The maximum maximum of a martingale with given nn marginals

We obtain bounds on the distribution of the maximum of a martingale with fixed marginals at finitely many intermediate times. The bounds are sharp and attained by a solution to $n$-marginal Skorokhod embedding problem in Ob{\l}\'oj and Spoida [An iterated Az\'ema-Yor type embedding for finitely many marginals (2013) Preprint]. It follows that their embedding maximizes the maximum among all other embeddings. Our motivating problem is superhedging lookback options under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We derive a pathwise inequality which induces the cheapest superhedging value, which extends the two-marginals pathwise inequality of Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558-578]. This inequality, proved by elementary arguments, is derived by following the stochastic control approach of Galichon, Henry-Labord\`ere and Touzi [Ann. Appl. Probab. 24 (2014) 312-336].Comment: Published at http://dx.doi.org/10.1214/14-AAP1084 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Robust pricing and hedging beyond one marginal

The robust pricing and hedging approach in Mathematical Finance, pioneered by Hobson (1998), makes statements about non-traded derivative contracts by imposing very little assumptions about the underlying financial model but directly using information contained in traded options, typically call or put option prices. These prices are informative about marginal distributions of the asset. Mathematically, the theory of Skorokhod embeddings provides one possibility to approach robust problems. In this thesis we consider mostly robust pricing and hedging problems of Lookback options (options written on the terminal maximum of an asset) and Convex Vanilla Options (options written on the terminal value of an asset) and extend the analysis which is predominately found in the literature on robust problems by two features: Firstly, options with multiple maturities are available for trading (mathematically this corresponds to multiple marginal constraints) and secondly, restrictions on the total realized variance of asset trajectories are imposed. Probabilistically, in both cases, we develop new optimal solutions to the Skorokhod embedding problem. More precisely, in Part I we start by constructing an iterated Azema-Yor type embedding (a solution to the n-marginal Skorokhod embedding problem, see Chapter 2. Subsequently, its implications are presented in Chapter 3. From a Mathematical Finance perspective we obtain explicitly the optimal superhedging strategy for Barrier/Lookback options. From a probability theory perspective, we find the maximum maximum of a martingale which is constrained by finitely many intermediate marginal laws. Further, as a by-product, we discover a new class of martingale inequalities for the terminal maximum of a cadlag submartingale, see Chapter 4. These inequalities enable us to re-derive the sharp versions of Doob's inequalities. In Chapter 5 a different problem is solved. Motivated by the fact that in some markets both Vanilla and Barrier options with multiple maturities are traded, we characterize the set of market models in this case. In Part II we incorporate the restriction that the total realized variance of every asset trajectory is bounded by a constant. This has been previously suggested by Mykland (2000). We further assume that finitely many put options with one fixed maturity are traded. After introducing the general framework in Chapter 6, we analyse the associated robust pricing and hedging problem for convex Vanilla and Lookback options in Chapters 7 and 8. Robust pricing is achieved through construction of appropriate Root solutions to the Skorokhod embedding problem. Robust hedging and pathwise duality are obtained by a careful development of dynamic pathwise superhedging strategies. Further, we characterize existence of market models with a suitable notion of arbitrage.</p

Robust pricing and hedging beyond one marginal

The robust pricing and hedging approach in Mathematical Finance, pioneered by Hobson (1998), makes statements about non-traded derivative contracts by imposing very little assumptions about the underlying financial model but directly using information contained in traded options, typically call or put option prices. These prices are informative about marginal distributions of the asset. Mathematically, the theory of Skorokhod embeddings provides one possibility to approach robust problems. In this thesis we consider mostly robust pricing and hedging problems of Lookback options (options written on the terminal maximum of an asset) and Convex Vanilla Options (options written on the terminal value of an asset) and extend the analysis which is predominately found in the literature on robust problems by two features: Firstly, options with multiple maturities are available for trading (mathematically this corresponds to multiple marginal constraints) and secondly, restrictions on the total realized variance of asset trajectories are imposed. Probabilistically, in both cases, we develop new optimal solutions to the Skorokhod embedding problem. More precisely, in Part I we start by constructing an iterated Azema-Yor type embedding (a solution to the n-marginal Skorokhod embedding problem, see Chapter 2. Subsequently, its implications are presented in Chapter 3. From a Mathematical Finance perspective we obtain explicitly the optimal superhedging strategy for Barrier/Lookback options. From a probability theory perspective, we find the maximum maximum of a martingale which is constrained by finitely many intermediate marginal laws. Further, as a by-product, we discover a new class of martingale inequalities for the terminal maximum of a cadlag submartingale, see Chapter 4. These inequalities enable us to re-derive the sharp versions of Doob's inequalities. In Chapter 5 a different problem is solved. Motivated by the fact that in some markets both Vanilla and Barrier options with multiple maturities are traded, we characterize the set of market models in this case. In Part II we incorporate the restriction that the total realized variance of every asset trajectory is bounded by a constant. This has been previously suggested by Mykland (2000). We further assume that finitely many put options with one fixed maturity are traded. After introducing the general framework in Chapter 6, we analyse the associated robust pricing and hedging problem for convex Vanilla and Lookback options in Chapters 7 and 8. Robust pricing is achieved through construction of appropriate Root solutions to the Skorokhod embedding problem. Robust hedging and pathwise duality are obtained by a careful development of dynamic pathwise superhedging strategies. Further, we characterize existence of market models with a suitable notion of arbitrage.This thesis is not currently available in OR

