Example Market-Consistent Scenarios Q1/214 Ltd 14214 wwwmodelitfi For marketing purposes only 1 / 68
Notice This document is proprietary and confidential For and client use only c 214 Ltd wwwmodelitfi For marketing purposes only 2 / 68
This report provides documentation regarding the risk-neutral (market-consistent, MC) economic scenarios, including Input data Models and assumptions and outputs Quality tests wwwmodelitfi For marketing purposes only 3 / 68
Market-Consistent Framework The scenarios produced are market-consistent, or risk-neutral Rationale: Market prices of financial instruments reflect all available information regarding the expected future development of the risk factors to which the instruments are exposed to Try to back out these expectations from the observed market prices (by postulating a model to calibrate) In practice: market-consistent traded instruments are priced correctly Market-consistent scenarios should be used for valuation (of traded assets) are used for valuation of (non-traded) options & guarantees embedded in technical reserves should not be used for any other purpose (such as risk management or planning, for which real-world scenarios are needed) Models are calibrated (ie, model parameters are estimated) by minimizing the (mean-squared) error between market prices and model-implied prices wwwmodelitfi For marketing purposes only 4 / 68
Scenario Generation Modeling Stages Goodness-of-Fit testing s from the Model of the Model of the Market prices Market Market error error wwwmodelitfi For marketing purposes only 5 / 68
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Swap Rates Interest Rate (%) 35 3 25 2 15 1 5 LIBOR rates as of 313214 Forward Spot 1 2 3 4 5 6 Time (years from Valuation Date) Source: Based on market data from SuperDerivatives wwwmodelitfi For marketing purposes only 7 / 68
Smith-Wilson Extrapolation Used initial curves 45 4 Smith Wilson Extrapolated Yield Curves as of 313214 Interest Rate (%) 35 3 25 2 15 1 5 S W Forward S W Spot Market Forward Market Spot 1 2 3 4 5 6 Time (years from Valuation Date) Assumptions Risk-free base curve: Swap Curve 1 bps Smith-Wilson (S-W) parameters: UFR = 42 %, LLP = 2 y, α = 1 wwwmodelitfi Source: Based on market data from For marketing SuperDerivatives, purposeseiopa only (Smith-Wilson approach) 8 / 68
Swaption Volatility Surface ATM Swaption Volatility Surface as of 313214 Volatility (%) 1 8 6 4 2 1 2 3 Swap Tenor (y) 4 5 5 4 1 2 3 Option Maturity (y) Source: SuperDerivatives wwwmodelitfi For marketing purposes only 9 / 68
Swaption Volatilities A Sample of Swap Tenors Volatility (%) 9 8 7 6 5 4 Swaption Volatilities for certain Swap Tenors as of 313214 1Y 5Y 1Y 2Y 3Y 3 2 1 5 1 15 2 25 3 35 4 45 5 Option Expiry (years) Source: SuperDerivatives wwwmodelitfi For marketing purposes only 1 / 68
Index Volatilities Volatility (%) 5 45 4 35 3 25 2 15 1 5 ATM Implied Volatilities for Indices as of 313214 1 2 3 4 5 6 7 8 9 1 Time (years) SPX FTSE N225 OMXS3 OMXH25 MIASJPUS BVSPUSD IRTS NIFTY HSCEI Source: SuperDerivatives wwwmodelitfi For marketing purposes only 11 / 68
