Option Pricing under Regime-Switching

José Enrique Figueroa-López

Department of StatisticsPurdue University

ECE Seminar

Purdue University

Nov. 9, 2011

(Joint work with Agostino Capponi and Jeff Nisen)

A quick glance at mathematical finance

Outline

1 A quick glance at mathematical finance

2 Regime switching market model

3 Option Pricing under Regime Switching

Simple claims on the regime

Self-decomposable claims

A quick glance at mathematical finance First concepts

What is mathematical finance?

1 Personal definition:

Interdisciplinary field concerned with the pricing of financial assets in

such a way that the market is free of arbitrage opportunity.

2 What are the assets?

• Bonds• Stocks• Contingent claims

A contract between two parties where one party is obligated to deliver a

payoff at a specified future time T called the maturity. The final payoff is

contingent to the value of a stock or another risky measurement (say,

interest rate), called the underlying.

A quick glance at mathematical finance First concepts

What is mathematical finance?

1 Personal definition:

Interdisciplinary field concerned with the pricing of financial assets in

such a way that the market is free of arbitrage opportunity.

2 What are the assets?

• Bonds• Stocks• Contingent claims

A contract between two parties where one party is obligated to deliver a

payoff at a specified future time T called the maturity. The final payoff is

contingent to the value of a stock or another risky measurement (say,

interest rate), called the underlying.

A quick glance at mathematical finance First concepts

What is mathematical finance?

1 Personal definition:

Interdisciplinary field concerned with the pricing of financial assets in

such a way that the market is free of arbitrage opportunity.

2 What are the assets?

• Bonds• Stocks• Contingent claims

A contract between two parties where one party is obligated to deliver a

payoff at a specified future time T called the maturity. The final payoff is

contingent to the value of a stock or another risky measurement (say,

interest rate), called the underlying.

A quick glance at mathematical finance First concepts

What is mathematical finance?

1 Personal definition:

Interdisciplinary field concerned with the pricing of financial assets in

such a way that the market is free of arbitrage opportunity.

2 What are the assets?

• Bonds• Stocks• Contingent claims

A contract between two parties where one party is obligated to deliver a

payoff at a specified future time T called the maturity. The final payoff is

contingent to the value of a stock or another risky measurement (say,

interest rate), called the underlying.

A quick glance at mathematical finance First concepts

What is mathematical finance?

1 Personal definition:

Interdisciplinary field concerned with the pricing of financial assets in

such a way that the market is free of arbitrage opportunity.

2 What are the assets?

• Bonds• Stocks• Contingent claims

A contract between two parties where one party is obligated to deliver a

payoff at a specified future time T called the maturity. The final payoff is

contingent to the value of a stock or another risky measurement (say,

interest rate), called the underlying.

A quick glance at mathematical finance First concepts

Put option on a stock in a two-state economy

• A 1-year bond pays an interest of 5% per year: B(0) = 1, B(1) = 1.05;• Current price per share is S(0) = $100;• At T = 1 year, price can go up to S(1) = $110 with probability 1/4 or go

down to S(1) = $95 with probability 3/4;• Put Option = Right to sell a share of the stock at T = 1 for $100;• Equivalent to a contingent claim with payoff:

Φ (S(1)) = max{100− S(1),0}.

A quick glance at mathematical finance First concepts

Put option on a stock in a two-state economy

• A 1-year bond pays an interest of 5% per year: B(0) = 1, B(1) = 1.05;• Current price per share is S(0) = $100;• At T = 1 year, price can go up to S(1) = $110 with probability 1/4 or go

down to S(1) = $95 with probability 3/4;• Put Option = Right to sell a share of the stock at T = 1 for $100;• Equivalent to a contingent claim with payoff:

Φ (S(1)) = max{100− S(1),0}.

A quick glance at mathematical finance First concepts

Put option on a stock in a two-state economy

• A 1-year bond pays an interest of 5% per year: B(0) = 1, B(1) = 1.05;• Current price per share is S(0) = $100;• At T = 1 year, price can go up to S(1) = $110 with probability 1/4 or go

down to S(1) = $95 with probability 3/4;• Put Option = Right to sell a share of the stock at T = 1 for $100;• Equivalent to a contingent claim with payoff:

Φ (S(1)) = max{100− S(1),0}.

A quick glance at mathematical finance First concepts

Put option on a stock in a two-state economy

• A 1-year bond pays an interest of 5% per year: B(0) = 1, B(1) = 1.05;• Current price per share is S(0) = $100;• At T = 1 year, price can go up to S(1) = $110 with probability 1/4 or go

down to S(1) = $95 with probability 3/4;• Put Option = Right to sell a share of the stock at T = 1 for $100;• Equivalent to a contingent claim with payoff:

Φ (S(1)) = max{100− S(1),0}.

A quick glance at mathematical finance First concepts

Put option on a stock in a two-state economy

• A 1-year bond pays an interest of 5% per year: B(0) = 1, B(1) = 1.05;• Current price per share is S(0) = $100;• At T = 1 year, price can go up to S(1) = $110 with probability 1/4 or go

down to S(1) = $95 with probability 3/4;• Put Option = Right to sell a share of the stock at T = 1 for $100;• Equivalent to a contingent claim with payoff:

Φ (S(1)) = max{100− S(1),0}.

A quick glance at mathematical finance First concepts

Pricing and replication

1 What is the arbitrage-free price P(0) of the derivative?

2 Key idea: Replicating trading strategy.• Consider the following trading strategy at time t = 0:

(i) Borrow (40)(104)(3)(21)

≈ 60.31 from the bond market.(ii) Buy 2

3 shares of stock

• The net initial investment at time t = 0 is

V0 ≈23· (100)− 60.31 ≈ 6.3492.

