Pricing Options on Scalar Diffusions: An Eigenfunction Expansion Approach

Давыдов, Д. А.; Linetsky, Vadim

doi:10.1287/opre.51.2.185.12782

Cited by 146 publications

(123 citation statements)

References 44 publications

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“…Therefore, Assumption 3 is satisfied with ζ = r. Next, we choose arbitrary δ > 0 and introduce where the constant η is given by (15), I is defined in (29), κ is given by Assumption 2, ρ is defined in (23), and ζ is given by Assumption 3.…”

Section: Lemma 24 Let the Sl Problem (12) Be Regular On The Right mentioning

confidence: 99%

“…The linear asymptotic behavior of the coefficients at zero, essentially, means that the process stays strictly positive at all times, which may or may not be a natural assumption, depending on whether we want to incorporate the default event in the model. (i) Assumption 1 is satisfied, and, hence, the SL problem (12) is regular on the right, with a corresponding constant η, given by (15).…”

Section: Q(z)dzc(y)dymentioning

confidence: 99%

“…However, the above models are rather limited due to the respective assumptions of independence of the time-change and symmetry of the local volatility function against the desired symmetries of the problem via the properties of the fundamental solutions to the associated SL equation. The idea of using the Laplace transform to represent option prices in time-homogeneous diffusion models is not new: for example, Davydov and Linetsky expressed the Laplace transform of the price of a European or barrier option in terms of the spectral expansion of the corresponding SL problem (see [14], [15], [24]). The problem of static hedging turns out to be quite different from the problem of pricing: one cannot simply go from the pricing formulas of Davydov and Linetsky to static hedges in a general time-homogeneous diffusion model.…”

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confidence: 99%

“…We need to find an "approximate solution" to (7), function g, such that g(x) = δ(x − K * ) + , for some δ ∈ R, and such that the asymptotic behavior of both sides of (7), as λ → ∞, is the same. Recall that the Laplace transforms (in the "time-to-maturity" variable) of out-of-the-money call and put prices in the CEV model can be computed explicitly (for example, one can consult with [15], or use the derivations in the proof of Proposition 3.1). Hence, in the notation…”

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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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Section: Lemma 24 Let the Sl Problem (12) Be Regular On The Right mentioning

confidence: 99%

Section: Q(z)dzc(y)dymentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Static Hedging under Time-Homogeneous Diffusions

Carr¹,

Nadtochiy²

2011

SIAM J. Finan. Math.

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“…The CEV model is usually applied to calculate the theoretical price, sensitivity and expected volatility of options (Emanuel & MacBeth, 1982;Fouqueet at al., 2000). In recent years, the problem of a pension fund investment is very urgent, it turns out that the CEV model has been successfully applied to study the effective investment strategy (Davydov & Linetsky, 2001; Davydov & Linetsky, 2003).…”

Section: Introductionmentioning

confidence: 99%

Spectral study of options based on CEV model with multidimensional volatility

Burtnyak

Malytska

2018

Investment Management and Financial Innovations

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This article studies the derivatives pricing using a method of spectral analysis, a theory of singular and regular perturbations. Using a risk-neutral assessment, the authors obtain the Cauchy problem, which allows to calculate the approximate price of derivative assets and their volatility based on the diffusion equation with fast and slow variables of nonlocal volatility, and they obtain a model with multidimensional stochastic volatility. Applying a spectral theory of self-adjoint operators in Hilbert space and a theory of singular and regular perturbations, an analytic formula for approximate asset prices is established, which is described by the CEV model with stochastic volatility dependent on l-fast variables and r-slowly variables, l ≥ 1, r ≥ 1, l ∈ N, r ∈ N and a local variable. Applying the Sturm-Liouville theory, Fredholm’s alternatives, as well as the analysis of singular and regular perturbations at different time scales, the authors obtained explicit formulas for derivatives price approximations. To obtain explicit formulae, it is necessary to solve 2l Poisson equations.

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