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Interest rate models: from theory to practice. Ka Lok Chau HKUST, June 2005. Popular models still being used. Black, Derman and Toy (1989) Hull and White (1993) two- or three-factor extensions Gaussian Markovian short rate model (with s i (t) and l i (t))

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Interest rate models:from theory to practiceKa Lok ChauHKUST, June 2005Popular models still being usedBlack, Derman and Toy (1989) Hull and White (1993) two- or three-factor extensions Gaussian Markovian short rate model (with si(t) and li(t)) Special cases of Heath, Jarrow and Morton (1992) different forward volatility functions, but mostly with Markovian state variable(s) Ritchken and Sankarasubramian (1995), Cheyette (1993) Brace, Gatarek and Musiela (1997) (MSS(1997), J(1997)) different implementation tricks, e.g. drift approximations CEV/displaced diffusion/stochastic volatility Others: Vasicek (1978), CIR (1985), BK (1991), Markov Functional (Hunt, Kennedy and Pelsser (1998)) etc. WHY so many models? The market parametersYield curve Money market rates for maturity < 1 year (e.g. LIBORs) Futures prices for some liquid currencies Swap rates, usually up to at least 10 years; sometimes longer Cap/floor volatilities For major currencies, prices at different strikes are available For most Asian currencies, only ATM prices are available Swaption volatilities For most currencies, only the ATM prices are available even if the smile is available, points could be sparse Typical number of data points on any date: 15 points on the yield curve, 10 cap prices (no smile) to 50 cap prices (with smile data), 49 swaption prices (no smile) The bond market is a totally separate market Yield curve dynamicsBackward looking Start with an analysis of historical data Principal Component Analysis Time series properties e.g. CKLS (1992), Buhler et al. (1999) Forward looking Implied from market variables volatility and correlation structures Empirical results non-stationary time series correlation exhibits more stable behavior parallel shift (could have different magnitudes)twisthumpYield curve movementsParallel shift include flattening/steepening, but all rates move in one direction Twist long and short end may move in different directions Hump e.g. long and short end both move down, but mid-range move up Historical yield curve movementsBased on USD Treasury rates data between 1989 and 1995, parallel shift explains 83.1% of the movements of the yield curve twist explains 10% of the movements hump explains 2.8% of the movements These three types of movements explain 95.9% of the movements of the yield curve Similar results for different periods and different currencies are obtained by many authors J. Frye (1997), Rebonato (1998), Martinelli and Priaulet (2000) etc. ST1DF1DF20Ta1, L1a2, L2Caps and swaptionsCaps (or caplets) and swaptions could have highly overlapped periods Example: swaption maturity T, swap tenor 1 year, semi-annual r is the instantaneous correlation between L1 and L2 between time 0 to T (assume constant) Instantaneous correlationExample: the market volatility of the 2yr-into-1yr swaption is 17%; volatility of 24x30 caplet is 20%, and the volatility of 30x36 caplet is also 20% Question: what is the instantaneous correlation between the 24x30 caplet and the 30x36 caplet? Solution 1 assume the caplet volatilities are flat, such that sL1(T) =sL1= 20%, sL2(T) =sL2= 20% using the equation in the previous page, we could work out the correlation r (all the other terms are known) since sS = 17% < 20%, r would be less than 1 sL10TT2sL21sL22Instantaneous correlation (2)Solution 2: we could assume non-flat volatility structure for sL2 if we choose sL21 < 20% and sL22 > 20%, we may be able to find a solution such that r = 1 We could have an infinite number of combinations of sL21 and r USD calibration exampleCalibrated using a 1-factor HJM model Data as of May 11, 2005 USD calibration example (2)Calibrated using a 2-factor BGM model Data as of May 11, 2005 SGD cap/swaption exampleWhat does it show?A 2-factor BGM model was calibrated to the ATM caplet volatilites It is observed that the model generated swaption volatilities are consistently “higher” than the market swaption volatilities Model wrong? not rich enough to capture the market dynamics? Market wrong? is there an arbitrage opportunity? is it possible to devise a trading strategy? USD and GBP markets in late 1998 to summer 1999 phenomenon could exist for long periods, and could worsen The limits of arbitrage (Shleifer and Vishny (1997)) What information is available?From the yield curve obtain the discount function for any point in time choice of interpolation methodology From the cap/floor prices conversion to caplet volatilities - could be a model dependent process From swaption prices these are like options on a basket of underlyings (although the weights are not exactly constant), hence some correlation information may be available These are separately traded markets banks are natural buyers of swaptions (due to bond issues) corporate customers are natural buyers of caps What information is not available?Information content in caps/swaption prices De Jong, Driessen and Pelsser (2002) difference between implied covariance matrix and realized movements Term structure of local volatility is not availableTherefore instantaneously correlation between forwards could be arbitrary Forward volatility could be arbitrary, especially at different strikes What are required from a model?Probability distribution of the whole yield curve at