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Interest Rate Bermudan
Swaption Valuation and
Risk
Dmitry Popov
FinPricing
http://www.finpricing.com
Bermudan Swaption
Summary
◆ Bermudan Swaption Definition
◆ Bermudan Swaption Payoffs
◆ Valuation Model Selection Criteria
◆ LGM Model
◆ LGM Assumption
◆ LGM calibration
◆ Valuation Implementation
◆ A real world example
Bermudan Swaption
Bermudan Swaption Definition
◆ An interest rate Bermudan swaption is an option on an interest rate
swap with predefined exercise schedules.
◆ A Bermudan swaption gives the holder the right but not the obligation
to enter an interest rate swap at predefined dates.
◆ Bermudan swaptions give the holders some flexibility to enter swaps.
◆ A comparison of European, American and Bermudan swaptions
◆ European swaption has only one exercise date at the maturity.
◆ American swaption has multiple exercise dates (daily)
◆ Bermudan swaption has multiple exercise dates (but not daily): such as
quarterly, monthly, etc.
Bermudan Swaption
Bermudan Swaption Payoffs
◆ At the maturity T, the payoff of a Bermudan swaption is given by
𝑃𝑎𝑦𝑜𝑓𝑓 𝑇 = max(0, 𝑉𝑠𝑤𝑎𝑝 𝑇 )
where 𝑉𝑠𝑤𝑎𝑝(𝑇) is the value of the underlying swap at T.
◆ At any exercise date 𝑇𝑖, the payoff of the Bermudan swaption is given by
𝑃𝑎𝑦𝑜𝑓𝑓 𝑇𝑖 = 𝑚𝑎𝑥 𝑉𝑠𝑤𝑎𝑝 𝑇𝑖 , 𝐼(𝑇𝑖)
where 𝑉𝑠𝑤𝑎𝑝(𝑇𝑖) is the exercise value of the Bermudan swap and 𝐼(𝑇𝑖) is the
intrinsic value.
Bermudan Swaption
Model Selection Criteria
◆ Given the complexity of Bermudan swaption valuation, there is no
closed form solution. Therefore, we need to select an interest rate term
structure model and a numeric solution to price Bermudan swaptions
numerically.
◆ The selection of interest rate term structure models
◆ Popular interest rate term structure models:
Hull-White, Linear Gaussian Model (LGM), Quadratic Gaussian Model (QGM),
Heath Jarrow Morton (HJM), Libor Market Model (LMM).
◆ HJM and LMM are too complex.
◆ Hull-White is inaccurate for computing sensitivities.
◆ Therefore, we choose either LGM or QGM.
Bermudan Swaption
Model Selection Criteria (Cont)
◆ The selection of numeric approaches
◆ After selecting a term structure model, we need to choose a numeric
approach to approximate the underlying stochastic process of the model.
◆ Commonly used numeric approaches are tree, partial differential equation
(PDE), lattice and Monte Carlo simulation.
◆ Tree and Monte Carlo are notorious for inaccuracy on sensitivity calculation.
◆ Therefore, we choose either PDE or lattice.
◆ Our decision is to use LGM plus lattice.
Bermudan Swaption
LGM Model
◆ The dynamics
𝑑𝑋 𝑡 = 𝛼 𝑡 𝑑𝑊
where X is the single state variable and W is the Wiener process.
◆ The numeraire is given by
𝑁 𝑡, 𝑋 = 𝐻 𝑡 𝑋 + 0.5𝐻2
𝑡 𝜁 𝑡 /𝐷(𝑡)
◆ The zero coupon bond price is
𝐵 𝑡, 𝑋; 𝑇 = 𝐷 𝑇 𝑒𝑥𝑝 −𝐻 𝑡 𝑋 − 0.5𝐻2
𝑡 𝜁 𝑡
Bermudan Swaption
LGM Assumption
◆ The LGM model is mathematically equivalent to the Hull-White model
but offers
◆ Significant improvement of stability and accuracy for calibration.
