Superhedging Strategy Research Articles

Neural network approximation for superhedging prices

AbstractThis article examines neural network‐based approximations for the superhedging price process of a contingent claim in a discrete time market model. First we prove that the α‐quantile hedging price converges to the superhedging price at time 0 for α tending to 1, and show that the α‐quantile hedging price can be approximated by a neural network‐based price. This provides a neural network‐based approximation for the superhedging price at time 0 and also the superhedging strategy up to maturity. To obtain the superhedging price process for , by using the Doob decomposition, it is sufficient to determine the process of consumption. We show that it can be approximated by the essential supremum over a set of neural networks. Finally, we present numerical results.

Robust hedging of options on a leveraged exchange traded fund

A leveraged exchange traded fund (LETF) is an exchange traded fund that uses financial derivatives to amplify the price changes of a basket of goods. In this paper, we consider the robust hedging of European options on a LETF, finding model-free bounds on the price of these options. To obtain an upper bound, we establish a new optimal solution to the Skorokhod embedding problem (SEP) using methods introduced in Beiglbock–Cox–Huesmann. This stopping time can be represented as the hitting time of some region by a Brownian motion, but unlike other solutions of, for example, Root, this region is not unique. Much of this paper is dedicated to characterising the choice of the embedding region that gives the required optimality property. Notably, this appears to be the first solution to the SEP where the solution is not uniquely characterised by its geometric structure, and an additional condition is needed on the stopping region to guarantee that it is the optimiser. An important part of determining the optimal region is identifying the correct form of the dual solution, which has a financial interpretation as a model-independent superhedging strategy.

Robust Valuation, Arbitrage Ambiguity and Profit & Loss Analysis

Model uncertainty is a type of inevitable financial risk. Mistakes on the choice of pricing model may cause great financial losses. In this paper we investigate financial markets with mean-volatility uncertainty. Models for stock market and option market with uncertain prior distributions are established by Peng’s G-stochastic calculus. On the hedging market, the upper price of an (exotic) option is derived following the Black–Scholes–Barenblatt equation. It is interesting that the corresponding Barenblatt equation does not depend on mean uncertainty of the underlying stocks. Appropriate definitions of arbitrage for super- and sub-hedging strategies are presented such that the super- and sub-hedging prices are reasonable. In particular, the condition of arbitrage for sub-hedging strategy fills the gap of the theory of arbitrage under model uncertainty. Finally we show that the term K of finite variance arising in the superhedging strategy is interpreted as the max Profit & Loss ( $$ \mathrm{P} \& \mathrm{L} $$ ) of shorting a delta-hedged option. The ask-bid spread is in fact an accumulation of the superhedging $$ \mathrm{P} \& \mathrm{L}$$ and the sub-hedging $$ \mathrm{P} \& \mathrm{L} $$ .

Partial super-hedging of derivatives with model risk

This paper studies a hedging problem in the market where an investor can observe a risky asset price process but he does not know true parameters of this process. The investor is assumed to hedge derivatives on the risky asset by a partial super-hedging strategy in order to cut the hedging cost. We show two partial differential equations which play an important role to solve the optimal hedging cost and an optimal strategy. Further we analyze the optimal hedging cost numerically by trinomial models.

Robust pricing and hedging around the globe

We consider the martingale optimal transport duality for cadlag processes with given initial and terminal laws. Strong duality and existence of dual optimizers (robust semistatic superhedging strategies) are proved for a class of payoffs that includes American, Asian, Bermudan and European options with intermediate maturity. We exhibit an optimal superhedging strategy for which the static part solves an auxiliary problem and the dynamic part is given explicitly in terms of the static part.

Robust Pricing and Hedging around the Globe

We consider the martingale optimal transport duality for cadlag processes with given initial and terminal laws. Strong duality and existence of dual optimizers (robust semi-static superhedging strategies) are proved for a class of payoffs that includes American, Asian, Bermudan, and European options with intermediate maturity. We exhibit an optimal superhedging strategy for which the static part solves an auxiliary problem and the dynamic part is given explicitly in terms of the static part. In the case of finitely supported marginal laws, solving for the static part reduces to a semi-infinite linear program.

Pricing and Hedging of European Plain Vanilla Options under Jump Uncertainty

This paper studies the pricing and hedging problem of European plain vanilla options in a modified Black–Scholes market. That is the price of the risky asset is allowed to jump, where the timing and the size of the jump is unknown (with no jump being possible as well). Using a superhedging approach, worst case pricing formulae, Greeks, and superhedging strategy for call and put options will be given in closed form (where the closed form is of the same level as the Black–Scholes solution) and will be discussed. Moreover, the worst case prices explain the volatility smile which can be observed in market data. Finally, the model is calibrated to market data.

