A reduced lattice model for option pricing under regime-switching

Abstract

We present a binomial approach for pricing contingent claims when the parameters governing the underlying asset process follow a regime-switching model. In each regime, the asset dynamics is discretized by a Cox–Ross–Rubinstein lattice derived by a simple transformation of the parameters characterizing the highest volatility tree, which allows a simultaneous representation of the asset value in all the regimes. Derivative prices are computed by forming expectations of their payoffs over the lattice branches. Quadratic interpolation is invoked in case of regime changes, and the switching among regimes is captured through a transition probability matrix. An econometric analysis is provided to pick reasonable volatility values for option pricing, for which we show some comparisons with the existing models to assess the goodness of the proposed approach.

Appendix

1.1 The quadratic interpolation scheme

The option price c ₀(i, j) in the high-volatility regime is computed as described in (12) by discounting, at the regime 0 risk-free rate r ₀ on the time interval $\Updelta t$, the option values associated with the asset values Su ^j+1 d ^i−j and Su ^j d ^i+1−j, in each one of the two regimes, taking into account the corresponding occurrence probabilities and the transition or persistence probabilities. Clearly, option prices are available on the lattice for regime 0 at time $(i+1)\Updelta t$, because Su ^j+1 d ^i−j and Su ^j d ^i+1−j are associated to the node (i + 1, j + 1) and (i + 1, j), respectively (see Fig. 3). It may also be the case that Su ^j+1 d ^i−j and Su ^j d ^i+1−j coincide with the asset values associated to two generic nodes of the low-volatility lattice and the corresponding option values would be immediately available. Contrary, if they do not coincide with any value at time $(i+1)\Updelta t$ of the lattice in regime 1, we consider a simple approximation of the option prices associated to the risky asset values Su ^j+1 d ^i−j and Su ^j d ^i+1−j, respectively. We propose a simple quadratic interpolation technique working on the known option prices at time $(i+1)\Updelta t$ of the lattice discretizing the asset value in regime 1. More in detail, consider the asset value Su ^j+1 d ^i−j. To apply the quadratic interpolation scheme, we need to select three values among the ones associated to the nodes lain at time $(i+1)\Updelta t$ in the lattice discretizing the asset value under regime 1, $S_1(i+1,l),l=0,\ldots,i+1.$ Two of such three values, S ₁(i + 1, l _u ) and S ₁(i + 1, l _u  + 1), are the closest ones to Su ^j+1 d ^i−j, and are such that S ₁(i + 1, l _u ) < Su ^j+1 d ^i−j ≤ S ₁(i + 1, l _u  + 1). They are identified by the integer l _u which satisfies the following relation,

$$ Su^{\varsigma l_u}d^{\varsigma(i+1-l_u)}<Su^{j+1}d^{i-j}\leq Su^{\varsigma l_u+1}d^{\varsigma(i-l_u)}, \quad \hbox{leading to}\quad l_u=\left\lfloor\frac{2j-i+1+(i+1)\varsigma}{2\varsigma}\right\rfloor,$$

(15)

where ⌊ x ⌋ indicates the greatest integer smaller than or equal to x. The third value is determined as follows:

• if l _u  ≤ i − 1, we choose between S ₁(i + 1, l _u  + 2) and S ₁(i + 1, l _u  − 1) the closest value to Su ^j+1 d ^i−j;

• if l _u  > i − 1, it is given by S ₁(i + 1, l _u  − 1) because in this case we are near to the upper hedge of the lattice where the maximum value assumed by the node index at the (i + 1)th time step is i + 1.