• The resulting wealth at time t = 1 is

V1 =

(23 · 110− 60.31 · (1.05) = 10 if stock goes up23 · 95− 60.31 · (1.05) = 0 if stock goes down

3 Conclusion: Arbitrage-free price P(0) = V0 = 6.3492.

A quick glance at mathematical finance First concepts

Pricing and replication

1 What is the arbitrage-free price P(0) of the derivative?

2 Key idea: Replicating trading strategy.• Consider the following trading strategy at time t = 0:

(i) Borrow (40)(104)(3)(21)

≈ 60.31 from the bond market.(ii) Buy 2

3 shares of stock

• The net initial investment at time t = 0 is

V0 ≈23· (100)− 60.31 ≈ 6.3492.

• The resulting wealth at time t = 1 is

V1 =

(23 · 110− 60.31 · (1.05) = 10 if stock goes up23 · 95− 60.31 · (1.05) = 0 if stock goes down

3 Conclusion: Arbitrage-free price P(0) = V0 = 6.3492.

A quick glance at mathematical finance First concepts

Pricing and replication

1 What is the arbitrage-free price P(0) of the derivative?

2 Key idea: Replicating trading strategy.• Consider the following trading strategy at time t = 0:

(i) Borrow (40)(104)(3)(21)

≈ 60.31 from the bond market.(ii) Buy 2

3 shares of stock

• The net initial investment at time t = 0 is

V0 ≈23· (100)− 60.31 ≈ 6.3492.

• The resulting wealth at time t = 1 is

V1 =

(23 · 110− 60.31 · (1.05) = 10 if stock goes up23 · 95− 60.31 · (1.05) = 0 if stock goes down

3 Conclusion: Arbitrage-free price P(0) = V0 = 6.3492.

A quick glance at mathematical finance First concepts

Pricing and replication

1 What is the arbitrage-free price P(0) of the derivative?

2 Key idea: Replicating trading strategy.• Consider the following trading strategy at time t = 0:

(i) Borrow (40)(104)(3)(21)

≈ 60.31 from the bond market.(ii) Buy 2

3 shares of stock

• The net initial investment at time t = 0 is

V0 ≈23· (100)− 60.31 ≈ 6.3492.

• The resulting wealth at time t = 1 is

V1 =

(23 · 110− 60.31 · (1.05) = 10 if stock goes up23 · 95− 60.31 · (1.05) = 0 if stock goes down

3 Conclusion: Arbitrage-free price P(0) = V0 = 6.3492.

A quick glance at mathematical finance First concepts

Pricing and replication

1 What is the arbitrage-free price P(0) of the derivative?

2 Key idea: Replicating trading strategy.• Consider the following trading strategy at time t = 0:

(i) Borrow (40)(104)(3)(21)

≈ 60.31 from the bond market.(ii) Buy 2

3 shares of stock

• The net initial investment at time t = 0 is

V0 ≈23· (100)− 60.31 ≈ 6.3492.

• The resulting wealth at time t = 1 is

V1 =

(23 · 110− 60.31 · (1.05) = 10 if stock goes up23 · 95− 60.31 · (1.05) = 0 if stock goes down

3 Conclusion: Arbitrage-free price P(0) = V0 = 6.3492.

A quick glance at mathematical finance First concepts

Pricing and replication

1 What is the arbitrage-free price P(0) of the derivative?

2 Key idea: Replicating trading strategy.• Consider the following trading strategy at time t = 0:

(i) Borrow (40)(104)(3)(21)

≈ 60.31 from the bond market.(ii) Buy 2

3 shares of stock

• The net initial investment at time t = 0 is

V0 ≈23· (100)− 60.31 ≈ 6.3492.

• The resulting wealth at time t = 1 is

V1 =

(23 · 110− 60.31 · (1.05) = 10 if stock goes up23 · 95− 60.31 · (1.05) = 0 if stock goes down

3 Conclusion: Arbitrage-free price P(0) = V0 = 6.3492.

A quick glance at mathematical finance First concepts

Pricing and replication

1 What is the arbitrage-free price P(0) of the derivative?

2 Key idea: Replicating trading strategy.• Consider the following trading strategy at time t = 0:

(i) Borrow (40)(104)(3)(21)

≈ 60.31 from the bond market.(ii) Buy 2

3 shares of stock

• The net initial investment at time t = 0 is

V0 ≈23· (100)− 60.31 ≈ 6.3492.

• The resulting wealth at time t = 1 is

V1 =

(23 · 110− 60.31 · (1.05) = 10 if stock goes up23 · 95− 60.31 · (1.05) = 0 if stock goes down

3 Conclusion: Arbitrage-free price P(0) = V0 = 6.3492.

A quick glance at mathematical finance First concepts

Risk-neutral valuation

A fundamental question:

Is the previous arbitrage-free price consistent with the expected discounted

payoff of the option?

Risk-neutral Valuation Formula1 Expected Discounted Payoff:

101.05

· +0

1.05· ;

with qu = 23 , qd = 1

3 .

2 The probabilities weights qu = 23 and qd = 1

3 are such that

EQ(

S(1)− S(0)

S(0)

)=

110− 100100

· qu +95− 100

100· qd = .05

Expected return on the stock = risk-free return

A quick glance at mathematical finance First concepts

Risk-neutral valuation

A fundamental question:

Is the previous arbitrage-free price consistent with the expected discounted

payoff of the option?

Risk-neutral Valuation Formula1 Expected Discounted Payoff:

101.05

· +0

1.05· ;

with qu = 23 , qd = 1

3 .

2 The probabilities weights qu = 23 and qd = 1

3 are such that

EQ(

S(1)− S(0)

S(0)

)=

110− 100100

· qu +95− 100

100· qd = .05

Expected return on the stock = risk-free return

A quick glance at mathematical finance First concepts

Risk-neutral valuation

A fundamental question:

Is the previous arbitrage-free price consistent with the expected discounted

payoff of the option?

Risk-neutral Valuation Formula1 Expected Discounted Payoff:

101.05

· +0

1.05· ;

with qu = 23 , qd = 1

3 .

2 The probabilities weights qu = 23 and qd = 1

3 are such that

EQ(

S(1)− S(0)

S(0)

)=

110− 100100

· qu +95− 100

100· qd = .05

Expected return on the stock = risk-free return

A quick glance at mathematical finance First concepts

Risk-neutral valuation

A fundamental question:

Is the previous arbitrage-free price consistent with the expected discounted

payoff of the option?

