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Robust pricing and hedging of double no-touch options

Alexander M. G. Cox, Jan Obloj

arXiv:0901.0674v1q-fin.PR

TL;DR

The paper asks how to price a double no-touch option from more liquid option prices without assuming an underlying probability model. It constructs super- and sub-hedges and uses Skorokhod embeddings to establish tight bounds, while examining which arbitrage conditions connect prices to a market model.

  • Problem

    The paper asks what call prices at a given maturity imply about the price of a double no-touch option on the same underlying and maturity, without assuming a given model.

    Source: §1 · pp. 1–2

  • Method

    The paper constructs super- and sub-hedges from traded options and uses Skorokhod embedding techniques to identify extremal processes and establish tight price bounds.

    Source: §4 · p. 11

  • Results

    The hedging bounds for double no-touch options are tight, and stronger no-arbitrage conditions are linked to the existence of a market model.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Call and digital-call prices can be used to bound double no-touch prices without specifying a probability model for the underlying.

    Source: §1 · p. 1, §4 · p. 11

    For the upper bound, continuity can be relaxed only slightly by requiring that the price not jump across either barrier; without continuity, the upper bound changes.

    Source: §4.2 · p. 15

Abstract

from arXiv · show

Double no-touch options, contracts which pay out a fixed amount provided an underlying asset remains within a given interval, are commonly traded, particularly in FX markets. In this work, we establish model-free bounds on the price of these options based on the prices of more liquidly traded options (call and digital call options). Key steps are the construction of super- and sub-hedging strategies to establish the bounds, and the use of Skorokhod embedding techniques to show the bounds are the best possible. In addition to establishing rigorous bounds, we consider carefully what is meant by arbitrage in settings where there is no {\it a priori} known probability measure. We discuss two natural extensions of the notion of arbitrage, weak arbitrage and weak free lunch with vanishing risk, which are needed to establish equivalence between the lack of arbitrage and the existence of a market model.

1 Introduction

The paper asks how vanilla option prices can bound a double no-touch option without assuming a model for the underlying, and uses Skorokhod embeddings to identify tight bounds from hedging strategies. It also sets out the market assumptions and arbitrage concepts used to relate prices to the existence of a market model.

1 Introduction

  • The paper derives information about double no-touch prices from call prices while assuming as little as possible about the underlying asset’s behavior.A double no-touch pays one unit at maturity if the asset stays strictly between two barriers.Source: §1 · pp. 1–2
  • Skorokhod embedding constructions identify extremal processes that guide optimal super- and sub-hedges, and the resulting price bounds are tight.The paper derives hedging strategies using constructions already present in the embedding literature.Source: §1 · p. 2
  • Without a specified probability measure, the paper introduces weak arbitrage and weak free lunch with vanishing risk to connect stronger no-arbitrage conditions with market-model existence.The required condition depends on the traded strikes: weak free lunch with vanishing risk for all strikes and weak arbitrage for finitely many.Source: §1 · p. 2

1.1 Market input and the modelling setup

  • The setup uses continuous nonnegative asset-price paths, zero cost of carry, linearly priced traded assets, calls at specified strikes, and zero-cost forwards.The paper does not require a given probability space, probability measure, or semimartingale price process.Source: §1.1 · pp. 3–4
  • The setup asks whether the pricing operator can be extended to include a barrier-option payoff while preserving the no-arbitrage property.The paper motivates this question by considering an additional asset payoff within the existing market input.Source: §1.1 · p. 4

2 Arbitrage-free prices of call and digital options

The section asks which call and digital-call prices are compatible with a market model when market prices are specified without a probability measure. It shows that model-free arbitrage alone is insufficient, motivating stronger arbitrage criteria.

