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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

Introduction

Interest Rates

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

Introduction

Interest Rates

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

Introduction

Interest Rates

Conclusion

Introduction

Interest Rates

Conclusion

Introduction

Interest Rates

Conclusion

Interest Rate and EconomicsFactors

• 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

Introduction

Interest Rates

Conclusion

Interest Rate and EconomicsFactors

• 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

Introduction

Interest Rates

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

Introduction

Interest Rates

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

Introduction

Interest Rates

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.

Introduction

Interest Rates

Conclusion

• 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:

• Swap rates:

Introduction

Interest Rates

Conclusion

• Interest rates in the market.• Before the economic crisis in 2007 and 2008:

• Swap rates:

Introduction

Interest Rates

Conclusion

• Swap rates:

Introduction

Interest Rates

Conclusion

• 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:

Introduction

Interest Rates

Conclusion

• 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:

Introduction

Interest Rates

Conclusion

• Swap rates:

Introduction

Interest Rates

Conclusion

• Swap rates:• Fixed rates are calculated from forward rates,• floating rates are calculated from OIS rates.

Introduction

Interest Rates

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.

Introduction

Interest Rates

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.

Introduction

Interest Rates

Conclusion

Money Market Account as aNumeriare

• Money Market Account

• Constant Interest Rates

B(t) =[

limn→∞

)n]t= ert , t ≥ 0.

• Stochastic Interest Rates

B(t) = exp

{∫ t

r(u)du

}, t ≥ 0,

r(t) is time-t instantaneous interest rate.

Introduction

Interest Rates

Conclusion

• Money Market Account• Constant Interest Rates

B(t) =[

limn→∞

)n]t= ert , t ≥ 0.

B(t) = exp

{∫ t

r(u)du

}, t ≥ 0,

Introduction

Interest Rates

Conclusion

• Money Market Account• Constant Interest Rates

B(t) =[

limn→∞

)n]t= ert , t ≥ 0.

B(t) = exp

{∫ t

r(u)du

}, t ≥ 0,

Introduction

Interest Rates

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

f (t, s)ds

}, t ≤ T .

• r(T ) = f (t,T ), t ≤ T . If r(t) is deterministic

v(t,T ) = exp

{−∫ T

r(s)ds

B(T ), t ≤ T .

Introduction

Interest Rates

Conclusion

• Forward Rates• Instantaneous forward rate

f (t,T ) = − ∂

∂Tln v(t,T ), t ≤ T .

v(t,T ) = exp

{−∫ T

f (t, s)ds

}, t ≤ T .

v(t,T ) = exp

{−∫ T

r(s)ds

B(T ), t ≤ T .

Introduction

Interest Rates

Conclusion

f (t,T ) = − ∂

∂Tln v(t,T ), t ≤ T .

v(t,T ) = exp

{−∫ T

f (t, s)ds

}, t ≤ T .

v(t,T ) = exp

{−∫ T

r(s)ds

B(T ), t ≤ T .

Introduction

Interest Rates

Conclusion

f (t,T ) = − ∂

∂Tln v(t,T ), t ≤ T .

v(t,T ) = exp

{−∫ T

f (t, s)ds

}, t ≤ T .

v(t,T ) = exp

{−∫ T

r(s)ds

B(T ), t ≤ T .

Introduction

Interest Rates

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 ))

∣∣∣∣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

)(T − t) + σ (WT −Wt)

Introduction

Interest Rates

Conclusion

π(t) = B(t)EQ

[h(S(T ))

∣∣∣∣Ft

], 0 ≤ t ≤ T .

ST = St exp

{(r − 1

)(T − t) + σ (WT −Wt)

Introduction

Interest Rates

Conclusion

π(t) = B(t)EQ

[h(S(T ))

∣∣∣∣Ft

], 0 ≤ t ≤ T .

ST = St exp

{(r − 1

)(T − t) + σ (WT −Wt)

Introduction

Interest Rates

Conclusion

Pricing under Forward-NeutralMethod

• 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

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 .

Introduction

Interest Rates

Conclusion

dv(t,T )

v(t,T )= r(t)dt + σv (t)dW ∗, 0 ≤ t ≤ T ,

S= r(t)dt + σ(t)dW ∗, 0 ≤ t ≤ T ,

∣∣Ft

], 0 ≤ t ≤ T .

