Consistency of option prices under bid-ask spreads

Consistency of option prices under bid-ask spreads Stefan Gerhold TU Wien Joint work with I. Cetin Gülüm MFO, Feb 2017 (TU Wien) MFO, Feb 2017 1 / 32

Introduction The consistency problem Overview Consistency problem: Do given call prices allow for arbitrage? Strassen s theorem and some new extensions Application to the consistency problem under bid-ask spreads (TU Wien) MFO, Feb 2017 2 / 32

Introduction The consistency problem The consistency problem Given a finite set of call prices Is there a model that generates them? Which conditions are needed? Carr, Madan (2005): non-negative price of calendar spreads, butterfly spreads Davis, Hobson (2007): model-free and model-independent arbitrage Cousot (2007): bid-ask spread for option prices We: bid-ask spread for options and the underlying (TU Wien) MFO, Feb 2017 3 / 32

Introduction The consistency problem Data (frictionless case) Positive deterministic bank account (B(t)) t T, B(0) = 1 (In this talk: usually B 1) Strikes 0 < K t,1 < K t,2 < < K t,nt, t T Corresponding call option prices (at time zero) r t,i 0, Price of the underlying S 0 > 0 (TU Wien) MFO, Feb 2017 4 / 32

Introduction The consistency problem Frictionless case For each maturity t the linear interpolation L t of the points (K i, r t,i ) has to be convex, decreasing and all slopes of L t have to be in [ 1, 0]. Intuition: for every random variable S t the function K E[(S t K) + ] has these properties. prices 0 2 4 6 8 Slopes in [ 1, 0] Stockprice Optionprices Lt 0 5 10 15 strike (TU Wien) MFO, Feb 2017 5 / 32

Introduction The consistency problem Frictionless case: intertemporal conditions For all strikes K i we have that r t,i r t+1,i. Intuition: for every martingale S = (S t ) t {0,...,T } the function t E[(S t K) + ] is increasing by Jensen s inequality. prices 0 2 4 6 8 Stockprice Maturity 1 Maturity 2 Maturity 3 Lt 0 5 10 15 (TU Wien) MFO, Feb 2017 6 / 32

Introduction The consistency problem Frictionless case: necessary and sufficient conditions For all maturities t 0 r t,i+1 r t,i K i+1 K i r t,i r t,i 1 K i K i 1 1, for i {1,..., N 1}, and r t,i = r t,i 1 implies r t,i = 0, for i {1,..., N}. Note that we set K 0 = 0 and r t,0 = S 0 for all t {1,..., T 1}. For all strikes K i r t,i r t+1,i, t {1,..., T 1}. It is possible to state arbitrage strategies if any of these conditions fails. (TU Wien) MFO, Feb 2017 7 / 32

Introduction The consistency problem Frictionless case Main tool for the proof: Strassen s theorem. Let µ 1 and µ 2 be two probability measures on R with finite mean (µ 1, µ 2 M). Then µ 1 is smaller in convex order than µ 2 (µ 1 c µ 2 ) if for all convex functions φ R R. R φ(x) dµ 1 (x) R φ(x) dµ 2 (x), It suffices to consider functions (x K) +, K > 0 Strassen s Theorem, 1965 Let (µ n ) n N be a sequence in M. Then there exists a martingale (M n ) n N such that M n µ n if and only if µ s c µ t for all s t. (TU Wien) MFO, Feb 2017 8 / 32

Bid-ask spreads Data (with bid-ask spreads) Positive deterministic bank account (B(t)) t T, B(0) = 1 (In this talk: usually B 1) Strikes 0 < K t,1 < K t,2 < < K t,nt, t T Corresponding call option bid and ask prices (at time zero) Bid and ask of the underlying r t,i > 0, r t,i > 0 0 < S 0 S 0 (TU Wien) MFO, Feb 2017 9 / 32