2 Arbitrage-free prices of call and digital options

  • WFLVR strengthens model-free arbitrage by also constraining pricing on a closure of the traded payoffs.The paper introduces it because model-free arbitrage alone does not ensure the expected link between arbitrage absence and a market model.Source: §2 · p. 5
  • For call prices at all strikes, no WFLVR is equivalent to a market model and to call prices satisfying the stated conditions, including non-negativity, convexity, and decrease.The conditions also require C(0) = S0 and a further derivative condition.Source: §2 · pp. 6–7
  • No model-free arbitrage requires condition (7), but if condition (8) fails, a market model does not exist despite the absence of model-free arbitrage.Thus the model-free criterion is weaker than the condition that characterizes model existence.Source: §2 · p. 6
  • With finitely many call strikes, no weak arbitrage is equivalent to a market model and to extending the quoted prices to a call-price function satisfying the stated conditions.Finite-strike prices can admit no model-free arbitrage yet still admit weak arbitrage.Source: §2 · p. 7

3 Robust hedging strategies

The section constructs super- and sub-hedging strategies for double no-touch options using traded calls, puts, digital calls, and forwards. Under no model-free arbitrage, the resulting payoff inequalities yield bounds on option prices.

3 Robust hedging strategies

  • The hedging section aims to produce simple super- and sub-hedging inequalities using the available assets, so their prices immediately bound the double no-touch payoff.The inequalities are framed under the assumption of no model-free arbitrage.Source: §3 · p. 8

3.1 Superhedges

  • The superhedges include static digital-option positions and strategies adapted from barrier-option hedges, with threshold-triggered forward purchases providing a martingale strategy.The three superhedge cases correspond to three types of extremal market-model behavior that maximize the double no-touch price.Source: §3.1 · p. 8

3.2 Subhedges

  • The subhedges comprise doing nothing or holding cash while selling a call and a put and entering a forward when the asset reaches a chosen level.The second strategy is constructed so its payoff has specified values across the barrier and strike regions.Source: §3.2 · pp. 9–10

3.3 Model-free bounds on double no-touch options

  • The hedging inequalities, together with linear pricing, yield bounds on double no-touch prices under no model-free arbitrage.The strategies use the traded assets specified for the hedges.Source: §3 · p. 8, §3.3 · p. 10

4 Model-free pricing of double no-touch options

This section develops model-free bounds for double no-touch prices, using Skorokhod embeddings to study their attainability, including upper bounds and approximating market models for finite strike sets. It relates no-arbitrage conditions to the existence of market models and distinguishes model-supported prices from boundary cases where weaker arbitrage may arise.

4.1 The Skorokhod embedding problem

  • Perkins embeddings minimize the stopped process’s maximum law and maximize its minimum law, while tilted-Jacka embeddings provide the reverse optimality for barrier-exit probabilities.Choosing the tilt parameter appropriately can maximize the probability that the stopped Brownian path leaves the barrier interval.Source: §4.1 · p. 12

4.2 Prices and hedges for the double no-touch option when K = R+

  • With call options at every strike and continuous paths, the no-WFLVR conditions are equivalent to the stated pricing conditions, and Perkins and tilted-Jacka models attain the upper and lower bounds.The bounds are therefore best possible within the stated setting.Source: §4.2 · p. 14

4.3 Pricing and hedging when K is finite

  • For finitely many traded strikes, with at least two strikes between the barriers, the lower bound is attained by a market model under the stated no-WA condition.The bound is expressed using traded strikes selected by the maximum in the theorem’s formula.Source: §4.3 · pp. 16–17
  • With finitely many strikes, the upper-bound analysis must account for digital-call prices and the payoff discontinuity, so in some cases the bound is only arbitrarily closely approximable.The discussion notes that this attainment issue would not arise for a modified double no-touch payoff.Source: §4.3 · p. 17
  • In the simpler traded-strike case, Theorem 4.8 establishes an upper bound on the double no-touch price and a sequence of market models approximating it.If the upper bound is attained, the theorem’s stated alternatives include weak arbitrage in the enlarged market.Source: §4.3 · p. 17
  • The non-attainable lower bound based on the barrier call-price slope is consistent with no model-free arbitrage, but not with no weak arbitrage; model prices can converge to the optimal values.The construction uses a sequence of market models.Source: §4.3 · p. 18
  • When the barriers are not traded strikes, Theorem 4.9 gives an upper bound using extended call and digital prices and a sequence of market models approximating it.The theorem assumes (A), no weak arbitrage on X, and no model-free arbitrage for the extended market.Source: §4.3 · pp. 17, 19
  • The results specify prices at which double no-touch options can trade without model-free arbitrage and identify when market models exist; at a boundary without a model, weaker arbitrage may arise.The two sets are generally exclusive and, except possibly at a boundary, cover all prices.Source: §4.3 · pp. 19–20

