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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

School of Education, Culture and Communication Division of Applied Mathematics

An Introduction to Modern Pricing of Interest Rate Derivatives

Master Thesis in Financial Engineering Author: Hossein Nohrouzian

M¨ alardalen University

June 5, 2015

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Outline

School of Education, Culture and Communication Division 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

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Risky Asset vs Risk-Less Asset

School of Education, Culture and Communication Division of Applied Mathematics

• Does exist two kind of investments?

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Risky Asset vs Risk-Less Asset

School of Education, Culture and Communication Division of Applied Mathematics

• Does exist two kind of investments?
• An example is pension salary vs inflation.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Risky Asset vs Risk-Less Asset

School of Education, Culture and Communication Division 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

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Interest Rate and Economics Factors

School of Education, Culture and Communication Division of Applied Mathematics

• Interest rate and monetary policy.
• Interest rate and international trading.
• Interest rate and economic growth.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Interest Rate and Economics Factors

School of Education, Culture and Communication Division 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

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Jump Diffusion

School of Education, Culture and Communication Division of Applied Mathematics

• On 15th of January 2015 SNB unexpectedly scrapped its

cap on the Euro value of the Franc.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Jump Diffusion

School of Education, Culture and Communication Division of Applied Mathematics

• On 15th of January 2015 SNB unexpectedly scrapped its

cap on the Euro value of the Franc.

• The result was 27.5% change in USD vs CHF and shake in

stock prices.

Figure: Exchange rate between USD and CHF

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Banks vs Market Rates

School of Education, Culture and Communication Division of Applied Mathematics

• Banks offered rates to individuals and companies.
• Interest rates in the market.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Banks vs Market Rates

School of Education, Culture and Communication Division of Applied Mathematics

• Banks offered rates to individuals and companies.
• Different banks use different rates for loans and savings.
• Interest rates in the market.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Banks vs Market Rates

School of Education, Culture and Communication Division 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:
• From and after the economic crisis in 2007 and 2008:

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Banks vs Market Rates

School of Education, Culture and Communication Division 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 the

international financial market.

• From and after the economic crisis in 2007 and 2008:

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Banks vs Market Rates

School of Education, Culture and Communication Division 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 the

international financial market.

• From and after the economic crisis in 2007 and 2008:
• Collateral rate:

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Banks vs Market Rates

School of Education, Culture and Communication Division 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 the

international 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.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Banks vs Market Rates

School of Education, Culture and Communication Division 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 the

international 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:

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Banks vs Market Rates

School of Education, Culture and Communication Division 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 the

international 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.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Risk-Neutral Evaluation

School of Education, Culture and Communication Division 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 an

• ption (or any other investment) is the risk-free rate.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Risk-Neutral Evaluation

School of Education, Culture and Communication Division 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 an

• ption (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.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Money Market Account as a Num´ eriare

School of Education, Culture and Communication Division of Applied Mathematics

• Money Market Account

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Money Market Account as a Num´ eriare

School of Education, Culture and Communication Division of Applied Mathematics

• Money Market Account
• Constant Interest Rates

B(t) =

• lim

n→∞

• 1 + r

n nt = ert, t ≥ 0.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Money Market Account as a Num´ eriare

School of Education, Culture and Communication Division of Applied Mathematics

• Money Market Account
• Constant Interest Rates

B(t) =

• lim

n→∞

• 1 + r

n nt = ert, t ≥ 0.

• Stochastic Interest Rates

B(t) = exp t r(u)du

• ,

t ≥ 0, r(t) is time-t instantaneous interest rate.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Discount Bond as a Num´ eriare

School of Education, Culture and Communication Division of Applied Mathematics

• Forward Rates

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Discount Bond as a Num´ eriare

School of Education, Culture and Communication Division of Applied Mathematics

• Forward Rates
• Instantaneous forward rate

f (t, T) = − ∂ ∂T ln v(t, T), t ≤ T.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Discount Bond as a Num´ eriare

School of Education, Culture and Communication Division of Applied Mathematics

• Forward Rates
• Instantaneous forward rate

f (t, T) = − ∂ ∂T ln v(t, T), t ≤ T.