Bid-ask spreads Bid-ask spread: How to define option payoff? Example: Call struck at e 100 bid-ask at maturity: S T = e 99, S T = e 101. Exercise? Yes! Get asset for e 1 less than in the market. No! Investing e 100 gives liquidation value e 99. Exercise cannot be decided without further assumptions. Typical solution in the literature: S t = (1 ε)s t, S t = (1 + ε)s t, mid-price S t triggers exercise decision, then physical settlement. (TU Wien) MFO, Feb 2017 10 / 32

Bid-ask spreads Option payoff under bid-ask spreads Assume that options are cash-settled, using a reference price S C t Payoff (S C T K)+ transferred to bank account We do not model a limit order book, and want to avoid ad-hoc definitions of St C Our approach: Any St C within the bid-ask spread will do. Fairly weak notion of consistency (TU Wien) MFO, Feb 2017 11 / 32

Bid-ask spreads Models with bid-ask spreads An arbitrage-free model consists of a filtered probability space (Ω, F, P) and four adapted non-negative processes S, S, S C, S. S C and S evolve in the bid-ask spread: S is a martingale S t S C t S t, S t S t S t S C is not a traded asset, hence S C does not have to be a martingale. (TU Wien) MFO, Feb 2017 12 / 32

Bid-ask spreads Models with bid-ask spreads Definition: The given prices are consistent with the absence of arbitrage, if there is an arbitrage-free model with E[(S C t K t,i ) + ] [r t,i, r t,i ], 1 i N t, t T. For each asset (underlying and options), we then have a martingale evolving in its bid-ask spread FTAP: Kabanov, Stricker (2001), Schachermayer (2004) (TU Wien) MFO, Feb 2017 13 / 32

Bid-ask spreads Consistency of call prices under bid-ask spreads If we allow models where the bid ask can get arbitrarily large than there are no intertemporal conditions. For all maturities t the following conditions are then necessary and sufficient for the existence of arbitrage-free models: 0 r t,i+1 r t,i K i+1 K i r t,i r t,i 1 K i K i 1 1, for i {2,..., N 1}, and r t,i = r t,i 1 implies r t,i = 0, for i {2,..., N}. Note that the initial bid and ask price of the underlying (S 0, S 0 ) do not appear! (TU Wien) MFO, Feb 2017 14 / 32

Bid-ask spreads Bounded Bid-Ask Spreads We focus on models where the bid-ask spread is bounded by a non-negative constant: S t S t ɛ. We then call the given prices ɛ-consistent. The option prices allow us to construct measures which correspond to the law of S C. Strassen s theorem is not applicable anymore since S C does not have to be a martingale. But, S C has to be close to a martingale. (TU Wien) MFO, Feb 2017 15 / 32

Variants of Strassen s Theorem Digression: Extending Strassen s theorem Let d be a metric on M and ɛ > 0. Formulation 1 Given a sequence (µ n ) n N in M, when does there exist a martingale (M n ) n N such that d(µ n, LM n ) ɛ, for all n N? Formulation 2 Given a sequence (µ n ) n N in M, when does there exist a sequence (ν n ) n N which is increasing in convex order (peacock) such that d(µ n, ν n ) ɛ, for all n N? (TU Wien) MFO, Feb 2017 16 / 32

Variants of Strassen s Theorem We solve this problem for different d: Infinity Wasserstein distance Modified Prokhorov distance Prokhorov distance, Lévy distance, modified Lévy distance, stop-loss distance (TU Wien) MFO, Feb 2017 17 / 32

Variants of Strassen s Theorem Definitions The modified Prokhorov distance with parameter p [0, 1] is the mapping d P p M M [0, ], defined by d P p (µ, ν) = inf{h > 0 ν(a) µ(a h ) + p, for all closed sets A R} where A h = {x S inf a A x a h}. The modified Prokhorov distance is not a metric in general The infinity Wasserstein distance W is defined by W (µ, ν) = d P 0 (µ, ν). (TU Wien) MFO, Feb 2017 18 / 32

