--- title: Option Pricing Functions to Accompany *Derivatives Markets* author: Robert McDonald date: "`r Sys.Date()`" output: bookdown::pdf_document2: toc: true number_sections: true bookdown::html_document2: toc: true number_sections: true # pdf_document: # toc: true # number_sections: true md_document: variant: gfm toc: true bibliography: "derivmkts.bib" header-includes: \usepackage{tikz} \usetikzlibrary{positioning} vignette: > %\VignetteEncoding{UTF-8} %\VignetteIndexEntry{Option Pricing Functions to Accompany *Derivatives Markets*} %\VignetteEngine{knitr::rmarkdown} geometry: 'margin=1.25in' fontsize: 12pt --- ```{r echo=FALSE, message=FALSE, warning=FALSE} rm(list=ls()) library(bookdown) library(knitr) library(ggplot2) library(dplyr) library(tidyr) library(kableExtra) ##homedir <- '/home/rmcd/tex/d67/Rtutorial/' options(digits=4) figsize <- 4.5 opts_chunk$set(size='footnotesize', prompt=FALSE, comment=NA ##,fig.align='center', ## fig.width = figsize, ## fig.height=figsize, ## out.width='3.75in' ) opts_knit$set(#eval.after='fig.cap', prompt=TRUE, #renderer=renderer_latex(document=FALSE), size='footnotesize') curr <- function(amt) formatC(amt, format='f', digits=2) ``` ```{r echo=FALSE} library(derivmkts) library(mnormt) library(markdown) opts_chunk$set(collapse=TRUE) ``` # Introduction This vignette is an overview to the functions in the *derivmkts* package, which was conceived as a companion to my book *Derivatives Markets* [@mcdonald:derivs:13]. The material has an educational focus. There are other option pricing packages for R, but this package has several distinguishing features: * function names (mostly) correspond to those in \emph{Derivatives Markets}. * vectorized Greek calculations are convenient both for individual options and for portfolios * the `simprice` function produces simulated asset prices * the `quincunx` function illustrates the workings of a quincunx (Galton board). * binomial functions include a plotting function that provides a visual depiction of early exercise # Pricing functions and greeks ## European Calls and Puts Table \@ref(tab:bslist) lists the Black-Scholes related functions in the package.^[See @black/scholes:73 and @merton:73-bell.] The functions `bscall`, `bsput`, and `bsopt` provide basic pricing of European calls and puts. There are also options with binary payoffs: cash-or-nothing and asset-or-nothing options. All of these functions are vectorized. The function `bsopt` by default provides option greeks. Here are some examples: ```{r } s <- 100; k <- 100; r <- 0.08; v <- 0.30; tt <- 2; d <- 0 bscall(s, k, v, r, tt, d) bsput(s, c(95, 100, 105), v, r, tt, d) ``` ```{r bslist, echo=FALSE, eval=TRUE} bstbl <- data.frame( Function = c('bscall', 'bsput', 'bsopt', 'assetcall', 'assetput', 'cashcall', 'cashput'), Description = c('European call', 'European put', 'European call and put and associated Greeks: delta, gamma, vega, theta, rho, psi, and elasticity', 'Asset-or-nothing call', 'Asset-or-nothing put', 'Cash-or-nothing call', 'Cash-or-nothing put')) kbl(bstbl, caption = 'Black-Scholes related option pricing functions', booktabs=TRUE) |> kable_styling() ``` ```{block2, include=FALSE} \begin{table}[tp] \centering \begin{tabular}{cp{4in}} Function& Description \\ \hline bscall & European call\\ bsput & European put\\ bsopt & European call and put and associated Greeks: delta, gamma, vega, theta, rho, psi, and elasticity \\ assetcall & Asset-or-nothing call\\ assetput & Asset-or-nothing put\\ cashcall & Cash-or-nothing call\\ cashput & Cash-or-nothing put \end{tabular} \caption{Black-Scholes related option pricing functions} \label{tab:bslist} \end{table} ``` ## Barrier Options There are pricing functions for the following barrier options:^[See @merton:73-bell, p. 175, for the first derivation of a barrier option pricing formula and @mcdonald:derivs:13, Chapter 14, for an overview.] * down-and-in and down-and-out barrier binary options * up-and-in and up-and-out barrier binary options * more standard down- and up- calls and puts, constructed using the barrier binary options Naming for the barrier options generally follows the convention ```` [u|d][i|o][call|put] ```` which means that the option is ``up'' or ``down'', ``in'' or ``out'', and a call or put.