Robust superhedging with jumps and diffusion

We establish a nondominated version of the optional decomposition theorem in a setting that includes jump processes with nonvanishing diffusion as well as general continuous processes. This result is used to derive a robust superhedging duality and the existence of an optimal superhedging strategy for general contingent claims. We illustrate the main results in the framework of nonlinear Lévy processes.

Open-End Knock-Outs on Bond Futures: Valuation, Properties and Estimation of Hidden Profit Drivers

This paper is the first on retail Structured Financial Products (SFPs) from the fixed-income area, namely open-end knock-outs (OEKOs) on government bond futures. We develop and apply a valuation algorithm for OEKOs to analyze the issuers' profit. On the basis of a simple superhedging strategy we disclose the profit drivers for the issuers. The crucial profit drivers, the 'margin factor' and the 'rollover fee', are set by the issuers and can be regarded directly as costs to the investors. But even disregarding these drivers there exists a 'hidden' interest gain. However, asymmetric jumps in the interest rate dynamics may lead to significant gap risks which can considerably lower the issuers' benefit, even below zero. Since the crucial profit drivers of OEKOs are not directly observable, we estimate them by analyzing an extensive data set of OEKO prices. Inter alia, depending on the issuer we find that the profit drivers can take values of up to 60% p.a. on average.

AN ALGORITHM FOR CALCULATING THE SET OF SUPERHEDGING PORTFOLIOS IN MARKETS WITH TRANSACTION COSTS

We study the explicit calculation of the set of superhedging portfolios of contingent claims in a discrete-time market model for d assets with proportional transaction costs. The set of superhedging portfolios can be obtained by a recursive construction involving set operations, going backward in the event tree. We reformulate the problem as a sequence of linear vector optimization problems and solve it by adapting known algorithms. The corresponding superhedging strategy can be obtained going forward in the tree. Examples are given involving multiple correlated assets and basket options. Furthermore, we relate existing algorithms for the calculation of the scalar superhedging price to the set-valued algorithm by a recent duality theory for vector optimization problems. The main contribution of the paper is to establish the connection to linear vector optimization, which allows to solve numerically multi-asset superhedging problems under transaction costs.

Superhedging and Dynamic Risk Measures under Volatility Uncertainty

We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a càdlàg nonlinear martingale which is also the value process of a superhedging problem. The superhedging strategy is obtained from a representation similar to the optional decomposition. Furthermore, we prove an optional sampling theorem for the nonlinear martingale and characterize it as the solution of a second order backward SDE. The uniqueness of dynamic extensions of static sublinear expectations is also studied.

Superhedging and Dynamic Risk Measures Under Volatility Uncertainty

We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a cadlag nonlinear martingale which is also the value process of a superhedging problem. The superhedging strategy is obtained from a representation similar to the optional decomposition. Furthermore, we prove an optional sampling theorem for the nonlinear martingale and characterize it as the solution of a second order backward SDE. The uniqueness of dynamic extensions of static sublinear expectations is also studied.

Static super-replicating strategies for a class of exotic options

In this paper, we investigate static super-replicating strategies for European-type call options written on a weighted sum of asset prices. This class of exotic options includes Asian options and basket options among others. We assume that there exists a market where the plain vanilla options on the different assets are traded and hence their prices can be observed in the market. Both the infinite market case (where prices of the plain vanilla options are available for all strikes) and the finite market case (where only a finite number of plain vanilla option prices are observed) are considered. We prove that the finite market case converges to the infinite market case when the number of observed plain vanilla option prices tends to infinity. We show how to construct a portfolio consisting of the plain vanilla options on the different assets, whose pay-off super-replicates the pay-off of the exotic option. As a consequence, the price of the super-replicating portfolio is an upper bound for the price of the exotic option. The super-hedging strategy is model-free in the sense that it is expressed in terms of the observed option prices on the individual assets, which can be e.g. dividend paying stocks with no explicit dividend process known. This paper is a generalization of the work of Simon et al. [Simon, S., Goovaerts, M., Dhaene, J., 2000. An easy computable upper bound for the price of an arithmetic Asian option. Insurance Math. Econom. 26 (2–3), 175–184] who considered this problem for Asian options in the infinite market case. Laurence and Wang [Laurence, P., Wang, T.H., 2004. What’s a basket worth? Risk Mag. 17, 73–77] and Hobson et al. [Hobson, D., Laurence, P., Wang, T.H., 2005. Static-arbitrage upper bounds for the prices of basket options. Quant. Fin. 5 (4), 329–342] considered this problem for basket options, in the infinite as well as in the finite market case. As opposed to Hobson et al. [Hobson, D., Laurence, P., Wang, T.H., 2005. Static-arbitrage upper bounds for the prices of basket options. Quant. Fin. 5 (4), 329–342] who use Lagrange optimization techniques, the proofs in this paper are based on the theory of integral stochastic orders and on the theory of comonotonic risks.