Variants of Strassen s Theorem The infinity Wasserstein distance For µ M and x R we define call function resp. distribution function R µ (x) = R (y x) + µ(dy), F µ (x) = µ((, x]) W has the following representation in terms of call functions: W (µ, ν) = inf{h > 0 R µ(x h) R ν(x) R µ(x + h), x R} Moreover: = inf{h > 0 F µ (x h) F ν (x) F µ (x + h), x R} W (µ, ν) = inf X Y, where the inf is over all probability spaces and random pairs with marginals (µ, ν) (TU Wien) MFO, Feb 2017 19 / 32

Variants of Strassen s Theorem Minimal distance coupling Theorem (Strassen 1965, Dudley 1968) Given measures µ, ν on R, p [0, 1], and ɛ > 0 there exists a probability space (Ω, F, P) with random variables X µ and Y ν such that P( X Y > ɛ) p, if and only if d P p (µ, ν) ɛ. Application to consistency: consider models where P( S C t S t > ɛ) p. (TU Wien) MFO, Feb 2017 20 / 32

Variants of Strassen s Theorem Strassen s theorem for the modified Prokhorov distance Theorem Given a sequence (µ n ) n N in M, p (0, 1) and ɛ > 0 there always exists a peacock (ν n ) n N such that d P p (µ n, ν n ) ɛ, for all n N. (TU Wien) MFO, Feb 2017 21 / 32

Variants of Strassen s Theorem Strassen s theorem for W (p = 0), Part 1 Let B (µ, ɛ) be the closed ball wrt. W with center µ and radius ɛ. Let M m be the set of all probability measures on R with mean m R. Given ɛ > 0, a measure µ M and m R such that B (µ, ɛ) M m there exist unique measures S(µ), T (µ) B (µ, ɛ) M m such that S(µ) c ν c T (µ) for all ν B (µ, ɛ) M m. The call functions of S(µ) and T (µ) are given by R min µ (x; m, ɛ) = R S(µ) (x) = (m + R µ (x ɛ) (Eµ + ɛ)) R µ (x + ɛ), R max µ (x; m, ɛ) = R T (µ) (x) = conv(m + R µ ( + ɛ) (Eµ ɛ), R µ ( ɛ))(x (TU Wien) MFO, Feb 2017 22 / 32

Variants of Strassen s Theorem Strassen s theorem for W (p = 0), Part 2 Question Given a sequence (µ n ) n N in M and ɛ > 0 when does there exist a peacock (ν n ) n N such that W (µ n, ν n ) = d P 0 (µ, ν) ɛ, for all n N? Answer: if and only if I = [Eµ n ɛ, Eµ n + ɛ], n N and there exists m I such that for all N N, x 1,..., x N R, we have R min N µ 1 (x 1 ; m, ɛ)+ n=2 (R µn (x n +ɛσ n ) R µn (x n 1 +ɛσ n )) R max µ N+1 (x N ; m, ɛ), where σ n = sgn(x n 1 x n ). If ɛ = 0 this simplifies to R µ1 (x) R µ2 (x) R µn+1 (x).... (TU Wien) MFO, Feb 2017 23 / 32

Application of the new results Our results on the consistency problem under bid-ask spreads: overview Single maturity, spread bounded by ɛ: Necessary and sufficient conditions Multiple maturities, spread bounded by ɛ with probability 1 p: Necessary and sufficient conditions. Apply our Strassen-type thm for d P p Multiple maturities, spread bounded by ɛ: Necessary conditions Necessary and sufficient conditions under simplified assumptions. Apply our Strassen-type thm for W (TU Wien) MFO, Feb 2017 24 / 32