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Component Models The following models are used for generating the scenarios: Interest rates: Libor Market Model (LMM) Calibrated to swaption volatilities (and initial forward curve) Index assets: geometric Brownian motion (GBM) Equities: Calibrated to ATM equity option prices : Historical volatility assumption is used (no liquid option markets for real-estate) : Cox-Ingersoll-Ross square-root diffusion (CIR) Calibrated to historical inflation (Finland) wwwmodelitfi For marketing purposes only 13 / 68
Dependency The dependencies between random processes (risk factors) are modeled using Gaussian copula with correlation matrix ρ: Correlations between driving processes / risk factors IR_level 1 2 2 2 2 2 1 1 1 1 1 IR_twist 1 IR_butterfly 1 SPX 2 1 8 8 8 8 8 8 8 8 8 2 FTSE 2 8 1 8 9 9 8 8 8 8 8 3 N225 2 8 8 1 8 9 8 8 8 8 8 2 OMXS3 2 8 9 8 1 9 8 8 8 8 8 3 OMXH25 2 8 9 9 9 1 8 8 8 8 8 3 MIASJPUS 1 8 8 8 8 8 1 8 8 8 8 1 BVSPUSD 1 8 8 8 8 8 8 1 8 8 8 1 IRTS 1 8 8 8 8 8 8 8 1 8 8 1 NIFTY 1 8 8 8 8 8 8 8 8 1 8 1 HSCEI 1 8 8 8 8 8 8 8 8 8 1 1 2 3 2 3 3 1 1 1 1 1 1 1 9 8 7 6 5 4 3 2 1 IR_level IR_twist IR_butterfly SPX FTSE N225 OMXS3 OMXH25 wwwmodelitfi For marketing purposes only 14 / 68 MIASJPUS BVSPUSD IRTS NIFTY HSCEI 1
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Libor Market Model Libor Market Model is used for modeling interest rates Instead of hypothetical continuous interest rates, model discrete forward rates actually observed in the market Intuitive, possible to capture (more) accurately the prices of market instruments Libor Market Model takes the initial forward rates (fwd curve) as input, and therefore calibrates to the initial term structure automatically consists of estimating the forward rate volatility and correlation (function) parameters from prices of actively traded instruments In this case, ATM Swaptions are used For more information and details, see eg [BME6] wwwmodelitfi For marketing purposes only 16 / 68
Libor Market Model Model Specification Basic quantities Tenor dates: = T < T 1 < < T M < T M+1 Interval lengths: δ j = T j+1 T j Forward rates: L j (t) := L(t, T j ) L j (t) is the forward rate, at time t, for accrual period [T j, T j+1 ] The development of forward rates is describe by the stochastic differential equation dl j (t) = µ j (t) dt + σ j (t) dw j (t), (1) L j (t) t T j, j = 1,, M, where W = (W j ) is (M-dimensional) Brownian motion with correlation matrix ρ(t) (ie, E[ dw i (t) dw j (t)] = ρ i,j (t) dt with the common notation) wwwmodelitfi For marketing purposes only 17 / 68
LMM Calibrating correlations between forward rates 1 Estimate historical correlation matrix 2 Fit a smooth functional form to historical correlations 3 Perform a principal components analysis on the fitted correlation matrix Calibrating volatilities of forward rates 1 Specify a functional form for the volatilities of forward rates 1 Here piece-wise linear, continuous 2 Pivot points as decision variables 2 Minimize the weighted sum-of-squared differences (SSE) between market and model swaption volatilities wwwmodelitfi For marketing purposes only 18 / 68
LMM Calibrating Volatilities The swap option volatility which functions as the calibration target can be approximated as a function of the forward rate volatilities (see [RJA3]): ν 2 α,β = w i () = β i,j=α+1 w i ()w j ()L i ()L j () S α,β () 2 T α δ i P(, T i ) β j=α+1 δ jp(, T j ), Tα σ i (t)σ j (t)ρ i,j (t) dt, where S α,β (t) is the forward swap-rate at time t for a swap with payment times T α+1,, T β, and P(t, T i ) is the time-t price of a zero-coupon bond maturing at T i wwwmodelitfi For marketing purposes only 19 / 68
