Pitfalls of the Fourier Transform Method in Affine Models, and Remedies

2019

SSRN Journal

In this paper, we argue that, once the costs of maintaining the hedging portfolio are properly taken into account, semi-static portfolios should more properly be thought of as separate classes of derivatives, with non-trivial, model-dependent payoff structures. We derive new integral representations for payoffs of exotic European options in terms of payoffs of vanillas, different from 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 varianceminimizing portfolios. We explain why the exact semi-static hedging of barrier options is impossible for processes with jumps, and derive general formulas for variance-minimizing semistatic portfolio. 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.

Section: 3mentioning

confidence: 96%

Static and Semi-Static Hedging as Contrarian or Conformist Bets

2019

SSRN Journal

“…Assuming that

{\overset{G}{true}}_{j}, j = 0, 1, \dots, N

, are calculated, calculation of the mean hedging error

\begin{matrix} true \\ right double-struckE ([] E r r (n; x, X_{T})) & = & left - V_{0} (T, x) + \sum_{j = 1}^{N} n_{j} V_{j} (T, x) \end{matrix}

is reducible to the Fourier inversion. As it is explained in Boyarchenko and Levendorskiĭ (2014), Levendorskiĭ (2016), and Boyarchenko and Levendorskiĭ (2019), if (a)

{\hat{G}}_{j}

is of the form

{\overset{G}{true}}_{j} false(ξ false) = e^{- i k_{j} ξ} {\overset{G}{true}}_{j 0} false(ξ false)

, where

k_{j} \in R

and

{\hat{G}}_{j 0}

is a rational functions, and (b) ψ is of the form

ψ false(ξ false) = - i μ ξ + ψ^{0} false(ξ false)

, where

μ \in R

and

prefix Re ψ^{0} false(ξ false) \to + \infty

ξ \to \infty

remaining in a cone around the real axis, then it is advantageous to represent

V_{j} false(T, x false)

in the form

V_{j} false(T; x false) = \frac{1}{2 π} \int_{prefix Im ξ = ω} ...

…”

Section: Static Hedging Of European Optionsmentioning

confidence: 70%

Static and semistatic hedging as contrarian or conformist bets

2020

Mathematical Finance

Self 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.

“…The bound can be simplified (at the expense of some loss in accuracy) as Given the error tolerance ǫ > 0, one can find an approximation to Λ 1 satisfying Err tr (Λ 1 ) < ǫ quite easily (see [7,28,30]), and then define…”

Section: 3mentioning

confidence: 99%

Sinh-Acceleration: Efficient Evaluation of Probability Distributions, Option Pricing, and Monte Carlo Simulations

Int. J. Theor. Appl. Finan.

2019

Self Cite

Characteristic functions of several popular classes of distributions and processes admit analytic continuation into unions of strips and open coni around R ⊂ C. The Fourier transform techniques reduces calculation of probability distributions and option prices to evaluation of integrals whose integrands are analytic in domains enjoying these properties. In the paper, we suggest to use changes of variables of the form ξ = √ −1ω1 + b sinh( √ −1ω + y) and the simplified trapezoid rule to evaluate the integrals accurately and fast. We formulate the general scheme, and apply the scheme for calculation probability distributions and pricing European options in Lévy models, the Heston model, the CIR model, and a Lévy model with the CIR-subordinator. We outline applications to fast and accurate calibration procedures and Monte Carlo simulations in Lévy models, regime switching Lévy models that can account for stochastic drift, volatility and skewness, and the Heston model. For calculation of quantiles in the tails using the Newton or bisection method, it suffices to precalculate several hundred of values of the characteristic exponent at points of an appropriate grid (conformal principal components) and use these values in formulas for cpdf and pdf.