bb nn,mm tt + 1 bb SS xx SS nn,mm tt ddyy nn,mm SS tt xx FF nn,mm tt ddyy nn,mm FF tt OIS volatility: FF ddzz nn,mm tt = κκ FF FF FF ff αα tt zz nn,mm tt dddd + ηη nn,mm FF zz nn,mm tt ddzz FF nn,mm tt

28 Fitted Term Parameters: Euro 1/9/2011 OIS Per Smile Parameters FF Volatility λλ nn,mm FF Skew bb nn,mm Maturity (yrs) Expiry 3 Months year Year Year Year Year FF Volatility of Variance ηη nn,mm Maturity (yrs) Expiry 3 Months year Year Year Year Year

29 Fitted Term Parameters: Euro 1/9/2011 Global Parameters OIS stochastic volatility mean reversion rate κκ FF 0.66 Spread deterministic volatility λλ SS 0.47 Spread Skew bb SS 1.00 Markov Chain Summary ff αα tt Occupation Time % % % % %

30 Pricing error short end IV basis points

31 Pricing error long end IV basis points Average absolute error 22bp of IV

32 Summary New SV TS model from new volatility dynamics First additive OIS+Spread Heston-based TS model First 2 curve TS with Markov switching Heston Smile persistence between Heston and SABR Heston-level mathematical rigour and tractability Solve Heston moment explosion Excellent fit to swaption cube FRA dynamics not conditionally lognormal

33 Further research Collateralisation, default and funding liquidity adjustments Impact on exotics prices Same volatility different TS models Jumps in Spread connected with volatility regime switches Times series of calibrations Greeks