Hedging via Perpetual Derivatives: Trinomial Option Pricing and Implied Parameter Surface Analysis

Despite known limitations – log-normal prices driven by Brownian motion; absence of the drift term of the underlying in its option price; assumption of the abilities to borrow any monetary amount at the risk-free rate and trade assets of any monetary amount continuously in time with no transaction costs; – the Black-Scholes-Merton (BSM) model [7, 23] continues to serve as a fundamental reference tool in option pricing. Our interest here is in discrete tree models, which address option pricing without dealing with the machinery of stochastic integration theory. As real trading occurs over (perhaps very short, but nonetheless) discrete time intervals, such models avoid continuous-time assumptions and engender a more realistic pricing model.

As introduced by [27] and formalized by [9], the basic discrete model, employing a recombining binomial tree, was specifically designed to converge to the BSM model as $\Delta t\downarrow 0$. Binomial pricing models have undergone continued development, including providing faster rates of convergence [20] and efficient computation of the “Greeks” [30], as well as addressing the inclusion of stochastic volatility [13, 2], skewness and kurtosis [26], and jump processes [5, 2]. [18] extended the basic Cox-Ross-Rubenstein model to a new version with time-dependent parameters. [14] further extended the [18] binomial option pricing model to allow for variable-spaced time increments.

Trinomial trees for option pricing were introduced by [5]. As with original formulations of binomial models [9, 16], trinomial trees were developed specifically to converge to the BSM option price formula in the continuous-time limit. By adding a third option to the pricing tree (that of no price change over a discrete time interval), trinomial tree models provide a richer state space and the potential for an improved rate of convergence to the BSM solution (compared to binomial models).¹¹1 See, however, the results of [8] which show that the Tian third-order moment binomial tree model outperforms eight other trinomial tree models. A number of trinomial (and, by natural extension, multinomial) tree models have been developed subsequently [6, 3, 24, 17, 4, 12, 10, 31, 25, 19, 18]. Convergence rate studies of trinomial models have been examined theoretically and numerically [1, 25, 15, 22].

A fundamental problem with the published trinomial (and multinomial) trees is that they are defined directly in the risk-neutral world. The free parameters of the model - the directional price change factors and probabilities - are fit to the risk-neutral BSM model. Consequently connection to crucial natural world parameters (the price drift and directional change probabilities) are lost. This connection is lost because no hedging is performed (i.e., no replicating portfolio is developed). This issue was first addressed in [18] for the specific case of the Cox-Ross-Rubenstein model. However, their trinomial model is not market-complete. The purpose of this paper is to address the market-completeness issue in the context of a fairly general trinomial model. To ensure market completeness, we work within a market consisting of two risky assets, a riskless asset and a European contingent claim (call or put option). The market uncertainty is driven by a single Brownian motion. To ensure this, the two risky assets consist of a stock and a derivative based on that stock. As we wish to price the option for any possible maturity date, the stock derivative is chosen to be a perpetual derivative. We develop the replicating portfolio producing risk-neutral pricing. The resulting unique relationship between the risk-neutral and real world parameters enables computation of real world implied parameter values.

In Section 2, we briefly review the price dynamics of the stock perpetual derivative [29, 21]. In Section 3 we develop our general trinomial tree model, establishing the unique relationship between risk-neutral and real-world parameters. In Section 4 we discuss calibration of the model’s natural-world parameters to empirical data in the cases in which asset returns are either arithmetic or continuous (i.e., log-returns). Application of the model to empirical data is presented in Section 5. We consider historical stock and option prices for three large cap stocks and develop implied surfaces for the following parameters: volatility, price drift, price change probabilities, and the risk-free rate. Conclusions are presented in Section 6.