LMM Input Data 5 ATM Swaption Volatility Surface as of 313214 45 4 Swap Tenor (y) 35 3 25 2 15 1 5 5 1 15 2 25 3 35 4 45 5 Option Maturity (y) (See also p9) wwwmodelitfi For marketing purposes only 2 / 68
LMM Correlations & Principal Components 1 8 Correlation between 1 year fwd rates Principal component weights for 1 year fwd rates 1 8 6 4 PC1 PC2 PC3 6 2 4 4 2 Time to maturity for 1y fwd rate 2 4 2 4 1 2 3 4 5 6 Time to maturity (y) wwwmodelitfi For marketing purposes only 21 / 68
LMM Forward Rate Volatility Functional Form Volatility multiplier 4 35 3 25 2 15 Forward Rate Volatility (function) time maturity residual maturity 1 5 5 1 15 2 25 3 35 4 45 5 Time (years) wwwmodelitfi For marketing purposes only 22 / 68
LMM Swaption Volatility Surfaces Model volatilities Market volatilities Relative difference 1 5 1 5 5 Swap Option Maturity Expiry Model volatilities ABS 1 5 1 5 5 Swap Option Maturity Expiry Market volatilities ABS 1 1 5 5 5 Swap Option Maturity Expiry Absolute difference x 1 4 5 5 Option Expiry 5 5 Swap Maturity Option Expiry wwwmodelitfi For marketing purposes only 23 / 68 5 5 Swap Maturity 5 Option Expiry 5 5 Swap Maturity
LMM Volatilities of Forward Rates Forward Rate Maturity 6 55 5 45 4 35 3 25 2 15 1 5 696 616 535 776 776 776 455 455 616 616 937 937 616 696616 616 696616 776 616 696 776 616 696 696 616 696 616 696 616 696 616 696 616 616 857 776 857 696 616 616 776 616 535 616 535 776 535 616 535 616 535 455 535 455 535 455 535 455 455 535 455 535 696 535 776 535 776 535 857 535 616 616 535 535 616 535 616 696 616 857 696 776 937 12 776 696 696 118 776 616 776 776 Absolute Volatilities x 1 3 375 375 12 118 126 12 118 126 126 118 12 126 118 12 295 295 12 118 126 12 118 126 455 455 126 118 12 535 126 118 12 535 12 118 126 12 118 126 696 696 126 118 12 126 118 12 16 12 118 126 214 12 118 126 214 174 11 12 12 166 12 166 11 158 11 12 158 15 15 857 126 142857 126 142 696 118 134 134 696 118 134 134 5 1 15 2 25 3 35 4 45 5 55 6 Time (years) wwwmodelitfi For marketing purposes only 24 / 68 11 616 11 11 118 118 174 616 937 11 11 118 118 937 14 12 1 8 6 4 2
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Model The equity indices follow geometric Brownian motion with stochastic interest rates and deterministic volatility, ds i (t) S i (t) = r(t) dt + σ i (t) dw i (t), i = 1,, N, (2) where N is the number of indices modeled, and the (instantaneous) correlation between equity processes is E[ dw i (t) dw j (t)] = ρ i,j dt wwwmodelitfi For marketing purposes only 26 / 68
Indices ID Index Name Market/Region SPX S&P 5 US FTSE FTSE 1 UK N225 NIKKEI 225 Japan OMXS3 OMX 3 Sweden OMXH25 HEX 25 Finland MIASJPUS ishares MSCI All Country Asia ex Japan Asia ex Japan BVSPUSD Ibovespa Index Sao Paolo SE Brazil IRTS RTSI Index Russia NIFTY S&P CNX NIFTY India HSCEI Hang Seng China Enterprise China wwwmodelitfi For marketing purposes only 27 / 68
Implied Volatilities Spot Vol The implied volatilities for different maturities are derived from ATM equity options for the equity indices: Volatility (%) 5 45 4 35 3 25 2 15 1 5 ATM Implied Volatilities for Indices as of 313214 SPX FTSE N225 OMXS3 OMXH25 MIASJPUS BVSPUSD IRTS NIFTY HSCEI 1 2 3 4 5 6 7 8 9 1 Time (years) wwwmodelitfi For marketing purposes only 28 / 68