• Default-free discount bond

v(t, T) = exp

• −

T

t

f (t, s)ds

• ,

t ≤ T.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Discount Bond as a Num´ eriare

School of Education, Culture and Communication Division of Applied Mathematics

• Forward Rates
• Instantaneous forward rate

f (t, T) = − ∂ ∂T ln 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.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Pricing under Risk-Neutral Method

School of Education, Culture and Communication Division 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)).

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Pricing under Risk-Neutral Method

School of Education, Culture and Communication Division 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.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Pricing under Risk-Neutral Method

School of Education, Culture and Communication Division 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)
• .

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Pricing under Forward-Neutral Method

School of Education, Culture and Communication Division 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,

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Pricing under Forward-Neutral Method

School of Education, Culture and Communication Division 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,

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Pricing under Forward-Neutral Method

School of Education, Culture and Communication Division 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.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Term-Structure Models

School of Education, Culture and Communication Division of Applied Mathematics

• Spot-Rate (Equilibrium) Models

dr = a(m − r)dt + σrγdW , t ≥ 0,

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Term-Structure Models

School of Education, Culture and Communication Division 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,

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Term-Structure Models

School of Education, Culture and Communication Division 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.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Term-Structure Models

School of Education, Culture and Communication Division of Applied Mathematics

• Spot-Rate (No Arbitrage) Models

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Term-Structure Models

School of Education, Culture and Communication Division 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 Lognormal version of Ho–Lee Model,

• Black–Karasinski Model, Extension of Black–Derman–Toy

Model,

• Hull–White (Two-Factor) Model.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Pricing Discount Bond via Vasicek Model

School of Education, Culture and Communication Division of Applied Mathematics

• Market price of risk λ(t) = λ and SDE under Q

dr = a(¯ r − r)dt + σdW ∗,

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Pricing Discount Bond via Vasicek Model

School of Education, Culture and Communication Division 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 a λ,

• r.a. drift m(r, t) = a¯

r − ar & diffusion σ(r, t) = σ.

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Pricing Discount Bond via Vasicek Model

School of Education, Culture and Communication Division 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 a λ,

• r.a. drift m(r, t) = a¯

r − 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)(a2¯ r − σ2/2) a2 − σ2H2

2(t)

4a

• .

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Pricing Discount Bond via CIR Model

School of Education, Culture and Communication Division 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,

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Pricing Discount Bond via CIR Model

School of Education, Culture and Communication Division 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) = a¯

r − ar & diffusion σ(r, t) = σ2r.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Pricing Discount Bond via CIR Model

School of Education, Culture and Communication Division 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) = a¯

r − 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 2a¯

r/σ2

, H2(t) = 2(eγt − 1) (a + γ)(eγt − 1) + 2γ .

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Forward LIBOR and Black’s Formula

School of Education, Culture and Communication Division of Applied Mathematics

• Ti-forward LIBOR Li(t) under QTi+1 is a martingale

Li(t) = EQTi+1 Li(τ)

• Ft
• ,

t ≤ τ ≤ T,

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Forward LIBOR and Black’s Formula

School of Education, Culture and Communication Division 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

dLi Li = σi(t)dW Ti+1, 0 ≤ t ≤ Ti, {W Ti+1(t)} is a standard Brownian motion under QTi+1.

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Cap and Caplets

School of Education, Culture and Communication Division 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 + ςi 2 , ςi > 0. and ς2

i =

Ti

t

σ2

i (s)ds is accumulated variance.

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Cap and Caplets

School of Education, Culture and Communication Division 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 + ςi 2 , ςi > 0. and ς2

i =

Ti

t

σ2

i (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.

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Cap and Caplets

School of Education, Culture and Communication Division 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 + ςi 2 , ςi > 0. and ς2

i =

Ti

t

σ2

i (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.