Consider a market containing a stock $\mathcal{S}$ having price dynamics

$$dS_{t}=\mu_{t}S_{t}dt+\sigma_{t}S_{t}dW_{t},$$

(1)

where $W_{t}$ is a standard Brownian motion, $\mu_{t}$ is a drift and $\sigma_{t}$ is a volatility. The dynamics of a riskless asset (bond) ${\mathcal{B}}$ is

$$dB_{t}=r_{f,t}B_{t}dt,$$

(2)

where $r_{f,t}$ is a risk-free rate. Let $\mathcal{D}$ denote a perpetual derivative of $\mathcal{S}$ whose price, $g(S_{t},t)$, is governed by the Itô process

$$dg_{t}=\left(\frac{\partial g_{t}}{\partial t}+\mu_{t}S_{t}\frac{\partial g_{t}}{\partial S_{t}}+\frac{\sigma^{2}_{t}S_{t}^{2}}{2}\frac{\partial^{2}g_{t}}{\partial S_{t}^{2}}\right)dt+\sigma_{t}S_{t}\frac{\partial g_{t}}{\partial S_{t}}dW_{t},$$

(3)

ensuring that uncertainty in the prices of $\mathcal{S}$ and $\mathcal{D}$ are driven by the same Brownian motion. To ensure that $\mathcal{D}$ can be priced, form a replicating portfolio $\pi_{t}^{(\mathcal{D})}=a_{t}S_{t}+b_{t}B_{t}-g_{t}$. Requiring $\pi_{t}^{(\mathcal{D})}=0$ and $d\pi_{t}^{(\mathcal{D})}=0$ leads, in the standard way, to the BSM PDE for the price dynamics of $\mathcal{D}$,

$$r_{f,t}g_{t}=\frac{\partial g_{t}}{\partial t}+r_{f,t}S_{t}\frac{\partial g_{t}}{\partial S_{t}}+\frac{\sigma^{2}_{t}S_{t}^{2}}{2}\frac{\partial^{2}g_{t}}{\partial S_{t}^{2}},$$

(4)

with initial data $g_{0}(S_{0},0)$. [21] investigated separable solutions to (4) and showed the existence of a one-parameter family of solutions. Of these solutions, the price process

$$g_{t}(S_{t},t)=S_{t}^{-\delta_{t}},\qquad\delta_{t}=\frac{2r_{f,t}}{\sigma^{2}_{t}},$$

(5)

for the perpetual derivative $\mathcal{D}$ has the dynamics

$$dg_{t}=\mu_{\delta}g_{t}dt+\sigma_{\delta}g_{t}dW_{t},\qquad\mu_{\delta}=(1+\delta_{t})r_{f,t}-\mu_{t}\delta_{t},\quad\sigma_{\delta}=-\delta_{t}\sigma_{t},$$

(6)

ensuring that the uncertainty in $S_{t}$ and $g_{t}$ are driven by the same Brownian motion. This is also the form of the perpetual derivative (assuming the time-independent parameters) used by [29].²²2 Let $\xi\in\mathbb{R}$ be a parameter and $\cal{V}^{\xi}$ denote a perpetual derivative having the price process $\boldsymbol{V}_{t}^{\xi}=S_{t}^{\xi}\beta_{t}^{\gamma},t\geq 0$, where $\gamma=\frac{1-\xi}{r_{f}}\left[r_{f}+\frac{1}{2}\xi\sigma^{2}\right]$. Then the price process $\boldsymbol{V}_{t}^{\xi}$ discounted by a riskless bond rate is a martingale under the EMM $\mathbb{Q}\sim\mathbb{P}$ and thus the security $\cal{V}^{\xi}$ can be traded within the BSM market model. The log-return of this perpetual derivative is a linear combination of the log-returns of the underlying stock $\cal{S}$ and the bond $\cal{B}$. When $\xi=-2r_{f}/\sigma^{2}$, then $\gamma=0$ and the perpetual derivative price becomes independent of the bond price.

Consider the market $\{\mathcal{S},\mathcal{D},\mathcal{B},\mathcal{C}\}$, where $\mathcal{C}$ is a European contingent claim (option). We model the price development of $\mathcal{S}$, $\mathcal{D}$ and $\mathcal{B}$ on a trinomial tree and use a replicating portfolio under no-arbitrage conditions to determine the price of $\mathcal{C}$. We develop a general trinomial pricing model first, and then consider the two special cases in which returns are treated either as arithmetic and logarithmic. For simplicity we assume a constant time increment $\Delta t=T/N$ for the lattice, where $T$ is the maturity date of the option. The notation for the general trinomial pricing tree is summarized in Fig. 1.