◆ Significant improvement of stability and accuracy for sensitivity calculation.
◆ The state variable is normally distributed under the appropriate
measure.
◆ The LGM model has only one stochastic driver (one-factor), thus
changes in rates are perfected correlated.
Bermudan Swaption
LGM calibration
◆ Match today’s curve
At time t=0, X(0)=0 and H(0)=0. Thus Z(0,0;T)=D(T). In other words, the
LGM automatically fits today’s discount curve.
◆ Select a group of market swaptions.
◆ Solve parameters by minimizing the relative error between the market
swaption prices and the LGM model swaption prices.
Bermudan Swaption
Valuation Implementation
◆ Calibrate the LGM model.
◆ Create the lattice based on the LGM: the grid range should cover at least
3 standard deviations.
◆ Calculate the underlying swap value at each final note.
◆ Conduct backward induction process iteratively rolling back from final
dates until reaching the valuation date.
◆ Compare exercise values with intrinsic values at each exercise date.
◆ The value at the valuation date is the price of the Bermudan swaption.
Bermudan Swaption
A real world exampleSwaption definition
Counterparty xxx
Buy or sell Sell
Payer or receiver Receiver
Currency USD
Settlement Cash
Trade date 9/12/2012
Underlying swap definition Leg 1 Leg2
Day Count dcAct360 dcAct360
Leg Type Fixed Float
Notional 250000 250000
Payment Frequency 1 1
Pay Receive Receive Pay
Start Date 9/14/2012 9/14/2012
End Date 9/14/2022 9/14/2022
Fix rate 0.0398 NA
Index Type NA LIBOR
Index Tenor NA 1M
Index Day Count NA dcAct360
Exercise Schedules
Exercise Type Notification Date Settlement Date
Call 1/12/2017 1/14/2017
Call 1/10/2018 1/14/2018
Thanks!

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Bermudan Swaption Valuation

  • 1. Interest Rate Bermudan Swaption Valuation and Risk Dmitry Popov FinPricing http://www.finpricing.com
  • 2. Bermudan Swaption Summary ◆ Bermudan Swaption Definition ◆ Bermudan Swaption Payoffs ◆ Valuation Model Selection Criteria ◆ LGM Model ◆ LGM Assumption ◆ LGM calibration ◆ Valuation Implementation ◆ A real world example
  • 3. Bermudan Swaption Bermudan Swaption Definition ◆ An interest rate Bermudan swaption is an option on an interest rate swap with predefined exercise schedules. ◆ A Bermudan swaption gives the holder the right but not the obligation to enter an interest rate swap at predefined dates. ◆ Bermudan swaptions give the holders some flexibility to enter swaps. ◆ A comparison of European, American and Bermudan swaptions ◆ European swaption has only one exercise date at the maturity. ◆ American swaption has multiple exercise dates (daily) ◆ Bermudan swaption has multiple exercise dates (but not daily): such as quarterly, monthly, etc.
  • 4. Bermudan Swaption Bermudan Swaption Payoffs ◆ At the maturity T, the payoff of a Bermudan swaption is given by 𝑃𝑎𝑦𝑜𝑓𝑓 𝑇 = max(0, 𝑉𝑠𝑤𝑎𝑝 𝑇 ) where 𝑉𝑠𝑤𝑎𝑝(𝑇) is the value of the underlying swap at T. ◆ At any exercise date 𝑇𝑖, the payoff of the Bermudan swaption is given by 𝑃𝑎𝑦𝑜𝑓𝑓 𝑇𝑖 = 𝑚𝑎𝑥 𝑉𝑠𝑤𝑎𝑝 𝑇𝑖 , 𝐼(𝑇𝑖) where 𝑉𝑠𝑤𝑎𝑝(𝑇𝑖) is the exercise value of the Bermudan swap and 𝐼(𝑇𝑖) is the intrinsic value.