Introduction

Interest Rates

Conclusion

dv(t,T )

v(t,T )= r(t)dt + σv (t)dW ∗, 0 ≤ t ≤ T ,

S= r(t)dt + σ(t)dW ∗, 0 ≤ t ≤ T ,

∣∣Ft

], 0 ≤ t ≤ T .

Introduction

Interest Rates

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.

Introduction

Interest Rates

Conclusion

• 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.

Introduction

Interest Rates

Conclusion

• 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.

Introduction

Interest Rates

Conclusion

• 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.

Introduction

Interest Rates

Conclusion

• 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.

Introduction

Interest Rates

Conclusion

Pricing Discount Bond via VasicekModel

• 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

• 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

H1(t) = exp

{(H2(t)− t)(a2r − σ2/2)

a2− σ2H2

Introduction

Interest Rates

Conclusion

r = m − σ

• 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

H1(t) = exp

{(H2(t)− t)(a2r − σ2/2)

a2− σ2H2

Introduction

Interest Rates

Conclusion

r = m − σ

• 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

H1(t) = exp

{(H2(t)− t)(a2r − σ2/2)

a2− σ2H2

Introduction

Interest Rates

Conclusion

Pricing Discount Bond via CIRModel

• Market price of risk λ(t) = a(m−r)

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.

Introduction

Interest Rates

Conclusion

dr = a(r − r)dt + σ√r(t)dW ∗, 0 ≤ t ≤ T ,

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.

Introduction

Interest Rates

Conclusion

dr = a(r − r)dt + σ√r(t)dW ∗, 0 ≤ t ≤ T ,

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.

Introduction

Interest Rates

Conclusion

Forward LIBOR and Black’sFormula

• 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

= σi (t)dW Ti+1 , 0 ≤ t ≤ Ti ,

{W Ti+1(t)} is a standard Brownian motion under QTi+1 .

Introduction

Interest Rates

Conclusion

Forward LIBOR and Black’sFormula

• 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

= σi (t)dW Ti+1 , 0 ≤ t ≤ Ti ,

{W Ti+1(t)} is a standard Brownian motion under QTi+1 .

Introduction

Interest Rates

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.

Introduction

Interest Rates

Conclusion

• Caplet price

di =ln(Li (t)/K )

ςi+ςi2, ςi > 0.

and ς2i =

∫ Ti

Cap(t) =n−1∑i=0

Introduction

Interest Rates

Conclusion

• Caplet price

di =ln(Li (t)/K )

ςi+ςi2, ςi > 0.

and ς2i =

∫ Ti

Cap(t) =n−1∑i=0

Introduction

Interest Rates

Conclusion

Swap Rate and SwaptionsSchool of Education, Culture and CommunicationDivision of Applied Mathematics

• Swap rate

S(t) =VFL

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

S= σs(t)dWQTi+1

, 0 ≤ t ≤ τ

• Swaption price is approximated by Black’s formula.

Introduction

Interest Rates

Conclusion

• Swap rate

S(t) =VFL

v(t,T0)− v(t,Tn)

δ∑n

i=1 v(t,Ti ), 0 ≤ t ≤ T0.

S= σs(t)dWQTi+1

, 0 ≤ t ≤ τ

Introduction

Interest Rates

Conclusion

• Swap rate

S(t) =VFL

v(t,T0)− v(t,Tn)

δ∑n

i=1 v(t,Ti ), 0 ≤ t ≤ T0.

S= σs(t)dWQTi+1

, 0 ≤ t ≤ τ

Introduction

Interest Rates

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

σN =σB (f0 − K)

ln(f0/K)

120[ln(f0/K)]2

)σ2Bτ +

5760σ4Bτ2

] , f0

K> 0, f0 6= K .

τ exercise date in years.• The alternative formula

σB√

24[ln(f0/K)]2

24σ2Bτ +

5760σ4Bτ2

∣∣∣∣ f0 − K

∣∣∣∣ < 0.001.

Numerical methods (Newton-Raphson method) to get σBknowing σN .

Introduction

Interest Rates

Conclusion

df = σB fdW

σN =σB (f0 − K)

ln(f0/K)

120[ln(f0/K)]2

)σ2Bτ +

5760σ4Bτ2

] , f0

K> 0, f0 6= K .

