On robust pricing and hedging and the resulting notions of weak - PowerPoint PPT Presentation

On robust pricing and hedging and the resulting notions of weak arbitrage Jan Ob� l´ oj University of Oxford obloj@maths.ox.ac.uk based on joint works with Alexander Cox (University of Bath) 5 th Oxford–Princeton Workshop , Princeton, 27–28 March 2009

Principal Questions and Answers Double barrier options Theoretical framework and arbitrages Outline Principal Questions and Answers Financial Problem (2 questions) Methodology (2 answers) Double barrier options Introduction and types of barriers Double no–touch example Theoretical framework and arbitrages Pricing operators and arbitrages No arbitrage vs existence of a model

Principal Questions and Answers Double barrier options Theoretical framework and arbitrages Robust techniques in quantitative finance Oxford–Man Institute of Quantitative Finance 18–19 March 2010

Principal Questions and Answers Double barrier options Theoretical framework and arbitrages Robust methods: principal ideas Model risk: • Any given model is unlikely to capture the reality. • Strategies which are sensitive to model assumptions or changes in parameters are questionable. • We look for strategies which are robust w.r.t. departures from the modelling assumptions. Market input: • We want to start by taking information from the market. E.g. prices of liquidly traded instruments should be treated as an input. • We can then add modelling assumptions and try to see how these affect, for example, admissible prices and hedging techniques.

Principal Questions and Answers Double barrier options Theoretical framework and arbitrages Robust methods: principal ideas Model risk: • Any given model is unlikely to capture the reality. • Strategies which are sensitive to model assumptions or changes in parameters are questionable. • We look for strategies which are robust w.r.t. departures from the modelling assumptions. Market input: • We want to start by taking information from the market. E.g. prices of liquidly traded instruments should be treated as an input. • We can then add modelling assumptions and try to see how these affect, for example, admissible prices and hedging techniques.

Principal Questions and Answers Double barrier options Theoretical framework and arbitrages Robust pricing and hedging: 2 questions The general setting and challenge is as follows: • Observe prices of some liquid instruments which admit no arbitrage. ( � interesting questions!) • Q1: (very) robust pricing Given a new product, determine its feasible price, i.e. range of prices which do not introduce an arbitrage in the market. • Q2: (very) robust hedging Furthermore, derive tight super-/sub- hedging strategies which always work. E.g.: Put-Call parity, Up-and-in put

Principal Questions and Answers Double barrier options Theoretical framework and arbitrages Robust pricing and hedging: 2 questions The general setting and challenge is as follows: • Observe prices of some liquid instruments which admit no arbitrage. ( � interesting questions!) • Q1: (very) robust pricing Given a new product, determine its feasible price, i.e. range of prices which do not introduce an arbitrage in the market. • Q2: (very) robust hedging Furthermore, derive tight super-/sub- hedging strategies which always work. E.g.: Put-Call parity, Up-and-in put

Principal Questions and Answers Double barrier options Theoretical framework and arbitrages Robust pricing and hedging: 2 questions The general setting and challenge is as follows: • Observe prices of some liquid instruments which admit no arbitrage. ( � interesting questions!) • Q1: (very) robust pricing Given a new product, determine its feasible price, i.e. range of prices which do not introduce an arbitrage in the market. • Q2: (very) robust hedging Furthermore, derive tight super-/sub- hedging strategies which always work. E.g.: Put-Call parity, Up-and-in put

Principal Questions and Answers Double barrier options Theoretical framework and arbitrages Robust pricing and hedging: 2 questions The general setting and challenge is as follows: • Observe prices of some liquid instruments which admit no arbitrage. ( � interesting questions!) • Q1: (very) robust pricing Given a new product, determine its feasible price, i.e. range of prices which do not introduce an arbitrage in the market. • Q2: (very) robust hedging Furthermore, derive tight super-/sub- hedging strategies which always work. E.g.: Put-Call parity, Up-and-in put

Principal Questions and Answers Double barrier options Theoretical framework and arbitrages Q1 and the Skorokhod Embedding Problem Q1: What is the range of no-arbitrage prices of an option O T given prices of European calls? • Suppose: • ( S t ) is a continuous martingale under P = Q , • we see market prices C T ( K ) = E ( S T − K ) + , K ≥ 0. • Equivalently ( S t : t ≤ T ) is a UI martingale, S T ∼ µ , µ ( dx ) = C ′′ ( x ) dx . • Via Dubins-Schwarz S t = B τ t is a time-changed Brownian motion. Say we have O T = O ( S ) T = O ( B ) τ T . • We are led then to investigate the bounds LB = inf τ E O ( B ) τ , UB = sup τ E O ( B ) τ , and for all stopping times τ : B τ ∼ µ and ( B t ∧ τ ) a UI martingale, i.e. for all solutions to the Skorokhod Embedding problem. • The bounds are tight: the process S t := B τ ∧ T − t defines an t asset model which matches the market data.