Risk-neutral Valuation Formula1 Expected Discounted Payoff:

EP(

Φ(S(1))

B(1)

)=

101.05

· 14

+0

1.05· 3

4;

with qu = 23 , qd = 1

3 .

2 The probabilities weights qu = 23 and qd = 1

3 are such that

EQ(

S(1)− S(0)

S(0)

)=

110− 100100

· qu +95− 100

100· qd = .05

Expected return on the stock = risk-free return

A quick glance at mathematical finance First concepts

Risk-neutral valuation

A fundamental question:

Is the previous arbitrage-free price consistent with the expected discounted

payoff of the option?

Risk-neutral Valuation Formula1 Expected Discounted Payoff:

EP(

Φ(S(1))

B(1)

)=

101.05

· 14

+0

1.05· 3

46= 6.3492;

with qu = 23 , qd = 1

3 .

2 The probabilities weights qu = 23 and qd = 1

3 are such that

EQ(

S(1)− S(0)

S(0)

)=

110− 100100

· qu +95− 100

100· qd = .05

Expected return on the stock = risk-free return

A quick glance at mathematical finance First concepts

Risk-neutral valuation

A fundamental question:

Is the previous arbitrage-free price consistent with the expected discounted

payoff of the option?

Risk-neutral Valuation Formula1 Expected Discounted Payoff:

EQ(

Φ(S(1))

B(1)

)=

101.05

· qu +0

1.05· qd = 6.3492;

with qu = 23 , qd = 1

3 .

2 The probabilities weights qu = 23 and qd = 1

3 are such that

EQ(

S(1)− S(0)

S(0)

)=

110− 100100

· qu +95− 100

100· qd = .05

Expected return on the stock = risk-free return

A quick glance at mathematical finance First concepts

Risk-neutral valuation

A fundamental question:

Is the previous arbitrage-free price consistent with the expected discounted

payoff of the option?

Risk-neutral Valuation Formula1 Expected Discounted Payoff:

EQ(

Φ(S(1))

B(1)

)=

101.05

· qu +0

1.05· qd = 6.3492;

with qu = 23 , qd = 1

3 .

2 The probabilities weights qu = 23 and qd = 1

3 are such that

EQ(

S(1)− S(0)

S(0)

)=

110− 100100

· qu +95− 100

100· qd = .05

Expected return on the stock = risk-free return

A quick glance at mathematical finance First concepts

Contingent claim in a defaultable market

1 Market:

• Zero-coupon bond exposed to default risk, with maturity T = 1• Recovery rate 1− δ = 60%

• Historical default probability p = 1%

• Current price of defaultable bond is .941.• A “risk-free" default-free zero-coupon bond with interest rate 5%.

2 Contingent claim on the default process:

• This contract pays 1 dollar if the bond defaults and pays 0 otherwise.

A quick glance at mathematical finance First concepts

Pricing and replication

1 Replicating portfolio:

• Go short 2.5 dollars in the defaultable bond• Put 50/21 ≈ 2.38 dollars in the default-free bond• Time t = 1 value of the portfolio

V1 =

((−2.5)(1) + ( 50

21 )(1.05) = 0 if no default

(−2.5)(.6) + ( 5021 )(1.05) = 1 if default

• Time t = 0 initial investment:

V0 = (−2.5)(.941) +5021≈ .0285.

2 Arbitrage-free price of the option:

P(0) = V0 = .0285.

A quick glance at mathematical finance First concepts

Pricing and replication

1 Replicating portfolio:

• Go short 2.5 dollars in the defaultable bond• Put 50/21 ≈ 2.38 dollars in the default-free bond• Time t = 1 value of the portfolio

V1 =

((−2.5)(1) + ( 50

21 )(1.05) = 0 if no default

(−2.5)(.6) + ( 5021 )(1.05) = 1 if default

• Time t = 0 initial investment:

V0 = (−2.5)(.941) +5021≈ .0285.

2 Arbitrage-free price of the option:

P(0) = V0 = .0285.

A quick glance at mathematical finance First concepts

Pricing and replication

1 Replicating portfolio:

• Go short 2.5 dollars in the defaultable bond• Put 50/21 ≈ 2.38 dollars in the default-free bond• Time t = 1 value of the portfolio

V1 =

((−2.5)(1) + ( 50

21 )(1.05) = 0 if no default

(−2.5)(.6) + ( 5021 )(1.05) = 1 if default

• Time t = 0 initial investment:

V0 = (−2.5)(.941) +5021≈ .0285.

2 Arbitrage-free price of the option:

P(0) = V0 = .0285.

A quick glance at mathematical finance First concepts

Pricing and replication

1 Replicating portfolio:

• Go short 2.5 dollars in the defaultable bond• Put 50/21 ≈ 2.38 dollars in the default-free bond• Time t = 1 value of the portfolio

V1 =

((−2.5)(1) + ( 50

21 )(1.05) = 0 if no default

(−2.5)(.6) + ( 5021 )(1.05) = 1 if default

• Time t = 0 initial investment:

V0 = (−2.5)(.941) +5021≈ .0285.

2 Arbitrage-free price of the option:

P(0) = V0 = .0285.

A quick glance at mathematical finance First concepts

Risk-neutral valuation

Fundamental question:

Is the price consistent with the “expected discounted payoff" of the claim?

Risk-Neutral Valuation Formula:1 Calibration: Determine a measure Q so that all traded assets have the

same rate of return:

Risk-free rate of return = EQ {Rate of return of the traded assets}

0.05 =1− 0.941

1.05· q +

0.6− 0.9411.05

· (1− q) =⇒ q = .03

2 Risk-neutral pricing:

P(0) = EQ {Discounted payoff} =⇒ P(0) =1

1.05· 03 +

01.05

· .97.

A quick glance at mathematical finance First concepts

Risk-neutral valuation

Fundamental question:

Is the price consistent with the “expected discounted payoff" of the claim?