any time t in the future For some products we need the evolution of the yield curve and volatilities from time 0 to time t Forward spot volatilities (at different strikes) Forward forward volatilities (at different strikes) Terminal correlation of different rates in the future We see that the last two items are not available, and assumptions have to be made these are not hedgeable parameters wrong/naïve views could lead to losses though! What do we want to achieve?Explanatory power vs exact fitting many degrees of freedom -> exact fit parametric function -> identify trading opportunities Price of derivative structure only depends on “intuitive” inputs e.g. pricing of 3-year Bermudan swaption if a global calibration is performed, it may depend on the price of a 5-year option into a 5-year swap traders usually feel uncomfortable with this kind of approach Transparency between model parameters and market prices what is a “short rate” or an “instantaneous forward”? what is s(t)? A “ruler” to express the derivative price as a combination of vanilla instruments as close as possible in terms of characteristics Able to identify a static/dynamic replication strategy for exotic products Properties of a good modelBe arbitrage free i.e. it should not be possible to find an arbitrage within the pricing model, e.g. by constructing some long-short strategies to earn arbitrage profits Be well-calibrated correctly price as many relevant liquid instruments as possible Stability in the model parameters Be realistic and transparent in its properties will it give rise to all possible yield curve shapes that affect the pricing of a particular product? is there a direct relationship between the model parameters and the market prices? what additional properties would be implied by the model? is it easy to express a view on certain parameters which affect pricing? Allow an efficient implementation accurate calculation of prices and Greeks Based on Hunt, Kennedy and Pelsser (1998)Calculating vegasNaïve method is to calculate if global calibration is performed via an optimization process, there is no one-to-one correspondence between the price of the derivative product and the input option prices calculating the vega depends critically on the calibration strategy A universal model?Brace, Dun and Barton (1998) proposed to use the BGM model (especially the lognormal LIBOR version) as “the” model Lognormal in LIBORs (same as the market standard for Caps) approximately lognormal in Swap rates (same as the market standard for swaptions) Easy to express the views of volatility term structure Easy to express the views of correlation between forward rates However, is the world so simple? Smiles? Jumps? Stochastic volatility? other unexplained factors? “God does not care about our mathematical difficulties; he integrates empirically” - Albert Einstein Model complexityBlack (1976) European caps and swaptions One-factor model use mean reversion to control auto-correlation e.g. Hull & White, Multi-factor model terminal correlation between rates Smile (local volatility model, CEV) volatility sensitivity at different strikes Stochastic volatility products which depend on the volatility process Increasing complexity usually means more parameters Case studiesNon-path dependent products Bermudan swaptions Callable range accrual notes/swaps (CRANs) Callable CMS spread range accrual options (CASOs) Path dependent products Ratchet caps CRANs with varying coupons Enhanced Target Redemption Notes (Enhanced TARNs) ST1,TST3,T0T1T2T3…TN-2TN-1ST2,TBermudan swaptionTypical structure Maturity: 10 years Fixed rate: 5% Floating rate: USD 6-month LIBOR Option: At each reset date on or after 1 year, Party A has the right to enter into a swap which it receives fixed and pays floating; final maturity of the swap is 10 years from trade date; the option could be exercised only once This is often known as 10NC1, which reads “10-yr non-callable 1-yr” TBermudan swaption: analysisCritical factors The model should decide the exercise boundary If we link the state variable to swap rate, we need a model to capture the auto-correlation of the state variable Need to have the ability to price the underlying swaptions correctly The critical volatility parameters are the volatilities of swaptions with the same terminal maturity, e.g. 1Y-9Y, 2Y-8Y, 3Y-7Y etc. Other volatilities have much less influence many of these could be deeply out-of-the-money given the shape of the yield curve For pricing purpose, a 1-factor model calibrated to the underlying swaptions (properly adjusted for mean reversion) may suffice Longstaff and Schwartz (2001) vs Andersen and Andreasen (2001) For hedging, a richer model may be required e.g. a multi-factor model LIBOR-spreadSwap(non-callable)TimeExotic couponBermudan optionExotic couponTimeLIBOR - spreadTypical exotic structuresThe above represents the seller’s position Initial cost of swap + Bermudan option = 0 (after fees) The difficulty is usually in evaluating the fair value of the Bermudan option Some non-path dependent productsExamples: Bermudan swaption Exotic coupon = fixed rate, say 4% Callable Range Accrual swap Exotic coupon = 6.5% x n / N where N = no. of days in the payment period, e.g. every 6 months n = no. of days where 0 < 6-month LIBOR < 7% Callable CMS Spread Range Accrual swap Exotic coupon = 6.5% x n / N where N = no. of days in the payment period, e.g. every 6 months n = no. of days where 30-year swap rate > 10-year swap rate In each of these structures, the option is to exchange the exotic leg by the LIBOR leg (= swap rate), i.e. idea similar to an