Post on 16-Aug-2015
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
School of Education, Culture and CommunicationDivision of Applied Mathematics
An Introduction to ModernPricing of Interest Rate Derivatives
Master Thesis in Financial Engineering
Author: Hossein Nohrouzian
Malardalen University
June 5, 2015
1/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
OutlineSchool of Education, Culture and CommunicationDivision of Applied Mathematics
1 Introduction
2 Interest Rates
3 Security Market Models
4 Term-Structure Models
5 Pricing Interest Rate Derivatives
6 HJM Framework and LIIBOR Market Model
7 Collateral Agreement (CSA)
8 Conclusion
2/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Risky Asset vs Risk-Less AssetSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Does exist two kind of investments?
• An example is pension salary vs inflation.
• NASDQ value increased by almost 150% in 5 years.
Figure: Price behavior of the NASDAQ from 2010 to 2015
3/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Risky Asset vs Risk-Less AssetSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Does exist two kind of investments?
• An example is pension salary vs inflation.
• NASDQ value increased by almost 150% in 5 years.
Figure: Price behavior of the NASDAQ from 2010 to 2015
3/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Risky Asset vs Risk-Less AssetSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Does exist two kind of investments?
• An example is pension salary vs inflation.
• NASDQ value increased by almost 150% in 5 years.
Figure: Price behavior of the NASDAQ from 2010 to 2015
3/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Interest Rate and EconomicsFactors
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Interest rate and monetary policy.• Interest rate and international trading.• Interest rate and economic growth.
• 0.0%, -0.1% and -0.25%
Figure: The exchange rate between USD and SEK
4/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Interest Rate and EconomicsFactors
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Interest rate and monetary policy.• Interest rate and international trading.• Interest rate and economic growth.• 0.0%, -0.1% and -0.25%
Figure: The exchange rate between USD and SEK
4/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Jump DiffusionSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• On 15th of January 2015 SNB unexpectedly scrapped itscap on the Euro value of the Franc.
• The result was 27.5% change in USD vs CHF and shake instock prices.
Figure: Exchange rate between USD and CHF
5/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Jump DiffusionSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• On 15th of January 2015 SNB unexpectedly scrapped itscap on the Euro value of the Franc.
• The result was 27.5% change in USD vs CHF and shake instock prices.
Figure: Exchange rate between USD and CHF
5/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Banks vs Market RatesSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Banks offered rates to individuals and companies.
• Different banks use different rates for loans and savings.
• Interest rates in the market.
• Before the economic crisis in 2007 and 2008:
• XIBOR was reference of interest rate for loans in theinternational financial market.
• From and after the economic crisis in 2007 and 2008:
• Collateral rate:• is used in the collateral agreement or CSA,• calculated daily on the overnight index swaps.
• Swap rates:
• Fixed rates are calculated from forward rates,• floating rates are calculated from OIS rates.
6/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Banks vs Market RatesSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Banks offered rates to individuals and companies.• Different banks use different rates for loans and savings.
• Interest rates in the market.
• Before the economic crisis in 2007 and 2008:
• XIBOR was reference of interest rate for loans in theinternational financial market.
• From and after the economic crisis in 2007 and 2008:
• Collateral rate:• is used in the collateral agreement or CSA,• calculated daily on the overnight index swaps.
• Swap rates:
• Fixed rates are calculated from forward rates,• floating rates are calculated from OIS rates.
6/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Banks vs Market RatesSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Banks offered rates to individuals and companies.• Different banks use different rates for loans and savings.
• Interest rates in the market.• Before the economic crisis in 2007 and 2008:
• XIBOR was reference of interest rate for loans in theinternational financial market.
• From and after the economic crisis in 2007 and 2008:
• Collateral rate:• is used in the collateral agreement or CSA,• calculated daily on the overnight index swaps.
• Swap rates:
• Fixed rates are calculated from forward rates,• floating rates are calculated from OIS rates.
6/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Banks vs Market RatesSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Banks offered rates to individuals and companies.• Different banks use different rates for loans and savings.
• Interest rates in the market.• Before the economic crisis in 2007 and 2008:
• XIBOR was reference of interest rate for loans in theinternational financial market.
• From and after the economic crisis in 2007 and 2008:
• Collateral rate:• is used in the collateral agreement or CSA,• calculated daily on the overnight index swaps.
• Swap rates:
• Fixed rates are calculated from forward rates,• floating rates are calculated from OIS rates.
6/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Banks vs Market RatesSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Banks offered rates to individuals and companies.• Different banks use different rates for loans and savings.
• Interest rates in the market.• Before the economic crisis in 2007 and 2008:
• XIBOR was reference of interest rate for loans in theinternational financial market.
• From and after the economic crisis in 2007 and 2008:• Collateral rate:
• is used in the collateral agreement or CSA,• calculated daily on the overnight index swaps.
• Swap rates:
• Fixed rates are calculated from forward rates,• floating rates are calculated from OIS rates.
6/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Banks vs Market RatesSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Banks offered rates to individuals and companies.• Different banks use different rates for loans and savings.
• Interest rates in the market.• Before the economic crisis in 2007 and 2008:
• XIBOR was reference of interest rate for loans in theinternational financial market.
• From and after the economic crisis in 2007 and 2008:• Collateral rate:• is used in the collateral agreement or CSA,• calculated daily on the overnight index swaps.
• Swap rates:
• Fixed rates are calculated from forward rates,• floating rates are calculated from OIS rates.
6/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Banks vs Market RatesSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Banks offered rates to individuals and companies.• Different banks use different rates for loans and savings.
• Interest rates in the market.• Before the economic crisis in 2007 and 2008:
• XIBOR was reference of interest rate for loans in theinternational financial market.
• From and after the economic crisis in 2007 and 2008:• Collateral rate:• is used in the collateral agreement or CSA,• calculated daily on the overnight index swaps.
• Swap rates:
• Fixed rates are calculated from forward rates,• floating rates are calculated from OIS rates.
6/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Banks vs Market RatesSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Banks offered rates to individuals and companies.• Different banks use different rates for loans and savings.
