Motivation General Itˆ o processes Good deals by BSDEs Problem Solution Bounds on optional growth From bounds on optimal growth towards a theory of good-deal hedging Dirk Becherer, Humboldt-Universit¨ at Tamerza, Tunesia, Oct.2010 Dirk Becherer, Humboldt-Universit¨ at Berlin Good-deal hedging
Motivation General Itˆ o processes Good deals by BSDEs Problem Solution Bounds on optional growth Problem Complete Market (e.g Black-Scholes) unique martingale measure Q for asset prices S any claim X ≥ 0 is priced by replication � ¯ T t ≤ ¯ E Q X = t [ X ] + ϑ dS , T � �� � t � �� � replication cost hedging Incomplete Market infinitely many martingale measures Q ∈ M ( S ) No-arbitrage valuations bounds Q ∈M E Q E Q inf t [ X ] and sup t [ X ] Q ∈M are the super-replication costs � notion of hedging Problem: The bounds are typically too wide! Dirk Becherer, Humboldt-Universit¨ at Berlin Good-deal hedging
Motivation General Itˆ o processes Good deals by BSDEs Problem Solution Bounds on optional growth Problem Complete Market (e.g Black-Scholes) unique martingale measure Q for asset prices S any claim X ≥ 0 is priced by replication � ¯ T t ≤ ¯ E Q X = t [ X ] + ϑ dS , T � �� � t � �� � replication cost hedging Incomplete Market infinitely many martingale measures Q ∈ M ( S ) No-arbitrage valuations bounds Q ∈M E Q E Q inf t [ X ] and sup t [ X ] Q ∈M are the super-replication costs � notion of hedging Problem: The bounds are typically too wide! Dirk Becherer, Humboldt-Universit¨ at Berlin Good-deal hedging
Motivation General Itˆ o processes Good deals by BSDEs Problem Solution Bounds on optional growth “Solution” Ad-hoc Solution Get tighter bounds by using smaller subset Q ngd ⊂ M Q ∈Q ngd E Q Q ∈Q ngd E Q inf t [ X ] and sup t [ X ] Questions Which subset Q ngd to choose ? ... for good mathematical dynamical valuation properties ? ... for financial meaning of such valuation bounds ? Can one associate to such bounds any notion of hedging ? Dirk Becherer, Humboldt-Universit¨ at Berlin Good-deal hedging
Refs: Cochrane/Saa Reqquejo 2000 and Hodges/Cerny 2000
Motivation General Itˆ o processes Good deals by BSDEs Problem Solution Bounds on optional growth Outline Bounds for Optimal Growth for Semimartingales by Duality 1 An Itˆ o process model 2 Good-deal valuation and hedging via BSDE 3 Dirk Becherer, Humboldt-Universit¨ at Berlin Good-deal hedging
Motivation General Itˆ o processes Good deals by BSDEs Problem Solution Bounds on optional growth Bounds on Optimal Growth discounted asset prices processes: Semimartingales S ≥ 0 positive (normalized) wealth processes = tradable numeraires � t ≤ ¯ N t = 1 + ϑ dS > 0 , T 0 cond. expected growth over any period ] ] T , τ ] ] is � � log N τ (1) E T N T Question : Can we choose the set Q ngd such that a pre-specified bound for growth (1) is ensured for any market extension ¯ S = ( S , S ′ ) by derivative price processes t [ X ] for X ≥ 0 computed by Q ∈ Q ngd ? S ′ t = E Q Dirk Becherer, Humboldt-Universit¨ at Berlin Good-deal hedging
Motivation General Itˆ o processes Good deals by BSDEs Problem Solution Bounds on optional growth Bounds on Optimal Growth discounted asset prices processes: Semimartingales S ≥ 0 positive (normalized) wealth processes = tradable numeraires � t ≤ ¯ N t = 1 + ϑ dS > 0 , T 0 cond. expected growth over any period ] ] T , τ ] ] is � � log N τ (1) E T N T Question : Can we choose the set Q ngd such that a pre-specified bound for growth (1) is ensured for any market extension ¯ S = ( S , S ′ ) by derivative price processes t [ X ] for X ≥ 0 computed by Q ∈ Q ngd ? S ′ t = E Q Dirk Becherer, Humboldt-Universit¨ at Berlin Good-deal hedging
Motivation General Itˆ o processes Good deals by BSDEs Bounds for optimal growth Dynamic valuation properties Ensuring Bounds for Optimal Growth by defining a suitable set Q ngd of pricing measures Def : Measures with finite (reverse) relative entropy � � � � E [ − log Z ¯ Q ∈ M e ( S ) Q := T ] < ∞ Fix some predictable and bounded process h = ( h t ) > 0, and Def : let Q ngd contain Q ∈ Q iff density process Z satisfies � � �� τ � − log Z τ ≤ 1 for all T ≤ τ ≤ ¯ h 2 E T 2 E T u du T , Z T T ... equivalently with only deterministic times � � �� t � − log Z t ≤ 1 h 2 for all s ≤ t ≤ ¯ E s 2 E s u du T Z s s Dirk Becherer, Humboldt-Universit¨ at Berlin Good-deal hedging