Risk-Neutral Valuation Formula:1 Calibration: Determine a measure Q so that all traded assets have the

same rate of return:

Risk-free rate of return = EQ {Rate of return of the traded assets}

0.05 =1− 0.941

1.05· q +

0.6− 0.9411.05

· (1− q) =⇒ q = .03

2 Risk-neutral pricing:

P(0) = EQ {Discounted payoff} =⇒ P(0) =1

1.05· 03 +

01.05

· .97.

A quick glance at mathematical finance First concepts

Risk-neutral valuation

Fundamental question:

Is the price consistent with the “expected discounted payoff" of the claim?

Risk-Neutral Valuation Formula:1 Calibration: Determine a measure Q so that all traded assets have the

same rate of return:

Risk-free rate of return = EQ {Rate of return of the traded assets}

0.05 =1− 0.941

1.05· q +

0.6− 0.9411.05

· (1− q) =⇒ q = .03

2 Risk-neutral pricing:

P(0) = EQ {Discounted payoff} =⇒ P(0) =1

1.05· 03 +

01.05

· .97.

A quick glance at mathematical finance First concepts

Risk-neutral valuation

Fundamental question:

Is the price consistent with the “expected discounted payoff" of the claim?

Risk-Neutral Valuation Formula:1 Calibration: Determine a measure Q so that all traded assets have the

same rate of return:

Risk-free rate of return = EQ {Rate of return of the traded assets}

0.05 =1− 0.941

1.05· q +

0.6− 0.9411.05

· (1− q) =⇒ q = .03

2 Risk-neutral pricing:

P(0) = EQ {Discounted payoff} =⇒ P(0) =1

1.05· 03 +

01.05

· .97.

A quick glance at mathematical finance First concepts

Risk-neutral valuation

Fundamental question:

Is the price consistent with the “expected discounted payoff" of the claim?

Risk-Neutral Valuation Formula:1 Calibration: Determine a measure Q so that all traded assets have the

same rate of return:

Risk-free rate of return = EQ {Rate of return of the traded assets}

0.05 =1− 0.941

1.05· q +

0.6− 0.9411.05

· (1− q) =⇒ q = .03

2 Risk-neutral pricing:

P(0) = EQ {Discounted payoff} =⇒ P(0) =1

1.05· 03 +

01.05

· .97.

Regime switching market model

Outline

1 A quick glance at mathematical finance

2 Regime switching market model

3 Option Pricing under Regime Switching

Simple claims on the regime

Self-decomposable claims

Regime switching market model

Black-Scholes model for stock prices

1 The log return of the stock price S during [s, t ] is given by

R(t ,u) := log{

Su

St

}.

2 Principles of the Black-Scholes model:(i) Log returns on disjoint time periods are independent of one another:

Corr (R(s, t),R(t , u)) = 0, s < t < u

(ii) Log returns on time periods of equal size have the same distribution:

R(s, t) D= R(t , u), t − s = u − t

(iii) The information flow is continuous:

t −→ St is continuous;

3 The model:

(iv) log{

St+∆t

St

}∼ N

(m ∆t , σ2 ∆t

)m = Mean rate of return σ = Volatility or measure of variability

Regime switching market model

Black-Scholes model for stock prices

1 The log return of the stock price S during [s, t ] is given by

R(t ,u) := log{

Su

St

}.

2 Principles of the Black-Scholes model:(i) Log returns on disjoint time periods are independent of one another:

Corr (R(s, t),R(t , u)) = 0, s < t < u

(ii) Log returns on time periods of equal size have the same distribution:

R(s, t) D= R(t , u), t − s = u − t

(iii) The information flow is continuous:

t −→ St is continuous;

3 The model:

(iv) log{

St+∆t

St

}∼ N

(m ∆t , σ2 ∆t

)m = Mean rate of return σ = Volatility or measure of variability

Regime switching market model

Black-Scholes model for stock prices

1 The log return of the stock price S during [s, t ] is given by

R(t ,u) := log{

Su

St

}.

2 Principles of the Black-Scholes model:(i) Log returns on disjoint time periods are independent of one another:

Corr (R(s, t),R(t , u)) = 0, s < t < u

(ii) Log returns on time periods of equal size have the same distribution:

R(s, t) D= R(t , u), t − s = u − t

(iii) The information flow is continuous:

t −→ St is continuous;

3 The model:

(iv) log{

St+∆t

St

}∼ N

(m ∆t , σ2 ∆t

)m = Mean rate of return σ = Volatility or measure of variability

Regime switching market model

Black-Scholes model for stock prices

1 The log return of the stock price S during [s, t ] is given by

R(t ,u) := log{

Su

St

}.

2 Principles of the Black-Scholes model:(i) Log returns on disjoint time periods are independent of one another:

Corr (R(s, t),R(t , u)) = 0, s < t < u

(ii) Log returns on time periods of equal size have the same distribution:

R(s, t) D= R(t , u), t − s = u − t

(iii) The information flow is continuous:

t −→ St is continuous;

3 The model:

(iv) log{

St+∆t

St

}∼ N

(m ∆t , σ2 ∆t

)m = Mean rate of return σ = Volatility or measure of variability

Regime switching market model

Black-Scholes model for stock prices

1 The log return of the stock price S during [s, t ] is given by

R(t ,u) := log{

Su

St

}.

2 Principles of the Black-Scholes model:(i) Log returns on disjoint time periods are independent of one another:

Corr (R(s, t),R(t , u)) = 0, s < t < u

(ii) Log returns on time periods of equal size have the same distribution:

R(s, t) D= R(t , u), t − s = u − t

(iii) The information flow is continuous:

t −→ St is continuous;

3 The model:

(iv) log{

St+∆t

St

}∼ N

(m ∆t , σ2 ∆t

)m = Mean rate of return σ = Volatility or measure of variability

Regime switching market model

Black-Scholes model for stock prices

1 The log return of the stock price S during [s, t ] is given by

R(t ,u) := log{

Su

St

}.