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Swap Rate and Swaptions

School of Education, Culture and Communication Division 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 an

• ption so called swaptions.

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Swap Rate and Swaptions

School of Education, Culture and Communication Division 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 an

• ption so called swaptions.
• Swaption’s SDE

dS S = σs(t)dW QTi +1, 0 ≤ t ≤ τ

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Swap Rate and Swaptions

School of Education, Culture and Communication Division 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 an

• ption so called swaptions.
• Swaption’s SDE

dS S = σs(t)dW QTi +1, 0 ≤ t ≤ τ

• Swaption price is approximated by Black’s formula.

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Black’s Volatility

School of Education, Culture and Communication Division of Applied Mathematics

• Dynamic of forward rates (cap/floor/swap rate)
• Lognormally distributed, i.e. Black’s Model

df = σBfdW

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Black’s Volatility

School of Education, Culture and Communication Division of Applied Mathematics

• Dynamic of forward rates (cap/floor/swap rate)
• Lognormally distributed, i.e. Black’s Model

df = σBfdW

• Let πCN(t) = πCB(t)

σN = σB (f0 − K) ln(f0/K)

• 1 +

1 24

• 1 −

1 120 [ln(f0/K)]2

• σ2

B τ +

1 5760 σ4

B τ2

, f0 K > 0, f0 = K.

τ exercise date in years.

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Black’s Volatility

School of Education, Culture and Communication Division of Applied Mathematics

• Dynamic of forward rates (cap/floor/swap rate)
• Lognormally distributed, i.e. Black’s Model

df = σBfdW

• Let πCN(t) = πCB(t)

σN = σB (f0 − K) ln(f0/K)

• 1 +

1 24

• 1 −

1 120 [ln(f0/K)]2

• σ2

B τ +

1 5760 σ4

B τ2

, f0 K > 0, f0 = K.

τ exercise date in years.

• The alternative formula

σN = σB

• f0K
• 1 +

1 24 [ln(f0/K)]2

• 1 +

1 24 σ2

B τ +

1 5760 σ4

B τ2

, for

• f0 − K

K

• < 0.001.

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

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Bachelier’s Model

School of Education, Culture and Communication Division of Applied Mathematics

• Bachelier’s SDE

dS(τ) = S(t)σdW (τ), 0 ≤ t ≤ τ ≤ T.

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Bachelier’s Model

School of Education, Culture and Communication Division 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))] .

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Bachelier’s Model

School of Education, Culture and Communication Division 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)

.

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Bachelier’s Model

School of Education, Culture and Communication Division 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).

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Black’s Model vs Normal Model

School of Education, Culture and Communication Division of Applied Mathematics

• Black’s model
• Normal model

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Black’s Model vs Normal Model

School of Education, Culture and Communication Division of Applied Mathematics

• Black’s model
• Black’s SDE

df = σnfdW , 0 ≤ t ≤ T.

• Normal model

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Black’s Model vs Normal Model

School of Education, Culture and Communication Division 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 fT ft

• = σn(WT − Wt),

fT ft > 0, ft = 0.

• Normal model

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Black’s Model vs Normal Model

School of Education, Culture and Communication Division 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 fT ft

• = σn(WT − Wt),

fT ft > 0, ft = 0.

• Normal model
• Normal SDE

df = σndW , 0 ≤ t ≤ T.

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Black’s Model vs Normal Model

School of Education, Culture and Communication Division 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 fT ft

• = σn(WT − Wt),

fT ft > 0, ft = 0.

• Normal model
• Normal SDE

df = σndW , 0 ≤ t ≤ T.

• Normal forward price

fT = ft + σn(WT − Wt).

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Generating Sample Paths

School of Education, Culture and Communication Division of Applied Mathematics

• Heath–Jarrow–Morton (HJM) Framework

df (t, T) = µ(t, T)dt + σ σ σ(f , t, T)⊤dW W W (t). 22/30

An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Generating Sample Paths

School of Education, Culture and Communication Division of Applied Mathematics

• Heath–Jarrow–Morton (HJM) Framework

df (t, T) = µ(t, T)dt + σ σ σ(f , t, T)⊤dW W W (t).