Application of the new results Necessary and Sufficient Conditions for single maturities The following conditions are necessary and sufficient for ɛ-consistency (S t S t ɛ): and 0 r t,i+1 r t,i K i+1 K i r t,i r t,i 1 K i K i 1 1, for i {2,..., N 1}, ( ) r t,i = r t,i 1 implies r t,i = 0, for i {2,..., N}. r t,2 r t,1 r t,1 S 0 K 2 K 1 K 1 ɛ and r t,1 S 0 K 1 + ɛ 1. (TU Wien) MFO, Feb 2017 25 / 32

Application of the new results Model-independent and weak arbitrage Model-independent arbitrage: Arbitrage strategy works for any model Weak arbitrage: For any model, there is an arbitrage strategy (depending on the null sets of the model). E.g.: Use a different strategy according to whether P(S T > K) = 0 or not Terminology from Davis and Hobson (2007) If the condition ( ) fails, then there is a weak arbitrage opportunity. If any of the other conditions is violated, then there is model-independent arbitrage. (TU Wien) MFO, Feb 2017 26 / 32

Application of the new results Application of our result on the Prokhorov distance Theorem Given a sequence (µ n ) n N in M, p (0, 1) and ɛ > 0 there always exists a peacock (ν n ) n N such that d P p (µ n, ν n ) ɛ, for all n N. Corollary If we allow models where P(S t S t > ɛ) p, for p (0, 1), then the following conditions are necessary and sufficient for the existence of arbitrage-free models: 0 r t,i+1 r t,i K i+1 K i r t,i r t,i 1 K i K i 1 1, for i {2,..., N t 1}, and r t,i = r t,i 1 implies r t,i = 0, for i {2,..., N t }. (TU Wien) MFO, Feb 2017 27 / 32

Application of the new results Corollary: proof idea Necessity: From our result on unbounded bid-ask spread Sufficiency: Get peacock from theorem. Yields processes S, S C with P( S t S C t ɛ) p. Define S t = S t S C t and S t = S t S C t. (TU Wien) MFO, Feb 2017 28 / 32

Application of the new results ɛ-consistency What about applying our main extension of Strassen s theorem, the one with W? Should be useful for constructing models with S t S t ɛ Necessary and sufficient conditions seem to be difficult to find (see next slide) We found necessary and sufficient conditions under simplified assumptions (TU Wien) MFO, Feb 2017 29 / 32

Necessary conditions Necessary Conditions for multiple maturities If we restrict ourselves to models where P(S t S t > ɛ) = 0 then we get the following intertemporal conditions: If K i + ɛ < K j ɛσ s < K l + ɛ, s t and s u then the following conditions are necessary: where r CV s B (σ s, K j ) r t,i (K j ɛσ s ) (K i + ɛ) r u,l r CV B r CV s B (σ s, K j ) r t,i 0, and (K j ɛσ s ) (K i + ɛ) r u,l r CV B s (σ s, K j ) K l + ɛ (K s ɛσ s ) 1 r CV B s s (σ s, K j ) K l + ɛ (K s ɛσ s ), s = r 1,j1 + (r t,jt r t,it 1 ) + 2ɛ1 {σ1 = 1}. t=2 (TU Wien) MFO, Feb 2017 30 / 32

Conclusion Conclusion If there are no transaction costs on the underlying then necessary sufficient conditions can be derived from Strassen s theorem (Carr and Madan 2005, Davis and Hobson 2007). If there is no bound on the bid-ask spread on the underlying, then there are no intertemporal conditions, and there is no relation between option prices and price of the underlying. If the bid-ask spread satisfies some boundedness conditions, we can apply our generalizations of Strassen s theorem to derive consistency conditions. (TU Wien) MFO, Feb 2017 31 / 32

References References Carr, Madan: A note on sufficient conditions for no arbitrage. Finance Research Letters 2005 Davis, Hobson: The range of traded option prices. Math. Finance 2007 Gerhold, Gülüm: A variant of Strassen s theorem: Existence of martingales within a prescribed distance. Preprint Gerhold, Gülüm: Consistency of option prices under bid-ask spreads. Preprint (TU Wien) MFO, Feb 2017 32 / 32