2 Principles of the Black-Scholes model:(i) Log returns on disjoint time periods are independent of one another:

Corr (R(s, t),R(t , u)) = 0, s < t < u

(ii) Log returns on time periods of equal size have the same distribution:

R(s, t) D= R(t , u), t − s = u − t

(iii) The information flow is continuous:

t −→ St is continuous;

3 The model:

(iv) log{

St+∆t

St

}∼ N

(m ∆t , σ2 ∆t

)m = Mean rate of return σ = Volatility or measure of variability

Regime switching market model

Black-Scholes model for stock prices

1 The log return of the stock price S during [s, t ] is given by

R(t ,u) := log{

Su

St

}.

2 Principles of the Black-Scholes model:(i) Log returns on disjoint time periods are independent of one another:

Corr (R(s, t),R(t , u)) = 0, s < t < u

(ii) Log returns on time periods of equal size have the same distribution:

R(s, t) D= R(t , u), t − s = u − t

(iii) The information flow is continuous:

t −→ St is continuous;

3 The model:

(iv) log{

St+∆t

St

}∼ N

(m ∆t , σ2 ∆t

)m = Mean rate of return σ = Volatility or measure of variability

Regime switching market model

The log price process and Brownian Motion

1 It is convenient to express S in terms of the log price process Xt :

Xt := log {St/S0} ⇐⇒ St = S0 eXt

Clearly, X inherits the following properties:

(i) Corr (Xt − Xs,Xu − Xt ) = 0, s < t < u

(ii) Xt − XsD= Xu − Xt , t − s = u − t

(iii) t −→ Xt is continuous.

(iv) Xt ∼ N (tµ, σ2t). is continuous.

2 A process t →Wt satisfying (i)-(iii) with EWt = 0 and Var(Wt ) = t is said

to be a standard Brownian motion or Wiener process.

3 Common representation:

St = S0eσ·Wt +µ·t ,

Regime switching market model

The log price process and Brownian Motion

1 It is convenient to express S in terms of the log price process Xt :

Xt := log {St/S0} ⇐⇒ St = S0 eXt

Clearly, X inherits the following properties:

(i) Corr (Xt − Xs,Xu − Xt ) = 0, s < t < u

(ii) Xt − XsD= Xu − Xt , t − s = u − t

(iii) t −→ Xt is continuous.

(iv) Xt ∼ N (tµ, σ2t). is continuous.

2 A process t →Wt satisfying (i)-(iii) with EWt = 0 and Var(Wt ) = t is said

to be a standard Brownian motion or Wiener process.

3 Common representation:

St = S0eσ·Wt +µ·t ,

Regime switching market model

The log price process and Brownian Motion

1 It is convenient to express S in terms of the log price process Xt :

Xt := log {St/S0} ⇐⇒ St = S0 eXt

Clearly, X inherits the following properties:

(i) Corr (Xt − Xs,Xu − Xt ) = 0, s < t < u

(ii) Xt − XsD= Xu − Xt , t − s = u − t

(iii) t −→ Xt is continuous.

(iv) Xt ∼ N (tµ, σ2t). is continuous.

2 A process t →Wt satisfying (i)-(iii) with EWt = 0 and Var(Wt ) = t is said

to be a standard Brownian motion or Wiener process.

3 Common representation:

St = S0eσ·Wt +µ·t ,

Regime switching market model

The log price process and Brownian Motion

1 It is convenient to express S in terms of the log price process Xt :

Xt := log {St/S0} ⇐⇒ St = S0 eXt

Clearly, X inherits the following properties:

(i) Corr (Xt − Xs,Xu − Xt ) = 0, s < t < u

(ii) Xt − XsD= Xu − Xt , t − s = u − t

(iii) t −→ Xt is continuous.

(iv) Xt ∼ N (tµ, σ2t). is continuous.

2 A process t →Wt satisfying (i)-(iii) with EWt = 0 and Var(Wt ) = t is said

to be a standard Brownian motion or Wiener process.

3 Common representation:

St = S0eσ·Wt +µ·t ,

Regime switching market model

Black-Scholes-Merton’s fundamental results

1 The market consisting of• a stock following the Black-Scholes model• borrowing and lending of money at a constant interest rate r :

Equivalently, there exists a (risk-free) asset with price process B(t) = ert ;• “all" the claims on the stock

is arbitrage-free.

2 There exists a trading strategy on the stock and the money market

account whose final wealth replicates the payoff of any given derivative

3 The arbitrage-free price of a derivative equals its expected discounted

payoff with certain probability weights Q function:

P(0) = EQ [e−rT Φ(ST )].

4 The probability weight function Q is such that, under Q,

St = S0eσ·Wt +µQ ·t , µQ = r − σ2/2.

Regime switching market model

Black-Scholes-Merton’s fundamental results

1 The market consisting of• a stock following the Black-Scholes model• borrowing and lending of money at a constant interest rate r :

Equivalently, there exists a (risk-free) asset with price process B(t) = ert ;• “all" the claims on the stock

is arbitrage-free.

2 There exists a trading strategy on the stock and the money market

account whose final wealth replicates the payoff of any given derivative

3 The arbitrage-free price of a derivative equals its expected discounted

payoff with certain probability weights Q function:

P(0) = EQ [e−rT Φ(ST )].

4 The probability weight function Q is such that, under Q,

St = S0eσ·Wt +µQ ·t , µQ = r − σ2/2.

Regime switching market model

Black-Scholes-Merton’s fundamental results

1 The market consisting of• a stock following the Black-Scholes model• borrowing and lending of money at a constant interest rate r :

Equivalently, there exists a (risk-free) asset with price process B(t) = ert ;• “all" the claims on the stock

is arbitrage-free.

2 There exists a trading strategy on the stock and the money market

account whose final wealth replicates the payoff of any given derivative

3 The arbitrage-free price of a derivative equals its expected discounted

payoff with certain probability weights Q function:

P(0) = EQ [e−rT Φ(ST )].

4 The probability weight function Q is such that, under Q,

St = S0eσ·Wt +µQ ·t , µQ = r − σ2/2.

Regime switching market model

Black-Scholes-Merton’s fundamental results

1 The market consisting of• a stock following the Black-Scholes model• borrowing and lending of money at a constant interest rate r :

Equivalently, there exists a (risk-free) asset with price process B(t) = ert ;• “all" the claims on the stock

is arbitrage-free.