• Risk-neutral valuation under Q

df (t, T) =

• σ

σ σ(f , t, T)⊤ T

t

σ σ σ(f , t, u)du

• dt + σ(f , t, T)⊤dW

W W (t),

None of forward rates become martingale.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Generating Sample Paths

School of Education, Culture and Communication Division of Applied Mathematics

• Heath–Jarrow–Morton (HJM) Framework

df (t, T) = µ(t, T)dt + σ σ σ(f , t, T)⊤dW W W (t).

• Risk-neutral valuation under Q

df (t, T) =

• σ

σ σ(f , t, T)⊤ T

t

σ σ σ(f , t, u)du

• dt + σ(f , t, T)⊤dW

W W (t),

None of forward rates become martingale.

• Forward-neutral valuation under QTF

df (t, T) = −σ σ σ(f , t, T)⊤ TF

T

σ σ σ(f , t, u)du

• dt + σ

σ σ(t, T)⊤dW W W TF (t), t ≤ T ≤ TF . 22/30

An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Generating Sample Paths

School of Education, Culture and Communication Division of Applied Mathematics

• Heath–Jarrow–Morton (HJM) Framework

df (t, T) = µ(t, T)dt + σ σ σ(f , t, T)⊤dW W W (t).

• Risk-neutral valuation under Q

df (t, T) =

• σ

σ σ(f , t, T)⊤ T

t

σ σ σ(f , t, u)du

• dt + σ(f , t, T)⊤dW

W W (t),

None of forward rates become martingale.

• Forward-neutral valuation under QTF

df (t, T) = −σ σ σ(f , t, T)⊤ TF

T

σ σ σ(f , t, u)du

• dt + σ

σ σ(t, T)⊤dW W W TF (t), t ≤ T ≤ TF .

• LIBOR Market Model (LMM)

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Generating Sample Paths

School of Education, Culture and Communication Division of Applied Mathematics

• Heath–Jarrow–Morton (HJM) Framework

df (t, T) = µ(t, T)dt + σ σ σ(f , t, T)⊤dW W W (t).

• Risk-neutral valuation under Q

df (t, T) =

• σ

σ σ(f , t, T)⊤ T

t

σ σ σ(f , t, u)du

• dt + σ(f , t, T)⊤dW

W W (t),

None of forward rates become martingale.

• Forward-neutral valuation under QTF

df (t, T) = −σ σ σ(f , t, T)⊤ TF

T

σ σ σ(f , t, u)du

• dt + σ

σ σ(t, T)⊤dW W W TF (t), t ≤ T ≤ TF .

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

dLn(t) Ln(t) =

n

• j=η(t)

δj Lj (t)σ σ σn(t)⊤σ σ σj (t) 1 + δj Lj (t) dt + σ σ σn(t)⊤dW W W (t), 0 ≤ t ≤ Tn, n = 1, . . . , M. 22/30

An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Generating Sample Paths

School of Education, Culture and Communication Division of Applied Mathematics

• Heath–Jarrow–Morton (HJM) Framework

df (t, T) = µ(t, T)dt + σ σ σ(f , t, T)⊤dW W W (t).

• Risk-neutral valuation under Q

df (t, T) =

• σ

σ σ(f , t, T)⊤ T

t

σ σ σ(f , t, u)du

• dt + σ(f , t, T)⊤dW

W W (t),

None of forward rates become martingale.

• Forward-neutral valuation under QTF

df (t, T) = −σ σ σ(f , t, T)⊤ TF

T

σ σ σ(f , t, u)du

• dt + σ

σ σ(t, T)⊤dW W W TF (t), t ≤ T ≤ TF .