• Interest rates in the market.• Before the economic crisis in 2007 and 2008:
• XIBOR was reference of interest rate for loans in theinternational financial market.
• From and after the economic crisis in 2007 and 2008:• Collateral rate:• is used in the collateral agreement or CSA,• calculated daily on the overnight index swaps.
• Swap rates:• Fixed rates are calculated from forward rates,• floating rates are calculated from OIS rates.
6/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Risk-Neutral EvaluationSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Risk-Neutral world
1 The expected return on a stock (or any other investment)is the risk-free rate,
2 The discount rate used for the expected payoff on anoption (or any other investment) is the risk-free rate.
• Under Risk-neutral P∗ equivalent to the P
1 The discounted price of a derivative is martingale,2 The discounted expected value under the P∗ or Q of a
derivative, gives its no-arbitrage price.
7/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Risk-Neutral EvaluationSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Risk-Neutral world
1 The expected return on a stock (or any other investment)is the risk-free rate,
2 The discount rate used for the expected payoff on anoption (or any other investment) is the risk-free rate.
• Under Risk-neutral P∗ equivalent to the P
1 The discounted price of a derivative is martingale,2 The discounted expected value under the P∗ or Q of a
derivative, gives its no-arbitrage price.
7/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Money Market Account as aNumeriare
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Money Market Account
• Constant Interest Rates
B(t) =[
limn→∞
(1 +
r
n
)n]t= ert , t ≥ 0.
• Stochastic Interest Rates
B(t) = exp
{∫ t
0
r(u)du
}, t ≥ 0,
r(t) is time-t instantaneous interest rate.
8/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Money Market Account as aNumeriare
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Money Market Account• Constant Interest Rates
B(t) =[
limn→∞
(1 +
r
n
)n]t= ert , t ≥ 0.
• Stochastic Interest Rates
B(t) = exp
{∫ t
0
r(u)du
}, t ≥ 0,
r(t) is time-t instantaneous interest rate.
8/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Money Market Account as aNumeriare
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Money Market Account• Constant Interest Rates
B(t) =[
limn→∞
(1 +
r
n
)n]t= ert , t ≥ 0.
• Stochastic Interest Rates
B(t) = exp
{∫ t
0
r(u)du
}, t ≥ 0,
r(t) is time-t instantaneous interest rate.
8/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Discount Bond as a NumeriareSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Forward Rates
• Instantaneous forward rate
f (t,T ) = − ∂
∂Tln v(t,T ), t ≤ T .
• Default-free discount bond
v(t,T ) = exp
{−∫ T
t
f (t, s)ds
}, t ≤ T .
• r(T ) = f (t,T ), t ≤ T . If r(t) is deterministic
v(t,T ) = exp
{−∫ T
t
r(s)ds
}=
B(t)
B(T ), t ≤ T .
9/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Discount Bond as a NumeriareSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Forward Rates• Instantaneous forward rate
f (t,T ) = − ∂
∂Tln v(t,T ), t ≤ T .
• Default-free discount bond
v(t,T ) = exp
{−∫ T
t
f (t, s)ds
}, t ≤ T .
• r(T ) = f (t,T ), t ≤ T . If r(t) is deterministic
v(t,T ) = exp
{−∫ T
t
r(s)ds
}=
B(t)
B(T ), t ≤ T .
9/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Discount Bond as a NumeriareSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Forward Rates• Instantaneous forward rate
f (t,T ) = − ∂
∂Tln v(t,T ), t ≤ T .
• Default-free discount bond
v(t,T ) = exp
{−∫ T
t
f (t, s)ds
}, t ≤ T .
• r(T ) = f (t,T ), t ≤ T . If r(t) is deterministic
v(t,T ) = exp
{−∫ T
t
r(s)ds
}=
B(t)
B(T ), t ≤ T .
9/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Discount Bond as a NumeriareSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Forward Rates• Instantaneous forward rate
f (t,T ) = − ∂
∂Tln v(t,T ), t ≤ T .
• Default-free discount bond
v(t,T ) = exp
{−∫ T
t
f (t, s)ds
}, t ≤ T .
• r(T ) = f (t,T ), t ≤ T . If r(t) is deterministic
v(t,T ) = exp
{−∫ T
t
r(s)ds
}=
B(t)
B(T ), t ≤ T .
9/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing under Risk-Neutral MethodSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Price of European Option under Q
π(t) = B(t)EQ
[h(S(T ))
B(T )
∣∣∣∣Ft
], 0 ≤ t ≤ T .
where π(T ) = h(S(T )).
• Samuelson price process{dS(t) = µS(t)dt + σS(t)dW ,S(0) = S0.
• Black–Scholes-Merton Lognormal Price
ST = St exp
{(r − 1
2σ2
)(T − t) + σ (WT −Wt)
}.
10/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing under Risk-Neutral MethodSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Price of European Option under Q
π(t) = B(t)EQ
[h(S(T ))
B(T )
∣∣∣∣Ft
], 0 ≤ t ≤ T .
where π(T ) = h(S(T )).
• Samuelson price process{dS(t) = µS(t)dt + σS(t)dW ,S(0) = S0.
• Black–Scholes-Merton Lognormal Price
ST = St exp
{(r − 1
2σ2
)(T − t) + σ (WT −Wt)
}.
10/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing under Risk-Neutral MethodSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Price of European Option under Q
π(t) = B(t)EQ
[h(S(T ))
B(T )
∣∣∣∣Ft
], 0 ≤ t ≤ T .
where π(T ) = h(S(T )).
• Samuelson price process{dS(t) = µS(t)dt + σS(t)dW ,S(0) = S0.
• Black–Scholes-Merton Lognormal Price
ST = St exp
{(r − 1
2σ2
)(T − t) + σ (WT −Wt)
}.
10/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing under Forward-NeutralMethod
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Price process of discount bond under Q
dv(t,T )
v(t,T )= r(t)dt + σv (t)dW ∗, 0 ≤ t ≤ T ,
• Price process of security under Q
dS
S= r(t)dt + σ(t)dW ∗, 0 ≤ t ≤ T ,
• Price of European claim under QT
πC (t) = v(t,T )EQT [h(S(T ))
∣∣Ft
], 0 ≤ t ≤ T .