2 There exists a trading strategy on the stock and the money market

account whose final wealth replicates the payoff of any given derivative

3 The arbitrage-free price of a derivative equals its expected discounted

payoff with certain probability weights Q function:

P(0) = EQ [e−rT Φ(ST )].

4 The probability weight function Q is such that, under Q,

St = S0eσ·Wt +µQ ·t , µQ = r − σ2/2.

Regime switching market model

Black-Scholes-Merton’s fundamental results

1 The market consisting of• a stock following the Black-Scholes model• borrowing and lending of money at a constant interest rate r :

Equivalently, there exists a (risk-free) asset with price process B(t) = ert ;• “all" the claims on the stock

is arbitrage-free.

2 There exists a trading strategy on the stock and the money market

account whose final wealth replicates the payoff of any given derivative

3 The arbitrage-free price of a derivative equals its expected discounted

payoff with certain probability weights Q function:

P(0) = EQ [e−rT Φ(ST )].

4 The probability weight function Q is such that, under Q,

St = S0eσ·Wt +µQ ·t , µQ = r − σ2/2.

Regime switching market model

Black-Scholes-Merton’s fundamental results

1 The market consisting of• a stock following the Black-Scholes model• borrowing and lending of money at a constant interest rate r :

Equivalently, there exists a (risk-free) asset with price process B(t) = ert ;• “all" the claims on the stock

is arbitrage-free.

2 There exists a trading strategy on the stock and the money market

account whose final wealth replicates the payoff of any given derivative

3 The arbitrage-free price of a derivative equals its expected discounted

payoff with certain probability weights Q function:

P(0) = EQ [e−rT Φ(ST )].

4 The probability weight function Q is such that, under Q,

St = S0eσ·Wt +µQ ·t , µQ = r − σ2/2.

Regime switching market model

Examples

1 Digital Option: Φ(ST ) = 1{ST≥K}

P(0) = e−rT EQ [Φ(ST )] = e−rT EQ [1{ST≥s}]

= e−rT Q(ST ≥ K ) = e−rT Q(S0eσWT +µQT ≥ K )

= e−rT Q(σN (0,T ) + µQT ≥ logKS0

)

= e−rT N(d), d =log K

S0− µQT

σ√

T.

2 Call Option: Φ(ST ) = max{ST − K ,0}

P(0) = e−rT EQ [Φ(ST )] = S0N(d1)− Ke−rT N(d2),

d1 =log K

S0+(r + 1

2σ2)

T

σ√

T, d2 = d1 − σ

√T .

Regime switching market model

Examples

1 Digital Option: Φ(ST ) = 1{ST≥K}

P(0) = e−rT EQ [Φ(ST )] = e−rT EQ [1{ST≥s}]

= e−rT Q(ST ≥ K ) = e−rT Q(S0eσWT +µQT ≥ K )

= e−rT Q(σN (0,T ) + µQT ≥ logKS0

)

= e−rT N(d), d =log K

S0− µQT

σ√

T.

2 Call Option: Φ(ST ) = max{ST − K ,0}

P(0) = e−rT EQ [Φ(ST )] = S0N(d1)− Ke−rT N(d2),

d1 =log K

S0+(r + 1

2σ2)

T

σ√

T, d2 = d1 − σ

√T .

Regime switching market model

Examples

1 Digital Option: Φ(ST ) = 1{ST≥K}

P(0) = e−rT EQ [Φ(ST )] = e−rT EQ [1{ST≥s}]

= e−rT Q(ST ≥ K ) = e−rT Q(S0eσWT +µQT ≥ K )

= e−rT Q(σN (0,T ) + µQT ≥ logKS0

)

= e−rT N(d), d =log K

S0− µQT

σ√

T.

2 Call Option: Φ(ST ) = max{ST − K ,0}

P(0) = e−rT EQ [Φ(ST )] = S0N(d1)− Ke−rT N(d2),

d1 =log K

S0+(r + 1

2σ2)

T

σ√

T, d2 = d1 − σ

√T .

Regime switching market model

Black-Scholes Model with Regime-Switching

1 There are N finitely many market “regimes";

2 In the i th regime,

• The risk-free rate to borrow and lend is ri ;• The mean rate of return on the stock is µi ;• The volatility of the stock is σi ;

3 The regime in place is determined by the state of a continuous-time

Markov chain Ct with values in {1, . . . ,N};

4 Asset prices:

• (Risk-free asset): Bt = exp“R t

0 rCudu”⇐⇒ dBt = Bt rCt

dt ;

• (Risky asset): St = S0 exp“R t

0 σCudWu +

R t0 µCu

du”

, µi = ri − σ2i /2;

Regime switching market model

Black-Scholes Model with Regime-Switching

1 There are N finitely many market “regimes";

2 In the i th regime,

• The risk-free rate to borrow and lend is ri ;• The mean rate of return on the stock is µi ;• The volatility of the stock is σi ;

3 The regime in place is determined by the state of a continuous-time

Markov chain Ct with values in {1, . . . ,N};

4 Asset prices:

• (Risk-free asset): Bt = exp“R t

0 rCudu”⇐⇒ dBt = Bt rCt

dt ;

• (Risky asset): St = S0 exp“R t

0 σCudWu +

R t0 µCu

du”

, µi = ri − σ2i /2;

Regime switching market model

Black-Scholes Model with Regime-Switching

1 There are N finitely many market “regimes";

2 In the i th regime,

• The risk-free rate to borrow and lend is ri ;• The mean rate of return on the stock is µi ;• The volatility of the stock is σi ;

3 The regime in place is determined by the state of a continuous-time

Markov chain Ct with values in {1, . . . ,N};

4 Asset prices:

• (Risk-free asset): Bt = exp“R t

0 rCudu”⇐⇒ dBt = Bt rCt

dt ;

• (Risky asset): St = S0 exp“R t

0 σCudWu +

R t0 µCu

du”

, µi = ri − σ2i /2;