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

dLn(t) Ln(t) =

n

• j=η(t)

δj Lj (t)σ σ σn(t)⊤σ σ σj (t) 1 + δj Lj (t) dt + σ σ σn(t)⊤dW W W (t), 0 ≤ t ≤ Tn, n = 1, . . . , M.

• Ln = (vn − vn+1)/(δnvn+1), bond price is martingale when

it is deflated (rather discounted) by the num´ eriare asset.

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Lehman Brothers Bankruptcy

School of Education, Culture and Communication Division of Applied Mathematics

Biggest bankruptcy in the US history, Sep 15,2008

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Lehman Brothers Bankruptcy

School of Education, Culture and Communication Division 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).

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Lehman Brothers Bankruptcy

School of Education, Culture and Communication Division 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.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Lehman Brothers Bankruptcy

School of Education, Culture and Communication Division 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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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Unsecure vs Secure Trade

School of Education, Culture and Communication Division of Applied Mathematics

LIBOR(Short Tenor) LIBOR(Long Tenor) Spread(Wave)

Figure: A 3-month floating against a 6-month floating rate

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Unsecure vs Secure Trade

School of Education, Culture and Communication Division 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 to

zero,

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Unsecure vs Secure Trade

School of Education, Culture and Communication Division 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 to

zero,

• After, it represents the difference in risk levels and it can

be quite significant.

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Unsecure vs Secure Trade

School of Education, Culture and Communication Division of Applied Mathematics

R B Cash = PV Option Payment LIBOR 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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Collateral Agreement (CSA)

School of Education, Culture and Communication Division of Applied Mathematics

Base Currency USD Eligible Currency USD, EUR, GBP Independent Amount 5 Million Haircuts [Schedule] Threshold 50 Million Minimum Transfer Amount 500,000 Rounding Nearest 100,000 USD Valuation Agent Red Firm Valuation Date Daily, New York Business Day Notification Time 2:00 PM, New York Business Day Interest Rate OIS, EONIA, SONIA Day Count Act/360

Figure: Data in a collateral agreement.

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Multiple Currency Bootstrapping

School of Education, Culture and Communication Division 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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Pricing under CSA

School of Education, Culture and Communication Division of Applied Mathematics

• Assumptions

1 Full collateralization (zero threshold) by cash, 2 Adjusted continuously with zero MTA.

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Pricing under CSA

School of Education, Culture and Communication Division of Applied Mathematics

• Assumptions

1 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).

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Pricing under CSA

School of Education, Culture and Communication Division of Applied Mathematics

• Assumptions

1 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 + dW Q(t), 0 ≤ t ≤ T.

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Pricing under CSA

School of Education, Culture and Communication Division of Applied Mathematics

• Assumptions

1 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 + dW Q(t), 0 ≤ t ≤ T.
• Forward-Neutral measure

π(d)(t) = EQ(d)

• exp
• −

T

t

r(d)(u)du + T

t

R(f )(u)du

• π(d)(T)
• Ft
• = v(d)(t, T)E

QT (d)

• exp

T

t

R(d,f )(u)du

• π(d)(T)
• Ft
• ,

0 ≤ t ≤ T. 28/30

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Pricing Derivatives Under CSA

School of Education, Culture and Communication Division of Applied Mathematics

• Curve construction in single currency

1 Choose the calibration instrument to adjust the starting

point of simulation,

2 Bootstrap a forward curve, 3 Find the discount factor.

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Pricing Derivatives Under CSA

School of Education, Culture and Communication Division of Applied Mathematics

• Curve construction in single currency

1 Choose the calibration instrument to adjust the starting

point 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.

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Conclusion

School of Education, Culture and Communication Division 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.

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Conclusion

School of Education, Culture and Communication Division 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?

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An Introduction to Modern Pricing of Interest Rate Derivatives Introduction Interest Rates Security Market Models Term- Structure Models Pricing Interest Rate Derivatives HJM Framework and LIIBOR Market Model Collateral Agreement (CSA) Conclusion

Conclusion

School of Education, Culture and Communication Division 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!

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