11/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing under Forward-NeutralMethod
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Price process of discount bond under Q
dv(t,T )
v(t,T )= r(t)dt + σv (t)dW ∗, 0 ≤ t ≤ T ,
• Price process of security under Q
dS
S= r(t)dt + σ(t)dW ∗, 0 ≤ t ≤ T ,
• Price of European claim under QT
πC (t) = v(t,T )EQT [h(S(T ))
∣∣Ft
], 0 ≤ t ≤ T .
11/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing under Forward-NeutralMethod
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Price process of discount bond under Q
dv(t,T )
v(t,T )= r(t)dt + σv (t)dW ∗, 0 ≤ t ≤ T ,
• Price process of security under Q
dS
S= r(t)dt + σ(t)dW ∗, 0 ≤ t ≤ T ,
• Price of European claim under QT
πC (t) = v(t,T )EQT [h(S(T ))
∣∣Ft
], 0 ≤ t ≤ T .
11/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Term-Structure ModelsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Spot-Rate (Equilibrium) Models
dr = a(m − r)dt + σrγdW , t ≥ 0,
• Rendleman–Bartter Model, (+) Rates,• Vasicek Model, (-) Rates,• Cox–Ingersoll–Ross (CIR) Model, (+) Rates,• Longstaff–Schwartz Stochastic Volatility Model, (-) Rates,• Problem: Do not fit today’s term structure of interest rate.
12/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Term-Structure ModelsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Spot-Rate (Equilibrium) Models
dr = a(m − r)dt + σrγdW , t ≥ 0,
• Rendleman–Bartter Model, (+) Rates,• Vasicek Model, (-) Rates,• Cox–Ingersoll–Ross (CIR) Model, (+) Rates,• Longstaff–Schwartz Stochastic Volatility Model, (-) Rates,
• Problem: Do not fit today’s term structure of interest rate.
12/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Term-Structure ModelsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Spot-Rate (Equilibrium) Models
dr = a(m − r)dt + σrγdW , t ≥ 0,
• Rendleman–Bartter Model, (+) Rates,• Vasicek Model, (-) Rates,• Cox–Ingersoll–Ross (CIR) Model, (+) Rates,• Longstaff–Schwartz Stochastic Volatility Model, (-) Rates,• Problem: Do not fit today’s term structure of interest rate.
12/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Term-Structure ModelsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Spot-Rate (No Arbitrage) Models
• Ho–Lee Model, Developed from Lattice approximation(Binomial Tree),
• Hull–White (One-Factor) Model, Application in pricingAmerican option via trinomial tree,
• Black–Derman–Toy Model, Developed from binomial treemodel for lognormal spot rate, Identical to Lognormalversion of Ho–Lee Model,
• Black–Karasinski Model, Extension of Black–Derman–ToyModel,
• Hull–White (Two-Factor) Model.
13/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Term-Structure ModelsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Spot-Rate (No Arbitrage) Models• Ho–Lee Model, Developed from Lattice approximation
(Binomial Tree),• Hull–White (One-Factor) Model, Application in pricing
American option via trinomial tree,• Black–Derman–Toy Model, Developed from binomial tree
model for lognormal spot rate, Identical to Lognormalversion of Ho–Lee Model,
• Black–Karasinski Model, Extension of Black–Derman–ToyModel,
• Hull–White (Two-Factor) Model.
13/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing Discount Bond via VasicekModel
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Market price of risk λ(t) = λ and SDE under Q
dr = a(r − r)dt + σdW ∗,
• risk-adjusted (r.a.) mean reverting level
r = m − σ
aλ,
• r.a. drift m(r , t) = ar − ar & diffusion σ(r , t) = σ.• default-free discount bond price
v(t,T ) = H1(T − t)e−H2(T−t)r(t), 0 ≤ t ≤ T ,
H2(t) =1− e−at
a,
H1(t) = exp
{(H2(t)− t)(a2r − σ2/2)
a2− σ2H2
2 (t)
4a
}.
14/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing Discount Bond via VasicekModel
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Market price of risk λ(t) = λ and SDE under Q
dr = a(r − r)dt + σdW ∗,
• risk-adjusted (r.a.) mean reverting level
r = m − σ
aλ,
• r.a. drift m(r , t) = ar − ar & diffusion σ(r , t) = σ.
• default-free discount bond price
v(t,T ) = H1(T − t)e−H2(T−t)r(t), 0 ≤ t ≤ T ,
H2(t) =1− e−at
a,
H1(t) = exp
{(H2(t)− t)(a2r − σ2/2)
a2− σ2H2
2 (t)
4a
}.
14/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing Discount Bond via VasicekModel
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Market price of risk λ(t) = λ and SDE under Q
dr = a(r − r)dt + σdW ∗,
• risk-adjusted (r.a.) mean reverting level
r = m − σ
aλ,
• r.a. drift m(r , t) = ar − ar & diffusion σ(r , t) = σ.• default-free discount bond price
v(t,T ) = H1(T − t)e−H2(T−t)r(t), 0 ≤ t ≤ T ,
H2(t) =1− e−at
a,
H1(t) = exp
{(H2(t)− t)(a2r − σ2/2)
a2− σ2H2
2 (t)
4a
}.
14/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing Discount Bond via CIRModel
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Market price of risk λ(t) = a(m−r)
σ√
r(t), and SDE under Q
dr = a(r − r)dt + σ√r(t)dW ∗, 0 ≤ t ≤ T ,
• r.a. drift m(r , t) = ar − ar & diffusion σ(r , t) = σ2r .
• Letγ =√a2 + 2σ2, then price of d.f.d.b. is
v(t,T ) = H1(T − t)e−H2(T−t)r(t), 0 ≤ t ≤ T ,
H1(t) =
(2γe(a+γ)t/2
(a + γ)(eγt − 1) + 2γ
)2ar/σ2
,
H2(t) =2(eγt − 1)
(a + γ)(eγt − 1) + 2γ.