Regime switching market model

Black-Scholes Model with Regime-Switching

1 There are N finitely many market “regimes";

2 In the i th regime,

• The risk-free rate to borrow and lend is ri ;• The mean rate of return on the stock is µi ;• The volatility of the stock is σi ;

3 The regime in place is determined by the state of a continuous-time

Markov chain Ct with values in {1, . . . ,N};

4 Asset prices:

• (Risk-free asset): Bt = exp“R t

0 rCudu”⇐⇒ dBt = Bt rCt

dt ;

• (Risky asset): St = S0 exp“R t

0 σCudWu +

R t0 µCu

du”

, µi = ri − σ2i /2;

Regime switching market model

Black-Scholes Model with Regime-Switching

1 There are N finitely many market “regimes";

2 In the i th regime,

• The risk-free rate to borrow and lend is ri ;• The mean rate of return on the stock is µi ;• The volatility of the stock is σi ;

3 The regime in place is determined by the state of a continuous-time

Markov chain Ct with values in {1, . . . ,N};

4 Asset prices:

• (Risk-free asset): Bt = exp“R t

0 rCudu”⇐⇒ dBt = Bt rCt

dt ;

• (Risky asset): St = S0 exp“R t

0 σCudWu +

R t0 µCu

du”

, µi = ri − σ2i /2;

Regime switching market model

Some useful background on Markov processes

1 Infinitesimal generator: A = [ai,j ]i,j=1,...,N

ai,j (t) = limh→0

1h

P(Ct+h = j |Ct = i), (i 6= j), ai,i (t) = −∑j 6=i

ai,j (t).

2 Important Example: Time-invariant generator A(t) ≡ A = [ai,j ].

• Stay in C0 =: i an exponential time τ1 with rate −ai,i ;• At time τ1, jump into state ` 6= i with prob. ai,`/

Pk 6=i ai,k ;

• Stay in Cτ1 =: j an exponential time τ2 with rate −aj,j ;• At time τ2, jump into state ` 6= j with prob. aj,`/

Pk 6=j aj,k ;

• So forth...

Option Pricing under Regime Switching

Outline

1 A quick glance at mathematical finance

2 Regime switching market model

3 Option Pricing under Regime Switching

Simple claims on the regime

Self-decomposable claims

Option Pricing under Regime Switching Simple claims on the regime

Simple claims on the regime

1 Goal: Compute the price of a contingent claim with payoff Φ(CT );

2 Arbitrage-free price:

Pi (0) = EQ(

e−R T

0 rCudu

Φ(CT )∣∣∣C0 = i

).

3 Assume that under Q, t → Ct is a continuous Markov Chain with

generator A(t) = [ai,j (t)]i,j=1,...,N ;

Option Pricing under Regime Switching Simple claims on the regime

A new numerical methodStep 1: Change of Probability Measure

• Simplifying assumption: A = [ai,j ] is time-invariant.

• Key idea:

• There is a probability measure eQ such that, under eQ, the process t → Ct is a

continuous Markov process with generator eA(t) = [ai,j (t)];• Take eQ such that ai,j = 1/(N − 1), for all i 6= j .

• The following formula follows:

Pi (0) = EeQ[ Φ(CT ) exp

{−∫ T

0rCs

ds −∑

{s≤T :∆Cs 6=0}

K (Cs− ,Cs)

}∣∣∣∣∣∣C0 = i],

where

K (i , j) := − log ((N − 1)ai,j ) 1i 6=j , ri := ri − 1− ai,i .

Option Pricing under Regime Switching Simple claims on the regime

A new numerical methodStep 2: Conditioning on number of transitions

• Given that, under Q, the transition time follows a homogeneous Poisson

process with unit 1, conditioning on the number of transitions by time T :

Pi (0) =∞∑

m=0

e−T T m

m!Pi,m(T ),

where, denoting E eQi [·] = E eQ[·|C0 = i],

Pi,m(T ) := EeQi

[Φ(Cm) exp

{− T

m∑n=0

(U(n+1) − U(n))rCn−

m∑n=1

K (Cn−1, Cn)

}]

and Cn is the embedded Markov chain and Ui are iid U([0,1]);

• Note that Q(Cn = j |Cn−1 = i) = 1N−1 , (i 6= j).

• V1 = U(1) − U(0), . . . ,Vm = Um−1 − Um have “Dirichlet distribution".

Option Pricing under Regime Switching Simple claims on the regime

A new numerical methodStep 3: Representation in terms of Dirichlet Laplace Transforms

• Conditioning on (C1, . . . , Cm),

Pi,m(T ) =1

(N − 1)m

∑(c1,...,cm)

Φ(cm) exp{− T rcm −

m∑n=1

K (cn−1, cn)

}× Lm(T (rc0 − rcm ), . . . ,T (rcm−1 − rcm )),

where c0 = i and the above summation is over all “paths" (c1, . . . , cm)

such that cj ∈ {1, . . . ,N} and cj 6= cj−1, for j = 1, . . . ,m.

• Here, Λm is the Laplace transform of the “symmetric" Dirichlet distribution:

,

where Tm := {(λ1, . . . , λm) ∈ Rm : λi ≥ 0,∑m

i=1 λi ≤ 1}.

Option Pricing under Regime Switching Simple claims on the regime

A new numerical methodStep 3: Representation in terms of Dirichlet Laplace Transforms

• Conditioning on (C1, . . . , Cm),

Pi,m(T ) =1

(N − 1)m

∑(c1,...,cm)

Φ(cm) exp{− T rcm −

m∑n=1

K (cn−1, cn)

}× Lm(T (rc0 − rcm ), . . . ,T (rcm−1 − rcm )),

where c0 = i and the above summation is over all “paths" (c1, . . . , cm)

such that cj ∈ {1, . . . ,N} and cj 6= cj−1, for j = 1, . . . ,m.

• Here, Λm is the Laplace transform of the “symmetric" Dirichlet distribution:

,

where Tm := {(λ1, . . . , λm) ∈ Rm : λi ≥ 0,∑m

i=1 λi ≤ 1}.