On the “fundamental unit” of the trinomial tree,³³3 This is often referred to as a “single period” tree. However, a single period tree would imply $k=0$. We prefer the designation fundamental unit, as the tree is assembled by replication of this unit. In the binomial tree literature, it has become convention to adopt $S_{k+1}^{(u)}$ and $S_{k+1}^{(d)}$ as the generic price changes. This convention does not extend naturally to multinomial trees. The level indexing $S_{k+1}^{(i+1)}$, $S_{k+1}^{(i)}$ and $S_{k+1}^{(i-1)}$ employed here does extend naturally to multinomial (including binomial) trees, and we prefer it. To be consistent with a general multinomial tree nomenclature, our price change probabilities should be written $p_{u,k}\rightarrow p_{+1,k}$, $p_{m,k}\rightarrow p_{0,k}$, and $p_{d,k}\rightarrow p_{-1,k}$. We confess to being inconsistent in adopting the most general notation. the stock price follows the discrete process,

$$S_{k+1}^{(i)}=\begin{cases}\begin{aligned} S_{k+1}^{(i+1)}&=S_{k}^{(i)}u_{k}&&\text{ w.p. }p_{u,k},\\ S_{k+1}^{(i)}&=S_{k}^{(i)}&&\text{ w.p. }p_{m,k},\\ S_{k+1}^{(i-1)}&=S_{k}^{(i)}d_{k}&&\text{ w.p. }p_{d,k},\end{aligned}\end{cases}$$

(7)

where we adopt the shortened notation $S_{k}^{(i)}=S_{k\Delta t}^{(i)}$, $u_{k}=u_{k\Delta t}$, $p_{u,k}=p_{u,k\Delta t}$, etc., $k=0,1,...,N$. The price change probabilities in the natural world are determined by independent, trinomially distributed random variables $\zeta_{k}$ satisfying: $p_{u,k}=P(\zeta_{k}=1)$, $p_{m,k}=P(\zeta_{k}=0)$, and $p_{d,k}=P(\zeta_{k}=-1)$ where $p_{u,k}+p_{m,k}+p_{d,k}=1$, $k=1,...,N$. The pricing trees in this trinomial model are adapted to the discrete filtration

$$\mathbb{F}^{(N)}=\left\{{\cal F}^{(N,k)}=\sigma(\zeta_{1},...,\zeta_{k}),\ k=1,...,N;\ {\cal F}^{(N,0)}=\{\emptyset,\Omega\}\right\}.$$

(8)

The probabilities $p_{u,k}$, $p_{m,k}$ and $p_{d,k}$ are ${\cal F}^{(N,k)}$-measurable. The perpetual derivative price follows the discrete process

$$\left(S_{k+1}^{(i)}\right)^{\gamma_{k}}=\begin{cases}\begin{aligned} \left(S_{k+1}^{(i+1)}\right)^{\gamma_{k}}&=\left(S_{k}^{(i)}\right)^{\gamma_{k}}u^{\gamma_{k}}_{k}&&\text{ w.p. }p_{u,k},\\ \left(S_{k+1}^{(i)}\right)^{\gamma_{k}}&=\left(S_{k}^{(i)}\right)^{\gamma_{k}}&&\text{ w.p. }p_{m,k},\\ \left(S_{k+1}^{(i-1)}\right)^{\gamma_{k}}&=\left(S_{k}^{(i)}\right)^{\gamma_{k}}d^{\gamma_{k}}_{k}&&\text{ w.p. }p_{s,k},\end{aligned}\end{cases}$$

(9)

where, for notational simplicity, we denote ${\gamma_{k}}=-\delta_{k}=-2r_{f,k}/\sigma^{2}_{k}$. The dynamics of the bond price is

$$B_{k+1}^{(i)}=\begin{cases}\begin{aligned} B_{k+1}^{(i+1)}&=B_{k}^{(i)}R_{f,k}&&\text{ w.p. }p_{u,k},\\ B_{k+1}^{(i)}&=B_{k}^{(i)}R_{f,k}&&\text{ w.p. }p_{m,k},\\ B_{k+1}^{(i-1)}&=B_{k}^{(i)}R_{f,k}&&\text{ w.p. }p_{d,k}.\end{aligned}\end{cases}$$

(10)