Implied Volatilities Forward Vol The implied volatilities (of the previous slide) for different maturities at t = are spot volatilities; these need to be converted to corresponding forward volatilities for simulation time steps: σt 2 σ T1 T 2 = 2 T 2 σt 2 1 T 1 T 2 T 1 wwwmodelitfi For marketing purposes only 29 / 68
Implied Volatilities Currency Volatility For indices that are not denominated in Euros, a simple volatility adjustment is applied to approximate the effect of currency volatility, ie, the additional return volatility arising from changes in exchange rates: σ tot = σeq 2 + 2ρ eq,fx σ eq σ fx + σfx 2, where σ eq is the equity index volatility, σ fx is the currency volatility (volatility associated with EUR/CCY exchange rate for currency CCY), and ρ eq,fx is the correlation between equity index and currency returns Here it is assumed that ρ eq,fx = for each index, ie, the movements in equity and its currency are uncorrelated The currency volatility adjustment is applied to each point in the forward vol curve as a simple add-on, for each equity index denominated in a currency other than EUR wwwmodelitfi For marketing purposes only 3 / 68
Implied Volatilities FX-adjusted Fwd Vols Volatility (%) 5 45 4 35 3 25 2 FX adjusted Forward Volatilities for Indices SPX FTSE N225 OMXS3 OMXH25 MIASJPUS BVSPUSD IRTS NIFTY HSCEI 15 1 5 1 2 3 4 5 6 Time (years) wwwmodelitfi For marketing purposes only 31 / 68
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Model (real-estate) index follows geometric Brownian motion with stochastic interest rates and constant volatility, ds(t) S(t) = r(t) dt + σ dw (t), (3) where the volatility is assumed to be 12 % pa (σ = 12) wwwmodelitfi For marketing purposes only 33 / 68
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Model rate follows Cox-Ingersoll-Ross (CIR) mean-reverting square-root diffusion, di(t) = α(θ i(t)) dt + σ i(t) dw (t), (4) where θ = mean-reversion level (long-term mean) α = mean-reversion speed σ = process volatility The long-term mean is taken to correspond to the inflation target of the European Central Bank (ECB), while the other parameters are estimated from historical data using maximum likelihood estimation, conditional on θ being fixed Data on Finnish inflation as provided by Statistics Finland Parameter estimates: ˆθ params = (ˆα, ˆθ, ˆσ) = (143, 2, 16) wwwmodelitfi For marketing purposes only 35 / 68
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Outputs Non-MATLAB users The following entries and simulated values are written to the Scenario output file (eg, csv, xlsx): Field Name simulation period short rate nom rate Tyr ba cpi real rate Tyr X eq index X eq return prop index prop return Description Scenario number time (in months), with corresponding to valuation date Short rate (nominal) Nominal T-year interest rate Cumulative value of bank account process (1/discount factor) rate Real T-year interest rate Cumulative value of equity index X {eur, fin, wexeur, em} 1-period return of equity index X {eur, fin, wexeur, em} Cumulative value of property index 1-period return of property index wwwmodelitfi For marketing purposes only 37 / 68
Outputs MATLAB users Alternatively, the Scenarios can be provided as a MATLAB structure for easy and convenient access to data directly from MATLAB This is the preferred method for cframe users wwwmodelitfi For marketing purposes only 38 / 68