^[This naming convention differs from that in @mcdonald:derivs:13, in which names are `callupin`, `callupout`, etc. Thus, I have made both names available for these functions.] An up-and-in call, for example, would be denoted by `uicall`. For binary options, we add the underlying, which is either the asset or \$1: cash: ```` [asset|cash][u|d][i|o][call|put] ```` ```{r } H <- 115 bscall(s, c(80, 100, 120), v, r, tt, d) ## Up-and-in call uicall(s, c(80, 100, 120), v, r, tt, d, H) bsput(s, c(80, 100, 120), v, r, tt, d) ## Up-and-out put uoput(s, c(80, 100, 120), v, r, tt, d, H) ``` ## Perpetual American Options The functions `callperpetual` and `putperetual` price infinitely-lived American options.^[@merton:73-bell derived the price of a perpetual American put.] The pricing formula assumes that all inputs (risk-free rate, volatility, dividend yield) are fixed. This is of course usual with the basic option pricing formulas, but it is more of a conceptual stretch for an infinitely-lived option than for a 3-month option. In order for the option to have a determined value, the dividend yield on the underlying asset must be positive if the option is a call. If this is not true, the call is never exercised and the price is undefined.^[A well-known result (see @merton:73-bell) is that a standard American call is never exercised before expiration if the dividend yield is zero and the interest rate is non-negative. A perpetual call with $\delta=0$ and $r>0$ would thus never be exercised. The limit of the option price as $\delta\to 0$ is $s$, so in this case the function returns the stock price as the option value.] Similarly, the risk-free rate must be positive if the option is a put. By default, the perpetual pricing formulas return the price. By setting `showbarrier=TRUE`, the function returns both the option price and the stock price at which the option is optimally exercised (the ``barrier''). Here are some examples: ```{r } s <- 100; k <- 100; r <- 0.08; v <- 0.30; tt <- 2; d <- 0.04 callperpetual(s, c(95, 100, 105), v, r, d) callperpetual(s, c(95, 100, 105), v, r, d, showbarrier=TRUE) ``` ## Option Greeks Options greeks are mathematical derivatives of the option price with respect to inputs; see @mcdonald:derivs:13, Chapters 12 and 13, for a discussion of the greeks for vanilla options. Greeks for vanilla and barrier options can be computed using the `greeks` function, which is a wrapper for any pricing function that returns the option price and which uses the default naming of inputs.^[In this version of the package, I have two alternative functions that return Greeks: a) The `bsopt` function by default produces prices and Greeks for European calls and puts, and b) The `greeks2` function takes as arguments the name of the pricing function and then inputs as a list. These may be deprecated in the future. `greeks2` is more cumbersome to use but may be more robust. I welcome feedback on these functions and what you find useful. ] ```{r } H <- 105 greeks(uicall(s, k, v, r, tt, d, H)) ``` The value of this approach is that you can easily compute Greeks for spreads and custom pricing functions. Here are two examples. First, the value at time 0 of a prepaid contract that pays $S_{T}^{a}$ at time $T$ is given by the `powercontract()` function: ```{r } powercontract <- function(s, v, r, tt, d, a) { price <- exp(-r*tt)*s^a* exp((a*(r-d) + 1/2*a*(a-1)*v^2)*tt) } ``` We can easily compute the Greeks for a power contract: ```{r } greeks(powercontract(s=40, v=.08, r=0.08, tt=0.25, d=0, a=2)) ``` Second, consider a bull spread in which we buy a call with a strike of $k_{1}$ and sell a call with a strike of $k_2$. We can create a function that computes the value of the spread, and then compute the greeks for the spread by using this newly-created function together with `greeks()`: ```{r } bullspread <- function(s, v, r, tt, d, k1, k2) { bscall(s, k1, v, r, tt, d) - bscall(s, k2, v, r, tt, d) } greeks(bullspread(39:41, .3, .08, 1, 0, k1=40, k2=45)) ``` \medskip The Greeks function is vectorized, so you can create vectors of greek values with a single call. This example plots, for a bull spread, the gamma as a function of the stock price; see Figure \@ref(fig:bullgamma). \medskip ```{r bullgamma, fig.cap='Gamma for a 40-45 bull spread'} sseq <- seq(1, 100, by=0.5) greeks(bullspread(sseq, .3, .08, 1, 0, k1=40, k2=45), initcaps = TRUE, long = TRUE) %>% filter(greek == 'Gamma' ) %>% ggplot(aes(x = s, y = value)) + geom_line() ``` \medskip This code produces the plots in Figure \@ref(fig:allgreeks): \medskip ```{r allgreeks, fig.cap='All option Greeks for a call and a put, computed using the greeks function'} k <- 100; r <- 0.08; v <- 0.30; tt <- 2; d <- 0 S <- seq(.5, 200, by=.5) Call <- greeks(bscall(S, k, v, r, tt, d), long = TRUE) Put <- greeks(bsput(S, k, v, r, tt, d), long = TRUE) ggplot(rbind(Call, Put), aes(x = s, y = value, color = funcname )) + geom_line() + facet_wrap(~ greek, scales = 'free_y') + scale_color_discrete(name = 'Option', labels = c('Call','Put' )) + scale_x_continuous('Stock', breaks =c(0, 100, 200) ) + scale_y_continuous('Value') ``` ## Binomial Pricing of European and American Options There are two functions related to binomial option pricing:^[See @cox/ross/rubinstein:79, @rendleman/bartter:79-jf, and @mcdonald:derivs:13, Chapter 11] * **binomopt** computes prices of American and European calls and puts. The function has three optional parameters that control output: * `returnparams=TRUE` will return as a vector the option pricing inputs, computed parameters, and risk-neutral probability. * `returngreeks=TRUE` will return as a vector the price, delta, gamma, and theta at the initial node. * `returntrees=TRUE` will return as a list the price, greeks, the full stock price tree, the exercise status (`TRUE` or `FALSE`) at each node, and the replicating portfolio at each node. * **binomplot** displays the asset price tree, the corresponding probability of being at each node, and whether or not the option is exercised at each node. This function is described in more detail in Section \@ref(sec:binomplot). Here are examples of pricing, illustrating the default of just returning the price, and the ability to return the price plus parameters, as well as the price, the parameters, and various trees: ```{r } s <- 100; k <- 100; r <- 0.08; v <- 0.30; tt <- 2; d <- 0.03 binomopt(s, k, v, r, tt, d, nstep=3) binomopt(s, k, v, r, tt, d, nstep=3, returnparams=TRUE) binomopt(s, k, v, r, tt, d, nstep=3, putopt=TRUE) binomopt(s, k, v, r, tt, d, nstep=3, returntrees=TRUE, putopt=TRUE) ``` ## Asian Options There are analytical functions for valuing geometric Asian options and Monte Carlo routines for valuing arithmetic Asian options.^[See @kemna/vorst:90.] Be aware that the `greeks()` function at this time will not work with the Monte Carlo valuation for arithmetic Asian options. I plan to address this in a future release.^[As the functions are currently written, each invocation of the pricing function will start with a different random number seed, resulting in price variation that is due solely to random variation. Moreover, random number generation changes the random number seed globally. In a future release I hope to address this by saving and restoring the seed within the greeks function.] ### Geometric Asian Options Geometric Asian options can be valued using the Black-Scholes formulas for vanilla calls and puts, with modified inputs. The functions return both call and put prices with a named vector: ```{r } s <- 100; k <- 100; r <- 0.08; v <- 0.30; tt <- 2; d <- 0.03; m <- 3 geomavgpricecall(s, 98:102, v, r, tt, d, m) geomavgpricecall(s, 98:102, v, r, tt, d, m, cont=TRUE) geomavgstrikecall(s, k, v, r, tt, d, m) ``` ### Arithmetic Asian Options Monte Carlo valuation is used to price arithmetic Asian options. For efficiency, the function `arithasianmc` returns call and put prices for average price and average strike options. By default the number of simulations is 1000. Optionally the function returns the standard deviation of each estimate ```{r } arithasianmc(s, k, v, r, tt, d, 3, numsim=5000, printsds=TRUE) ``` The function `arithavgpricecv` uses the control variate method to reduce the variance in the simulation. At the moment this function prices only calls, and returns both the price and the regression coefficient used in the control variate correction: ```{r } arithavgpricecv(s, k, v, r, tt, d, 3, numsim=5000) ``` ## Compound Options A compound option is an option where the underlying asset is an option.