At time $t=k\Delta t$, let $a_{k}$, $b_{k}$ and $c_{k}$ represent the number of respective shares of $\mathcal{S}$, $\mathcal{B}$ and $\mathcal{D}$ held in a portfolio used to replicate the price of the option ${\mathcal{C}}$ having ${\mathcal{S}}$ and ${\mathcal{D}}$ as underlying. Over the single time–step $k\rightarrow k+1$, the arbitrage–free, replicating portfolio obeys

	$\displaystyle a_{k}S_{k}^{(i)}+b_{k}B_{k}^{(i)}+c_{k}\left(S_{k}^{(i)}\right)^{\gamma_{k}}$	$\displaystyle=f_{k}^{(i)},$		(11a)
	$\displaystyle a_{k}S_{k}^{(i)}u_{k}+b_{k}B_{k}^{(i)}R_{f,k}+c_{k}\left(S_{k}^{(i)}\right)^{\gamma_{k}}u^{\gamma_{k}}_{k}$	$\displaystyle=f_{k+1}^{(i+1)},$		(11b)
	$\displaystyle a_{k}S_{k}^{(i)}+b_{k}B_{k}^{(i)}R_{f,k}+c_{k}\left(S_{k}^{(i)}\right)^{\gamma_{k}}$	$\displaystyle=f_{k+1}^{(i)},$		(11c)
	$\displaystyle a_{k}S_{k}^{(i)}d_{k}+b_{k}B_{k}^{(i)}R_{f,k}+c_{k}\left(S_{k}^{(i)}\right)^{\gamma_{k}}d^{\gamma_{k}}_{k}$	$\displaystyle=f_{k+1}^{(i-1)}.$		(11d)

Solution of the system (11b) - (11d), determines the terms $a_{k}S_{k}^{(i)}$, $b_{k}B_{k}^{(i)}$ and $c_{k}\left(S_{k}^{(i)}\right)^{\gamma_{k}}$. From (11a), the recursive formula for the option price is then

$$f_{k}^{(i)}=R_{f,k}^{-1}(q_{u,k}f_{k+1}^{(i+1)}+q_{m,k}f_{k+1}^{(i)}+q_{d,k}f_{k+1}^{(i-1)}),$$

(12)

where the risk-neutral probabilities are

	$\displaystyle q_{u,k}$	$\displaystyle=\frac{(d^{\gamma_{k}}_{k}-d_{k})(R_{f,k}-1)}{D_{1,k}},$		(13a)
	$\displaystyle q_{m,k}$	$\displaystyle=1-q_{u,k}-q_{d,k}=\frac{u^{\gamma_{k}}_{k}(R_{f,k}-d_{k})+R_{f,k}(d_{k}-u_{k})+d^{\gamma_{k}}_{k}(u_{k}-R_{f,k})}{D_{1,k}},$		(13b)
	$\displaystyle q_{d,k}$	$\displaystyle=\frac{(u_{k}-u_{k}^{\gamma_{k}})(R_{f,k}-1)}{D_{1,k}},$		(13c)
	$\displaystyle\text{where}\quad D_{1,k}$	$\displaystyle=(u_{k}-1)d_{k}^{\gamma_{k}}-(u_{k}-d_{k})+(1-d_{k})u_{k}^{\gamma_{k}}.$		(13d)

It is straightforward to show that $u_{k}q_{u,k}+q_{m,k}+d_{k}q_{d,k}=R_{f,k}$ and $u_{k}^{\gamma_{k}}q_{u,k}+q_{m,k}+d_{k}^{\gamma_{k}}q_{d,k}=R_{f,k}$. Consequently

$$\pi_{u,k}\stackrel{{\scriptstyle\rm def}}{{=}}\frac{q_{u,k}}{R_{f,k}},\qquad\pi_{m,k}\stackrel{{\scriptstyle\rm def}}{{=}}\frac{q_{m,k}}{R_{f,k}},\qquad\pi_{d,k}\stackrel{{\scriptstyle\rm def}}{{=}}\frac{q_{d,k}}{R_{f,k}}$$

(14)

are the risk-neutral, single time step, price deflators:

