Static Hedging under Time-Homogeneous Diffusions

Abstract

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

Model uncertainty and the pricing of American options

“…There is a closely related body of literature on robust hedging as exemplified by Carr, Ellis and Gupta [7] and Carr and Nadtochiy [10] that seeks hedging strategies for exotic options that work well across a wide range of models. These strategies do require restrictions on the underlying process, such as the symmetry of the implied volatility surface or the requirement that instantaneous volatility be a deterministic function of the price level, which are shared by a broad range of standard models.…”

Robust Trading of Implied Skew

“…The above result also applies to the processes X obtained as an independent continuous time change of a timehomogeneous diffusion, such as the PCLVG processes. However, strictly speaking, the PCLVG process does not satisfy the assumptions made in [17]. Indeed, the coefficient σ is discontinuous: it is piecewise constant, taking values σ 1 , for x ∈ (0, U ), and σ 2 , for x ≥ U .…”

“…Third, any model from this family has an explicit static hedging strategy, constructed via the weak reflection principle (cf. [17], [5]). We believe that these results are of independent interest.…”

Optimal static quadratic hedging

“…Also, there is a wealth of static hedging results specifically for barrier options under one-dimensional diffusion models; see Derman et al (1995); Carr and Chou (1997); ; Carr and Lee (2009) ;Carr and Nadtochiy (2011); Bardos et al (2010), among others. In contrast to these works, our framework applies to other exotic derivatives and multidimensional diffusion models.…”