^[See @geske:79 and @mcdonald:derivs:13, Chapter 14.] The terminology associated with compound options can be confusing, so it may be easiest to start with an example. Figure \@ref(fig:compoundopt) is a timeline for a compound option that is an option to buy a put. The compound option expires at $t_{1}$ and the put expires at $t_{2}$. The owner of the compound option only acquires the put if at time $t_{1}$ it is worth at least $k_{co}$, and only exercises the put if at time $t_{2}$ the stock price is less than $k_{uo}$. ```{r, include=FALSE} compound.caption <- 'The timeline for a compound option: a call to buy a put. The compound option expires at time $t_{1}$ and the underlying asset is a put option that expires at time $t_{2}$. At time $t_{1}$, the owner decides whether to pay $k_{co}$ to buy a put option which has time to expiration $t_{2} - t_{1}$. At time $t_{2}$ the owner decides whether to exercise the put, selling the stock for the strike price of $k_{uo}$.' # \label{fig:compoundopt} #\end{figure} ``` ```{r compoundopt, engine='tikz', fig.cap='The timeline for a compound option: a call to buy a put. The compound option expires at time $t_{1}$ and the underlying asset is a put option that expires at time $t_{2}$. At time $t_{1}$, the owner decides whether to pay $k_{co}$ to buy a put option which has time to expiration $t_{2} - t_{1}$. At time $t_{2}$ the owner decides whether to exercise the put, selling the stock for the strike price of $k_{uo}$.', echo=FALSE} \begin{tikzpicture} \newcommand{\start}{0} \newcommand{\finish}{8} \newcommand{\midpt}{(\finish/2-\start/2} \newcommand{\tickheight}{0.3} \draw (\start,0) -- (\finish,0); \draw (0,\tickheight) -- (0, -\tickheight) node[below] {\begin{tabular}{c} Time 0 \\ Buy compound \\option \end{tabular}}; \draw (\midpt,\tickheight) -- (\midpt, -\tickheight) node[below] {\begin{tabular}{c} Time $t_{1}$ \\ Compound exercise \\ decision: Pay $k_{co}$ \\ to buy put? \end{tabular} }; \draw (\finish, \tickheight) -- (\finish, -\tickheight) node[below] { \begin{tabular}{c} Time $t_{2}$ \\ Put exercise decision\\ Sell stock for $k_{uo}$? \end{tabular} }; \end{tikzpicture} ``` ### Definition of a Compound Option Based on the example, you can see that there are three prices associated with a compound option: * The price of an underlying asset. * The price of the underlying option, which is an option to buy or sell the underlying asset (we will refer to this as the price of the underlying option) * The price of the compound option, which gives us the right to buy or sell the underlying option The definition of a compound option therefore requires that we specify * whether the underlying option is a put or a call * whether the compound option is a put or a call * the strike price at which you can exercise the underlying option ($k_{uo}$) * the strike price at which you can exercise the compound option ($k_{co}$) * the date at which you can exercise the compound option (first exercise date, $t_{1}$) * the date at which you can exercise the underlying option expires, $t_{2}>t_{1}$. Given these possibilities, you can have a call on a call, a put on a call, a call on a put, and a put on a put. The valuation procedure require calculating the underlying asset price at which you are indifferent about acquiring the underlying option. The price calculation requires computing the stock price above or below which you would optimally exercise the option at time $t_{1}$. ### Examples As an example, consider the following inputs for a call option to buy a call option: ```{r } s <- 100; kuo <- 95; v <- 0.30; r <- 0.08; t1 <- 0.50; t2 <- 0.75; d <- 0 kco <- 3.50 calloncall(s, kuo, kco, v, r, t1, t2, d, returnscritical=TRUE) ``` With these parameters, after 6 months ($t_{1}=0.5$), the compound option buyer decides whether to pay \$`r formatC(kco, format='f', digits=2)` to acquire a 3-month call on the underlying asset. (The volatility of the underlying asset is `r v`.) It will be worthwhile to pay the compound strike, \$`r curr(kco)`, as long as the underlying asset price exceeds `r calloncall(s, kuo, kco, v, r, t1, t2, d, returnscritical=TRUE)['scritical']`. Similarly, there is a put on the call, and a call and put on the corresponding put: ```{r } putoncall(s, kuo, kco, v, r, t1, t2, d, returnscritical=TRUE) callonput(s, kuo, kco, v, r, t1, t2, d, returnscritical=TRUE) putonput(s, kuo, kco, v, r, t1, t2, d, returnscritical=TRUE) ``` ## Jumps and Stochastic Volatility \label{sec:jumps} The `mertonjump` function returns call and put prices for a stock that can jump discretely.