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Valuing American-style options under the CEV model: an integral representation based method

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Abstract

This article derives a new integral representation of the early exercise boundary for valuing American-style options under the constant elasticity of variance (CEV) model. An important feature of this novel early exercise boundary characterization is that it does not involve the usual (time) recursive procedure that is commonly employed in the so-called integral representation approach well known in the literature. Our non-time recursive pricing method is shown to be analytically tractable under the local volatility CEV process and the numerical experiments demonstrate its robustness and accuracy.

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Notes

  1. A common interpretation for this stylized fact is that when an asset price declines, the associated firm becomes more leveraged since its debt to equity ratio becomes larger. Therefore, the risk of the asset, namely its volatility, should become higher. Another possible economic rationale for this phenomenon is that the forecast of an increase in the volatility should be compensated by a higher rate of return, which can only be obtained via a decrease in the asset value.

  2. The valuation of American-style contingent claims has a long list of relevant contributions and an exhaustive literature review would be prohibitive. However, a general overview of the most important developments on this subject may be found, for example, in Myneni (1992), Broadie and Detemple (2004) and Barone-Adesi (2005).

  3. As usual, \({\mathbb {E}}_{{\mathbb {Q}}}[ \left. X \right| {\mathcal {F}}_{t}]\) denotes the (time-t) expected value of the random variable X, conditional on \({\mathcal {F}}_{t}\), and computed under the equivalent martingale measure \({\mathbb {Q}}\). Moreover, for any two real numbers x and y, we denote by \(x \vee y\) and \(x \wedge y\), respectively, their maximum and minimum.

  4. For the sake of completeness, we notice that Nunes (2009, Proposition 1) also decomposes the price of an American-style option into the sum of the two aforementioned sources of value, as stated in Eq. (5). However, instead of using the integral representation (7), he proposes the use of an alternative characterization for the early exercise premium that requires an efficient valuation formula for the European-style counterpart and the knowledge of the underlying asset price transition density function.

  5. As usual, the risk-neutral measure \({\mathbb {Q}}\) is associated to the money market account numéraire, while the equivalent martingale measure \({\mathbb {Q}}^S\) takes as numéraire of the economy the underlying asset price.

  6. The existence and uniqueness of solutions of the nonlinear integral equation describing the early exercise boundary have been analyzed in Jacka (1991) and Peskir (2005) under the GBM diffusion and in Detemple and Tian (2002) for the CEV process. More recently, the existence and uniqueness of the early exercise boundary attached to American-style standard options under a jump to default CEV process—which nests the CEV diffusion as a special case—have been proven by Nunes et al. (2018).

  7. \({\mathcal {C}}:=\left\{ \left( S,t\right) \in [0,\infty [ \times \left[ t_{0},T\right] :\phi S_{t}>\phi B_t \right\} \) is defined as the corresponding continuation (or holding) region.

  8. Note that, for any \(T>t\), equation (18) holds for every \(S_t\) below (resp., above) the boundary value \(B_t\) of the put (resp., call). While applications under the CEV model using the usual integral representation approach—e.g., Kim and Yu (1996) and Detemple and Tian (2002)—look at the integral representation (18) only for \(S_t = B_t\), thus neglecting the region \(\phi S_{t} < \phi B_t\) and leading to the convolution type integral equation representation (17), the early exercise representation exploited here—as well as in Little et al. (2000) and Kim et al. (2013)—considers all the exercise region \(\phi S_{t} \le \phi B_t\).

  9. Note that Eq. (32) is simply a compact formula of Larguinho et al. (2013, Equations A7-A10).

  10. For completeness, we note also that the required noncentral chi-square distribution functions are computed through Benton and Krishnamoorthy (2003, Algorithm 7.3), which has been also used in many recent articles involving the CEV model, e.g. in Larguinho et al. (2013), Ruas et al. (2013), Dias et al. (2015), Nunes et al. (2015) and Cruz and Dias (2017).

  11. As expected, most of the computational burden of the proposed iterative method is due to the required numerical integrations. Even though other numerical integration schemes might be applied to diminish the computational effort, such efficiency considerations are outside the main scope of the present paper and, hence, speed-accuracy trade-off analysis are not exploited here.

  12. We have also tested the benchmark considered in Nunes (2009, Table 2) and Ruas et al. (2013, Table 1)—i.e., the Crank–Nicolson finite difference scheme with 15,000 time intervals and 10,000 space steps—and, as expected, the corresponding MAPE values are similar to the ones reported in Tables 1 and 2.

  13. Such proof of convergence of the non-time recursive iterative method is also missing even under the simpler GBM setup, as mentioned in a footnote in Kim et al. (2013, p. 889).

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  1. Business Research Unit (BRU-IUL), Instituto Universitário de Lisboa (ISCTE-IUL), Edifício II, Av. Prof. Aníbal Bettencourt, 1600-189, Lisbon, Portugal

    Aricson Cruz & José Carlos Dias

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Correspondence to José Carlos Dias.

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The authors thank the editor and the reviewers for the suggestions that have improved this manuscript. They also gratefully acknowledge the financial support provided by the Fundação para a Ciência e Tecnologia (Grant UID/GES/00315/2013). Aricson Cruz also thanks the financial support provided by the Fundação Millenium BCP.

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Cruz, A., Dias, J.C. Valuing American-style options under the CEV model: an integral representation based method. Rev Deriv Res 23, 63–83 (2020). https://doi.org/10.1007/s11147-019-09157-w

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