LMM Technicalities 1/2 Recall from (1) that in the Libor market model for discretely compounded forward rates, each of the modeled forward rates L j evolves according to the SDE dl j (t) = µ j (L, t) dt + σ j (t) dw j (t), L j (t) where the drift terms depend on the measure used, and are determined from no-arbitrage arguments For generating economic interest rate scenarios, the spot (-Libor) measure is used; that is, the discretely compounded bank acount is taken as the numéraire asset: Measure, Q Numéraire, N(t) Drift, µ j(t) Spot Measure B(t) = P η(t) (t) η(t) 1 i= [1 + δ j δ j L j (t) il i(t i)] i=η(t) ρi,j(t)σi(t)σj(t) 1+δ j L j (t) where η(t) is the index of the next forward rate to reset (η(t) = j if T j 1 t < T j ), and P j (t) is the value at time t of a zero-coupon bond maturing at T j wwwmodelitfi For marketing purposes only 39 / 68
LMM Technicalities 2/2 Greater accuracy is achieved when the LMM implementation is done using log-forward rate rather than directly the forward itself (see, eg, [GLA4]); simulating the log-process has also the benefit that the diffusion coefficient becomes state-independent Using Itó s lemma, d ln L = dl L 1 2 σ2 dt, and we get, for a time step from t to u u ln L j (u) = ln L j (t) + µ j (s) ds 1 u u σj 2 t 2 (s) ds + σ j (s) dw j (s), (5) t t Note that the drifts µ j (t) of the forward rates L j (t) are indirectly stochastic, as they depend explicitly on the stochastic L themselves, also continuously evolving from t to u This state-dependency makes discretization of the drift for simulation purposes hard The most straightforward (and standard) way is to use Euler stepping, µ j = j i=η(t) δ i L i (t) 1 + δ i L i (t) ρ i,j (t)σ i (t)σ j (t) (6) to approximate the integral (ie, the rates L are freezed to be constant at their (known) starting values over the interval [t, u]) wwwmodelitfi For marketing purposes only 4 / 68
LMM More technicalities: Predictor-Corrector The accuracy of the simple Euler-stepping for drift can be increased by using the so-called predictor-corrector (PC) scheme (see [BDJ8]) The idea is to simulate the forward rates one step ahead keeping all the state variables constant, recalculate the drift term using these evolved rates, and take the average of the two drifts The actual time step is then performed using the same random numbers and this PC drift More precisely, to implement the PC scheme, we use single Euler step (with µ) to get initial estimates of the 1-step ahead fwd rates, ˆL(u), which are used to calculate new drift terms, ˆµ The adjusted drifts are then obtained as µ [t,u] j := [ u µ j (s) ds 1 j t 2 i=η(t) δ i L i (t) 1 + δ i L i (t) + δ i ˆL i (u) 1 + δ i ˆL i (u) ] ρ i,j (t)σ i (t)σ j (t), and these µ are used to generate the actual vector of fwd rates L(u) using the same set of random numbers used to generate the estimates ˆL(u) in the first step The accuracy of the PC method has been shown to be good in the literature wwwmodelitfi For marketing purposes only 41 / 68
Nominal & Real For the economic scenario output file, spot rates are derived from the forward rates simulated using LMM These are nominal interest rates The real rates are obtained using the relationship: 1 + r Re = 1 + r Nom, 1 + i where subscripts Re and Nom refer to real and nominal interest rates, respectively, and i is the inflation rate wwwmodelitfi For marketing purposes only 42 / 68
