Mathematical Foundations for Finance Exercise 4 Martin Stefanik - PowerPoint PPT Presentation

Mathematical Foundations for Finance Exercise 4 Martin Stefanik ETH Zurich

Arbitrage Opportunity Self-financing and with zero initial investment at time k 0 so that we stand a chance of making a gain. 0 P V T 0 P -a.s. so that we do not lose money P -a.s. V T external financing is needed. 0 so that no carry out anyway (such as the doubling strategy). Definition 1 (Arbitrage opportunity) Admissible so that we exclude strategies that we would not be able to satisfies (NA). there exist no arbitrage opportunities. Sometimes one also says that S 1 / 6 An arbitrage opportunity is an admissible self-financing strategy ϕ � = ( 0 , ϑ ) with zero initial wealth, with V T ( ϕ ) ≥ 0 P -a.s. and with P [ V T ( ϕ ) > 0 ] > 0. The financial market (Ω , F , F , P , S 0 , S 1 ) or shortly S is called arbitrage-free if

Arbitrage Opportunity Self-financing and with zero initial investment at time k 0 so that we stand a chance of making a gain. 0 P V T 0 P -a.s. so that we do not lose money P -a.s. V T external financing is needed. 0 so that no carry out anyway (such as the doubling strategy). Definition 1 (Arbitrage opportunity) satisfies (NA). there exist no arbitrage opportunities. Sometimes one also says that S 1 / 6 An arbitrage opportunity is an admissible self-financing strategy ϕ � = ( 0 , ϑ ) with zero initial wealth, with V T ( ϕ ) ≥ 0 P -a.s. and with P [ V T ( ϕ ) > 0 ] > 0. The financial market (Ω , F , F , P , S 0 , S 1 ) or shortly S is called arbitrage-free if • Admissible so that we exclude strategies that we would not be able to

Arbitrage Opportunity Definition 1 (Arbitrage opportunity) 0 so that we stand a chance of making a gain. 0 P V T 0 P -a.s. so that we do not lose money P -a.s. V T external financing is needed. carry out anyway (such as the doubling strategy). satisfies (NA). there exist no arbitrage opportunities. Sometimes one also says that S 1 / 6 An arbitrage opportunity is an admissible self-financing strategy ϕ � = ( 0 , ϑ ) with zero initial wealth, with V T ( ϕ ) ≥ 0 P -a.s. and with P [ V T ( ϕ ) > 0 ] > 0. The financial market (Ω , F , F , P , S 0 , S 1 ) or shortly S is called arbitrage-free if • Admissible so that we exclude strategies that we would not be able to • Self-financing and with zero initial investment at time k = 0 so that no

Arbitrage Opportunity Definition 1 (Arbitrage opportunity) 0 so that we stand a chance of making a gain. 0 P V T external financing is needed. carry out anyway (such as the doubling strategy). 1 / 6 satisfies (NA). there exist no arbitrage opportunities. Sometimes one also says that S An arbitrage opportunity is an admissible self-financing strategy ϕ � = ( 0 , ϑ ) with zero initial wealth, with V T ( ϕ ) ≥ 0 P -a.s. and with P [ V T ( ϕ ) > 0 ] > 0. The financial market (Ω , F , F , P , S 0 , S 1 ) or shortly S is called arbitrage-free if • Admissible so that we exclude strategies that we would not be able to • Self-financing and with zero initial investment at time k = 0 so that no • V T ( ϕ ) ≥ 0 P -a.s. so that we do not lose money P -a.s.

Arbitrage Opportunity Definition 1 (Arbitrage opportunity) there exist no arbitrage opportunities. Sometimes one also says that S satisfies (NA). carry out anyway (such as the doubling strategy). external financing is needed. 1 / 6 An arbitrage opportunity is an admissible self-financing strategy ϕ � = ( 0 , ϑ ) with zero initial wealth, with V T ( ϕ ) ≥ 0 P -a.s. and with P [ V T ( ϕ ) > 0 ] > 0. The financial market (Ω , F , F , P , S 0 , S 1 ) or shortly S is called arbitrage-free if • Admissible so that we exclude strategies that we would not be able to • Self-financing and with zero initial investment at time k = 0 so that no • V T ( ϕ ) ≥ 0 P -a.s. so that we do not lose money P -a.s. • P [ V T ( ϕ ) > 0 ] > 0 so that we stand a chance of making a gain.