15/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing Discount Bond via CIRModel
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Market price of risk λ(t) = a(m−r)
σ√
r(t), and SDE under Q
dr = a(r − r)dt + σ√r(t)dW ∗, 0 ≤ t ≤ T ,
• r.a. drift m(r , t) = ar − ar & diffusion σ(r , t) = σ2r .
• Letγ =√a2 + 2σ2, then price of d.f.d.b. is
v(t,T ) = H1(T − t)e−H2(T−t)r(t), 0 ≤ t ≤ T ,
H1(t) =
(2γe(a+γ)t/2
(a + γ)(eγt − 1) + 2γ
)2ar/σ2
,
H2(t) =2(eγt − 1)
(a + γ)(eγt − 1) + 2γ.
15/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing Discount Bond via CIRModel
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Market price of risk λ(t) = a(m−r)
σ√
r(t), and SDE under Q
dr = a(r − r)dt + σ√r(t)dW ∗, 0 ≤ t ≤ T ,
• r.a. drift m(r , t) = ar − ar & diffusion σ(r , t) = σ2r .
• Letγ =√a2 + 2σ2, then price of d.f.d.b. is
v(t,T ) = H1(T − t)e−H2(T−t)r(t), 0 ≤ t ≤ T ,
H1(t) =
(2γe(a+γ)t/2
(a + γ)(eγt − 1) + 2γ
)2ar/σ2
,
H2(t) =2(eγt − 1)
(a + γ)(eγt − 1) + 2γ.
15/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Forward LIBOR and Black’sFormula
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Ti -forward LIBOR Li (t) under QTi+1 is a martingale
Li (t) = EQTi+1 [Li (τ)
∣∣Ft
], t ≤ τ ≤ T ,
• SDE Ti -forward LIBOR under QTi+1
dLiLi
= σi (t)dW Ti+1 , 0 ≤ t ≤ Ti ,
{W Ti+1(t)} is a standard Brownian motion under QTi+1 .
16/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Forward LIBOR and Black’sFormula
School of Education, Culture and CommunicationDivision of Applied Mathematics
• Ti -forward LIBOR Li (t) under QTi+1 is a martingale
Li (t) = EQTi+1 [Li (τ)
∣∣Ft
], t ≤ τ ≤ T ,
• SDE Ti -forward LIBOR under QTi+1
dLiLi
= σi (t)dW Ti+1 , 0 ≤ t ≤ Ti ,
{W Ti+1(t)} is a standard Brownian motion under QTi+1 .
16/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Cap and CapletsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Caplet price
Cpli (t) = δiv(t,Ti+1) [Li (t)Φ(di )− KΦ(di − ςi )] , (1)
where δi are interval between tenor dates and
di =ln(Li (t)/K )
ςi+ςi2, ςi > 0.
and ς2i =
∫ Ti
t σ2i (s)ds is accumulated variance.
• Cap (portfolio of caplets) price
Cap(t) =n−1∑i=0
δiv(t,Ti+1) [Li (t)Φ(di )− KΦ(di − ςi )] , t < T0.
• Same procedure for floor and floorlets. If δi = 1, then (1)is identical to Black’s formula.
17/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Cap and CapletsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Caplet price
Cpli (t) = δiv(t,Ti+1) [Li (t)Φ(di )− KΦ(di − ςi )] , (1)
where δi are interval between tenor dates and
di =ln(Li (t)/K )
ςi+ςi2, ςi > 0.
and ς2i =
∫ Ti
t σ2i (s)ds is accumulated variance.
• Cap (portfolio of caplets) price
Cap(t) =n−1∑i=0
δiv(t,Ti+1) [Li (t)Φ(di )− KΦ(di − ςi )] , t < T0.
• Same procedure for floor and floorlets. If δi = 1, then (1)is identical to Black’s formula.
17/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Cap and CapletsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Caplet price
Cpli (t) = δiv(t,Ti+1) [Li (t)Φ(di )− KΦ(di − ςi )] , (1)
where δi are interval between tenor dates and
di =ln(Li (t)/K )
ςi+ςi2, ςi > 0.
and ς2i =
∫ Ti
t σ2i (s)ds is accumulated variance.
• Cap (portfolio of caplets) price
Cap(t) =n−1∑i=0
δiv(t,Ti+1) [Li (t)Φ(di )− KΦ(di − ςi )] , t < T0.
• Same procedure for floor and floorlets. If δi = 1, then (1)is identical to Black’s formula.
17/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Swap Rate and SwaptionsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Swap rate
S(t) =VFL
VFIX=
v(t,T0)− v(t,Tn)
δ∑n
i=1 v(t,Ti ), 0 ≤ t ≤ T0.
Swap rates can be used as an underlying asset for anoption so called swaptions.
• Swaption’s SDE
dS
S= σs(t)dWQTi+1
, 0 ≤ t ≤ τ
• Swaption price is approximated by Black’s formula.
18/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Swap Rate and SwaptionsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Swap rate
S(t) =VFL
VFIX=
v(t,T0)− v(t,Tn)
δ∑n
i=1 v(t,Ti ), 0 ≤ t ≤ T0.
Swap rates can be used as an underlying asset for anoption so called swaptions.
• Swaption’s SDE
dS
S= σs(t)dWQTi+1
, 0 ≤ t ≤ τ
• Swaption price is approximated by Black’s formula.
18/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Swap Rate and SwaptionsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Swap rate
S(t) =VFL
VFIX=
v(t,T0)− v(t,Tn)
δ∑n
i=1 v(t,Ti ), 0 ≤ t ≤ T0.
Swap rates can be used as an underlying asset for anoption so called swaptions.
• Swaption’s SDE
dS
S= σs(t)dWQTi+1
, 0 ≤ t ≤ τ
• Swaption price is approximated by Black’s formula.
18/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Black’s VolatilitySchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Dynamic of forward rates (cap/floor/swap rate)
• Lognormally distributed, i.e. Black’s Model
df = σB fdW
• Let πCN(t) = πCB
(t)
σN =σB (f0 − K)
ln(f0/K)
[1 +
1
24
(1−
1
120[ln(f0/K)]2
)σ2Bτ +
1
5760σ4Bτ2
] , f0
K> 0, f0 6= K .
