Can High-Order Convergence of European Option Prices be Achieved with Common CRR-Type Binomial Trees?

Abstract

Considering European call options, we prove that CRR-type binomial trees systematically underprice the value of these options, when the spot price is not near the money. However, we show that, with a volatility premium to compensate this mispricing, any arbitrarily high order of convergence can be achieved, within the common CRR-type binomial tree framework.

Appendix

We prove here Theorem 1. Apart from the fact the coefficient of $\lambda _{\ell }$ in $\mathcal {P}_{\ell }$ is given by (1.5), the rest is a textual application of the method described in Diener and Diener [4] with simple observations. The method is briefly summarized here.

Note first that, obviously, $\mathrm {u}(n,\,\overrightarrow{\lambda })$ and $\mathrm {d}(n,\,\overrightarrow{\lambda })$ have a convergent expansion in powers of $n^{-1/2}$, hence our binomial schemes are part of the general class described in [4]. To see that the coefficients $C_{\ell }$ have the form (1.3) with $\mathcal {P}_{\ell }$ a multivariate polynomial in $\sigma ^{-1},\sigma ,\lambda _{2},\ldots ,\lambda _{\ell },\kappa$, we follow the method described in [4]. More specifically, the authors obtain the asymptotic expansion of C(n) by replacing the “frozen” parameter $\kappa$ by $\overline{\kappa } (n,\,\overrightarrow{\lambda })$ in the asymptotic expansion of function $C(n,\kappa )$ which, in the notation of [4], is defined as

$$\begin{aligned}&\displaystyle C(n,\kappa )\overset{def}{=}S_{0}c(n,\kappa )I^{q}(n,\kappa )-Ke^{-rT} c(n,\kappa )I^{p}(n,\kappa ),\\&\displaystyle c(n,\kappa )\overset{def}{=}\frac{2^{1-n}}{\sqrt{n}}\mathrm {k}\left( n,\kappa \right) \left( {\begin{array}{c}n\\ \mathrm {k}\left( n,\kappa \right) \end{array}}\right) ,\\&\displaystyle \mathrm {k}\left( n,\kappa \right) \overset{def}{=}\mathrm {a} (n,\,\overrightarrow{\lambda })+1-\kappa , \end{aligned}$$

and where $I^{p}$ is defined by [4, Eq. (3.9)] and $I^{q}$ is defined similarly. Diener and Diener write the expansion of $\mathrm {k}\left( n,\kappa \right)$ as

$$\begin{aligned} \mathrm {k}\left( n,\kappa \right) =\mathrm {k}_{-2}n+\mathrm {k}_{-1}\sqrt{n}+\mathrm {k}_{0}+\cdots +\mathrm {k}_{i_{0}-3}n^{\frac{i_{0}-3}{2}}, \end{aligned}$$

(6.1)

where the $\mathrm {k}_{j}$ are obtained from (1.2). Note that $\mathrm {k}_{-2}=1/2$ and $\mathrm {k}_{0}=1-\kappa$. Following Diener and Diener, we write

$$\begin{aligned} c(n,\kappa )&=\exp (\left( 1-n\right) \ln 2-\frac{1}{2}\ln n+\ln \left( \mathrm {k}\left( n,\kappa \right) \right) \\&\quad +\ln \left( n!\right) -\ln \left( \left( n-\mathrm {k}\left( n,\kappa \right) \right) !\right) -\ln \left( \mathrm {k}\left( n,\kappa \right) !\right) ) \end{aligned}$$