The equity price paths are generated in a standard way by simulating GBMs based on drawing correlated standard Normal (N(, 1)) random variables 1 Volatility is deterministic and depends on the simulation time The drift is the stochastic short interest rate Therefore, the interest rate (LMM) simulation has to be run simultaneously or (for efficiency) before the equity simulation in practice indices (see p27) are simulated jointly, and four baskets are formed from the simulated indices as follows: Baskets Weights Name SPX FTSE N225 OMXS3 OMXH25 MIASJPUS BVSPUSD IRTS NIFTY HSCEI Euro 85 15 Finland 1 World ex Euro 75 13 12 Emerging Markets 25 2 2 15 2 wwwmodelitfi 1 For marketing purposes only 43 / 68
The property index price paths are generated by simulating GBM, similarly to equity processes wwwmodelitfi For marketing purposes only 44 / 68
The inflation rate paths are generated by simulating the Cox-Ingersoll-Ross process The values are obtained exactly for each simulation point by sampling from the exact transition law of the CIR process (essentially, sampling from non-central χ 2 df; see [GLA4]) wwwmodelitfi For marketing purposes only 45 / 68
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Checking Market Consistency Market consistency tests: Yield curve replication Does the average simulated yield curve match the initial curve? Martingale (1 = 1) tests Do all the asset classes earn, on average, the risk-free return? Volatility replication Is the average scenario volatility of each asset equal to the market (input) volatility? Pricing test Are traded instruments priced correctly using the simulated scenarios? In practice, suitable variance reduction and adjustment techniques can be used to improve the fit to (the consistency with) market inputs/prices wwwmodelitfi For marketing purposes only 47 / 68
Checking the Market Consistency Note The most important criteria for market-consistent scenarios is that they price traded instruments correctly (by definition) Interest rate scenarios reproduce observed bond prices and swaption prices (volatilities) Index asset scenarios reproduce index asset option prices 2 Most other tests, such as martingale (1st moment) test and volatility (2nd moment) test, serve more as sanity checks Assuming the simulation procedures employed are correct, these tests concern the (possible) discretization error in simulating SDEs, and the (inevitable) sampling error in using Monte-Carlo with finite sample size In other words, these tests check that model output = model input 2 In the trading currency of the underlying, ie, before the discretionary fx vol adjustment of p3 wwwmodelitfi For marketing purposes only 48 / 68
Variance Reduction 1/2 When using a small number of scenarios (such as 1 or even 1 ), the observed sampling errors can be large Obvious solutions: use 1) more scenarios, or 2) quasi-random numbers/low-discrepancy sequences (such as Sobol) But: 1) raising scenario number not always feasible, and 2) use of quasi-random numbers becomes problematic in very high dimensional spaces (such as here) wwwmodelitfi For marketing purposes only 49 / 68