τ exercise date in years.• The alternative formula
σN =
σB√
f0K
(1 +
1
24[ln(f0/K)]2
)1 +
1
24σ2Bτ +
1
5760σ4Bτ2
, for
∣∣∣∣ f0 − K
K
∣∣∣∣ < 0.001.
Numerical methods (Newton-Raphson method) to get σBknowing σN .
19/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Black’s VolatilitySchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Dynamic of forward rates (cap/floor/swap rate)
• Lognormally distributed, i.e. Black’s Model
df = σB fdW
• Let πCN(t) = πCB
(t)
σN =σB (f0 − K)
ln(f0/K)
[1 +
1
24
(1−
1
120[ln(f0/K)]2
)σ2Bτ +
1
5760σ4Bτ2
] , f0
K> 0, f0 6= K .
τ exercise date in years.
• The alternative formula
σN =
σB√
f0K
(1 +
1
24[ln(f0/K)]2
)1 +
1
24σ2Bτ +
1
5760σ4Bτ2
, for
∣∣∣∣ f0 − K
K
∣∣∣∣ < 0.001.
Numerical methods (Newton-Raphson method) to get σBknowing σN .
19/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Black’s VolatilitySchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Dynamic of forward rates (cap/floor/swap rate)
• Lognormally distributed, i.e. Black’s Model
df = σB fdW
• Let πCN(t) = πCB
(t)
σN =σB (f0 − K)
ln(f0/K)
[1 +
1
24
(1−
1
120[ln(f0/K)]2
)σ2Bτ +
1
5760σ4Bτ2
] , f0
K> 0, f0 6= K .
τ exercise date in years.• The alternative formula
σN =
σB√
f0K
(1 +
1
24[ln(f0/K)]2
)1 +
1
24σ2Bτ +
1
5760σ4Bτ2
, for
∣∣∣∣ f0 − K
K
∣∣∣∣ < 0.001.
Numerical methods (Newton-Raphson method) to get σBknowing σN .
19/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Bachelier’s ModelSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Bachelier’s SDE
dS(τ) = S(t)σdW (τ), 0 ≤ t ≤ τ ≤ T .
• Normal price process
S(T ) = S(t) [1 + σ (W (T )−W (t))] .
• Bachelier’s price formula
πC (t) = [S(t)− K ]N(d) + S(t)σ√
(T − t)φ(d),
πP(t) = [K − S(t)]N(−d)− S(t)σ√
(T − t)φ(−d),
d =S(t)− K
S(t)σ√
(T − t).
• ATM S(t) = K and implied volatility
πC (t) = S(t)σ
√(T − t)
2π, σ =
πC (t)
S(t)
√2π
(T − t).
20/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Bachelier’s ModelSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Bachelier’s SDE
dS(τ) = S(t)σdW (τ), 0 ≤ t ≤ τ ≤ T .
• Normal price process
S(T ) = S(t) [1 + σ (W (T )−W (t))] .
• Bachelier’s price formula
πC (t) = [S(t)− K ]N(d) + S(t)σ√
(T − t)φ(d),
πP(t) = [K − S(t)]N(−d)− S(t)σ√
(T − t)φ(−d),
d =S(t)− K
S(t)σ√
(T − t).
• ATM S(t) = K and implied volatility
πC (t) = S(t)σ
√(T − t)
2π, σ =
πC (t)
S(t)
√2π
(T − t).
20/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Bachelier’s ModelSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Bachelier’s SDE
dS(τ) = S(t)σdW (τ), 0 ≤ t ≤ τ ≤ T .
• Normal price process
S(T ) = S(t) [1 + σ (W (T )−W (t))] .
• Bachelier’s price formula
πC (t) = [S(t)− K ]N(d) + S(t)σ√
(T − t)φ(d),
πP(t) = [K − S(t)]N(−d)− S(t)σ√
(T − t)φ(−d),
d =S(t)− K
S(t)σ√
(T − t).
• ATM S(t) = K and implied volatility
πC (t) = S(t)σ
√(T − t)
2π, σ =
πC (t)
S(t)
√2π
(T − t).
20/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Bachelier’s ModelSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Bachelier’s SDE
dS(τ) = S(t)σdW (τ), 0 ≤ t ≤ τ ≤ T .
• Normal price process
S(T ) = S(t) [1 + σ (W (T )−W (t))] .
• Bachelier’s price formula
πC (t) = [S(t)− K ]N(d) + S(t)σ√
(T − t)φ(d),
πP(t) = [K − S(t)]N(−d)− S(t)σ√
(T − t)φ(−d),
d =S(t)− K
S(t)σ√
(T − t).
• ATM S(t) = K and implied volatility
πC (t) = S(t)σ
√(T − t)
2π, σ =
πC (t)
S(t)
√2π
(T − t).
20/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Black’s Model vs Normal ModelSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Black’s model
• Black’s SDE
df = σnfdW , 0 ≤ t ≤ T .
• Black’s forward price
fT = ft exp {σn(WT −Wt)} ,equivalently
ln
(fTft
)= σn(WT −Wt),
fTft> 0, ft 6= 0.
• Normal model
• Normal SDE
df = σndW , 0 ≤ t ≤ T .
• Normal forward price
fT = ft + σn(WT −Wt).
21/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Black’s Model vs Normal ModelSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Black’s model• Black’s SDE
df = σnfdW , 0 ≤ t ≤ T .
• Black’s forward price
fT = ft exp {σn(WT −Wt)} ,equivalently
ln
(fTft
)= σn(WT −Wt),
fTft> 0, ft 6= 0.
• Normal model
• Normal SDE
df = σndW , 0 ≤ t ≤ T .
• Normal forward price
fT = ft + σn(WT −Wt).
21/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Black’s Model vs Normal ModelSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Black’s model• Black’s SDE
df = σnfdW , 0 ≤ t ≤ T .
• Black’s forward price
fT = ft exp {σn(WT −Wt)} ,equivalently
ln
(fTft
)= σn(WT −Wt),
fTft> 0, ft 6= 0.
• Normal model
• Normal SDE
df = σndW , 0 ≤ t ≤ T .
• Normal forward price
fT = ft + σn(WT −Wt).