  • 5. Bermudan Swaption Model Selection Criteria ◆ Given the complexity of Bermudan swaption valuation, there is no closed form solution. Therefore, we need to select an interest rate term structure model and a numeric solution to price Bermudan swaptions numerically. ◆ The selection of interest rate term structure models ◆ Popular interest rate term structure models: Hull-White, Linear Gaussian Model (LGM), Quadratic Gaussian Model (QGM), Heath Jarrow Morton (HJM), Libor Market Model (LMM). ◆ HJM and LMM are too complex. ◆ Hull-White is inaccurate for computing sensitivities. ◆ Therefore, we choose either LGM or QGM.
  • 6. Bermudan Swaption Model Selection Criteria (Cont) ◆ The selection of numeric approaches ◆ After selecting a term structure model, we need to choose a numeric approach to approximate the underlying stochastic process of the model. ◆ Commonly used numeric approaches are tree, partial differential equation (PDE), lattice and Monte Carlo simulation. ◆ Tree and Monte Carlo are notorious for inaccuracy on sensitivity calculation. ◆ Therefore, we choose either PDE or lattice. ◆ Our decision is to use LGM plus lattice.
  • 7. Bermudan Swaption LGM Model ◆ The dynamics 𝑑𝑋 𝑡 = 𝛼 𝑡 𝑑𝑊 where X is the single state variable and W is the Wiener process. ◆ The numeraire is given by 𝑁 𝑡, 𝑋 = 𝐻 𝑡 𝑋 + 0.5𝐻2 𝑡 𝜁 𝑡 /𝐷(𝑡) ◆ The zero coupon bond price is 𝐵 𝑡, 𝑋; 𝑇 = 𝐷 𝑇 𝑒𝑥𝑝 −𝐻 𝑡 𝑋 − 0.5𝐻2 𝑡 𝜁 𝑡
  • 8. Bermudan Swaption LGM Assumption ◆ The LGM model is mathematically equivalent to the Hull-White model but offers ◆ Significant improvement of stability and accuracy for calibration. ◆ Significant improvement of stability and accuracy for sensitivity calculation. ◆ The state variable is normally distributed under the appropriate measure. ◆ The LGM model has only one stochastic driver (one-factor), thus changes in rates are perfected correlated.
  • 9. Bermudan Swaption LGM calibration ◆ Match today’s curve At time t=0, X(0)=0 and H(0)=0. Thus Z(0,0;T)=D(T). In other words, the LGM automatically fits today’s discount curve. ◆ Select a group of market swaptions. ◆ Solve parameters by minimizing the relative error between the market swaption prices and the LGM model swaption prices.
  • 10. Bermudan Swaption Valuation Implementation ◆ Calibrate the LGM model. ◆ Create the lattice based on the LGM: the grid range should cover at least 3 standard deviations. ◆ Calculate the underlying swap value at each final note. ◆ Conduct backward induction process iteratively rolling back from final dates until reaching the valuation date. ◆ Compare exercise values with intrinsic values at each exercise date. ◆ The value at the valuation date is the price of the Bermudan swaption.
  • 11. Bermudan Swaption A real world exampleSwaption definition Counterparty xxx Buy or sell Sell Payer or receiver Receiver Currency USD Settlement Cash Trade date 9/12/2012 Underlying swap definition Leg 1 Leg2 Day Count dcAct360 dcAct360 Leg Type Fixed Float Notional 250000 250000 Payment Frequency 1 1 Pay Receive Receive Pay Start Date 9/14/2012 9/14/2012 End Date 9/14/2022 9/14/2022 Fix rate 0.0398 NA Index Type NA LIBOR Index Tenor NA 1M Index Day Count NA dcAct360 Exercise Schedules Exercise Type Notification Date Settlement Date Call 1/12/2017 1/14/2017 Call 1/10/2018 1/14/2018