τ exercise date in years.

• The alternative formula

σB√

24[ln(f0/K)]2

24σ2Bτ +

5760σ4Bτ2

∣∣∣∣ f0 − K

∣∣∣∣ < 0.001.

Introduction

Interest Rates

Conclusion

df = σB fdW

σN =σB (f0 − K)

ln(f0/K)

120[ln(f0/K)]2

)σ2Bτ +

5760σ4Bτ2

] , f0

K> 0, f0 6= K .

τ exercise date in years.• The alternative formula

σB√

24[ln(f0/K)]2

24σ2Bτ +

5760σ4Bτ2

∣∣∣∣ f0 − K

∣∣∣∣ < 0.001.

Introduction

Interest Rates

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)

√2π

(T − t).

Introduction

Interest Rates

Conclusion

dS(τ) = S(t)σdW (τ), 0 ≤ t ≤ τ ≤ T .

S(T ) = S(t) [1 + σ (W (T )−W (t))] .

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).

πC (t) = S(t)σ

√(T − t)

2π, σ =

πC (t)

√2π

(T − t).

Introduction

Interest Rates

Conclusion

dS(τ) = S(t)σdW (τ), 0 ≤ t ≤ τ ≤ T .

S(T ) = S(t) [1 + σ (W (T )−W (t))] .

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).

πC (t) = S(t)σ

√(T − t)

2π, σ =

πC (t)

√2π

(T − t).

Introduction

Interest Rates

Conclusion

dS(τ) = S(t)σdW (τ), 0 ≤ t ≤ τ ≤ T .

S(T ) = S(t) [1 + σ (W (T )−W (t))] .

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).

πC (t) = S(t)σ

√(T − t)

2π, σ =

πC (t)

√2π

(T − t).

Introduction

Interest Rates

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

)= 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).

Introduction

Interest Rates

Conclusion

• Black’s model• Black’s SDE

df = σnfdW , 0 ≤ t ≤ T .

)= σn(WT −Wt),

fTft> 0, ft 6= 0.

• Normal model

• Normal SDE

df = σndW , 0 ≤ t ≤ T .

Introduction

Interest Rates

Conclusion

df = σnfdW , 0 ≤ t ≤ T .

)= σn(WT −Wt),

fTft> 0, ft 6= 0.

• Normal model

• Normal SDE

df = σndW , 0 ≤ t ≤ T .

Introduction

Interest Rates

Conclusion

df = σnfdW , 0 ≤ t ≤ T .

)= σn(WT −Wt),

fTft> 0, ft 6= 0.

• Normal model• Normal SDE

df = σndW , 0 ≤ t ≤ T .

Introduction

Interest Rates

Conclusion

df = σnfdW , 0 ≤ t ≤ T .

)= σn(WT −Wt),

fTft> 0, ft 6= 0.

• Normal model• Normal SDE

df = σndW , 0 ≤ t ≤ T .

Introduction

Interest Rates

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 )>

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.

Introduction

Interest Rates

Conclusion

df (t,T ) =

(σσσ(f , t,T )>

tσσσ(f , t, u)du

None of forward rates become martingale.

• Forward-neutral valuation under QTF

df (t,T ) = −σσσ(f , t,T )>(∫ TF

Tσσσ(f , t, u)du

dLn(t)

Ln(t)=

n∑j=η(t)

Introduction

Interest Rates

Conclusion

df (t,T ) =

(σσσ(f , t,T )>

tσσσ(f , t, u)du

df (t,T ) = −σσσ(f , t,T )>(∫ TF

Tσσσ(f , t, u)du

dLn(t)

Ln(t)=

n∑j=η(t)

Introduction

Interest Rates

Conclusion

df (t,T ) =

(σσσ(f , t,T )>

tσσσ(f , t, u)du

df (t,T ) = −σσσ(f , t,T )>(∫ TF

Tσσσ(f , t, u)du

dLn(t)

Ln(t)=

n∑j=η(t)