21/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Black’s Model vs Normal ModelSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Black’s model• Black’s SDE
df = σnfdW , 0 ≤ t ≤ T .
• Black’s forward price
fT = ft exp {σn(WT −Wt)} ,equivalently
ln
(fTft
)= σn(WT −Wt),
fTft> 0, ft 6= 0.
• Normal model• Normal SDE
df = σndW , 0 ≤ t ≤ T .
• Normal forward price
fT = ft + σn(WT −Wt).
21/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Black’s Model vs Normal ModelSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Black’s model• Black’s SDE
df = σnfdW , 0 ≤ t ≤ T .
• Black’s forward price
fT = ft exp {σn(WT −Wt)} ,equivalently
ln
(fTft
)= σn(WT −Wt),
fTft> 0, ft 6= 0.
• Normal model• Normal SDE
df = σndW , 0 ≤ t ≤ T .
• Normal forward price
fT = ft + σn(WT −Wt).
21/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Generating Sample PathsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Heath–Jarrow–Morton (HJM) Framework
df (t,T ) = µ(t,T )dt + σσσ(f , t,T )>dWWW (t).
• Risk-neutral valuation under Q
df (t,T ) =
(σσσ(f , t,T )>
∫ T
tσσσ(f , t, u)du
)dt + σ(f , t,T )>dWWW (t),
None of forward rates become martingale.• Forward-neutral valuation under QTF
df (t,T ) = −σσσ(f , t,T )>(∫ TF
Tσσσ(f , t, u)du
)dt + σσσ(t,T )>dWWWTF (t), t ≤ T ≤ TF .
• LIBOR Market Model (LMM)
• Forward-LIBOR SDE (Spot measure)
dLn(t)
Ln(t)=
n∑j=η(t)
δjLj (t)σσσn(t)>σσσj (t)
1 + δjLj (t)dt + σσσn(t)>dWWW (t), 0 ≤ t ≤ Tn, n = 1, . . . ,M.
• Ln = (vn − vn+1)/(δnvn+1), bond price is martingale whenit is deflated (rather discounted) by the numeriare asset.
22/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Generating Sample PathsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Heath–Jarrow–Morton (HJM) Framework
df (t,T ) = µ(t,T )dt + σσσ(f , t,T )>dWWW (t).
• Risk-neutral valuation under Q
df (t,T ) =
(σσσ(f , t,T )>
∫ T
tσσσ(f , t, u)du
)dt + σ(f , t,T )>dWWW (t),
None of forward rates become martingale.
• Forward-neutral valuation under QTF
df (t,T ) = −σσσ(f , t,T )>(∫ TF
Tσσσ(f , t, u)du
)dt + σσσ(t,T )>dWWWTF (t), t ≤ T ≤ TF .
• LIBOR Market Model (LMM)
• Forward-LIBOR SDE (Spot measure)
dLn(t)
Ln(t)=
n∑j=η(t)
δjLj (t)σσσn(t)>σσσj (t)
1 + δjLj (t)dt + σσσn(t)>dWWW (t), 0 ≤ t ≤ Tn, n = 1, . . . ,M.
• Ln = (vn − vn+1)/(δnvn+1), bond price is martingale whenit is deflated (rather discounted) by the numeriare asset.
22/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Generating Sample PathsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Heath–Jarrow–Morton (HJM) Framework
df (t,T ) = µ(t,T )dt + σσσ(f , t,T )>dWWW (t).
• Risk-neutral valuation under Q
df (t,T ) =
(σσσ(f , t,T )>
∫ T
tσσσ(f , t, u)du
)dt + σ(f , t,T )>dWWW (t),
None of forward rates become martingale.• Forward-neutral valuation under QTF
df (t,T ) = −σσσ(f , t,T )>(∫ TF
Tσσσ(f , t, u)du
)dt + σσσ(t,T )>dWWWTF (t), t ≤ T ≤ TF .
• LIBOR Market Model (LMM)
• Forward-LIBOR SDE (Spot measure)
dLn(t)
Ln(t)=
n∑j=η(t)
δjLj (t)σσσn(t)>σσσj (t)
1 + δjLj (t)dt + σσσn(t)>dWWW (t), 0 ≤ t ≤ Tn, n = 1, . . . ,M.
• Ln = (vn − vn+1)/(δnvn+1), bond price is martingale whenit is deflated (rather discounted) by the numeriare asset.
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AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Generating Sample PathsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Heath–Jarrow–Morton (HJM) Framework
df (t,T ) = µ(t,T )dt + σσσ(f , t,T )>dWWW (t).
• Risk-neutral valuation under Q
df (t,T ) =
(σσσ(f , t,T )>
∫ T
tσσσ(f , t, u)du
)dt + σ(f , t,T )>dWWW (t),
None of forward rates become martingale.• Forward-neutral valuation under QTF
df (t,T ) = −σσσ(f , t,T )>(∫ TF
Tσσσ(f , t, u)du
)dt + σσσ(t,T )>dWWWTF (t), t ≤ T ≤ TF .
• LIBOR Market Model (LMM)
• Forward-LIBOR SDE (Spot measure)
dLn(t)
Ln(t)=
n∑j=η(t)
δjLj (t)σσσn(t)>σσσj (t)
1 + δjLj (t)dt + σσσn(t)>dWWW (t), 0 ≤ t ≤ Tn, n = 1, . . . ,M.
• Ln = (vn − vn+1)/(δnvn+1), bond price is martingale whenit is deflated (rather discounted) by the numeriare asset.
22/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Generating Sample PathsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Heath–Jarrow–Morton (HJM) Framework
df (t,T ) = µ(t,T )dt + σσσ(f , t,T )>dWWW (t).
• Risk-neutral valuation under Q
df (t,T ) =
(σσσ(f , t,T )>
∫ T
tσσσ(f , t, u)du
)dt + σ(f , t,T )>dWWW (t),
None of forward rates become martingale.• Forward-neutral valuation under QTF
df (t,T ) = −σσσ(f , t,T )>(∫ TF
Tσσσ(f , t, u)du
)dt + σσσ(t,T )>dWWWTF (t), t ≤ T ≤ TF .
