Valuation of segregated funds: shout options with maturity extensions

Windcliff, H.; Vetzal, Kenneth R.

doi:10.1016/s0167-6687(01)00072-5

Cited by 35 publications

(34 citation statements)

References 26 publications

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“…In [21] we provide an example using this boundary specification for a very complex path-dependent option.…”

Section: The Linear Boundary Conditionmentioning

confidence: 99%

“…An example of such a situation is in the full three-dimensional numerical valuation of multiple shout options described in [22] where the required size of the computational domain grows exponentially with the number of shout opportunities. Some popular insurance products offered in Canada [21] can offer the investor as many as 60 total shout opportunities (referred to as "resets" in these contracts) over the life of the contract. Hence, for path-dependent products it can be extremely important to have an accurate method for approximating the boundary condition on the truncated computational domain.…”

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confidence: 99%

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Analysis of the stability of the linear boundary condition for the Black–Scholes equation

Windcliff¹,

Forsyth²,

Vetzal³

2004

JCF

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Abstract. The linear asymptotic boundary condition, i.e. assuming that the second derivative of the value of the derivative security vanishes as the asset price becomes large, is commonly used in practice. To our knowledge, there have been no rigorous studies of the stability of these methods, despite the fact that the discrete matrix equations obtained using this boundary condition loses diagonal dominance for large timesteps. In this paper, we demonstrate that the discrete equations obtained using this boundary condition satisfy necessary conditions for stability for a finite difference discretization. Computational experiments also show that this boundary condition satisfies sufficient conditions for stability as well. [19,18,10] have recommended a linear asymptotic boundary condition (that the second derivative of the option value with respect to the underlying asset value be zero) as the asset price becomes large. Although this boundary specification is often applied, to our knowledge there have been no rigorous studies of the stability of this technique. Looking at the form of the discrete matrix equations obtained using this boundary condition, the resulting discrete equations lose diagonal dominance for large timesteps and the usual arguments cannot be applied to guarantee unconditional stability.In order to determine the ranges of parameters (risk-free rate, volatility, etc.) for which this asymptotic boundary condition could cause instability, we derive necessary conditions for the stability of the discrete equations based on the spectrum of the matrix representing the spatial discretization. Somewhat surprisingly, we find that a finite difference (FD) discretization always satisfies these necessary conditions for stability, despite the fact that the matrix equations are not unconditionally diagonally dominant.The eigenvalues can be used to determine necessary conditions for stability but are known to be unreliable for determining sufficient conditions for stability. For finite dimensional problems analysis of the spectrum can lead to sufficient conditions but in the PDE context, the size of the matrix becomes unbounded as the grid is refined. In our case the matrix is non-symmetric and non-normal, and for non-normal matrices, counterexamples can be given where, even if the eigenvalues are less than one in magnitude, instability results as the dimension of the matrix becomes large (see [11,7,9,17]). For some values of the market parameters we are able to show that sufficient conditions for stability are satisfied using numerical range arguments [7,8,16,3]. In other cases, these arguments cannot be applied and we follow [2] and *

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“…In [21] we provide an example using this boundary specification for a very complex path-dependent option.…”

Section: The Linear Boundary Conditionmentioning

confidence: 99%

mentioning

confidence: 99%

Analysis of the stability of the linear boundary condition for the Black–Scholes equation

Windcliff¹,

Forsyth²,

Vetzal³

2004

JCF

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“…In our subsequent analysis, we derive two necessary conditions for the commencement of optimal reset based on the specific nature of the exercise payoff in the reset option [see Eqs. (5)(6)]. Most of the interesting phenomena on the optimal reset policies are dictated by these two necessary conditions and the functional form of the terminal payoff.…”

Section: Linear Complementarity Formulationmentioning

confidence: 99%

“…Longstaff (1990) considers a wide variety of extendible options with applications including American options with stochastic dividends, shared equity mortgages and debt negotiation under financial distress of the issuer. Windcliff et al (2001) perform a comprehensive analysis of the Canadian segregated funds. These funds allow the holder to reset the guarantee level and the maturity date a preset number of times during the life of the fund contract.…”

Section: Introductionmentioning

confidence: 99%

Options with Combined Reset Rights on Strike and Maturity

Dai

Kwok

2004

SSRN Journal

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“…Thomas 6 and Cheuk and Vorst 7 consider pricing in the context of binomial and trinomial lattices. Windcliff et al 8 price and hedge more complicated reset structures using finite difference methods but approach the benchmarking to plain vanilla puts differently than is the case in this paper and do not consider value-at-risk. Dai et al 9 and Dai and Kwok 10 also consider options with combined reset rights on strike and maturity but focus primarily on optimal reset strategies in the presence of a continuous dividend yield, and on characterising the optimal reset frontier.…”

Section: Introductionmentioning

confidence: 99%

Pricing, value-at-risk and dynamic properties of re-settable strike-price puts

McIntyre

Jackson

2007

J Deriv Hedge Funds

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This paper provides details of the pricing and hedging characteristics of re-settable strike-price puts as compared to plain vanilla puts and makes explicit the differences between the two. Examples are provided of exposures that arise for issuers of re-settable strike-price puts, either as separate instruments or as instruments that are embedded in investment products, such as protected index notes. This paper also provides guidance to practitioners concerning the need to hedge re-settable strike-price puts differently than plain vanilla puts when the time to reset of the option is small and the underlying is near S*, a key value defined in the paper. Specifically, the gamma of the re-settable strike-price put is much greater than the gamma of the plain vanilla put, so a delta hedge for the former likely will need to be reset more frequently than for the latter, particularly near the reset date. AbstractThis paper considers a particular type of re-settable strike-price put contract. This class of contract is important because it has come into widespread use as an embedded feature of investment vehicles such as segregated-funds investments and protected index notes. Holders of short positions in these contracts face a potential liability that crystallises if contracts mature at a time when market prices are below strike prices. This leads naturally to questions concerning the hedging behaviour of these products, and the value-at-risk associated with positions in them. We address these questions in this paper.

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Valuation of segregated funds: shout options with maturity extensions

Cited by 35 publications

References 26 publications

Analysis of the stability of the linear boundary condition for the Black–Scholes equation

Analysis of the stability of the linear boundary condition for the Black–Scholes equation

Options with Combined Reset Rights on Strike and Maturity

Pricing, value-at-risk and dynamic properties of re-settable strike-price puts

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