Prices of Barrier and First-Touch Digital Options in Levy-Driven Models, Near Barrier

Boyarchenko, Mitya; Innocentis, Marco de; Levendorskiı̌, Sergei

doi:10.2139/ssrn.1514025

Cited by 12 publications

(20 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In more detail, on the RHS of (42), we obtain, first,

\frac{1}{2 π q} \int_{prefix Im ξ = ω_{+}} d ξ e^{i x ξ} φ_{q}^{-} false(ξ false) φ_{q}^{+} false(ξ false) \hat{G} false(ξ false),

and then apply the Wiener–Hopf factorization formula (25). The integrals on the RHS of (41)–(42) absolutely converge (or absolutely converge after integration by parts in the oscillatory integrals with respect to ξ) and their sum equals

{\overset{V}{true}}_{1} false(G; q, x false)

(see Boyarchenko & Levendorskiĭ, 2002a, 2002b; Boyarchenko et al., 2011).

□

…”

Section: Calculation Of No‐touch Options and Expectations Of No‐touchmentioning

confidence: 99%

“…We repeat the main steps of the initial proof in Boyarchenko and Levendorskiĭ (2002a) omitting technical details; they are the same as in Boyarchenko and Levendorskiĭ (2002a, 2002b) and Boyarchenko et al. (2011). Let

L = - ψ (D)

be the infinitesimal generator of X .…”

Section: Pricing First‐touch Options and Expectations Of First‐touch mentioning

confidence: 99%

“…and then apply the Wiener-Hopf factorization formula (25). The integrals on the RHS of ( 41)-( 42) absolutely converge (or absolutely converge after integration by parts in the oscillatory integrals with respect to 𝜉) and their sum equals Ṽ1 (𝐺; 𝑞, 𝑥) (see Boyarchenko & Levendorskiȋ, 2002a, 2002bBoyarchenko et al, 2011). □…”

Section: General Formulas For No-touch Optionsmentioning

confidence: 99%

“…First, we calculate the expectation 𝑉 (𝐺; 𝑇 ; 𝑥) = 𝔼 𝑥 [𝟏 𝜏<𝑇 𝐺(𝜏, 𝑋 𝜏 )] in the general form, and then make further steps for the special cases mentioned above. We repeat the main steps of the initial proof in Boyarchenko and Levendorskiȋ (2002a) omitting technical details; they are the same as in Levendorskiȋ (2002a, 2002b) and Boyarchenko et al (2011). Let 𝐿 = −𝜓(𝐷) be the infinitesimal generator of 𝑋.…”

Section: General Casementioning

confidence: 99%

“…In all popular models used in finance, the prices of vanilla options are infinitely smooth before the maturity date and up to the boundary but prices of barrier options in Lévy models are not smooth at the boundary, the exceptions being double jump diffusion model, hyper‐exponential jump diffusion model, and other models with rational characteristic functions. For wide classes of purely jump models, it is proved in Boyarchenko and Levendorskiĭ (2002b) and Boyarchenko, de Innocentis, and Levendorskiĭ (2011) that the price of an “out” barrier option near the barrier behaves as

{c (T) false| x - h false|}^{κ}

, where

κ \in [0, 1)

is independent of time to maturity T ,

c (T) > 0

, and

| x - h |

is the log‐distance from the barrier. For finite variation processes with the drift pointing from the boundary,

κ = 0

, and the limit of the price at the barrier is positive.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Static and semistatic hedging as contrarian or conformist bets

Boyarchenko

Levendorskiı̌

2020

Mathematical Finance

View full text Add to dashboard Cite

In this paper, we argue that, once the costs of maintaining the hedging portfolio are properly taken into account, semistatic portfolios should more properly be thought of as separate classes of derivatives, with nontrivial, model‐dependent payoff structures. We derive new integral representations for payoffs of exotic European options in terms of payoffs of vanillas, different from the Carr–Madan representation, and suggest approximations of the idealized static hedging/replicating portfolio using vanillas available in the market. We study the dependence of the hedging error on a model used for pricing and show that the variance of the hedging errors of static hedging portfolios can be sizably larger than the errors of variance‐minimizing portfolios. We explain why the exact semistatic hedging of barrier options is impossible for processes with jumps, and derive general formulas for variance‐minimizing semistatic portfolios. We show that hedging using vanillas only leads to larger errors than hedging using vanillas and first touch digitals. In all cases, efficient calculations of the weights of the hedging portfolios are in the dual space using new efficient numerical methods for calculation of the Wiener–Hopf factors and Laplace–Fourier inversion.

show abstract

“…In more detail, on the RHS of (42), we obtain, first,

\frac{1}{2 π q} \int_{prefix Im ξ = ω_{+}} d ξ e^{i x ξ} φ_{q}^{-} false(ξ false) φ_{q}^{+} false(ξ false) \hat{G} false(ξ false),

{\overset{V}{true}}_{1} false(G; q, x false)

(see Boyarchenko & Levendorskiĭ, 2002a, 2002b; Boyarchenko et al., 2011).

□

…”

Section: Calculation Of No‐touch Options and Expectations Of No‐touchmentioning

confidence: 99%

L = - ψ (D)

be the infinitesimal generator of X .…”

Section: Pricing First‐touch Options and Expectations Of First‐touch mentioning

confidence: 99%

Section: General Formulas For No-touch Optionsmentioning

confidence: 99%

Section: General Casementioning

confidence: 99%

{c (T) false| x - h false|}^{κ}

, where

κ \in [0, 1)

is independent of time to maturity T ,

c (T) > 0

, and

| x - h |

is the log‐distance from the barrier. For finite variation processes with the drift pointing from the boundary,

κ = 0

, and the limit of the price at the barrier is positive.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Static and semistatic hedging as contrarian or conformist bets

Boyarchenko

Levendorskiı̌

2020

Mathematical Finance

View full text Add to dashboard Cite

show abstract

Approximating Lévy processes with completely monotone jumps

Hackmann¹,

Kuznetsov

2016

Ann. Appl. Probab.

View full text Add to dashboard Cite

Lévy processes with completely monotone jumps appear frequently in various applications of probability. For example, all popular stock price models based on Lévy processes (such as the Variance Gamma, CGMY/KoBoL and Normal Inverse Gaussian) belong to this class. In this paper we continue the work started in [Int. J. Theor. Appl. Finance 13 (2010) 63-91, Quant. Finance 10 (2010) 629-644] and develop a simple yet very efficient method for approximating processes with completely monotone jumps by processes with hyperexponential jumps, the latter being the most convenient class for performing numerical computations. Our approach is based on connecting Lévy processes with completely monotone jumps with several areas of classical analysis, including Padé approximations, Gaussian quadrature and orthogonal polynomials.

show abstract

Convergence of Price and Sensitivities in Carr's Randomization Approximation Globally and Near Barrier

Levendorskiı̌

2009

SSRN Journal

View full text Add to dashboard Cite

Prices of Barrier and First-Touch Digital Options in Levy-Driven Models, Near Barrier

Cited by 12 publications

References 41 publications

Static and semistatic hedging as contrarian or conformist bets

Static and semistatic hedging as contrarian or conformist bets

Approximating Lévy processes with completely monotone jumps

Convergence of Price and Sensitivities in Carr's Randomization Approximation Globally and Near Barrier

Contact Info

Product

Resources

About