A copula-based approach for generating lattices

Abstract

Discrete approximations such as binomial and trinomial lattices have been developed to model the intertemporal dynamics of variables in a way that also allows contingent decisions to be included at the appropriate increments in time. In this paper we present an approach for developing these types of models based on copulas. In addition to ease of implementation, a primary benefit of this approach is its generality, and we show that various binomial and trinomial approximation methods for valuing contingent claim securities in the literature are special cases of this approach, each based on a choice of a particular set of probability and/or branching parameters. Because this approach encompasses these and other cases as feasible solutions, we also show how it can be used to optimize the construction of lattices so that discretization error is minimized, and we demonstrate its application for an option pricing example.

Appendices

Appendix 1

For convenience, we will prove Part b of Theorem 1 first:

We can show the moments of $x\left( t\right) =\ln \left( S\left( t\right) \right)$ are a function of the moments of W(t) by using moment generating functions. The moment generating function of the normal distribution $X\sim N\left( \mu , \sigma ^{2}\right)$ is given as $M_{X}\left( z \right) ={\mathrm {exp}}(\mu z+\frac{1}{2}\sigma ^{2}z^{2})$. If $W\left( t \right) \sim N(0,t)$, then $\sigma W\left( t\right) \sim N(0,\sigma ^{2}t)$, and therefore, $M_{W\left( t \right) }\left( z\right) =\exp \left( \frac{1}{2}tz^{2}\right) ,$ and $M_{\sigma W\left( t\right) }\left( z \right) ={\mathrm {exp}}(\frac{1}{2}\sigma ^{2}tz^{2})$

We now focus on the log-transformed asset price. For $x\left( t \right) =\ln \left( S\left( t\right) \right) \sim N(\ln {\left( S\left( 0 \right) \right) +\nu t,\sigma ^{2}t})$, we will provide the derivation of its moment generating function.

$$\begin{aligned} M_{x\left( t\right) }\left( z\right)= & {} {\mathrm {e}}^{\left( \ln \left( S\left( 0 \right) \right) +\nu t\right) z+\frac{1}{2}\sigma ^{2}tz^{2}}=\hbox {e}^{\left( \ln \left( S\left( 0\right) \right) + \nu \right) z}{\mathrm {e}}^{\frac{1}{2}\sigma ^{2}tz^{2}}\\= & {} \hbox {e}^{\left( \ln \left( S\left( 0 \right) \right) + \nu \right) z}{\mathrm {e}}^{\left( \frac{1}{2}tz^{2}\right) \sigma ^{2}} =\hbox {e}^{\left( \ln \left( S\left( 0\right) \right) + \nu \right) z}\left( M_{W\left( t \right) }\left( z\right) \right) ^{\sigma ^{2}}\\= & {} \hbox {e}^{\left( \ln \left( S\left( 0 \right) \right) + \nu \right) z}{M_{\sigma W\left( t\right) }\left( z\right) } \end{aligned}$$

The following equation can be obtained using the Taylor expansion operation on ${\mathrm {e}}^{[\ln {\left( S\left( 0\right) \right) +\nu t]}z}$:

$$\begin{aligned} {\mathrm {e}}^{(\ln \left( S\left( 0\right) \right) +\nu )z}=1+(\ln \left( S\left( 0\right) \right) +\nu )z+\frac{((\ln \left( S\left( 0\right) \right) +\nu )z)^{2}}{2!}+\cdots \end{aligned}$$

which simplifies to

$$\begin{aligned} M_{x\left( t\right) }\left( z\right) =\left[ 1+(\ln {\left( S\left( 0\right) \right) +\nu t)}z+\frac{{\left( \ln \left( S\left( 0\right) \right) +\nu t\right) }^{2}z^{2}}{2!}+\cdots \right] {\times M}_{\sigma W\left( t\right) }\left( z\right) \end{aligned}$$

If the moment generating function exists on an open interval around $\hbox {t} = 0$, then it is the exponential generating function of the moments of the probability distribution:

$$\begin{aligned} E\left( X^{n}\right) =M^{\left( n\right) }_{X}\left( 0 \right) =\frac{d^{n}M_{X}}{dz^{n}}\left( 0\right) \end{aligned}$$