• LIBOR Market Model (LMM)• Forward-LIBOR SDE (Spot measure)
dLn(t)
Ln(t)=
n∑j=η(t)
δjLj (t)σσσn(t)>σσσj (t)
1 + δjLj (t)dt + σσσn(t)>dWWW (t), 0 ≤ t ≤ Tn, n = 1, . . . ,M.
• Ln = (vn − vn+1)/(δnvn+1), bond price is martingale whenit is deflated (rather discounted) by the numeriare asset.
22/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Generating Sample PathsSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Heath–Jarrow–Morton (HJM) Framework
df (t,T ) = µ(t,T )dt + σσσ(f , t,T )>dWWW (t).
• Risk-neutral valuation under Q
df (t,T ) =
(σσσ(f , t,T )>
∫ T
tσσσ(f , t, u)du
)dt + σ(f , t,T )>dWWW (t),
None of forward rates become martingale.• Forward-neutral valuation under QTF
df (t,T ) = −σσσ(f , t,T )>(∫ TF
Tσσσ(f , t, u)du
)dt + σσσ(t,T )>dWWWTF (t), t ≤ T ≤ TF .
• LIBOR Market Model (LMM)• Forward-LIBOR SDE (Spot measure)
dLn(t)
Ln(t)=
n∑j=η(t)
δjLj (t)σσσn(t)>σσσj (t)
1 + δjLj (t)dt + σσσn(t)>dWWW (t), 0 ≤ t ≤ Tn, n = 1, . . . ,M.
• Ln = (vn − vn+1)/(δnvn+1), bond price is martingale whenit is deflated (rather discounted) by the numeriare asset.
22/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Lehman Brothers BankruptcySchool of Education, Culture and CommunicationDivision of Applied Mathematics
Biggest bankruptcy in the US history, Sep 15,2008
• Main reasons
1 Liquidity problem (Lender refused to roll over funding),2 High leverage (ratio 31:1),3 Risky investments (Large positions in mortgage
derivatives).
• Around 8,000 OTC contracts
1 Create systemic risk,2 OTC transaction cost can lead others to go to default,3 Governments bailed out some firms before they failed.
• Credit default swap (CDS)
1 $400 billions of CDS contracts,2 $155 billions out standing dept,3 Payout to buyers of CDS was 91.375% of principle.
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AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Lehman Brothers BankruptcySchool of Education, Culture and CommunicationDivision of Applied Mathematics
Biggest bankruptcy in the US history, Sep 15,2008
• Main reasons
1 Liquidity problem (Lender refused to roll over funding),2 High leverage (ratio 31:1),3 Risky investments (Large positions in mortgage
derivatives).
• Around 8,000 OTC contracts
1 Create systemic risk,2 OTC transaction cost can lead others to go to default,3 Governments bailed out some firms before they failed.
• Credit default swap (CDS)
1 $400 billions of CDS contracts,2 $155 billions out standing dept,3 Payout to buyers of CDS was 91.375% of principle.
23/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Lehman Brothers BankruptcySchool of Education, Culture and CommunicationDivision of Applied Mathematics
Biggest bankruptcy in the US history, Sep 15,2008
• Main reasons
1 Liquidity problem (Lender refused to roll over funding),2 High leverage (ratio 31:1),3 Risky investments (Large positions in mortgage
derivatives).
• Around 8,000 OTC contracts
1 Create systemic risk,2 OTC transaction cost can lead others to go to default,3 Governments bailed out some firms before they failed.
• Credit default swap (CDS)
1 $400 billions of CDS contracts,2 $155 billions out standing dept,3 Payout to buyers of CDS was 91.375% of principle.
23/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Lehman Brothers BankruptcySchool of Education, Culture and CommunicationDivision of Applied Mathematics
Biggest bankruptcy in the US history, Sep 15,2008
• Main reasons
1 Liquidity problem (Lender refused to roll over funding),2 High leverage (ratio 31:1),3 Risky investments (Large positions in mortgage
derivatives).
• Around 8,000 OTC contracts
1 Create systemic risk,2 OTC transaction cost can lead others to go to default,3 Governments bailed out some firms before they failed.
• Credit default swap (CDS)
1 $400 billions of CDS contracts,2 $155 billions out standing dept,3 Payout to buyers of CDS was 91.375% of principle.
23/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Unsecure vs Secure TradeSchool of Education, Culture and CommunicationDivision of Applied Mathematics
LIBOR(Short Tenor)
LIBOR(Long Tenor)
Spread(Wave)
Figure: A 3-month floating against a 6-month floating rate
• Before crisis, spread (wave) considered to be zero/close tozero,
• After, it represents the difference in risk levels and it canbe quite significant.
24/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Unsecure vs Secure TradeSchool of Education, Culture and CommunicationDivision of Applied Mathematics
LIBOR(Short Tenor)
LIBOR(Long Tenor)
Spread(Wave)
Figure: A 3-month floating against a 6-month floating rate
• Before crisis, spread (wave) considered to be zero/close tozero,
• After, it represents the difference in risk levels and it canbe quite significant.
24/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Unsecure vs Secure TradeSchool of Education, Culture and CommunicationDivision of Applied Mathematics
LIBOR(Short Tenor)
LIBOR(Long Tenor)
Spread(Wave)
Figure: A 3-month floating against a 6-month floating rate
• Before crisis, spread (wave) considered to be zero/close tozero,
• After, it represents the difference in risk levels and it canbe quite significant.
24/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Unsecure vs Secure TradeSchool of Education, Culture and CommunicationDivision of Applied Mathematics
R B
Cash = PV
Option PaymentLIBOR
Cash
Funding
Figure: Unsecured trade with external funding.
R B
Cash = PV
Option Payment
Collateral
Collatral Rate
Funding
Figure: Secured trade with external funding.
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AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Collateral Agreement (CSA)School of Education, Culture and CommunicationDivision of Applied Mathematics
Base Currency USDEligible Currency USD, EUR, GBPIndependent Amount 5 MillionHaircuts [Schedule]Threshold 50 MillionMinimum Transfer Amount 500,000Rounding Nearest 100,000 USDValuation Agent Red FirmValuation Date Daily, New York Business DayNotification Time 2:00 PM, New York Business DayInterest Rate OIS, EONIA, SONIADay Count Act/360
Figure: Data in a collateral agreement.