Principal Questions and Answers Double barrier options Theoretical framework and arbitrages Q1 and the Skorokhod Embedding Problem Q1: What is the range of no-arbitrage prices of an option O T given prices of European calls? • Suppose: • ( S t ) is a continuous martingale under P = Q , • we see market prices C T ( K ) = E ( S T − K ) + , K ≥ 0. • Equivalently ( S t : t ≤ T ) is a UI martingale, S T ∼ µ , µ ( dx ) = C ′′ ( x ) dx . • Via Dubins-Schwarz S t = B τ t is a time-changed Brownian motion. Say we have O T = O ( S ) T = O ( B ) τ T . • We are led then to investigate the bounds LB = inf τ E O ( B ) τ , UB = sup τ E O ( B ) τ , and for all stopping times τ : B τ ∼ µ and ( B t ∧ τ ) a UI martingale, i.e. for all solutions to the Skorokhod Embedding problem. • The bounds are tight: the process S t := B τ ∧ T − t defines an t asset model which matches the market data.

Principal Questions and Answers Double barrier options Theoretical framework and arbitrages Q1 and the Skorokhod Embedding Problem Q1: What is the range of no-arbitrage prices of an option O T given prices of European calls? • Suppose: • ( S t ) is a continuous martingale under P = Q , • we see market prices C T ( K ) = E ( S T − K ) + , K ≥ 0. • Equivalently ( S t : t ≤ T ) is a UI martingale, S T ∼ µ , µ ( dx ) = C ′′ ( x ) dx . • Via Dubins-Schwarz S t = B τ t is a time-changed Brownian motion. Say we have O T = O ( S ) T = O ( B ) τ T . • We are led then to investigate the bounds LB = inf τ E O ( B ) τ , UB = sup τ E O ( B ) τ , and for all stopping times τ : B τ ∼ µ and ( B t ∧ τ ) a UI martingale, i.e. for all solutions to the Skorokhod Embedding problem. • The bounds are tight: the process S t := B τ ∧ T − t defines an t asset model which matches the market data.

Principal Questions and Answers Double barrier options Theoretical framework and arbitrages Q1 and the Skorokhod Embedding Problem Q1: What is the range of no-arbitrage prices of an option O T given prices of European calls? • Suppose: • ( S t ) is a continuous martingale under P = Q , • we see market prices C T ( K ) = E ( S T − K ) + , K ≥ 0. • Equivalently ( S t : t ≤ T ) is a UI martingale, S T ∼ µ , µ ( dx ) = C ′′ ( x ) dx . • Via Dubins-Schwarz S t = B τ t is a time-changed Brownian motion. Say we have O T = O ( S ) T = O ( B ) τ T . • We are led then to investigate the bounds LB = inf τ E O ( B ) τ , UB = sup τ E O ( B ) τ , and for all stopping times τ : B τ ∼ µ and ( B t ∧ τ ) a UI martingale, i.e. for all solutions to the Skorokhod Embedding problem. • The bounds are tight: the process S t := B τ ∧ T − t defines an t asset model which matches the market data.

Principal Questions and Answers Double barrier options Theoretical framework and arbitrages Q2 and pathwise inequalities Q2: if we see a price outside the bounds ( LB , UB ) can we (and how) realise a risk-less profit? • Consider UB. The idea is to devise inequalities of the form O ( B ) t ≤ N t + F ( B t ) , t ≥ 0 , with equality for some τ ∗ with B τ ∗ ∼ µ , and where is a martingale (i.e. trading strategy), E N τ ∗ = 0. • Then UB = E F ( S T ) and + F ( S t ) is a valid superhedge. It involves dynamic trading and a static position in calls F ( S T ). • Furthermore, we want ( N τ t ) explicitly. We are naturally restricted to the family of martingales N t = N ( B t , A t ), for some process ( A t ) related to the option O t , e.g. maximum and minimum processes for barrier options.