Performing the $\hbox {n}{\mathrm{th}}$ derivatives on the moment generating function of $x\left( t\right)$

$$\begin{aligned} E\left( {x\left( t\right) }^{n}\right)= & {} M^{\left( n\right) }_{x\left( t \right) }\left( 0\right) \\= & {} \frac{d^{n}\left[ 1+\left( \ln \left( S\left( 0\right) \right) +\nu t\right) z+\frac{{(\ln {\left( S\left( 0\right) \right) +\nu t)}}^{2}z^{2}}{2!}+\cdots \right] }{dz^{n}}*M_{\sigma W\left( t \right) }\left( z\right) \vert _{z=0}\\&+\cdots \cdots +\left[ 1+(\ln {\left( S\left( 0\right) \right) +\nu t)}z+\frac{{(\ln {\left( S\left( 0\right) \right) +\nu t)}}^{2}z^{2}}{2!}+\cdots \right] \vert _{z=0}\\&\times \,M^{\left( n \right) }_{\sigma W\left( t\right) }\left( 0\right) \\= & {} cE\left( \left( \sigma W\left( t\right) \right) ^{n}\right) \\= & {} {(\ln {\left( S\left( 0\right) \right) +\nu t)}}^{n}+\left[ 1+(\ln {\left( S\left( 0\right) \right) +\nu t)}z\right. \\&\left. +\frac{{(\ln {\left( S\left( 0\right) \right) +\nu t)}}^{2}z^{2}}{2!}+\cdots \right] \vert _{z=0}\\&\times \,M^{\left( n \right) }_{\sigma W\left( t\right) }\left( 0\right) =(\ln \left( S\left( 0 \right) \right) +\nu t)^{n}+E\left( \left( \sigma W\left( t \right) \right) ^{n}\right) \\= & {} {\left( \ln \left( S\left( 0\right) \right) +\nu t\right) }^{n}+\sigma ^{n}E\left( {(W\left( t\right) )}^{n}\right) \end{aligned}$$

Notice that the Taylor expansion only serves as an auxiliary step in the proof. The equation is exact and no approximation is required.

Proof of Part a:

By the definition of a moment generating function, $M_{x\left( t \right) }\left( z\right) =E\left( {\mathrm {e}}^{x\left( t\right) z} \right)$. Since $x\left( t\right) =\ln \left( S\left( t\right) \right)$, $M_{x\left( t\right) }\left( z\right) =E \left( {\mathrm {e}}^{x\left( t\right) z}\right) =E\left( {\mathrm {e}}^{\ln \left( S\left( t\right) \right) z}\right) =E({S\left( t\right) }^{z})$. Therefore, $E\left( {S\left( t\right) }^{n}\right) =M_{x\left( t \right) }\left( n\right)$. From the proof of Part b, we know that

$$\begin{aligned} E\left( {S\left( t\right) }^{n}\right)= & {} M_{x\left( t\right) }\left( n \right) =\hbox {e}^{(\ln {\left( S\left( 0\right) \right) + \nu t)}n}{M_{\sigma W\left( t\right) }\left( n\right) }={S\left( 0 \right) }^{n}{\mathrm {e}}^{\nu tn}{\times \,M}_{\sigma W\left( t \right) }\left( n\right) \\= & {} {S\left( 0\right) }^{n}{\mathrm {e}}^{\nu tn}\times \left( M_{W\left( t\right) }\left( z\right) \right) ^{\sigma ^{2}} \end{aligned}$$

$\square$

Appendix 2

RMSE is used as a goodness-of-fit measure for the CDF of the constructed trinomial lattice at the final stage in comparison to the theoretical underlying distributions. In this case, it can be shown that for overall distribution fit based on optimization of RMSE, the optimal $\upalpha _{1}$ and thence the lattice structure is independent of parameters of the underlying GBM process.

$$\begin{aligned} {\mathrm {RMSE}}= & {} \left( {\mathrm {MSE}}\right) ^{\frac{1}{2}}\\ {\mathrm {MSE}}= & {} \sum \limits _j {p_{T,j}\left( {CDF}^{ lattice}\left( S_{T,j}\right) -{CDF}^{theoretical}\left( S_{T,j}\right) \right) ^{2}, j=0,\ldots ,2T} \end{aligned}$$

Since the lattice of the GBM process is transferred from the underlying Wiener process, ${{CDF}^{ lattice}(S}_{T,j})={{CDF}^{ lattice}(W}_{T,j})= \sum \nolimits _i^{W_{T,i}\le W_{T,j}} p_{T,i}$.