Arbitrage Results in Finite Discrete Time Lemma 2 Here it is starting to be clear why we work with discounted price processes. A similar result for undiscounted prices does not hold. Intuition: our (intuitive) definition of arbitrage requires investment into at least two assets at minimum at one point in time – arbitrage is connected to how prices move relative to each other. Note that it is if fact sufficient if there exists an equivalent local martingale measure (ELMM). This lemma actually holds for continuous time as well as infinite time horizon. The next big result shows that the converse holds as well. 2 / 6 Let (Ω , F , F , P , S 0 , S ) , or shortly S, with F = ( F k ) k = 0 , 1 ,..., T be a financial market in finite discrete time. If there exists a probability measure Q ≈ P on F T such that S is a Q-martingale, then the market S is arbitrage-free.

Arbitrage Results in Finite Discrete Time Lemma 2 processes. A similar result for undiscounted prices does not hold. Intuition: our (intuitive) definition of arbitrage requires investment into at least two assets at minimum at one point in time – arbitrage is connected to how prices move relative to each other. Note that it is if fact sufficient if there exists an equivalent local martingale measure (ELMM). This lemma actually holds for continuous time as well as infinite time horizon. The next big result shows that the converse holds as well. 2 / 6 Let (Ω , F , F , P , S 0 , S ) , or shortly S, with F = ( F k ) k = 0 , 1 ,..., T be a financial market in finite discrete time. If there exists a probability measure Q ≈ P on F T such that S is a Q-martingale, then the market S is arbitrage-free. • Here it is starting to be clear why we work with discounted price

Arbitrage Results in Finite Discrete Time Lemma 2 processes. A similar result for undiscounted prices does not hold. at least two assets at minimum at one point in time – arbitrage is connected to how prices move relative to each other. Note that it is if fact sufficient if there exists an equivalent local martingale measure (ELMM). This lemma actually holds for continuous time as well as infinite time horizon. The next big result shows that the converse holds as well. 2 / 6 Let (Ω , F , F , P , S 0 , S ) , or shortly S, with F = ( F k ) k = 0 , 1 ,..., T be a financial market in finite discrete time. If there exists a probability measure Q ≈ P on F T such that S is a Q-martingale, then the market S is arbitrage-free. • Here it is starting to be clear why we work with discounted price • Intuition: our (intuitive) definition of arbitrage requires investment into

Arbitrage Results in Finite Discrete Time Lemma 2 processes. A similar result for undiscounted prices does not hold. at least two assets at minimum at one point in time – arbitrage is connected to how prices move relative to each other. martingale measure (ELMM). This lemma actually holds for continuous time as well as infinite time horizon. The next big result shows that the converse holds as well. 2 / 6 Let (Ω , F , F , P , S 0 , S ) , or shortly S, with F = ( F k ) k = 0 , 1 ,..., T be a financial market in finite discrete time. If there exists a probability measure Q ≈ P on F T such that S is a Q-martingale, then the market S is arbitrage-free. • Here it is starting to be clear why we work with discounted price • Intuition: our (intuitive) definition of arbitrage requires investment into • Note that it is if fact sufficient if there exists an equivalent local

Arbitrage Results in Finite Discrete Time Lemma 2 processes. A similar result for undiscounted prices does not hold. at least two assets at minimum at one point in time – arbitrage is connected to how prices move relative to each other. martingale measure (ELMM). horizon. The next big result shows that the converse holds as well. 2 / 6 Let (Ω , F , F , P , S 0 , S ) , or shortly S, with F = ( F k ) k = 0 , 1 ,..., T be a financial market in finite discrete time. If there exists a probability measure Q ≈ P on F T such that S is a Q-martingale, then the market S is arbitrage-free. • Here it is starting to be clear why we work with discounted price • Intuition: our (intuitive) definition of arbitrage requires investment into • Note that it is if fact sufficient if there exists an equivalent local • This lemma actually holds for continuous time as well as infinite time

Arbitrage Results in Finite Discrete Time Lemma 2 processes. A similar result for undiscounted prices does not hold. at least two assets at minimum at one point in time – arbitrage is connected to how prices move relative to each other. martingale measure (ELMM). horizon. 2 / 6 Let (Ω , F , F , P , S 0 , S ) , or shortly S, with F = ( F k ) k = 0 , 1 ,..., T be a financial market in finite discrete time. If there exists a probability measure Q ≈ P on F T such that S is a Q-martingale, then the market S is arbitrage-free. • Here it is starting to be clear why we work with discounted price • Intuition: our (intuitive) definition of arbitrage requires investment into • Note that it is if fact sufficient if there exists an equivalent local • This lemma actually holds for continuous time as well as infinite time • The next big result shows that the converse holds as well.