^[See @merton/jfe/1976.] A poisson process controls the occurrence of a jump and the size of the jump is lognormally distributed. The parameter `lambda` is the mean number of jumps per year, the parameter `alphaj` is the log of the expected jump, and `sigmaj` is the standard deviation of the log of the jump. The jump amount is thus drawn from the distribution \begin{equation*} Y \sim \mathcal{N}(\alpha_{J} - 0.5\sigma^{2}_{J}, \sigma_{J}^{2} ) \end{equation*} ```{r } mertonjump(s, k, v, r, tt, d, lambda=0.5, alphaj=-0.2, vj=0.3) c(bscall(s, k, v, r, tt, d), bsput(s, k, v, r, tt, d)) ``` ## Bonds The simple bond functions provided in this version compute the present value of cash flows (`bondpv`), the IRR of the bond (`bondyield`), Macaulay duration (`duration`), and convexity (`convexity`). ```{r } coupon <- 8; mat <- 20; yield <- 0.06; principal <- 100; modified <- FALSE; freq <- 2 price <- bondpv(coupon, mat, yield, principal, freq) price bondyield(price, coupon, mat, principal, freq) duration(price, coupon, mat, principal, freq, modified) convexity(price, coupon, mat, principal, freq) ``` # Monte Carlo simulation of prices The function `simprice` provides a flexible way to generate prices that can be used for Monte Carlo valuation or just to illustrate sample paths.^[For more information on Monte Carlo see @mcdonald:derivs:13, Chapter 19, and @glasserman:mc:04, especially Chapter 4.] This function implements "naive" Monte Carlo, not using any variance reduction techniques. When supplied with a covariance matrix, `simprice` produces simulations of multiple assets with correlated returns. There are default values for all inputs. Other option pricing functions in this package assume that you specify volatility as the annualized return standard deviation. With a scalar volatility input, the `simprice` function also interprets the value as a standard deviation. A matrix, however, is interpreted as a covariance matrix, which means that the individual stock volatilities are square root of the diagonal elements. Because scalar and matrix inputs are treated differently, there is an option To change the behavior for a scalar input. To interpret the value as a variance, specify `scalar_v_is_stddev = FALSE`. ## Long vs wide output The function `simprice` can produce either long or wide output. Here are examples of each, using the default parameters.^[The `simprice` function save and restores the random number seed, so repeated invocations without intermediate seed-changing actions will produce the same result.] ```{r} args(simprice) simprice(long = TRUE) simprice(long = FALSE) ``` ## Simulated price paths A simple example is to generate 5 random sample paths of daily stock prices for a year; see Figure \@ref(fig:fivepaths): ```{r fivepaths, fig.cap='Five simulated paths for the same stock, no jumps.' } s0 <- 100; v <- 0.3; r <- 0.10; d <- 0; tt <- 1 trials <- 5; periods <- 365; set.seed(1) s <- simprice(s0 = s0, v = v, r = r, tt = tt, d = d, trials = trials, periods = periods, jump = FALSE, long = TRUE) ggplot(s, aes(x = period, y = price, color = trial, group = trial)) + geom_line() ``` We can do the same exercise adding jumps; see Figure \@ref(fig:fivejumpers). ```{r fivejumpers, fig.cap='Five simulated paths for the same stock, which can jump.'} s0 <- 100; v <- 0.3; r <- 0.10; d <- 0; tt <- 1 trials <- 5; periods <- 365; jump <- TRUE; lambda <- 2; alphaj <- -0.1; vj <- 0.2; set.seed(1) s <- simprice(s0 = s0, v = v, r = r, tt = tt, d = d, trials = trials, periods = periods, jump = jump, alphaj = alphaj, lambda = lambda, vj = vj, long = TRUE) ggplot(s, aes(x = period, y = price, color = trial, group = trial)) + geom_line() ``` \newpage ## Multiple correlated stocks When you supply an $n\times n$ covariance matrix as the volatility input, `simprice` produces simulations for $n$ stocks using the specified covariances. This makes it convenient to use an esimated covariance matrix to produce a simulation. ### Negatively correlated