exchange option (Margrabe (1978)) CRANs: analysisCorrect pricing of the exotic leg a series of digital option on LIBOR smile information is important Need to calculate the volatility of the combined underlying floating leg comes from vanilla swaption volatility exotic leg comes from caplet volatilities need to account for the correlation between the two legs Minimum requirement multi-factor model, to account for de-correlation correct calibration for the auto-correlation of state variables use both swaption and caplet volatilities for pricing the Bermudan option use the model or some external pricing tool for the correct valuation of the exotic leg (with smile volatilities) CASOs: analysisCorrect pricing of exotic leg spread option on swap rates depend strongly on the terminal correlation between the swap rates some dependency on swaption smile in calculating the forwards and the spread option price Volatility of the combined underlying all are based on swaption volatilities correlation between the exotic leg and the floating leg Minimum requirement multi-factor model, preferably with strong control of correlation correct calibration for the auto-correlation of state variables only need to calibrate on swaption volatilities take into account of both the swaptions spanning the underlying Bermudan option and the swaptions for the CMS spread (digital option, therefore smaller vega) swaption smile information required for the exotic leg Ratchet capsTypical structure Maturity: 5 years Floating rate: USD 6-month LIBOR Frequency: reset every 6 months Payoff: max( Li-Li-1-K, 0) where Li is the 6-month LIBOR fixed at time iApplication: one way floating rate note a floating rate note where the coupon is always equal to or higher than the previous coupon eg. coupon=max (Li,Li-1 + 0.20%) this could be written as: Li-1+ 0.20% + max( Li-Li-1-0.20%,0) Ratchet caps: analysisSimilar to forward starting options (cliquets) in equity correct modeling of the forward volatility structure is critical don’t know what strike should be referred to need the process of underlying due to smile information because we are looking at LIBORs observed at adjacent periods, the correlation between them would be high anyway, and correlation structure is not very important Minimum requirement 1-factor model, calibrated to caplet volatilities stochastic volatility model with smile information CRANs with varying couponsTypical structure Maturity: 10 years Floating rate: USD 3-month LIBOR Frequency: reset every 3 months Exotic coupon: Quarter 1: 8% x n / N Quarter 2 to 40: preceding coupon x n / NN = no. of days in the payment period, e.g. every 6 months n = no. of days where 6-month LIBOR is within the range Range: Year 1 - Year 5: 0 to 6% Year 6 - Year 10: 0 to 7%Callable feature: Callable every 3 months starting from Year 1 Selling point: Higher initial coupons than fixed rate CRANs Enhanced TARNsTypical structure Maturity: 10 years Floating rate: USD 6-month LIBOR Frequency: reset every 6 months Exotic coupon: first 6 months: 14% p.a. afterwards: Max(10% - 2 x 6-mth LIBOR,0)Target: total coupon not exceeding 8% Bonus coupon: If the note redeems early, an extra coupon is paid depending on when it is terminatedBonus coupon: 0% in year 1, 2% in year 2 and so on increased to 18% in year 10More leverage based on the expected termination time Model comparison exercisePricing of particular products pay attention to calibration results Is the difference in pricing caused by the calibration strategy? do the models require re-calibration on a daily basis? Hedging performance use a powerful model or historical simulation to generate real-world movements self consistency - should have small residual hedging error (both in terms of expected value and variance) stability of hedge ratios Some references Bakshi, Cao and Chen (1997), Driessen, Klassen and Melenberg (2002), Fan, Ritchken and Gupta (2001), Gupta and Subrahmanyam (2001) Model selection considerationsGlobal approach Find a model which describes yield curve movements, given the market inputs (e.g. curves, cap/floor/swaption prices) With such a model, we should be able to price ANY derivative product based on the yield curve Therefore we could apply the same model to risk manage a wide range of exotic products Most yield curve models could be used in this manner Problem: it is easier said than done …. Model selection considerations (2)Product-based approach Given the derivative product to be priced, gain an understanding of what features of the yield curve will have the most impact on the pricing Find a model which describes these yield curve movements with a selection of certain market inputs (e.g. curves, some cap/floor/swaption prices) Intuitively appealing; favored by many practitioners Disadvantage: need a model for each type of product; consistency issues arise if we have a portfolio of exotic deals “model arbitrage” becomes possible ConclusionsCurrent market may not provide all the relevant information (the market is not “complete”) especially true for Asian currencies Need to have a good understanding of the properties of each model instantaneous match to the “relevant” market inputs implications for future behavior Need to have a good understanding of the properties of the product to be priced would it be dependent on smile information or jumps? would stochastic volatility add any value or change the hedge? Finally, it is a tradeoff between accuracy and complexity some simple models may be slightly wrong, but we can concentrate on managing the main risks

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