Expansion Formulas for Bivariate Payoffs With Application to Best-of Options on Equity and Inflation

Gobet, Emmanuel; Hok, Julien

doi:10.1142/s0219024914500101

Cited by 3 publications

(4 citation statements)

References 11 publications

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“…• The correlation parameter ρ is typically chosen either by historical estimation or from occasionally observed prices of hybrid product involving interest rate and the underlying spot (see discussion in [11]).…”

Section: Calibrationmentioning

confidence: 99%

“…• For the option pricing, in [6,11], the authors developed expansion formulas by applying the perturbation method using a proxy introduced by [12]. A pricing framework via Partial Differential Equation (PDE) approach was studied in [13] and the Crank-Nicolson scheme also the Alternating Direction Implicit (ADI) method were applied to build the PDE solver.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we assume a Markovian setting with a stochastic differential equation (SDE) given by (2), i.e., the underlying S follows a local volatility type diffusion and the short rate r(t) is represented by a general form dynamic. This model covers a particular case, a local volatility diffusion for S associated with a Gaussian Hull-White dynamic for r (see [22]), which is widely used for pricing hybrid products (see e.g [6,11,7]). We focus on the model calibration and propose a methodology for an efficient implementation using a PDE approach.…”

Section: Introductionmentioning

confidence: 99%

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Calibration of local volatility model with stochastic interest rates by efficient numerical PDE methods

Hok

Tan

2019

Decisions Econ Finan

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Long maturity options or a wide class of hybrid products are evaluated using a local volatility type modelling for the asset price S(t) with a stochastic interest rate r(t). The calibration of the local volatility function is usually time-consuming because of the multi-dimensional nature of the problem. In this paper, we develop a calibration technique based on a partial differential equation (PDE) approach which allows an efficient implementation. The essential idea is based on solving the derived forward equation satisfied by P (t, S, r)Z(t, S, r), where P (t, S, r) represents the risk neutral probability density of (S(t), r(t)) and Z(t, S, r) the projection of the stochastic discounting factor in the state variables (S(t), r(t)). The solution provides effective and sufficient information for the calibration and pricing. The PDE solver is constructed by using ADI (Alternative Direction Implicit) method based on an extension of the Peaceman-Rachford scheme. Furthermore, an efficient algorithm to compute all the corrective terms in the local volatility function due to the stochastic interest rates is proposed by using the PDE solutions and grid points. Different numerical experiments are examined and compared to demonstrate the results of our theoretical analysis.

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Section: Calibrationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Calibration of local volatility model with stochastic interest rates by efficient numerical PDE methods

Hok

Tan

2019

Decisions Econ Finan

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“…This method has been applied and extended in many directions, see e.g. Miri (2010b, 2012), Gobet and Miri (2014), Gobet and Hok (2014) and Gobet and Bompis (2014). We derive expansion formulas, which are of Black-Scholes type, and develop the analysis using Malliavin calculus, to provide accurate estimations of the errors.…”

Section: Introductionmentioning

confidence: 99%

Expansion Formulas for European Quanto Options in a Local Volatility Fx-Libor Model

Hok¹,

Ngare

Papapantoleon

2018

Int. J. Theor. Appl. Finan.

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We develop an expansion approach for the pricing of European quanto options written on LIBOR rates (of a foreign currency). We derive the dynamics of the system of foreign LIBOR rates under the domestic forward measure and then consider the price of the quanto option. In order to take the skew/smile effect observed in fixed income and FX markets into account, we consider local volatility models for both the LIBOR and the FX rate. Because of the structure of the local volatility function, a closed form solution for quanto option prices does not exist. Using expansions around a proxy related to log-normal dynamics, we derive approximation formulas of Black-Scholes type for the price, that have the benefit of giving very rapid numerical procedures. Our expansion formulas have the major advantage that they allow for an accurate estimation of the error, using Malliavin calculus, which is directly related to the maturity of the option, the payoff, and the level and curvature of the local volatility function. These expansions also illustrate the impact of the quanto drift adjustment, while the numerical experiments show an excellent accuracy.Key words and phrases. European quanto derivatives, convexity adjustment, volatility skew/smile, local volatility FX-LIBOR model, expansion formula, analytical approximations, Malliavin calculus. 1 arXiv:1801.01205v2 [q-fin.PR] 3 Apr 2018The FX forward rate is, per definition, a Q i -martingale, and we assume it follows again a local volatility model of the form 5)where W i is the Q i -Brownian motion and σ i : [0, T i ] × R → R d + , i ∈ I, is a continuous, deterministic function satisfying a suitable linear growth condition, and represents the local volatility of the FX forward rate.The domestic and foreign forward Brownian motions are related viafor all i ∈ I, and this equation together with (2.3) determines also the relations between the domestic forward Brownian motions; see Schlögl (2002) for the details (in particular Fig. 2).

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ANALYTIC PRICING OF CoCo BONDS

Turfus

Shubert²

2017

Int. J. Theor. Appl. Finan.

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We present a new model for pricing contingent convertible (CoCo) bonds which facilitates the calculation of equity, credit and interest rate risk sensitivities. We assume a lognormal equity process and a Hull–White (normal) short rate process for the conversion intensity with a downward jump in the equity price on conversion. We are able to derive an approximate solution in closed form based on the assumption that the conversion intensity volatility is asymptotically small. The simple first-order approximation is seen to be accurate for a wide range of market conditions, although particularly for longer maturities higher order terms in the asymptotic expansion may be needed.

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Expansion Formulas for Bivariate Payoffs With Application to Best-of Options on Equity and Inflation

Cited by 3 publications

References 11 publications

Calibration of local volatility model with stochastic interest rates by efficient numerical PDE methods

Calibration of local volatility model with stochastic interest rates by efficient numerical PDE methods

Expansion Formulas for European Quanto Options in a Local Volatility Fx-Libor Model

ANALYTIC PRICING OF CoCo BONDS

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