Maximum maximum of martingales given marginals

We consider the problem of superhedging under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We present a general duality result which converts this problem into a min-max calculus of variations problem where the Lagrange multipliers correspond to the static part of the hedge. Following Galichon, Henry-Labordére and Touzi \cite{ght}, we apply stochastic control methods to solve it explicitly for Lookback options with a non-decreasing payoff function. The first step of our solution recovers the extended optimal properties of the Azéma-Yor solution of the Skorokhod embedding problem obtained by Hobson and Klimmek \cite{hobson-klimmek} (under slightly different conditions). The two marginal case corresponds to the work of Brown, Hobson and Rogers \cite{brownhobsonrogers}. The robust superhedging cost is complemented by (simple) dynamic trading and leads to a class of semi-static trading strategies. The superhedging property then reduces to a functional inequality which we verify independently. The optimality follows from existence of a model which achieves equality which is obtained in Ob\lój and Spoida \cite{OblSp}

Blocking retinal chloride co-transporters KCC2 and NKCC: impact on direction selective ON and OFF responses in the rat's nucleus of the optic tract.

In the present study we investigated in vivo the effects of pharmacological manipulation of retinal processing on the response properties of direction selective retinal slip cells in the nucleus of the optic tract and dorsal terminal nucleus (NOT-DTN), the key visuomotor interface in the pathway underlying the optokinetic reflex. Employing a moving visual stimulus consisting of either a large dark or light edge we could differentiate direction selective ON and OFF responses in retinal slip cells. To disclose the origin of the retinal slip cells' unexpected OFF response we selectively blocked the retinal ON channels and inactivated the visual cortex by cooling. Cortical cooling had no effect on the direction selectivity of the ON or the OFF response in NOT-DTN retinal slip cells. Blockade of the retinal ON channel with APB led to a loss of the ON and, to a lesser degree, of the OFF response and a reduction in direction selectivity. Subsequent blocking of GABA receptors in the retina with picrotoxin unmasked a vigorous albeit direction unselective OFF response in the NOT-DTN. Disturbing the retinal chloride homeostasis by intraocular injections of bumetanide or furosemide led to a loss of direction selectivity in both the NOT-DTN's ON and the OFF response due to a reduced response in the neuron's preferred direction under bumetanide as well as under furosemide and a slightly increased response in the null direction under bumetanide. Our results indicate that the direction specificity of retinal slip cells in the NOT-DTN of the rat strongly depends on direction selective retinal input which depends on intraretinal chloride homeostasis. On top of the well established input from ON center direction selective ganglion cells we could demonstrate an equally effective input from the retinal OFF system to the NOT-DTN

Effects of 2-amino-4-phosphonobutyrat (40–80 µM APB) on neuronal activity of single units.

<p>Responses of 3 units (mean and standard deviation of 10 measurements) to flashed and moving ON (A) and OFF (B) visual stimuli before (control) and after intravitreous drug injection (APB; intravitreous concentration 40–80 µM). Ordinate: neuronal activity in spikes per second, abscissa: experimental conditions. APB blocks responses to all ON stimuli but spares the OFF flash response and the responses to the moving dark edge, albeit moderately decreased. Grey boxes: flash responses; black boxes: responses to moving edges in preferred direction; white boxes: responses in non-preferred direction.</p

Schematic demonstration of the visual stimuli used in the present study.

<p>A: stationary flash stimulus, duration of the presentation of the bright (ON) and the dark (OFF) monitor lasted for 1s each. B: moving ON stimulus. A light edge moved across the dark monitor until it was fully bright in the four cardinal directions. C: moving OFF stimulus. A dark edge moved across the bright monitor until it was fully dark in the four cardinal directions (for further description see methods).</p

Effects of intravitreal bumetanide injections on the responses of NOT-DTN neurons to moving light (ON) and dark (OFF) edges.

<p>Comparison of the activity during ON (A) and OFF (B) moving edge stimulation in preferred (PD, grey bars) and non-preferred (NPD, white bars) directions prior to (control) and after intravitreous injection of the drug (bumetanide). Horizontal lines indicate the median, boxes the 25–75%, whiskers the 10–90%, and black dots the 5–95 percentiles of the non-parametric statistical comparison. Bumetanide reduces responses in preferred direction and enhances responses in the non preferred direction, especially in the responses to the OFF stimulus.</p

NOT-DTN neurons respond to flashed as well as moving ON- and OFF visual stimuli.

<p>A: Response to flash stimulation, light onset at 500 ms, light offset at 1500 ms. This neuron exhibits clear phasic responses to both ON and OFF stimulation, the tonic response is low during ON stimulation. Spontaneous activity is represented by the first 500 ms of the PSTH. Ordinate: activity in spikes per second (spikes/s), abscissa: time in milliseconds (ms). B: direction selective response to a moving light edge (ON). Movement onset at 1000 ms. PD: preferred direction, NPD: non-preferred direction (opposite to PD). C: direction selective response to a moving dark edge (OFF).</p