Option Pricing under Regime Switching Simple claims on the regime

A new numerical methodStep 3: Representation in terms of Dirichlet Laplace Transforms

• Conditioning on (C1, . . . , Cm),

Pi,m(T ) =1

(N − 1)m

∑(c1,...,cm)

Φ(cm) exp{− T rcm −

m∑n=1

K (cn−1, cn)

}× Lm(T (rc0 − rcm ), . . . ,T (rcm−1 − rcm )),

where c0 = i and the above summation is over all “paths" (c1, . . . , cm)

such that cj ∈ {1, . . . ,N} and cj 6= cj−1, for j = 1, . . . ,m.

• Here, Λm is the Laplace transform of the “symmetric" Dirichlet distribution:

Lm(λ1, . . . , λm) := m!

∫Tm

e−Pm

j=1 λj xj dx ,

where Tm := {(λ1, . . . , λm) ∈ Rm : λi ≥ 0,∑m

i=1 λi ≤ 1}.

Option Pricing under Regime Switching Simple claims on the regime

A new numerical methodStep 3: Representation in terms of Dirichlet Laplace Transforms

• Conditioning on (C1, . . . , Cm),

Pi,m(T ) =1

(N − 1)m

∑(c1,...,cm)

Φ(cm) exp{− T rcm −

m∑n=1

K (cn−1, cn)

}× Lm(T (rc0 − rcm ), . . . ,T (rcm−1 − rcm )),

where c0 = i and the above summation is over all “paths" (c1, . . . , cm)

such that cj ∈ {1, . . . ,N} and cj 6= cj−1, for j = 1, . . . ,m.

• Here, Λm is the Laplace transform of the “symmetric" Dirichlet distribution:

Lm(λ1, . . . , λm) ≈ 1−∑m

i=1 λi

m + 1+

∑mi=1 λ

2i + 1

2

∑i 6=j λiλj

(m + 1)(m + 2),

where Tm := {(λ1, . . . , λm) ∈ Rm : λi ≥ 0,∑m

i=1 λi ≤ 1}.

Option Pricing under Regime Switching Simple claims on the regime

A new numerical methodStep 4: Discretization of time

• Drawback: The previous method will perform well as far as T is small;• A possible approach for large T :

• Fix k ∈ N, δ := T/k , and

Iu,v := −Z v

urCs

ds −X

{u<s≤v :∆Cs 6=0}

eK (Cs− ,Cs)

• Then, by Tower property and Markov property,

Pi (0) = EeQ heI0,T Φ(CT )

˛C0 = i

i= E

eQ heI0,δEeQ hΦ(CT )eIδ,T

˛Cu, u ≤ δ

i˛C0 = i

i= E

eQ heI0,δEeQ hΦ(CT )eIδ,T

˛Cδi˛

C0 = ii

=: EeQ heI0,δ eΦ(Cδ)

˛C0 = i

i,

where eΦ(j) = EeQ heI0,T−δΦ(CT−δ)

˛C0 = j

i.

Option Pricing under Regime Switching Simple claims on the regime

Complexity and Error Analysis

• The complexity of the algorithm will be O(NM−1), when at most M

transitions of the chain are considered;

• The error can be bounded by T M/kM−1;

C0 = 1 C0 = 2 C0 = 3

T (yrs.) Rec ODE Rec ODE Rec ODE

0.50 0.9837 0.9837 0.9772 0.9772 0.9393 0.9393

1.00 0.9659 0.9659 0.9555 0.9555 0.8864 0.8864

2.00 0.9282 0.9281 0.9146 0.9146 0.7991 0.7990

5.00 0.8140 0.8136 0.8029 0.8031 0.6274 0.6273

10.0 0.6488 0.6484 0.6430 0.6431 0.4701 0.4701

20.0 0.4119 0.4116 0.4090 0.4090 0.2931 0.2930

25.0 0.3282 0.3280 0.3259 0.3259 0.2334 0.2333

30.0 0.2616 0.2613 0.2597 0.2597 0.1859 0.1858

50.0 0.1055 0.1053 0.1047 0.1047 0.0750 0.0749

Option Pricing under Regime Switching Self-decomposable claims

Self-decomposabble claims

1 Consider a family of payoffs {Σt,T}0≤t≤T , where for each 0 ≤ t ≤ T ,

2 Σt,T represents a payoff depending on the path C on [t ,T ]. We write

Σt,T := Σ({Cs}t≤s≤T ).

3 We say that the family {Σt,T}0≤t≤T is self-decomposable if, for any

0 < t < t ′ < T , the following decomposition holds true:

Σt,T = f (Σt,t′ ) + g(Σt,t′ )Σt′ ,T ,

for some measurable functions f ,g : R 7→ R.

Option Pricing under Regime Switching Self-decomposable claims

A numerical method for self-decomposable options

1 (Decomposition) Fix a δ = (T − t)/k for a positive integer k and apply the

following decomposition with t ′ := t + δ:

EeQ [eIt,T Σt,T

∣∣Ct = i]

= EeQ [eIt,t′ f (Σt,t′)E

eQ [eIt′,T∣∣Ft′

]∣∣∣C0 = i]

+ EeQ [eIt,t′g(Σt,t′)E

eQ [eIt′,T Σt′,T∣∣Ft′

]∣∣∣C0 = ei

]=: E

eQ [eIt,t′ f (Σt,t′)Φ1(Ct′)∣∣∣C0 = i

]+ E

eQ [eIt,t′g(Σt,t′)Φ2(Ct′)∣∣∣C0 = i

],

where Φ1(`) := E eQ [eIt′,T∣∣Ct′ = `

]and Φ2(`) := E eQ [eIt′,T Σt′,T

∣∣Ct′ = `].

2 (Near-expiration approximation)Apply an approximation to evaluate

options of the form E eQ [eIt,t′h(Σt,t′)Φ(Ct′)∣∣Ct = i

], when t ′ is close to t .

Option Pricing under Regime Switching Self-decomposable claims

An example

1 Barrier Options:

Σt,T := Φ(CT )1{ maxt≤s≤T

σCs≥ B

}, Σt,T := Φ(CT )1{ maxt≤s≤T

σCs< B

}.