MARTINGALES AND LOCAL MARTINGALES

Transcription

1 MARINGALES AND LOCAL MARINGALES If S t is a (discounted) securtity, the discounted P/L V t = need not be a martingale. t θ u ds u Can V t be a valid P/L? When? Winter 25 1 Per A. Mykland

2 ARBIRAGE WIH SOCHASIC INEGRALS: AN EXAMPLE Stock price: ds t = σs t dw t rading strategy: θ t = 1 σs t t P/L: V t = = t t 1 ds u σs u u 1 dw u u Determination of F : ime change: set V t = B F (t) [B, B] F (t) = [V, V ] t = t ( Choose: F (t) = log ( ) 1 du = log u t t ), so [B, B] t = t Since V t martingale on [, ), and F () =, F ( ) = : (B t ) t< is standard Brownian motion (Levy) Winter 25 2 Per A. Mykland

3 Investment scheme: ( P/L: V t = B F (t) with F (t) = log ) t We seek arbitrage profit of α > dollars Stopping rule: τ = min{t : V t α} τ = F ( 1) (τ ) where τ = min{t : B t α} Since B is Brownian motion: P (τ < ) = 1 Modified P/L: And so P (τ < ) = 1 V t τ = t θ u I {u τ} ds u HIS RADING SRAEGY EARNS PROFI α WIH PROBABILIY 1 BY IME WHA IS WRONG? Relates to absence of credit constraint, as in 39 Lecture 5, p Winter 25 3 Per A. Mykland

4 SANDARD WAY OF COPING WIH HE PROBLEM 1. HE CONCEP OF LOCAL MARINGALE (LOC MG): M t is a local martingale for t [, ] if there is a sequence τ 1 τ 2... of stopping times so that P (τ n = ) 1 as n, and so that M t τn is a martingale for each n 2. INVARIANCE OF LOC MGness UNDER SOCHAS- IC INEGRAION: If M t is a continuous loc MG, if θ t ia adapted, if θ2 t d[m, M] t < with probability 1, then t θ udm u is a continuous loc MG (his is the real theorem about stochastic integrals being martingales) 3. HE IMPAC OF A CREDI CONSRAIN: If M t is a loc MG, and M t K for all t with probability 1, then M t is a supermartingale RANSLAION: with a credit constraint, you cannot earn arbitrage profit, but you can have arbitrage loss Winter 25 4 Per A. Mykland

5 EXPLANAION OF RANSLAION If M t is a supermartingale, then by Doob-Meyer: M t = N t D t = martingale - dividend he dividend, if any, is the arbitrage loss Arbitrage loss (failure of individual trader) is more palatable assumption than arbitrage profit (failure of market) RELAIONSHIP O OUR EXAMPLE V τ t cannot be bounded below. Strategy requires infinite credit PROOF OF IEM 3: Use Fatou s lemma: if X n X a.s., and X n K a.s. for all n, then lim inf n E(X n G) E(X G) Application here: s < t, X n = M t τn, X = M t, G = F s : M s τn = E(M t τn F s ) (optional stopping) and so, as n : Q.E.D. M s = lim M s τn = lim E(M t τn F s ) E(lim M t τn F s ) = E(M t F s ) Winter 25 5 Per A. Mykland

6 QUANILE HEDGING (based on paper by Föllmer and Leukert, Sect 1-3) SEING: SOCK PRICE PROCESS X t, PAYOFF H A IME, COMPLEE MARKE (P exists and is unique), r = Hedging strategy: V t = V + t ξ sdx s ; admissible: V t Solvency set: A = {V H} HE PROBLEM: GAMBLING INELLIGENLY Initial capital: Ṽ < E (H). PROBLEM: find ξ to maximize P (A) (= the probability of staying out of jail?) subject to starting value V Ṽ ASPEC OF SOLUION Enough to hedge HI A. If you spend money to hedge payoffs outside A, you are misallocating resources if the goal is P (A) = max. If instead constraint V t K, then you wish to end with V = K outside A. You seek to hedge HI A KI A c (A c is the complement of A) Winter 25 6 Per A. Mykland

7 Formalization as proposition (2.8): Suppose à solves P (A) = max subject to E [HI A ] Ṽ. Let ξ be the perfect hedge for payoff HI A : HIà = E [HIÃ] + ξ s dx s (2.11) hen Ṽt = Ṽ + t ξ s dx s solves the original optimization problem, and {Ṽ H} = à up to set of proba zero. Proof: (1) Let V t be any admissible strategy, V Ṽ. A is success set for this V. hen P (A) P (Ã) (2.14) Subproof: V HI A since V. Since V t is supermg: Ṽ V E [V ] E [HI A ] Hence A satisfies the constraint in the theorem. Hence (2.14) (2) Let V t = V + t ξ s dx s for Ṽ V E [HIÃ] V t E [HI A ] + so V t is admissible ξ s dx s = E [HI A F t ] Winter 25 7 Per A. Mykland