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Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Multiple Currency BootstrappingSchool of Education, Culture and CommunicationDivision of Applied Mathematics
USD
USD(OIS)
USD(3m) USD(3m6m)
EUR
EONIA(OIS)
EUR(6m) EUR(3m)
USDEUR(3m3m)
GBP
SONIA(OIS)
GBP(6m) GBP(6m3m)
USDGBP(3m3m)
JPY
TONAR(OIS)
JPY (6m) JPY (6m3m)
USDJPY (3m3m)
Trade Cur-
rency(USD)
Collateral Type(Cash)
CTD Curve
USD(IOS)Implied EONIA(IOS)
in USD
Implied SONIA(IOS)
in USD
Implied TANOR(IOS)
in USD
Figure: An Example of Multiple Currencies Bootstrapping Amounts
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AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing under CSASchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Assumptions1 Full collateralization (zero threshold) by cash,2 Adjusted continuously with zero MTA.
• R instantaneous return (cost if it is negative)
R(f )(t) = r (f )(t)− c(f )(t).
• Risk-neutral measure
dπ(d)(t) =(r (d)(t)− R(f )(t)
)π(d)(t)dt + dWQ(t), 0 ≤ t ≤ T .
• Forward-Neutral measure
π(d)(t) = E
Q(d)
[exp
{−∫ T
tr (d)(u)du +
∫ T
tR(f )(u)du
}π
(d)(T )
∣∣∣∣Ft
]
= v (d)(t,T )EQT
(d)
[exp
{∫ T
tR(d,f )(u)du
}π
(d)(T )
∣∣∣∣Ft
], 0 ≤ t ≤ T .
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AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing under CSASchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Assumptions1 Full collateralization (zero threshold) by cash,2 Adjusted continuously with zero MTA.
• R instantaneous return (cost if it is negative)
R(f )(t) = r (f )(t)− c(f )(t).
• Risk-neutral measure
dπ(d)(t) =(r (d)(t)− R(f )(t)
)π(d)(t)dt + dWQ(t), 0 ≤ t ≤ T .
• Forward-Neutral measure
π(d)(t) = E
Q(d)
[exp
{−∫ T
tr (d)(u)du +
∫ T
tR(f )(u)du
}π
(d)(T )
∣∣∣∣Ft
]
= v (d)(t,T )EQT
(d)
[exp
{∫ T
tR(d,f )(u)du
}π
(d)(T )
∣∣∣∣Ft
], 0 ≤ t ≤ T .
28/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing under CSASchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Assumptions1 Full collateralization (zero threshold) by cash,2 Adjusted continuously with zero MTA.
• R instantaneous return (cost if it is negative)
R(f )(t) = r (f )(t)− c(f )(t).
• Risk-neutral measure
dπ(d)(t) =(r (d)(t)− R(f )(t)
)π(d)(t)dt + dWQ(t), 0 ≤ t ≤ T .
• Forward-Neutral measure
π(d)(t) = E
Q(d)
[exp
{−∫ T
tr (d)(u)du +
∫ T
tR(f )(u)du
}π
(d)(T )
∣∣∣∣Ft
]
= v (d)(t,T )EQT
(d)
[exp
{∫ T
tR(d,f )(u)du
}π
(d)(T )
∣∣∣∣Ft
], 0 ≤ t ≤ T .
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AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing under CSASchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Assumptions1 Full collateralization (zero threshold) by cash,2 Adjusted continuously with zero MTA.
• R instantaneous return (cost if it is negative)
R(f )(t) = r (f )(t)− c(f )(t).
• Risk-neutral measure
dπ(d)(t) =(r (d)(t)− R(f )(t)
)π(d)(t)dt + dWQ(t), 0 ≤ t ≤ T .
• Forward-Neutral measure
π(d)(t) = E
Q(d)
[exp
{−∫ T
tr (d)(u)du +
∫ T
tR(f )(u)du
}π
(d)(T )
∣∣∣∣Ft
]
= v (d)(t,T )EQT
(d)
[exp
{∫ T
tR(d,f )(u)du
}π
(d)(T )
∣∣∣∣Ft
], 0 ≤ t ≤ T .
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AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing Derivatives Under CSASchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Curve construction in single currency
1 Choose the calibration instrument to adjust the startingpoint of simulation,
2 Bootstrap a forward curve,3 Find the discount factor.
• Calibration instruments
1 Overnight indexed swap (OIS),2 Interest rate swap (IRS),3 Tenor swap and basis spread.
29/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
Pricing Derivatives Under CSASchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Curve construction in single currency
1 Choose the calibration instrument to adjust the startingpoint of simulation,
2 Bootstrap a forward curve,3 Find the discount factor.
• Calibration instruments
1 Overnight indexed swap (OIS),2 Interest rate swap (IRS),3 Tenor swap and basis spread.
29/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
ConclusionSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Deterministic and stochastic interest rates,
• Risk and forward neutral probability measure,
• Term-structure model and negative interest rate,
• Pricing interest rate derivatives,
• Creating sample paths,
• New framework under CSA.
• Questions?
• Thanks!
30/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
ConclusionSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Deterministic and stochastic interest rates,
• Risk and forward neutral probability measure,
• Term-structure model and negative interest rate,
• Pricing interest rate derivatives,
• Creating sample paths,
• New framework under CSA.
• Questions?
• Thanks!
30/30
AnIntroductionto ModernPricing of
Interest RateDerivatives
Introduction
Interest Rates
SecurityMarketModels
Term-StructureModels
PricingInterest RateDerivatives
HJMFrameworkand LIIBORMarket Model
CollateralAgreement(CSA)
Conclusion
ConclusionSchool of Education, Culture and CommunicationDivision of Applied Mathematics
• Deterministic and stochastic interest rates,
• Risk and forward neutral probability measure,
• Term-structure model and negative interest rate,
• Pricing interest rate derivatives,
• Creating sample paths,
• New framework under CSA.
• Questions?
• Thanks!
30/30