Bubbles and futures contracts in markets with short-selling constraints

Transcription

1 Bubbles and futures contracts in markets with short-selling constraints Sergio Pulido, Cornell University PhD committee: Philip Protter, Robert Jarrow 3 rd WCMF, Santa Barbara, California November 13 th, 2009 Bubbles and futures contracts in markets with short-selling constraints 1/19

2 Table of contents 1 Motivation 2 The FTAP with no short-selling 3 Hedging with no short-selling 4 Completing with futures Bubbles and futures contracts in markets with short-selling constraints 2/19

3 Table of contents 1 Motivation 2 The FTAP with no short-selling 3 Hedging with no short-selling 4 Completing with futures Bubbles and futures contracts in markets with short-selling constraints 3/19

4 Motivation The current financial crisis is, to a large extent, a product of the burst of the alleged real estate bubble. Massive short-selling after the burst of a financial bubble. Short-selling ban, September 2008: U.S. Securities and Exchange Commission (SEC) and U.K. Financial Services Authority (FSA). In most of the third world emerging markets the practice of short-selling is not allowed. Bubbles and futures contracts in markets with short-selling constraints 4/19

5 Local martingale approach to bubbles Jarrow, Protter and Shimbo (2006, 2008) and Cox and Hobson (2005): (NF LV R) strategies with bounded liabilities M loc (S) Valuation measure Q M loc (S)\M mar (S) Bubbles E Q [S T ] < S 0 Bubbles and futures contracts in markets with short-selling constraints 5/19

6 Table of contents 1 Motivation 2 The FTAP with no short-selling 3 Hedging with no short-selling 4 Completing with futures Bubbles and futures contracts in markets with short-selling constraints 6/19

7 The model Reference filtered probability space: (Ω, F, F, P ). Price process: (S t ) 0 t T a nonnegative locally bounded semi-martingale. Money market account: R t 1. The admissible strategies: { A := H L(S) : H 0 = 0, H 0, 0 } H S α, for some α > 0. Payoffs of zero initial value portfolios: { } T K := H s ds s : H A L 0 (Ω, F, P ). 0 Bounded payoffs dominated by elements of K: C := (K L 0 +(Ω, F, P )) L (Ω, F, P ) L (Ω, F, P ). Bubbles and futures contracts in markets with short-selling constraints 7/19

8 The FTAP under short-selling prohibition Theorem Let M sup (S) be the set of probability measures Q P such that S is a Q-supermartingale. Then Related results: (NFLVR) C L + (Ω, F, P ) = {0} M sup (S). L 2 case for simple strategies: Jouini and Kallal (1995). Simple predictable strategies in L : Frittelli (1997). Bubbles and futures contracts in markets with short-selling constraints 8/19

9 A key observation Proposition (Extension of Ansel and Stricker, 1994) M sup (S) = { Q P : 0 } H ds is a Q-supermartingale for all H A. Bubbles and futures contracts in markets with short-selling constraints 9/19

10 Table of contents 1 Motivation 2 The FTAP with no short-selling 3 Hedging with no short-selling 4 Completing with futures Bubbles and futures contracts in markets with short-selling constraints 10/19

11 Replication under short selling prohibition Theorem (Extension of Ansel and Stricker, 1994) Suppose M sup (S). For f T L 0 +(Ω, F, P ) TFAE (i) f T = x + T 0 H s ds s with x constant and H A such that 0 H s ds s is a Q -martingale for some Q M sup (S). (ii) There exists Q M sup (S) such that sup E Q [f T ] = E Q [f T ] < Q M sup(s) This theorem is a corollary of a more general result proved by Föllmer & Kramkov (1997). Example If the price process is continuous (and nonconstant) the payoff f T = 1 (ST <S 0) cannot be perfectly replicated without short-selling. Bubbles and futures contracts in markets with short-selling constraints 11/19

12 Table of contents 1 Motivation 2 The FTAP with no short-selling 3 Hedging with no short-selling 4 Completing with futures Bubbles and futures contracts in markets with short-selling constraints 12/19

13 Futures contracts Purchase of S at time T via prearranged payment procedure. Importance: (1) Cash-flow depends on market valuation (2) Very liquid derivatives. Definition (Karatzas and Shreve, Methods of Mathematical Finance) A futures contract on S with maturity time T is a financial instrument with associated stream of cash-flows F t,t, such that (i) F t,t is a nonnegative F-adapted semi-martingale with F T,T = S T. (ii) The market price of the stream of cash-flows (F t,t ) t is zero at all times. F t,t is known as the futures price process. If this contract can be sold short, in the extended market (NFLVR) F t,t is a Q-local martingale for some Q M sup (S) Bubbles and futures contracts in markets with short-selling constraints 13/19

14 Completing with futures Theorem (Completing with futures - No interest rates) Suppose that S is positive and continuous, M loc (S) = {P } and Q M sup (S). If S = M A = E( B)E(N), with M, E(N) Q-martingales and A, B increasing, F t,t = E Q [S T F t ]. and B is deterministic then M loc (F,T ) = {Q}. Lemma Suppose that S is positive and continuous, M loc (S) = {P }, Q M sup (S) and S = M A is the Doob-Meyer decomposition of S under Q. Then M loc (M) = {Q}. Bubbles and futures contracts in markets with short-selling constraints 14/19

15 Pathological examples Cox and Hobson 2005 ( ) db s S = 1 + E. T s Strict local martingale S T 1, hence F t,t 1. Binary tree 0 Bubbles and futures contracts in markets with short-selling constraints 15/19

16 Open question Suppose that M loc (S) = 1. Find necessary and sufficient conditions on Q M sup (S) under which the futures (+ bonds) market is complete. Bubbles and futures contracts in markets with short-selling constraints 16/19

17 Thank you! Questions? Bubbles and futures contracts in markets with short-selling constraints 17/19

18 References Ansel J.-P. and Stricker C. Couverture des actifs contingents et prix maximum; Annales de l I.H.P. Probabilits et statistiques, vol. 30, no. 2, pp , Föllmer H. and Kramkov D. Optional decompositions under constraints; Probability Theory and Related Fields, vol. 109, no. 1, Frittelli M. Semimartingales and asset pricing under constraints; Mathematics of Derivative Securities, S. Pliska, M.A.H. Dempster eds., Newton Institute for Mathematical Science, Cambridge University Press, pp , Jarrow R. Protter P. and Shimbo K. Asset price bubbles in a complete market; Advances in Mathematical Finance, In Honor of Dilip B. Madan, pp , Jarrow R. Protter P. and Shimbo K. Asset price bubbles in incomplete markets; forthcoming, Jouini E. and Kallal H. Arbitrage in securities markets with short-sales constraints; Mathematical Finance, vol. 5, Issue 3, pp , Bubbles and futures contracts in markets with short-selling constraints 18/19

19 Current work and open questions Minimal entropy and minimal variance super-martingale measures. Specific models analysis: Stochastic volatility, models with jumps. More general conditions on Q to assure completeness. Liquidity aspects Air China Ltd: Shanghai Vs Hong Kong Bubbles and futures contracts in markets with short-selling constraints 19/19