Therefore, either $p_{T,j}$ or ${CDF}^{ lattice}(S_{T,j})$ is a function of parameters of the underlying GBM process. If ${CDF}^{ theoretical}(S_{T,j})$ is also independent of parameters of the underlying GBM process, then MSE hence RMSE is independent of parameters of the underlying GBM process.

Since the CDF of the lognormal distribution is the same as the CDF of the normal distribution, with log x substituted for x,

$$\begin{aligned}&\hbox {CDF}_{ lognormal}(x) = \hbox {CDF}_{ normal}(\hbox {log} x)\\&{CDF}^{ theoretical}(S_{T,j})= {CDF}^{ theoretical}({\mathrm {ln}}(S_{T,j}))\\&\ln \left( S\left( T\right) \right) \sim N(\ln {\left( S\left( 0\right) \right) +\nu T,\sigma ^{2}T}), \hbox {and}\\&{\mathrm {ln}}(S_{T,j})=S\left( 0\right) +\nu T+\sigma {\sqrt{T} {\mathrm {W}}}_{T,j}\\&{CDF}^{ theoretical}(S_{T,j})= {CDF}^{ theoretical}({\mathrm {ln}}(S_{T,j}))\\&\quad = {CDF}^{ theoretical}\left( \frac{S\left( 0\right) +\nu T+\sigma {\sqrt{T} {\mathrm {W}}}_{T,j}-\left( S\left( 0\right) +\nu T \right) }{\sigma \sqrt{T}} \right) \\&\quad ={CDF}^{ theoretical}({\mathrm {W}}_{T,j}). \end{aligned}$$

Therefore, RMSE of the CDF of the constructed trinomial lattice at the final stage in comparison to the theoretical underlying distributions is independent of parameters of the underlying GBM process, so is the optimal $\upalpha _{1}$ for overall distribution fit optimization of RMSE.

Appendix 3

If RMSE is used as a goodness-of-fit measure for option price (e.g, call option price) from the constructed trinomial lattice at the final stage in comparison to the theoretical BSM model, the optimal lattice structure is dependent on the parameters of the underlying GBM process.

$$\begin{aligned} {\mathrm {RMSE}}= & {} \left( {\mathrm {MSE}}\right) ^{\frac{1}{2}}\\ {\mathrm {MSE}}= & {} \sum \nolimits _j {p_{T,j}\left( {Max(0,S}_{T,j}-K)-{BS(S}_{0},K,r,\delta ,\sigma ,T)\right) ^{2}} , j=0,\ldots ,2T, \end{aligned}$$

and,

$$\begin{aligned} S_{T,j}=e^{S\left( 0\right) +\nu T+\sigma {\sqrt{T} {\mathrm {W}}}_{T,j}} \end{aligned}$$

Similarly, if ${\mathrm {RMSE}}_{\Delta }$ is used as a goodness-of-fit measure for Greeks of option price (e.g, Delta) from the constructed trinomial lattice at the final stage in comparison to the theoretical BSM model, the optimal lattice structure is dependent on the parameters of the underlying GBM process.

$$\begin{aligned} {\mathrm {RMSE}}_{\Delta }= & {} \left( {\mathrm {MSE}}_{\Delta } \right) ^{\frac{1}{2}}\\ {\mathrm {MSE}}_{\Delta }= & {} \frac{C_{T,2T}-C_{T,0}}{S_{T,2T} -S_{T,0}}-e^{-\delta T}N(d_{1})\\ \!= & {} \!\left( \frac{Max\left( 0,S_{T,2T}-K\right) \!-\!Max\left( 0,S_{T,0} -K\right) }{S_{T,2T}-S_{T,0}}\!-\!\Delta _{BS}\left( S_{0},K,r,\delta ,\sigma ,T\right) \right) ^{2} \end{aligned}$$

and,

$$\begin{aligned} d_{1}=\frac{\ln \left( \frac{S_{0}}{K} \right) +vT}{\sigma \sqrt{T}},S_{T,j}=e^{S\left( 0 \right) +\nu T+\sigma {\sqrt{T} {\mathrm {W}}}_{T,j}}, j=0,\ldots ,2T, \end{aligned}$$

Therefore, we know that RMSE $({\mathrm {RMSE}}_{\Delta })$ will be a function of the parameters of the underlying GBM process. The relationship between the optimal RMSE $({\mathrm {RMSE}}_{\Delta })$ and the parameters of the underlying GBM process are complex as illustrated in Sect. 4.