Cost-optimal static super-replication of barrier options: an optimization approach

Giese, Alexander; Maruhn, Jan H.

doi:10.21314/jcf.2007.176

Cited by 12 publications

(10 citation statements)

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“…Similarly, from (17) we conclude that the derivative of μ/σ is absolutely integrable at infinity, and, therefore, μ/σ has a finite limit at infinity, say, c 3 ∈ R, and we have…”

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confidence: 59%

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Static Hedging under Time-Homogeneous Diffusions

Carr¹,

Nadtochiy²

2011

SIAM J. Finan. Math.

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Abstract. We consider the problem of semistatic hedging of a single barrier option in a model where the underlying is a time-homogeneous diffusion, possibly running on an independent stochastic clock. The main result of the paper is an analytic expression for the payoff of a European-type contingent claim, which has the same price as the barrier option up to hitting the barrier. We then consider some examples, such as the Black-Scholes, constant elasticity of variance, and zero-correlation SABR models. Finally, we investigate an approximation of the static hedge with options of at most two different strikes. 1. Introduction. Barrier options are some of the most popular path-dependent derivatives traded on OTC markets. An up-and-out option, written on the underlying S, with terminal payoff F (S T ) at maturity T and (upper) barrier U pays F (S T ) at time T , provided the underlying has not hit the barrier U by that time; otherwise the option expires worthless. Similarly, the up-and-in option pays F (S T ) if and only if the underlying has hit the barrier by the time T . One can define lower barrier options analogously. The literature on pricing and hedging barrier options is vast and we, of course, cannot fully pay tribute to it. We, however, concentrate on the specific problem of static hedging of barrier options.The idea of dynamic hedging of financial derivatives by trading in the underlying asset has some obvious drawbacks. These drawbacks are largely due to the presence of transaction costs. The problem can, in principle, be solved by including other, more liquid, derivatives in the hedging portfolio: if one can construct a fixed portfolio of liquid derivatives (not necessarily the underlying process alone), whose price coincides with the price of the given barrier option at all times, up to the first hitting time of the barrier, then it is natural to hedge the sale of this barrier option by taking long positions in the corresponding liquid derivatives. Such a portfolio of liquid derivatives is then called a static hedge of the corresponding barrier option. We choose European-type options (the ones that have some fixed payoff G(S T ) at time T and no path dependence) as the liquid derivatives and try to find a static hedging portfolio for the up-and-out put (UOP). Recall that an UOP has the following payoff at the time of maturity

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“…Similarly, from (17) we conclude that the derivative of μ/σ is absolutely integrable at infinity, and, therefore, μ/σ has a finite limit at infinity, say, c 3 ∈ R, and we have…”

mentioning

confidence: 59%

“…Together with (17), condition (19) implies that there exist real constants c 1 and c 2 such that, for all z ≥ 0, we have…”

Section: Matching Prices In the Laplace Spacementioning

confidence: 99%

Static Hedging under Time-Homogeneous Diffusions

Carr¹,

Nadtochiy²

2011

SIAM J. Finan. Math.

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“…Extensions allow this idea to be used for stochastic volatility, and even to cover jumps, at the expense of needing possibly a very large portfolio of options. More recently, work of [NP06] unifies both these approaches, and allows a fairly general set of asset dynamics, as does [GM07], where the authors find an optimal portfolio by setting up an optimisation problem. Note however that all these strategies assume a known model for the underlying, and also that the hedging assets will be liquid enough for the portfolio to be liquidated at the price specified under the model.…”

Section: Model Riskmentioning

confidence: 99%

Robust Hedging of Double Touch Barrier Options

Cox¹,

Obłój²

2011

SIAM J. Finan. Math.

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We consider model-free pricing of digital options, which pay out if the underlying asset has crossed both upper and lower barriers. We make only weak assumptions about the underlying process (typically continuity), but assume that the initial prices of call options with the same maturity and all strikes are known. Under such circumstances, we are able to give upper and lower bounds on the arbitrage-free prices of the relevant options, and further, using techniques from the theory of Skorokhod embeddings, to show that these bounds are tight. Additionally, martingale inequalities are derived, which provide the trading strategies with which we are able to realise any potential arbitrages. We show that, depending of the risk aversion of the investor, the resulting hedging strategies can outperform significantly the standard delta/vega-hedging in presence of market frictions and/or model misspecification.

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“…We do not know in advance which (time to expiry, spot)-combinations we need to price, this is determined by when and how the barrier is crossed, so we use brute-force quadrature integration. 9 Both Nalholm and Poulsen (2006) and Giese and Maruhn (2007) also find the expiry-T component to be the most important one; in the later comparison we also restrict their hedges in that way. one-sided 10 risk-measures from Sect.…”

Section: Hedge Performance In the Bates Modelmentioning

confidence: 96%