8 V t is optimal From (2.11), and since V E [HIÃ]: HIà = E [HIÃ] + ξ s dx s V + ξ s dx s = V If we set A = {V H}, then: à A a.s. But from (2.14): P (A) P (Ã), so à = A a.s. In particular: Ṽ t is optimal. QED. Winter 25 8 Per A. Mykland

9 NEX PROBLEM: HOW O SOLVE HE PROBLEM FROM PROPOSIION (2.8)? P (A) = max, subject to E [HI A ] Ṽ Define Q : dq = H E [H] = H H Rewrite problem: P (A) = max, subject to Q (A) α where α = Ṽ /H. Note: Q << P, P, but maybe not reverse SOLUION O HIS PROBLEM: HE NEYMAN-PEARSON LEMMA { } { } Ã = > cdq = cdq B where c is constant and B is outcome of coin toss independent of everything else, so that { } { } α = Q (Ã) = Q > cdq +Q = cdq Q (B) B is only relevant if Q { = c dq } Winter 25 9 Per A. Mykland

10 c = min SPECIFIC DEFINIION { { }} c : Q cdq α Q (B) is defined to reach equality α = Q (Ã) Further mathematical formulation yields P (B) = P (B) = Q (B) ALERNAIVE DESCRIPION dq = H E [H] = H H and so à = { } > c H E [H] { } = c H E [H] B Winter 25 1 Per A. Mykland

11 HE DUAL PROBLEM Find V = min, subject to P (V H) 1 ɛ his is the same as: Find E [HI A ] = min, subject to P (A) 1 ɛ (We assume no liability if you actually use this...) Solution set à has the same Neyman-Pearson form as for the primal problem, but rewrite, with b = 1/c: { } { } à = > cdq = cdq B { } { } dq dq = > b = b B (Recall: Q << P ) Determine b and P (B) from P (Ã) = 1 ɛ When calculating Ṽ = E [HIÃ], use P (B) = P (B) Winter Per A. Mykland

12 HE BLACK SCHOLES MODEL and, as usual, dx t = mx t dt + σx t dw t log(x ) = log(x ) + σw + (m 1 2 σ) Risk neutral measure: = exp { m σ W 1 2 ( m σ ) 2 } ( = const X m/σ2 { 1 const = x m/σ2 exp 2 }) m (m σ) σ2 Winter Per A. Mykland

13 Since à = CALL PAYOFF: H = (X K) + { > c P can take (for some λ) = const X m/σ H E [H] } { } = c H E [H] à = {X m/σ2 { } = c H E [H] 2 = } > λh B PRIMAL PROBLEM: à maximizes success probability subject to initial capital Ṽ if λ is such that E [HIÃ] = Ṽ Winter Per A. Mykland

14 SOLUION à = {X m/σ2 > λ(x K) +} and E [(X K) + IÃ] = Ṽ If X K, then X à CASE (i): m σ 2 (the other case is similar) If X > K, then x x m/σ2 λ(x K) is decreasing, so x m/σ2 > λ(x K) is the same as x < c for some c Obviously, c K, and so à = {X < c} = {W < b} Since log(x ) = log(x ) + σw 1 2 σ2 and by c = x exp{σb 1 2 σ2 } Since W = W + m σ : Success probability: P (Ã) = Φ( b m σ ) Winter Per A. Mykland

15 Modified option: Initial capital (X K) + IÃ = (X K) + I {X >c} = (X c) + + (c K)I {X >c} = (X c) + + (c K)I {W >b} And so Ṽ = BS-price of call with strike c + (c K) BS-price of binary option = BS-price of call with strike c + (c K)Φ( b ) Winter Per A. Mykland

16 PRIMAL AND DUAL PROBLEM PRIMAL PROBLEM: Start with Ṽ, find c or b, then compute success probability P (Ã) DUAL PROBLEM: Start with P (Ã) Find b from P (Ã) = Φ( b m σ ) Find c = x exp{σb 1 2 σ2 } Compute Ṽ = BS-price of call with strike c+(c K)Φ( b ) Example in paper (bottom of p. 261): cost reduction of 41 % for failure probability of 5 % Quite possibly illegal Risk management by interval constraints can be subverted: For any constraint on P (Ã) (=, say, 95%), c is given, and Ṽ as K c Winter Per A. Mykland