5 Applications

The paper applies its results to pricing and robust hedging of double no-touch options. Although call prices yield robust price bounds, the authors say these are typically too wide for practical use; simulations illustrate the hedging approach.

5 Applications

  • Robust price bounds can be deduced from call prices, but are typically too wide for practical use; in major currency pairs, market prices should instead be treated as model inputs.Source: §5 · p. 20
  • Robust hedges are presented as an alternative to delta/vega hedging that avoids its listed problems: model risk, transaction costs, discrete monitoring, and gamma exposure.Source: §5 · p. 20
  • Average transaction costs were 0.32 and 0.13 for delta/vega and robust hedges respectively; an exponential utility trader would prefer the robust hedge, with utility −0.2755 against −0.4327.The comparison used 10,000 Monte Carlo runs and assumed proportional transaction costs of 1% for calls or puts and 0.02% for the underlying.Source: §5 · p. 21
  • The authors caution that single-barrier hedging is impractical because of transaction and operational costs, and banks hedge portfolios rather than single options.They say adapting the techniques would be required before implementation in real markets.Source: §5 · p. 22

A Weak Arbitrage

The appendix studies model-free bounds for double no-touch probabilities and the joint maximum-and-minimum law of uniformly integrable martingales with fixed terminal law, including attainability and continuity. It also characterizes arbitrage conditions for added call and digital prices and connects the absence of weak arbitrage and the tightness of the upper bound to market models and hedging strategies.

A Weak Arbitrage

  • Prices satisfying (40)–(43) admit an extension with no model-free arbitrage; failure of any condition produces a model-free arbitrage.Source: A Weak Arbitrage · p. 22
  • If no weak arbitrage exists, a (P, XD)-market model exists; it can have piecewise-linear C(·), with kinks only at the listed strikes and additional strikes arbitrarily close to barriers.Source: A Weak Arbitrage · p. 23
  • If the pricing operator has no model-free arbitrage, its prices can be perturbed by less than any ε to obtain a pricing operator admitting a market model.Source: A Weak Arbitrage · p. 24
  • Superhedges built only from traded options bound the double no-touch price above by (37), and suitable market models approximate that upper bound.The construction addresses the absence of calls and digitals traded at the barrier; when needed, ε-perturbations support the approximating models.Source: A Weak Arbitrage · pp. 24–25

B On the joint law of the maximum and minimum of a UI martingale

  • The double-no-touch probability is non-increasing in the upper barrier, and Proposition B.1 gives boundary properties and upper and lower bounds for the probability surface.The surface is defined over lower barriers at or below M0 and upper barriers at or above M0.Source: B On the joint law of the maximum and… · p. 28
  • The Perkins construction attains the upper bound for all barrier pairs, whereas the Jacka construction attains the lower bound only in a neighborhood of a selected interior pair.The lower bound is not attained by any single martingale for all barrier pairs.Source: B On the joint law of the maximum and… · pp. 28–29
  • Any discontinuity of the probability function requires an atom of the terminal law at one of the barriers and falls into at least one of the theorem’s two stated cases.Source: B On the joint law of the maximum and… · p. 29
  • The lower bound is continuous in the interior and at the boundary unless the terminal law has an atom at either support endpoint; the upper bound is discontinuous in correspondence with every atom.Source: B On the joint law of the maximum and… · p. 29
  • The appendix identifies describing possible joint laws of the terminal value, maximum, and minimum as a natural further question.Source: B On the joint law of the maximum and… · p. 30
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