	$\displaystyle B_{k}^{(i)}$	$\displaystyle=(\pi_{u,k}+\pi_{m,k}+\pi_{d,k})R_{f,k}B_{k}^{(i)},$
	$\displaystyle S_{k}^{(i)}$	$\displaystyle=(\pi_{u,k}u_{k}+\pi_{m,k}+\pi_{d,k}d_{k})S_{k}^{(i)},$
	$\displaystyle\left(S_{k}^{(i)}\right)^{\gamma_{k}}$	$\displaystyle=(\pi_{u,k}u_{k}^{\gamma_{k}}+\pi_{m,k}+\pi_{d,k}d_{k}^{\gamma_{k}})\left(S_{k}^{(i)}\right)^{\gamma_{k}}.$

In order to calibrate the parameters to real data, it is necessary to assume a form for the price change parameters $u_{k}$, $d_{k}$ and $R_{f,k}$. These forms must be self-consistent. For arithmetic returns, the self-consistent modeling of the parameters is

$$u_{k}=1+U_{k},\qquad d_{k}=1+D_{k},\qquad R_{f,k}=1+r_{f,k}\Delta t,$$

(15)

while for log-returns the parameters are modeled as

$$u_{k}=e^{U_{k}},\qquad d_{k}=e^{D_{k}},\qquad R_{f,k}=e^{r_{f,k}\Delta t}.$$

(16)

In either case, the no-arbitrage condition requires $D_{k}<r_{f,k}\Delta t<U_{k}$. From (7), the returns (whether arithmetic or logarithmic) are given by

$$r_{k}=\left\{\begin{aligned} U_{k},&\text{ w.p. }p_{u,k},\\ 0,&\text{ w.p. }p_{m,k},\\ D_{k},&\text{ w.p. }p_{d,k}.\end{aligned}\right.$$

(17)

We consider first the calibration of the natural world price change probabilities to historical data. Let $\{r_{j},j=k-L+1,...,k\}$ denote a historical record (i.e. a “window of length L”) of return (arithmetic or logarithmic) as appropriate data for $\cal{S}$. Consider the threshold values $r_{\text{thr}}^{+}\gtrsim 0$ and $r_{\text{thr}}^{-}\lesssim 0$. Denote by: $L_{u,k}$ the number of these historical instances when $r_{j}\geq r_{\text{thr}}^{+}$; $L_{m,k}$ the number when $r_{\text{thr}}^{-}<r_{j}<r_{\text{thr}}^{+}$; and $L_{d,k}$ the number when $r_{j}\leq r_{\text{thr}}^{-}$. The natural probabilities can then be estimated from the historical data as

$$p_{u,k}=\frac{L_{u,k}}{L},\qquad p_{m,k}=\frac{L_{m,k}}{L},\qquad p_{d,k}=1-p_{u,k}-p_{m,k}.$$

(18)

The parameters $U_{k}$ and $D_{k}$ are estimated by setting the conditional first and second moments of $r_{k}$ to the instantaneous mean and variance of the historical return series,

	$\displaystyle E\left[r_{k+1}|S_{k}^{(i)}\right]$	$\displaystyle=U_{k}p_{u,k}+D_{k}p_{d,k}=\mu_{k}^{(r)}\Delta t,$		(19a)
	$\displaystyle\text{Var}\left[r_{k+1}|S_{k}^{(i)}\right]$	$\displaystyle=U_{k}^{2}p_{u,k}+D_{k}^{2}p_{d,k}-\left(E\left[r_{k+1}|S_{k}^{(i)}\right]\right)^{2}=\left(\sigma_{k}^{(r)}\right)^{2}\Delta t.$		(19b)

The instantaneous mean and variance are estimated using the same historical window as for the price change probabilities. Evaluating (19a) and (19b) from (17) produces

