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Interest rate models: from theory to practice

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 practice

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  1. Interest rate models:from theory to practice Ka Lok Chau HKUST, June 2005

  2. 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 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?

  3. The market parameters • Yield 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

  4. Yield curve dynamics • Backward 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

  5. parallel shift (could have different magnitudes) twist hump Yield curve movements • Parallel 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

  6. Historical yield curve movements • Based 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.

  7. ST1 DF1 DF2 0 T a1, L1 a2, L2 Caps and swaptions • Caps (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)

  8. Instantaneous correlation • Example: 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

  9. sL1 0 T T2 sL21 sL22 Instantaneous 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

  10. USD calibration example • Calibrated using a 1-factor HJM model • Data as of May 11, 2005

  11. USD calibration example (2) • Calibrated using a 2-factor BGM model • Data as of May 11, 2005

  12. SGD cap/swaption example

  13. What 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))

  14. 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

  15. 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 available • Therefore instantaneously correlation between forwards could be arbitrary • Forward volatility could be arbitrary, especially at different strikes

  16. 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!

  17. 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

  18. Properties of a good model • Be 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)

  19. Calculating vegas • Naïve 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

  20. 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

  21. Model complexity • Black (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

  22. Case studies • Non-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)

  23. ST1,T ST3,T 0 T1 T2 T3 … TN-2 TN-1 ST2,T Bermudan swaption • Typical 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” T

  24. Bermudan swaption: analysis • Critical 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

  25. LIBOR-spread Swap (non-callable) Time Exotic coupon Bermudan option Exotic coupon Time LIBOR - spread Typical exotic structures • The 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

  26. Some non-path dependent products • Examples: • 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))

  27. CRANs: analysis • Correct 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)

  28. CASOs: analysis • Correct 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

  29. Ratchet caps • Typical 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 i • Application: 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)

  30. Ratchet caps: analysis • Similar 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

  31. CRANs with varying coupons • Typical 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 / N • N = 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

  32. Enhanced TARNs • Typical 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 terminated • Bonus coupon: 0% in year 1, 2% in year 2 and so on increased to 18% in year 10 • More leverage based on the expected termination time

  33. Model comparison exercise • Pricing 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)

  34. Model selection considerations • Global 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 ….

  35. 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

  36. Conclusions • Current 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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