Introduction

Interest Rates

Conclusion

df (t,T ) =

(σσσ(f , t,T )>

tσσσ(f , t, u)du

df (t,T ) = −σσσ(f , t,T )>(∫ TF

Tσσσ(f , t, u)du

• LIBOR Market Model (LMM)• Forward-LIBOR SDE (Spot measure)

dLn(t)

Ln(t)=

n∑j=η(t)

Introduction

Interest Rates

Conclusion

df (t,T ) =

(σσσ(f , t,T )>

tσσσ(f , t, u)du

df (t,T ) = −σσσ(f , t,T )>(∫ TF

Tσσσ(f , t, u)du

• LIBOR Market Model (LMM)• Forward-LIBOR SDE (Spot measure)

dLn(t)

Ln(t)=

n∑j=η(t)

Introduction

Interest Rates

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.

Introduction

Interest Rates

Conclusion

• Main reasons

derivatives).

Introduction

Interest Rates

Conclusion

• Main reasons

derivatives).

Introduction

Interest Rates

Conclusion

• Main reasons

derivatives).

Introduction

Interest Rates

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.

Introduction

Interest Rates

Conclusion

LIBOR(Short Tenor)

LIBOR(Long Tenor)

Spread(Wave)

Introduction

Interest Rates

Conclusion

LIBOR(Short Tenor)

LIBOR(Long Tenor)

Spread(Wave)

Introduction

Interest Rates

Conclusion

Cash = PV

Option PaymentLIBOR

Funding

Figure: Unsecured trade with external funding.

Cash = PV

Option Payment

Collateral

Collatral Rate

Funding

Figure: Secured trade with external funding.

Introduction

Interest Rates

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.

Introduction

Interest Rates

Conclusion

Multiple Currency BootstrappingSchool of Education, Culture and CommunicationDivision of Applied Mathematics

USD(OIS)

USD(3m) USD(3m6m)

EONIA(OIS)

EUR(6m) EUR(3m)

USDEUR(3m3m)

SONIA(OIS)

GBP(6m) GBP(6m3m)

USDGBP(3m3m)

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

Introduction

Interest Rates

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

{−∫ T

tr (d)(u)du +

tR(f )(u)du

(d)(T )

∣∣∣∣Ft

= v (d)(t,T )EQT

{∫ T

tR(d,f )(u)du

(d)(T )

∣∣∣∣Ft

], 0 ≤ t ≤ T .

Introduction

Interest Rates

Conclusion

R(f )(t) = r (f )(t)− c(f )(t).

dπ(d)(t) =(r (d)(t)− R(f )(t)

)π(d)(t)dt + dWQ(t), 0 ≤ t ≤ T .

π(d)(t) = E

{−∫ T

tr (d)(u)du +

tR(f )(u)du

(d)(T )

∣∣∣∣Ft

= v (d)(t,T )EQT

{∫ T

tR(d,f )(u)du

(d)(T )

∣∣∣∣Ft

], 0 ≤ t ≤ T .

Introduction

Interest Rates

Conclusion

R(f )(t) = r (f )(t)− c(f )(t).

dπ(d)(t) =(r (d)(t)− R(f )(t)

)π(d)(t)dt + dWQ(t), 0 ≤ t ≤ T .

π(d)(t) = E

{−∫ T

tr (d)(u)du +

tR(f )(u)du

(d)(T )

∣∣∣∣Ft

= v (d)(t,T )EQT

{∫ T

tR(d,f )(u)du

(d)(T )

∣∣∣∣Ft

], 0 ≤ t ≤ T .

Introduction

Interest Rates

Conclusion

R(f )(t) = r (f )(t)− c(f )(t).

dπ(d)(t) =(r (d)(t)− R(f )(t)

)π(d)(t)dt + dWQ(t), 0 ≤ t ≤ T .

π(d)(t) = E

{−∫ T

tr (d)(u)du +

tR(f )(u)du

(d)(T )

∣∣∣∣Ft

= v (d)(t,T )EQT

{∫ T

tR(d,f )(u)du

(d)(T )

∣∣∣∣Ft

], 0 ≤ t ≤ T .

Introduction

Interest Rates

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.

Introduction

Interest Rates

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.

Introduction

Interest Rates

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!

Introduction

Interest Rates

Conclusion

• Questions?

• Thanks!

Introduction

Interest Rates

Conclusion

• Questions?

• Thanks!

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