	$\displaystyle U_{k}$	$\displaystyle=\frac{1}{1-p_{m,k}}\left\{E\left[r_{k}|S_{k-1}^{(i)}\right]+\sqrt{\frac{p_{d,k}}{p_{u,k}}}\sqrt{\text{Var}\left[r_{k}|S_{k-1}^{(i)}\right]-p_{m,k}E\left[r_{k}^{2}|S_{k-1}^{(i)}\right]}\right\}$
		$\displaystyle=\frac{1}{1-p_{m,k}}\left\{\mu_{k}^{(r)}\Delta t+\sqrt{\frac{p_{d,k}}{p_{u,k}}}\sqrt{(1-p_{m,k})\left(\sigma_{k}^{(r)}\right)^{2}\Delta t-p_{m,k}\left(\mu_{k}^{(r)}\Delta t\right)^{2}}\right\},$		(20a)
	$\displaystyle D_{k}$	$\displaystyle=\frac{1}{1-p_{m,k}}\left\{E\left[r_{k}|S_{k-1}^{(i)}\right]-\sqrt{\frac{p_{u,k}}{p_{d,k}}}\sqrt{\text{Var}\left[r_{k}|S_{k-1}^{(i)}\right]-p_{m,k}E\left[r_{k}^{2}|S_{k-1}^{(i)}\right]}\right\}$
		$\displaystyle=\frac{1}{1-p_{m,k}}\left\{\mu_{k}^{(r)}\Delta t-\sqrt{\frac{p_{u,k}}{p_{d,k}}}\sqrt{(1-p_{m,k})\left(\sigma_{k}^{(r)}\right)^{2}\Delta t-p_{m,k}\left(\mu_{k}^{(r)}\Delta t\right)^{2}}\right\}.$		(20b)

When $p_{m,k}=0$, $p_{u,k}\equiv p_{k}$, $p_{d,k}=1-p_{k}$, and (20a), (20b) reduce to the binomial tree solutions,

$$U_{k}=\mu_{k}^{(r)}\Delta t+\left[\frac{1-p_{k}}{p_{k}}\right]^{1/2}\sigma_{k}^{(r)}\sqrt{\Delta t},\qquad D_{k}=\mu_{k}^{(r)}\Delta t-\left[\frac{p_{k}}{1-p_{k}}\right]^{1/2}\sigma_{k}^{(r)}\sqrt{\Delta t}.$$

The risk-neutral probabilities are computed from (13a) to (13d) using (20a), (20b) and either (15) or (16), as appropriate.

From (7), for arithmetic returns, the conditional mean and the variance of the stock price are

	$\displaystyle\mathbb{E}\left[S_{k+1}|S_{k}^{(i)}\right]$	$\displaystyle=S_{k}^{(i)}\left(1+E\left[r_{k+1}|S_{k}^{(i)}\right]\right)=S_{k}^{(i)}\left(1+\mu_{k}^{(r)}\Delta t\right),$
	$\displaystyle\mathbb{V}\left[S_{k+1}|S_{k}^{(i)}\right]$	$\displaystyle=\left(S_{k}^{(i)}\right)^{2}\text{Var}\left[r_{k+1}|S_{k}^{(i)}\right]=\left(S_{k}^{(i)}\right)^{2}\left(\sigma_{k}^{(r)}\right)^{2}\Delta t,$

and the instantaneous drift and variance of the risky asset price and arithmetic return processes are identical,

$$\mu_{k}=\mu_{k}^{(r)},\qquad\sigma_{k}^{2}=\left(\sigma_{k}^{(r)}\right)^{2}.$$

(21)

However, for log-returns, there is no simple relation between conditional mean and the variance of the stock price

	$\displaystyle\mathbb{E}\left[S_{k+1}|S_{k}^{(i)}\right]$	$\displaystyle=S_{k}^{(i)}\left(p_{u,k}e^{U_{k}}+p_{m,k}+p_{d,k}e^{D_{k}}\right),$		(22a)
	$\displaystyle\mathbb{V}\left[S_{k+1}|S_{k}^{(i)}\right]$	$\displaystyle=\left(S_{k}^{(i)}\right)^{2}\left(p_{u,k}e^{2U_{k}}+p_{m,k}+p_{d,k}e^{2D_{k}}\right)-\left(E\left[S_{k+1}|S_{k}^{(i)}\right]\right)^{2},$		(22b)

and the conditional mean and variance of the log-return (19a), (19b). Under the assumption that terms of $o(\Delta t)$ can be neglected, the exponentials in (22a) and (22b) can be expanded producing the results

	$\displaystyle\mathbb{E}\left[S_{k+1}|S_{k}^{(i)}\right]$	$\displaystyle=S_{k}^{(i)}\left\{1+\left(\mu_{k}^{(r)}+\frac{\left(\sigma_{k}^{(r)}\right)^{2}}{2}\right)\Delta t\right\},$
	$\displaystyle\mathbb{V}\left[S_{k+1}|S_{k}^{(i)}\right]$	$\displaystyle=\left(S_{k}^{(i)}\right)^{2}\left(\sigma_{k}^{(r)}\right)^{2}\Delta t.$