Fig. 3

c ₀(i + 1, j + 1) and c ₀(i + 1, j) are the option prices associated with the asset values Su ^j+1 d ^i−j and Su ^j d ^i+1−j in the regime 0 lattice, respectively. Consequently, no interpolation is required if regime 0 persists

Then, the option price, $\overline{c}_1(i+1,\,j^u),$ associated to Su ^j+1 d ^i−j is computed by interpolating (see Fig. 4) the option prices c ₁(i + 1, l _u ), c ₁(i + 1, l _u  + 1) and c ₁(i + 1, l _u  + 2) (or c ₁(i + 1, l _u  − 1)) available when the asset values are S ₁(i + 1, l _u ), S ₁(i + 1, l _u  + 1) and S ₁(i + 1, l _u  + 2) (or S ₁(i + 1, l _u  − 1)), that is

$$ \begin{aligned} \overline{c}_1(i+1,\,j^u)&=c_1(i+1,l_u)+\frac{Su^{j+1}d^{i-j}-S_1(i+1,l_u)}{S_1(i+1,l_u+1)-S_1(i+1,l_u)}[c_1(i+1,l_u+1)-c_1(i+1,l_u)]\\ &\quad+\frac{[Su^{j+1}d^{i-j}-S_1(i+1,l_u)][Su^{j+1}d^{i-j}-S_1(i+1,l_u+1)]}{[S_1(i+1,l_u+2)-S_1(i+1,l_u)][S_1(i+1,l_u+2)-S_1(i+1,l_u+1)]}[c_1(i+1,l_u+2)-c_1(i+1,l_u)]\\ &\quad-\frac{[Su^{j+1}d^{i-j}-S_1(i+1,l_u)][Su^{j+1}d^{i-j}-S_1(i+1,l_u+1)]}{[S_1(i+1,l_u+1)-S_1(i+1,l_u)][S_1(i+1,l_u+2)-S_1(i+1,l_u+1)]}[c_1(i+1,l_u+1)-c_1(i+1,l_u)]. \end{aligned} $$

(16)

Fig. 4

$\overline{c}_1(i+1,\,j^u)$ and $\overline{c}_1(i+1,\,j^d)$ are the option prices associated with the asset values Su ^j+1 d ^i−j and Su ^j d ^i+1−j, respectively, if the regime switches from regime 0 to regime 1. In general, both such asset values are not among the ones associated to the nodes of the regime 1 lattice at the (i + 1)-th time step, $S_1(i+1,\,j),\,j=0,\ldots,i+1.$ Consequently, the option values $\overline{c}_1(i+1,\,j^u)$ and $\overline{c}_1(i+1,\,j^d)$ are approximated by using a quadratic interpolation technique

We remark that for some values of i and j, Su ^j+1 d ^i−j would be larger than S ₁(i + 1, i + 1) or smaller than S ₁(i + 1, 0) because we transit from the high-volatility regime to the low-volatility one which, on the upper hedge, is characterized by lattice node values smaller than the ones characterizing the upper hedge of the lattice for regime 0, and on the lower hedge it is characterized by lattice node values larger than the ones characterizing the lower hedge of the lattice for regime 0. In these cases, $\overline{c}_1(i+1,\,j^u)$ is computed by quadratic extrapolation rather than interpolation. As an example, if Su ^j+1 d ^i−j > S ₁(i + 1, i + 1), we apply (16) on the asset values S ₁(i + 1, i + 1), S ₁(i + 1, i), and S ₁(i + 1, i − 1) and the corresponding option values c ₁(i + 1, i + 1), c ₁(i + 1, i), and c ₁(i + 1, i − 1).

A similar procedure is used to compute $\overline{c}_1(i+1,\,j^d),$ at first by selecting three asset values on the regime 1 lattice closest to Su ^j d ^i+1−j, i.e., S ₁(i + 1, l _d ), S ₁(i + 1, l _d  + 1), and S ₁(i + 1, l _d  + 2) (or S ₁(i + 1, l _d  − 1)), and then by interpolating the option prices c ₁(i + 1, l _d ), c ₁(i + 1, l _d  + 1), and c ₁(i + 1, l _d  + 2) (or c ₁(i + 1, l _d  − 1)) associated with them (see Fig. 4). For some values of i and j, Su ^j d ^i+1−j would be larger than S ₁(i + 1, i + 1) or smaller than S ₁(i + 1, 0) because, once again, we transit from the high-volatility regime to the low-volatility one. In these cases, $\overline{c}_1(i+1,\,j^d)$ is computed by quadratic extrapolation.