Variance Reduction 2/2 Variance reduction techniques used in practice with economic scenario data: Sampling selection: run several simulations and select the most market-consistent (as measured by the tests) In effect: try to find the set of pseudo-random numbers (or the seed of your rng) that produces best convergence Moment-matching: use simple linear transformations to ensure the market-consistency of (index asset) scenarios When generating economic scenarios, true model values are known/given by market: to minimize the size of sampling error, moment-matching 3 is used 3 The term moment matching is also used by Towers Watson, while Barrie&Hibbert calls this path adjustment wwwmodelitfi For marketing purposes only 5 / 68
Initial Yield Curve Replication Spot Rate (%) 35 3 25 2 15 1 5 Initial Spot Curve vs Average Simulated Spot Curve Initial Sim Mean 1 2 3 4 5 6 Time (years) wwwmodelitfi For marketing purposes only 51 / 68
Zero-Coupon Bond Price Replication Initial Discount Factor Curve Replication 1 9 Zero Coupon Bond Prices Initial Sim Mean P(,T) 8 7 6 5 4 3 2 1 1 2 3 4 5 6 Time (years) wwwmodelitfi For marketing purposes only 52 / 68
Indices Martingale (1=1) Test & Volatility Replication 15 14 13 12 11 1 99 98 97 96 Martingale Test: Indices Line 1=1 SPX FTSE N225 OMXS3 OMXH25 MIASJPUS BVSPUSD IRTS NIFTY HSCEI 95 2 4 6 Time (years) 5 4 3 2 1 EQ Index Volatilities; Input vs Scenario 1 2 3 4 5 6 Time (years) wwwmodelitfi For marketing purposes only 53 / 68
Baskets Martingale (1=1) Test 15 14 13 12 11 1 99 98 97 96 Martingale Test: Baskets Line 1=1 Euro_EQ FI_EQ World_ex_Euro_EQ EM_EQ 95 1 2 3 4 5 6 Time (years) wwwmodelitfi For marketing purposes only 54 / 68
Martingale (1=1) Test & Volatility Replication 15 14 13 12 11 1 99 98 97 96 Martingale Test: Index Line 1=1 95 2 4 6 Time (years) 125 12 Volatility; Input vs Scenario Input Scenario 115 1 2 3 4 5 6 Time (years) wwwmodelitfi For marketing purposes only 55 / 68
& Errors LMM - Swaption Prices Mean absolute relative errors: error (Market Model): 144 % error (Model Scenario): 139 % Total error 4 (Market Scenario): 22 % Total error s from the Model of the Model of the Market prices Market Market error error 4 Note that the simulation and calibration errors are generally independent wwwmodelitfi For marketing purposes only 56 / 68
Swaption Prices Relative Differences Payer Swaptions 5 Relative difference between swaption scenario price and market price; Payer swaptions (tenors (x axis) expiries (y axis)) 6 3 3 2 11 1 1 1 1 NaN NaN NaN NaN NaN NaN NaN NaN NaN 4 45 4 35 3 25 2 15 12 1 9 8 4 3 3 3 4 3 4 5 5 6 5 2NaN NaN NaN NaN NaN NaN NaN 1 1 1 1 1 1 1 1 1 12 1 NaN NaN NaN NaN NaN NaN 1 1 1 1 2 2 3 4 5 5 4 1 1 4NaN NaN NaN NaN NaN 1 3 2 1 1 1 1 1 2 3 3 12 2NaN NaN NaN NaN 2 2 1 2 2 3 4 5 6 5 21 2NaN NaN NaN 1 3 4 3 2 2 1 1 1 2 4 3 3 3 1 NaN NaN 1 2 1 1 2 3 4 5 4 2 1 1 2 1 NaN 11 3 2 2 2 2 1 1 2 3 3 2 2 1 NaN 2 2 1 1 1 1 2 2 1 1 1 3 2 1 2 2 3 2 2 2 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 3 2 2 7 6 5 4 3 2 1 1 1 1 1 1 2 2 2 2 1 1 1 21 2 2 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 2 4 4 3 3 3 4 4 4 4 5 4 3 3 2 1 1 1 3 2 1 1 1 1 1 1 1 1 2 3 4 3 5 31 2 1 2 2 2 2 2 4 3 5 5 6 6 7 6 7 4 2 61 1 1 1 1 1 1 2 2 4 2 4 4 4 4 5 4 5 6 1 4 2 1 2 2 2 3 4 4 4 7 5 6 6 5 4 4 3 3 1 2 3 4 5 6 7 8 9 1 12 wwwmodelitfi For marketing purposes only 57 / 68 15 2 25 3 35 4 45 5
