We prove a version of the Fundamental Theorem of Asset Pricing, which applies to Kabanov's modeling of foreign exchange markets under transaction costs. The financial market is described by a d×d ...matrix‐valued stochastic process (Πt)Tt=0 specifying the mutual bid and ask prices between d assets. We introduce the notion of “robust no arbitrage,” which is a version of the no‐arbitrage concept, robust with respect to small changes of the bid‐ask spreads of (Πt)Tt=0. The main theorem states that the bid‐ask process (Πt)Tt=0 satisfies the robust no‐arbitrage condition iff it admits a strictly consistent pricing system. This result extends the theorems of Harrison‐Pliska and Kabanov‐Stricker pertaining to the case of finite Ω, as well as the theorem of Dalang, Morton, and Willinger and Kabanov, Rásonyi, and Stricker, pertaining to the case of general Ω. An example of a 5 × 5‐dimensional process (Πt)2t=0 shows that, in this theorem, the robust no‐arbitrage condition cannot be replaced by the so‐called strict no‐arbitrage condition, thus answering negatively a question raised by Kabanov, Rásonyi, and Stricker.
Famously, mathematical finance was started by Bachelier in his 1900 PhD thesis where – among many other achievements – he also provided a formal derivation of the Kolmogorov forward equation. This ...also forms the basis for Dupire’s (again formal) solution to the problem of finding an arbitrage-free model calibrated to a given volatility surface. The latter result has rigorous counterparts in the theorems of Kellerer and Lowther. In this survey article, we revisit these hallmarks of stochastic finance, highlighting the role played by some optimal transport results in this context.
Understanding the structure of financial markets deals with suitably determining the functional relation between financial variables. In this respect, important variables are the trading activity, ...defined here as the number of trades
N
, the traded volume
V
, the asset price
P
, the squared volatility
σ
2
, the bid-ask spread
S
and the cost of trading
C
. Different reasonings result in simple proportionality relations (“scaling laws”) between these variables. A basic proportionality is established between the trading activity and the squared volatility, i.e.,
N
∼
σ
2
. More sophisticated relations are the so called 3/2-law
N
3
/
2
∼
σ
P
V
/
C
and the intriguing scaling
N
∼
(
σ
P
/
S
)
2
. We prove that these “scaling laws” are the only possible relations for considered sets of variables by means of a well-known argument from physics: dimensional analysis. Moreover, we provide empirical evidence based on data from the NASDAQ stock exchange showing that the sophisticated relations hold with a certain degree of universality. Finally, we discuss the time scaling of the volatility
σ
, which turns out to be more subtle than one might naively expect.
Affine processes are regular Keller-Ressel, Martin; Schachermayer, Walter; Teichmann, Josef
Probability theory and related fields,
12/2011, Volume:
151, Issue:
3-4
Journal Article
Peer reviewed
Open access
We show that stochastically continuous, time-homogeneous affine processes on the canonical state space
are always regular. In the paper of Duffie et al. (Ann Appl Probab 13(3):984–1053, 2003) ...regularity was used as a crucial basic assumption. It was left open whether this regularity condition is automatically satisfied for stochastically continuous affine processes. We now show that the regularity assumption is indeed superfluous, since regularity follows from stochastic continuity and the exponentially affine form of the characteristic function. For the proof we combine classic results on the differentiability of transformation semigroups with the method of the moving frame which has been recently found to be useful in the theory of SPDEs.
In a market with one safe and one risky asset, an investor with a long horizon, constant investment opportunities and constant relative risk aversion trades with small proportional transaction costs. ...We derive explicit formulas for the optimal investment policy, its implied welfare, liquidity premium, and trading volume. At the first order, the liquidity premium equals the spread, times share turnover, times a universal constant. The results are robust to consumption and finite horizons. We exploit the equivalence of the transaction cost market to another frictionless market, with a shadow risky asset, in which investment opportunities are stochastic. The shadow price is also found explicitly.
We prove that c-cyclically monotone transport plans \pi optimize the Monge-Kantorovich transportation problem under an additional measurability condition. This measurability condition is always ...satisfied for finitely valued, lower semi-continuous cost functions. In particular, this yields a positive answer to Problem 2.25 in C. Villani's book. We emphasize that we do not need any regularity conditions as were imposed in the previous literature.
Duality for Borel measurable cost functions BEIGLBOCK, Mathias; SCHACHERMAYER, Walter
Transactions of the American Mathematical Society,
08/2011, Volume:
363, Issue:
8
Journal Article
Peer reviewed
Open access
We consider the Monge-Kantorovich transport problem in an abstract measure theoretic setting. Our main result states that duality holds if c: X × Y → 0, ∞) is an arbitrary Borel measurable cost ...function on the product of Polish spaces X, Y. In the course of the proof we show how to relate a non-optimal transport plan to the optimal transport costs via a "subsidy" function and how to identify the dual optimizer. We also provide some examples showing the limitations of the duality relations.
We consider the problem of portfolio optimisation with general càdlàg price processes in the presence of proportional transaction costs. In this context, we develop a general duality theory. In ...particular, we prove the existence of a dual optimiser as well as a shadow price process in an appropriate generalised sense. This shadow price is defined by means of a "sandwiched" process consisting of a predictable and an optional strong supermartingale, and pertains to all strategies that remain solvent under transaction costs. We provide examples showing that, in the general setting we study, the shadow price processes have to be of such a generalised form.
A version of the fundamental theorem of asset pricing is proved for continuous asset prices with small proportional transaction costs. Equivalence is established between: (a) the absence of arbitrage ...with general strategies for arbitrarily small transaction costs
, (b) the absence of free lunches with bounded risk for arbitrarily small transaction costs
, and (c) the existence of
-consistent price systems—the analogue of martingale measures under transaction costs—for arbitrarily small
. The proof proceeds through an explicit construction, as opposed to the usual separation arguments. The paper concludes comparing numéraire-free and numéraire-based notions of admissibility, and the corresponding martingale and local martingale properties for consistent price systems.
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