Thus to terms of $O(\Delta t)$, for log–returns, the instantaneous drift and variance of the price of the risky asset $\cal{S}$ are

$$\mu_{k\Delta t}=\mu_{k\Delta t}^{(r)}+\frac{(\sigma_{k\Delta t}^{(r)})^{2}}{2},\qquad\sigma_{k\Delta t}^{2}=(\sigma_{k\Delta t}^{(r)})^{2}.$$

We consider the continuous time $\Delta t\downarrow 0$ limits of the trinomial tree price processes. Let $\lim_{\Delta t\downarrow 0}k\Delta t=t\in[0,T]$. As $\Delta t\downarrow 0$, $\mu_{k}^{(r)}\rightarrow\mu_{t}^{(r)}$, $\sigma_{k}^{(r)}\rightarrow\sigma_{t}^{(r)}$, $r_{f,k}^{(r)}\rightarrow r_{f,t}$; and $\gamma_{k}\rightarrow\gamma_{t}$, where we assume that the second derivatives of $\mu_{t}^{(r)}$, $\sigma_{t}^{(r)}$, $r_{f,t}$, and $\gamma_{t}$ are continuous on $[0,T]$.⁴⁴4 We impose sufficient conditions. A non-standard invariance principle [11] can be used to show that, under arithmetic returns, the pricing tree (7) generates a stochastic process which converges weakly in $D[0,T]$ topology [28] to the cumulative return process $R_{t}$ determined by

$$dR_{t}=dS_{t}/S_{t}=\mu_{t}^{(r)}dt+\sigma_{t}^{(r)}dW_{t}.$$

Under log-returns, the pricing tree (7) generates a càdlàg process which converges weakly in $D[0,T]$ topology to a continuous diffusion process governed by the stochastic differential equation

$$dS_{t}=\left(\mu_{t}^{(r)}+\frac{1}{2}\left(\sigma_{t}^{(r)}\right)^{2}\right)S_{t}dt+\sigma_{t}^{(r)}S_{t}dW_{t}.$$

In either case, the deterministic bond pricing tree (10) converges uniformly to

$$B_{t}=B_{0}e^{\int_{0}^{t}r_{f,s}\,ds}.$$

Using a window $\{t-L+1,...,t\}$ of historical data, we utilize the trinomial tree model with arithmetic returns to compute call option price surfaces for day $t$ (Section 5.1).⁹⁹9 We compute prices for European options. When we compute implied parameter values, we will use empirical prices for American options. As call option prices for European and American options are identical, but put option prices differ, we consider only call option prices and implied parameter values based on call options. Using published option price data for $t$, we compute implied parameter surfaces, specifically for volatility (Section 5.2), mean (Section 5.3), risk-free rate (Section 5.4), and price change probabilities (Section 5.5). Implied surfaces for $p_{d}$ and $p_{m}$ based on the extreme thresholds are provided in Section 5.6. The historical window covered the trading days from 01/16/2020 through 01/16/2024. We considered three stocks, Apple (AAPL), Amazon (AMZN) and Microsoft (MSFT).¹⁰¹⁰10 Stock and option price data obtained from Yahoo Finance. Accessed on 01/17/2024. The risk-free rate $r_{f,t}$ for the date 01/16/2024 was taken from the US Treasury 10-year yield curve. For each stock, the initial price used in computing options was the adjusted closing price on 01/16/2024. To eliminate dividend artifacts, all returns were computed from adjusted closing prices.

As noted in Section 4.1, estimation of values for $r_{\text{thr}}^{-}$ and $r_{\text{thr}}^{+}$ depends on the significance level $\alpha$ used in hypothesis tests. Using $\delta p=1$ basis point, Table A.1 in the appendix shows how $r_{\text{thr}}^{-}$ and $r_{\text{thr}}^{+}$ vary for these three stocks for values of $\alpha\in\{0.05,0.01,0.005,0.001\}$. We use the values $r_{\text{thr}}^{-}$ and $r_{\text{thr}}^{+}$ obtained for $\alpha=0.001$, which establishes a strong criterion for rejecting the null hypothesis.