Convergence of Utility Indifference Prices to the Superreplication Price: the Whole Real Line Case

A discrete-time financial market model is considered with a sequence of investors whose preferences are described by concave strictly increasing functions defined on the whole real line. Under suitable conditions we prove that, whenever their absolute risk-aversion tends to infinity, the respective utility indifference prices of a given bounded contingent claim converge to the superreplication price. We also prove that there exists an accumulation point of the optimal strategies’ sequence which is a superhedging strategy.

Risk Averse Asymptotics and the Optional Decomposition

We consider the problem of maximizing expected utility for a general utility function on R when the agent becomes increasingly risk averse. The limiting strategy will be shown to be a special, unique superhedging strategy, the so-called balanced strategy. The connections to the optional decomposition and the class of minimal hedging strategies described in [D. O. Kramkov, Probab. Theory Related Fields, 105 (1996), pp. 459–479] are examined.

Pitfalls in static superhedging of barrier options

If the cost of carry is non-zero and only finitely many options are traded, static hedging of barrier options is in general impossible. Alternatively, one can set up a static superhedging strategy. We demonstrate that such a superhedge may perform poorly close to the maturity of the barrier option if the strikes of the options used to superhedge the barrier option are not carefully chosen. Model risk amplifies this effect.

Bounds for the price of a European-style Asian option in a binary tree model

Inspired by the ideas of Rogers and Shi [J. Appl. Prob. 32 (1995) 1077], Chalasani et al. [J. Comput. Finance 1(4) (1998) 11] derived accurate lower and upper bounds for the price of a European-style Asian option with continuous averaging over the full lifetime of the option, using a discrete-time binary tree model. In this paper, we consider arithmetic Asian options with discrete sampling and we generalize their method to the case of forward starting Asian options. In this case with daily time steps, the method of Chalasani et al. is still very accurate but the computation can take a very long time on a PC when the number of steps in the binomial tree is high. We derive analytical lower and upper bounds based on the approach of Kaas et al. [Insurance: Math. Econ. 27 (2000) 151] for bounds for stop-loss premiums of sums of dependent random variables, and by conditioning on the value of underlying asset at the exercise date. The comonotonic upper bound corresponds to an optimal superhedging strategy. By putting in less information than Chalasani et al. the bounds lose some accuracy but are still very good and they are easily computable and moreover the computation on a PC is fast. We illustrate our results by different numerical experiments and compare with bounds for the Black and Scholes model [J. Pol. Econ. 7 (1973) 637] found in another paper [Bounds for the price of discretely sampled arithmetic Asian options, Working paper, Ghent University, 2002]. We notice that the intervals of Chalasani et al. do not always lie within the Black and Scholes intervals. We have proved that our bounds converge to the corresponding bounds in the Black and Scholes model. Our numerical illustrations also show that the hedging error is small if the Asian option is in the money. If the option is out of the money, the price of the superhedging strategy is not as adequate, but still lower than the straightforward hedge of buying one European option with the same exercise price.

Local Expected Shortfall-Hedging in Discrete Time *

This paper proposes a self-financing trading strategy that minimizes the expected shortfall locally when hedging a European contingent claim. A positive shortfall occurs if the hedger is not willing to follow a perfect hedging or a superhedging strategy. In contrast to the classical variance criterion, the expected shortfall criterion depends only on undesirable outcomes where the terminal value of the written option exceeds the terminal value of the hedge portfolio. Searching a strategy which minimizes the expected shortfall is equivalent to the iterative solution of linear programs whose number increases exponentially with respect to the number of trading dates. Therefore, we partition this complex overall problem into several one-period problems and minimize the expected shortfall only locally, i.e., only over the next trading period. This approximation is quite accurate and the number of linear programs to be solved increases only linearly with respect to the number of trading dates.

On superhedging under delta constraints

The superhedging problem of derivative securities under the constraint of portfolio amounts is revisited. This paper considers more general forms of constraints, characterizes the minimal superhedging cost using a ‘dual’ maximization problem, and shows that a replicating strategy of the so-called ‘face-lifted’ claim gives a minimal superhedging strategy in the European option case. Also, as hinted by the static-replication technique, a superhedging strategy is computed for a knockout option in closed form.

Superreplication of European multiasset derivatives with bounded stochastic volatility

In this paper we analyze the superreplication approach in stochastic volatility models in the case of European multiasset derivatives. We prove that the Black-Scholes-Barenblatt (BSB) equation gives a superhedging strategy even if its solution is not twice differentiable. This is done under convexity assumptions on the final payoff h that are verified in some applications presented here.