assets To illustrate the use of a covariance matrix, Figure \@ref(fig:negcoorsim) plots two stocks which are highly negatively correlated. ```{r negcoorsim, fig.cap='Two stocks for which the returns have a correlation of -.99.' } set.seed(1) s0 <- 100; r <- 0.08; tt <- 1; d <- 0; jump <- FALSE trials <- 1; periods <- 52; v <- .3^2*matrix(c(1, -.99, -.99, 1), nrow = 2) s <- simprice(s0 = s0, v = v, r = r, tt = tt, d = d, trials = trials, periods = periods, jump = jump, long = TRUE) ggplot(s, aes(x = period, y = price, group = asset, color = asset)) + geom_line() ``` ### Three correlated assets Here is an example using a three-asset covariance matrix, showing that the simulated series have the correct variance structure. We have three correlated stocks with standard deviations of 60%, 20%, and 45%., with the covariance matrix given by `v`. We can compare the estimated covariance matrix with the input,`v`: ```{r, eval=TRUE} set.seed(1) tt <- 2; periods <- tt*365 vc <- vols <- diag(3) diag(vols) <- c(.6, .2, .45) ## volatilities corrs <- c(.4, -.3, .25) vc[lower.tri(vc)] <- corrs ## correlations vc <- t(vc) ## lower triangular becomes upper triangular vc[lower.tri(vc)] <- corrs v <- vols %*% vc %*% vols v s <- simprice(s0 = s0, v = v, r = r, tt = tt, d = d, trials = 1, periods = periods, jump = FALSE, long = TRUE) threestocks <- s %>% filter(trial == 1) %>% group_by(asset) %>% mutate(ret = log(price/lag(price)), row = row_number()) %>% select(asset, period, ret) %>% spread(key = asset, value = ret ) var(threestocks[2:4], na.rm = TRUE)*365 ``` # Functions with Graphical Output Several functions provide visual illustrations of some aspects of the material. ## Quincunx or Galton Board The quincunx is a physical device the illustrates the central limit theorem. A ball rolls down a pegboard and strikes a peg, falling randomly either to the left or right. As it continues down the board it continues to strike a series of pegs, randomly falling left or right at each. The balls collect in bins and create an approximate normal distribution. The quincunx function allows the user to simulate a quincunx, observing the path of each ball and watching the height of each bin as the balls accumulate. More interestingly, the quincunx function permits altering the probability that the ball will fall to the right. Figure \@ref(fig:quincunx) illustrates the function after dropping 200 balls down 20 levels of pegs with a 70\% probability that each ball will fall right: ```{r quincunx, fig.cap='Output from the Quincunx function'} par(mar=c(2,2,2,2)) quincunx(n=20, numballs=200, delay=0, probright=0.7) ``` ## Plotting the Solution to the Binomial Pricing Model {#sec:binomplot} \label{sec:binomplot} The `binomplot` function calls `binomopt` to compute the option price and the various trees, which it then uses in plotting: The first plot, figure \@ref(fig:binomplot1), is basic: ```{r binomplot1, fig.cap='Basic option plot showing stock prices and nodes at which the option is exercised.'} s0 <- 100; k <- 100; v <- 0.3; r <- 0.08; tt <- 2; d <- 0 binomplot(s0, k, v, r, tt, d, nstep=6, american=TRUE, putopt=TRUE) ``` The second plot, figure \@ref(fig:binomplot2), adds a display of stock prices and arrows connecting the nodes. ```{r binomplot2, fig.cap='Same plot as Figure \\@ref(fig:binomplot1) except that values and arrows are added to the plot.'} binomplot(s0, k, v, r, tt, d, nstep=6, american=TRUE, putopt=TRUE, plotvalues=TRUE, plotarrows=TRUE) ``` As a final example, consider an American call when the dividend yield is positive and `nstep` has a larger value. Figure \@ref(fig:binomplot3) shows the plot, with early exercise evident. ```{r binomplot3, fig.cap="Binomial plot when nstep is 40."} d <- 0.06 binomplot(s0, k, v, r, tt, d, nstep=40, american=TRUE) ``` The large value of `nstep` creates a high maximum terminal stock price, which makes details hard to discern in the boundary region where exercise first occurrs. We can zoom in on that region by selecting values for `ylimval`; the result is in Figure \@ref(fig:binomplot4). ```{r binomplot4, fig.cap="Binomial plot when nstep is 40 using the argument ylimval to focus on a subset."} d <- 0.06 binomplot(s0, k, v, r, tt, d, nstep=40, american=TRUE, ylimval=c(75, 225)) ``` ## References