Interest rate models: from theory to practice

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Interest rate modelsfrom theory to practice

• Ka Lok Chau
• HKUST, June 2005

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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?

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The market parameters

• Yield curve
• Money market rates for maturity lt 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

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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

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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

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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.

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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)

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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 lt 20, r would be less than 1

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Instantaneous correlation (2)

• Solution 2 we could assume non-flat volatility
  structure for sL2

• if we choose sL21 lt 20 and sL22 gt 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

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USD calibration example

• Calibrated using a 1-factor HJM model
• Data as of May 11, 2005

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USD calibration example (2)

• Calibrated using a 2-factor BGM model
• Data as of May 11, 2005

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SGD cap/swaption example
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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))

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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

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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

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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!

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What do we want to achieve?

• Explanatory power vs exact fitting
• many degrees of freedom -gt exact fit
• parametric function -gt 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

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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)

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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

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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

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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

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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)

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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
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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

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Typical exotic structures

• The above represents the sellers position
• Initial cost of swap Bermudan option 0 (after
  fees)
• The difficulty is usually in evaluating the fair
  value of the Bermudan option

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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 lt 6-month LIBOR lt 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 gt 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))

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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)

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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

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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. couponmax (Li , Li-1 0.20)
• this could be written as
• Li-1 0.20 max( Li - Li-1 - 0.20, 0)

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Ratchet caps analysis

• Similar to forward starting options (cliquets) in
  equity
• correct modeling of the forward volatility
  structure is critical
• dont 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

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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

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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

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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)

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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 .

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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

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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