Motivation General Itˆ o processes Good deals by BSDEs Bounds for optimal growth Dynamic valuation properties Ensuring Bounds for Optimal Growth by defining a suitable set Q ngd of pricing measures Def : Measures with finite (reverse) relative entropy � � � � E [ − log Z ¯ Q ∈ M e ( S ) Q := T ] < ∞ Fix some predictable and bounded process h = ( h t ) > 0, and Def : let Q ngd contain Q ∈ Q iff density process Z satisfies � � �� τ � − log Z τ ≤ 1 h 2 for all T ≤ τ ≤ ¯ E T 2 E T u du T , Z T T Example : For h = const e.g. � � − log Z t s ≤ t ≤ ¯ ≤ const ( t − s ) , E s T Z s Dirk Becherer, Humboldt-Universit¨ at Berlin Good-deal hedging
Motivation General Itˆ o processes Good deals by BSDEs Bounds for optimal growth Dynamic valuation properties Ensuring Bounds for Optimal Growth Convex duality yields: When pricing with Q ∈ Q ngd , any extended market ¯ S t = ( S t , E Q t [ X ]) satisfies the bounds for expected growth of wealth � � � � ¯ N τ − log Z τ E T log ≤ E T (2) ¯ N T Z T for all stopping times T ≤ τ ≤ ¯ T . That is, derivatives price processes are taken such that there arise no dynamic trading opportunities which offer deals that are ‘too good’! Dirk Becherer, Humboldt-Universit¨ at Berlin Good-deal hedging
Motivation General Itˆ o processes Good deals by BSDEs Bounds for optimal growth Dynamic valuation properties Ensuring Bounds for Optimal Growth Convex duality yields: When pricing with Q ∈ Q ngd , any extended market ¯ S t = ( S t , E Q t [ X ]) satisfies the bounds for expected growth of wealth � � � � �� τ � ¯ ≤ 1 N τ − log Z τ h 2 E T log ≤ E T 2 E T u du (2) ¯ N T Z T T for all stopping times T ≤ τ ≤ ¯ T . That is, derivatives price processes are taken such that there arise no dynamic trading opportunities which offer deals that are ‘too good’! Dirk Becherer, Humboldt-Universit¨ at Berlin Good-deal hedging
Motivation General Itˆ o processes Good deals by BSDEs Bounds for optimal growth Dynamic valuation properties Ensuring Bounds for Optimal Growth Convex duality yields: When pricing with Q ∈ Q ngd , any extended market ¯ S t = ( S t , E Q t [ X ]) satisfies the bounds for expected growth of wealth � � � � �� τ � ¯ ≤ 1 N τ − log Z τ h 2 E T log ≤ E T 2 E T u du (2) ¯ N T Z T T for all stopping times T ≤ τ ≤ ¯ T . That is, derivatives price processes are taken such that there arise no dynamic trading opportunities which offer deals that are ‘too good’! Dirk Becherer, Humboldt-Universit¨ at Berlin Good-deal hedging
Motivation General Itˆ o processes Good deals by BSDEs Bounds for optimal growth Dynamic valuation properties Multiplicative Stability For any Q ∈ Q , we have a Doob-Meyer decomposition − log Z t = M t + A t with M = UI-martingale, A = predictable, increasing, integrable � � − log Z τ Additive functional for T ≤ τ : E T = E T [ A τ − A T ] Z T � Q ngd is multiplicative stable � Dynamic good-deal valuation bounds π u Q ∈Q ngd E Q π ℓ Q ∈Q ngd E Q t [ X ] = − π u t ( X ) = sup t [ X ] and t ( X ) = inf t ( − X ) have good dynamic behavior over time.... Dirk Becherer, Humboldt-Universit¨ at Berlin Good-deal hedging
Motivation General Itˆ o processes Good deals by BSDEs Bounds for optimal growth Dynamic valuation properties Good Dynamic Valuation Bound Properties T ) from L ∞ → L ∞ ( F t ) t ( X ) ( t ≤ ¯ Thm : Mappings X �→ π u satisfies ( nice paths ) For any X ∈ L ∞ there is an RCLL-version of ( π u t ( X )) t ≤ ¯ T π u E Q for all stopping times T ≤ ¯ T ( X ) = ess sup T [ X ] T . Q ∈S ( recursiveness ) For any stopping times T ≤ τ ≤ ¯ T holds that π u T ( X ) = π u T ( π u τ ( X )) . (Stopping-time consistency ) For stopping times T ≤ τ ≤ ¯ T the inequality π u τ ( X 1 ) ≥ π u τ ( X 2 ) implies π u T ( X 1 ) ≥ π u T ( X 2 ). Dirk Becherer, Humboldt-Universit¨ at Berlin Good-deal hedging
Motivation General Itˆ o processes Good deals by BSDEs Bounds for optimal growth Dynamic valuation properties Good Valuation Bound properties (cont.) Thm (cont.) ( dynamic coherent risk measure ) For any stopping time T ≤ ¯ T and m T , α T , λ T ∈ L ∞ ( F T ) with 0 ≤ α T ≤ 1, λ T ≥ 0, the mapping X �→ π u T ( X ) satisfies the properties: monotonicity: X 1 ≥ X 2 implies π u T ( X 1 ) ≥ π u T ( X 2 ) translation invariance: π u T ( X + m T ) = π u T ( X ) + m T convexity: T ( α T X 1 + (1 − α T ) X 2 ) ≤ α T π u π u T ( X 1 ) + (1 − α T ) π u T ( X 2 ) positive homogeneity: π u T ( λ T X ) = λ T π u T ( X ) No arbitrage consistency : π u T ( X ) = x + ϑ · S T for any X = x + ϑ · S ¯ T with (( ϑ · S t ) t ≤ ¯ T ) being uniformly bounded. Dirk Becherer, Humboldt-Universit¨ at Berlin Good-deal hedging
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