Consistent price systems and face-lifting pricing under

transaction costs

Miklós Rásonyi

Computer & Automation Institute of H.A.S.

and

Vienna University of Technology

joint work with Paolo Guasoni and Walter Schachermayer

Advances in Mathematics of Finance, Bedlewo, 30th April- 5th May, 2007

Model description I

(Ω, F, (F_{t})_{t∈[0,T]}, P), F_{0} trivial Stock price: S adapted (càdlàg, locally bounded) Bank account: B ≡ 1 Contingent claim: random variable G E.g. G = g(S_{T}) European-type

Model description II

Trading strategies: cumulative transfers up to t from stock to bond and vice versa: L_{t}, K_{t}. Adapted, monotone nondecreasing and càdlàg. Proportional transaction costs: transferring 1 euro costs ε euro. Transaction cost is subtracted from the bank account. Stock position at time t: θ_{t} := −L_{t} + K_{t}.

Portfolios and trading

θ_{t} is an adapted, càdlàg, finite variation process. x: initial capital Suppose θ_{0} = θ_{T} = 0, then the portfolio value is

V_{T} (θ) = x + ∫_{0}^{T} θ_{u}dS_{u} − ε ∫_{0}^{T} S_{u}dVar_{u}(θ).

Define V_{t}(θ) := V_{T} (θ1_{(0,t)} ).

Absence of arbitrage I

Admissibility: V_{t}(θ) ≥ −M(1 + S_{t}), t ∈ [0, T] for some M > 0. In frictionless markets: absence of arbitrage (free lunches) ⟺ Q ∼ P s.t. S is a (local) Q-martingale

Price systems I

Analogue of equivalent martingale densities: consistent price systems. (Jouini, Kallal, Kabanov, Stricker, Rásonyi, Schachermayer) In the present one-dimensional case: (S̃_{t}, Q) is a CPS if

1. Q ∼ P

2. (1 − ε)S_{t} ≤ S̃_{t} ≤ (1 + ε)S_{t}, t ∈ [0, T]

3. S̃ is a Q-martingale

Price systems II

Family of CPSs: Z Strictly consistent price system: inequalities are strict (Z^{s})

Absence of arbitrage II

Arbitrage: θ s.t. V_{T} (θ) ≥ 0 a.s. and P (V_{T} (θ) > 0) > 0. Proposition. Z ≠ ∅ implies absence of arbitrage. In discrete time models converse is also true (Grigoriev). Multidimensional extension (i.e. several assets) fails. More involved notion of absence of arbitrage is needed. Continuous-time models: a lot to be done.

Hedging contingent claims

Theorem. If Z^{s} is large enough then there exists an admissible θ such that G ≤ V_{T} (θ) with initial endowment x iff

x ≥ sup_{(S̃,Q)∈Z} E_{Q}G.

(Cvitanič, Karatzas, Kabanov, Last, Stricker, Campi, Schachermayer)

Model classes

In which classes of models do such theorems apply ? Sufficient condition: existence of equivalent martingale measures. OK, but how to go further ? New fancy models for markets with transaction costs (which perhaps admit arbitrage if there is no friction) ?

Conditional full support

S: process with continuous positive trajectories C_{z}^{+} [u, v]: the family of positive continuous functions f on [u, v] starting at f(u) = z Definition. S has full support if supp P_{S} = C_{S_{0}}^{+}[0, T]. Definition. S has conditional full support if, for all t ∈ [0, T],

supp P (S|_{[t,T]}|F_{t}) = C_{S_{t}}^{+}[t, T] a.s.

Examples

Diffusion processes. (Stroock-Varadhan theorem) Processes with smooth trajectories. (E.g. S_{t} = exp {∫_{0}^{t} W_{u}du}.) Geometric fractional Brownian motion: S_{t} = exp (B_{t}).

Main result I

Predecessor: Guasoni, 2005, MF Theorem. If S has conditional full support then there exist consistent price systems under ε-transaction costs, for each ε > 0. Multidimensional counterpart: also true.

Continued

Theorem. If S has conditional full support on C_{0}[0, T] (i.e. on the set of all continuous functions on [0, T] starting from 0) then there is Q ∼ P and a Q-martingale M such that for all t ∈ [0, T]

S_{t} − ε ≤ M_{t} ≤ S_{t} + ε.

Face-lifting pricing

Clark, Davis, Levental-Skorokhod and Soner-Shreve-Cvitanič: dS_{t} = μS_{t}dt + σS_{t}dW_{t} Theorem. Let ε > 0 be an arbitrary level of transaction costs. If g(x) = (x − K)^{+} and G = g(S_{T}) then the superreplication price of G is S_{0}. Concave envelope of g: ĝ In other terms: superreplication price of g(S_{T}) is ĝ(S_{0}).

Main result II

Theorem. Let ε > 0 be an arbitrary level of transaction costs. If S has conditional full support and g is lower semicontinuous and bounded from below then the superreplication price of g(S_{T}) is at least ĝ(S_{0}).

European call: same conclusion as above, but for a much wider class of models.

Future

Exploiting the duality theory developed. Pricing techniques extended to previously untractable processes. Thank you for your kind attention.