and use Stirling’s formula and (6.1), to see that $c(n,\kappa )$ has an asymptotic expansion in powers of $\sqrt{n}^{-1}$, where the coefficients are an exponential term, $\exp \left( -2\mathrm {k}_{-1} ^{2}\right)$, multiplying multivariate polynomials in $\mathrm {k} _{-1},\ldots ,\mathrm {k}_{i_{0}-3}$. The latter translate into polynomials in $\sigma ^{-1},\sigma ,\lambda _{2},\ldots \,,\lambda _{i_{0}},\kappa$. This asymptotic expansion is uniform not only over $0\le \kappa \le 1$ as pointed out in [4], but also over $\mathrm {k}_{-1},\ldots ,\mathrm {k} _{i_{0}-3}$ in any compact set and, therefore, over $\mathcal {L} ^{-1}\le \sigma \le \mathcal {L}$, and $\left| \lambda _{\ell }\right| \le \mathcal {L}$, $\ell =2,\ldots ,i_{0}$ for any $\mathcal {L} >0$. As for the asymptotic expansions of $I^{p}(n,\kappa )$ and $I^{q} (n,\kappa )$ in powers of $\sqrt{n}^{-1}$, they are treated in an analogous manner, resulting in a trivial extension of [4, Theorem 3.4]. The explicit expression of each $\mathcal {P}_{\ell }$ needs to be calculated using a computational algebra system such as Diener and Diener’s Maple worksheet, available at http://math.unice.fr/~diener, which we adapted to calculate the $\mathcal {P}_{\ell }$’s needed for our numerical illustration.

In order to show that $\mathcal {P}_{\ell }$ is of degree one when seen as functions of $\lambda _{\ell }$, we introduce a new “frozen” parameter $\mu$. More specifically, we write $C(n)=C(n,\mu \left( n\right) )$, where $C(n,\mu )$ is the value of the call option in the binomial scheme

$$\begin{aligned} \mathrm {u}(n,\lambda _{2},\mu )&=\exp \left( \sigma \sqrt{\frac{T}{n} }+\left( \lambda _{2}\,\sigma ^{2}+\mu \right) \frac{T}{n}\right) ,\\ \mathrm {d}(n,\lambda _{2},\mu )&=\exp \left( -\sigma \sqrt{\frac{T}{n} }+\left( \lambda _{2}\,\sigma ^{2}+\mu \right) \frac{T}{n}\right) , \end{aligned}$$

and where

$$\begin{aligned} \mu \left( n\right) =\sum _{\ell =3}^{i_{0}}\lambda _{\ell }\frac{2\,\sigma \,}{T}\sqrt{\frac{T}{n}}^{\ell -2}. \end{aligned}$$

Note that $C(n,\mu )$ is a particular case of $C(n,\,\overrightarrow{\lambda })$ with $\lambda _{2}$ replaced by $\lambda _{2}+\mu /\sigma ^{2}$, and $\lambda _{\ell }$ replaced by 0 for $\ell \ge 3$. Proceeding just as in [4], we get a “new” asymptotic expansion of C(n) by substituting our “frozen” parameter $\mu$ by $\mu \left( n\right)$ in the asymptotic expansion of $C(n,\mu )$. Obviously our “old” expansion of C(n) given in (1.3) can be obtained by collecting together, from the “new” expansion, all the factors of $n^{-\ell /2}$ for $\ell =2,\ldots ,i_{0}$. Now fix $\ell \ge 3$ and concentrate on $\lambda _{\ell }$. Obviously, any polynomial in $\lambda _{2}+\mu \left( n\right) /\sigma ^{2}$ translates into a polynomial in $\lambda _{\ell }n^{-(\ell -2)/2}$. Therefore, collecting the factors of $n^{-\ell /2}$ from the “new” expansion of C(n) involves only the terms for which the degree of $\lambda _{\ell }$ is one in the “new” $\mathcal {P}_{2}$, since for every $j\ge 2$, $\mathcal {P}_{j}$ is itself a factor of $n^{-j/2}$. Doing so, one easily gets that the coefficient of $\lambda _{\ell }$ in the “old” $\mathcal {P}_{\ell }$ is given by (1.5). Note that, clearly, for $j<\ell$, $\lambda _{\ell }$ does not appear in the “old” $\mathcal {P}_{j}$.