Swaption Prices Relative Differences Receiver Swaptions 5 Relative difference between swaption scenario price and market price; Receiver swaptions (tenors (x axis) expiries (y axis)) 8 5 5 4 3 1 1 1 2 3NaN NaN NaN NaN NaN NaN NaN NaN NaN 6 45 4 35 3 25 2 15 12 1 9 4 3 6 7 8 8 9 9 9 9 8 6NaN NaN NaN NaN NaN NaN NaN 4 4 6 6 7 6 7 8 9 1 1 8 9NaN NaN NaN NaN NaN NaN 1 1 1 1 3 4 5 6 6 4 6 1NaN NaN NaN NaN NaN 2 2 1 2 3 4 5 6 6 5 4 6 8NaN NaN NaN NaN 2 3 4 6 6 7 8 9 9 9 6 4 5 7 8NaN NaN NaN 2 3 4 5 6 7 8 9 9 8 6 4 6 6 7 9NaN NaN 1 2 1 1 3 4 5 7 7 6 4 3 2 2 2 3 4NaN 12 3 3 3 2 2 1 1 1 2 2 1 2NaN 2 5 5 5 5 5 4 3 2 2 2 2 2 2 1 1 2 2 3 4 4 4 4 4 4 3 2 2 2 1 1 1 1 3 4 2 2 8 7 6 13 4 3 4 4 4 3 3 2 2 2 2 2 2 1 1 1 22 3 3 2 2 2 2 2 1 1 2 1 1 1 2 13 4 4 4 3 3 3 4 3 3 4 4 3 3 3 2 2 4 5 4 3 4 5 5 4 4 4 5 5 5 5 5 5 4 4 3 2 2 1 32 4 4 3 3 3 3 4 4 3 3 1 1 2 1 3 23 5 5 4 3 3 3 3 2 2 2 3 3 4 3 4 6 2 62 2 2 1 1 1 2 4 3 5 5 5 5 6 5 6 8 1 4 11 1 2 2 2 1 2 2 3 3 3 4 5 5 6 6 1 2 3 4 5 6 7 8 9 1 12 wwwmodelitfi For marketing purposes only 58 / 68 15 2 25 3 35 4 45 5
Swaption Prices Option Price Surfaces 2 Payer: Sim 2 Payer: Mkt 1 Relative diff Payer 1 1 5 Swap Maturity Option Expiry Receiver: Sim 5 5 Swap Maturity Option Expiry Receiver: Mkt 5 1 5 Swap Option Maturity Expiry Relative diff Receiver 5 2 2 1 1 5 Swap Maturity Option Expiry 5 1 5 Swap Maturity Option Expiry wwwmodelitfi For marketing purposes only 59 / 68 5 1 5 Swap Maturity Option Expiry 5
Stressed Interest Rate Scenarios Spot Curves 45 4 Spot Curves in Stress Scenarios Interest Rate (%) 35 3 25 2 15 1 5 Up Initial Up Sim Mean Base Initial Base Sim Mean Down Initial Down Sim Mean 1 2 3 4 5 6 Time (years) wwwmodelitfi For marketing purposes only 6 / 68
Stressed Interest Rate Scenarios Forward Curves 6 5 Forward Curves in Stress Scenarios Interest Rate (%) 4 3 2 1 Up Initial Up Sim Mean Base Initial Base Sim Mean Down Initial Down Sim Mean 1 2 3 4 5 6 Time (years) Note: Stressed forwards calculated from yearly spot rates wwwmodelitfi For marketing purposes only 61 / 68
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This document is an example of an actual calibration report, with one set of models used for illustrative purposes The report and its contents are to be used for marketing purposes only The actual Scenarios provided to each client are tailored to meet the specific needs of the client wwwmodelitfi For marketing purposes only 63 / 68
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Ltd Information Ltd Unioninkatu 13, 7th Floor 13 Helsinki, FINLAND wwwmodelitfi For more information, contact Matias Leppisaari matiasleppisaari@modelitfi Lasse Koskinen lassekoskinen@modelitfi Timo Salminen timosalminen@modelitfi wwwmodelitfi For marketing purposes only 65 / 68
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I [BME6] D Brigo and F Mercurio (26): Interest Rate Models Theory and Practice, 2nd Ed Springer Finance, Heidelberg [RJA3] R Rebonato and P Jäckel (23): Linking Caplet and Swaption Volatilities in a BGM Framework: Approximate Solutions Journal of Computational Finance, Vol6, No4, pp41 6 [GLA4] P Glasserman (24): Monte Carlo Methods in Financial Engineering Springer [BDJ8] C Beveridge, N Denson and M Joshi (28): Comparing Discretization of the LIBOR Market Model in the Spot Measure Working paper, University of Melbourne wwwmodelitfi For marketing purposes only 67 / 68
II [SAL7] T Salminen (27): of a Forward Rate Market Model Master s Thesis, Degree Programme in Engineering Physics and Mathematics, Helsinki University of Technology wwwmodelitfi For marketing purposes only 68 / 68