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Parsiad Azimzadeh Parsiad Azimzadeh Selected publications Blog Menu Curriculum vitae Selected publications Blog Log in Closed-form expressions for perpetual and finite-maturity American binary options March 1, 2015 Parsiad Azimzadeh Introduction In this post, the reader is shown how to derive closed-form expressions for perpetual and finite-maturity American binary (a.k.a. digital) options written on a stock pursuit geometric Brownian motion. These expressions are closed-form in that they consist of a finite number of arithmetic operations and well-known functions (e.g. $\operatorname{erf}$). These follow immediately from expressions for the Laplace transform of the distribution of the provisionary first passage time of Brownian motion to a particular level; derived herein. A binary option is a type of option in which the payoff can take only two possible outcomes, either some stock-still monetary value of some windfall or nothing at all. The two main types of binary options are the cash-or-nothing binary option and the asset-or-nothing binary option. The cash-or-nothing binary option pays some stock-still value of mazuma if the option expires in-the-money while the asset-or-nothing pays the value of the underlying security. They are moreover tabbed all-or-nothing options, digital options (more worldwide in forex/interest rate markets), and stock-still return options (FROs) (on the American Stock Exchange). A European option may be exercised only at the expiration stage of the option (i.e. at a single pre-defined point in time). An American option, on the other hand, may be exercised at any time surpassing the expiration date. For an American binary put (resp. call), it is sufficient to consider the cash-or-nothing specimen with an option paying off a single unit in the numéraire. The asset-or-nothing specimen is a simple scaling (by the strike price) of the cash-or-nothing case. Option prices Closed-form expressions for the prices of American binary options written on an windfall pursuit geometric Brownian motion as per $$S_t=x\exp\left(\left(r-\delta-\frac{1}{2}\sigma^{2}\right)t+\sigma W_{t}\right)$$ are presented in this section. Both the *perpetual* and *finite-maturity* cases are considered. It is unsupportable that $r$ is real, $\delta\geq0$, and $\sigma>0$. $K>0$ is used to denote the strike price of an option. It will be helpful to pinpoint the pursuit symbols: (it is readily verified that $\xi^{2}+2r\geq0$; i.e. $b$ is real) $$a=\frac{1}{\sigma}\log\frac{K}{x},\text{ }\xi=\frac{r-\delta}{\sigma}-\frac{\sigma}{2},\text{ and }b=\sqrt{\xi^{2}+2r}.$$ Because a binary option is exercised as soon as it is in the money, it is unsupportable $K < x$ (resp. $K > x$) for a put (resp. call). Otherwise, the option is exercised upon inception and worth exactly one unit. The expressions are summarized below: The price of a cash-or-nothing American binary put (resp. call) with strike $K < x$ (resp. $K > x$) and time-to-expiry $T$ is $$\frac{1}{2}e^{a\left(\xi-b\right)}\left\{ 1+\text{sgn}\left(a\right)\text{erf}\left(\frac{bT-a}{\sqrt{2T}}\right)+e^{2ab}\left[1-\text{sgn}\left(a\right)\text{erf}\left(\frac{bT+a}{\sqrt{2T}}\right)\right]\right\}.$$ The price of a cash-or-nothing, perpetual American binary put (resp. call) with strike $K < x$ (resp. $K > x$) is $e^{a\xi-\left|a\right|b}$. These expressions follow immediately from the Laplace transform of the distribution of the provisionary first passage time of Brownian motion to a particular level (see below). For the specimen of an (ordinary) perpetual American call, when the interest rate is nonnegative (more generally, $2r\geq-\sigma^{2}$) and the dividend rate is zero, it is never optimal to exercise and the option is worth the initial stock price. A respective result for the perpetual American binary undeniability is as follows: When $2r\geq-\sigma^{2}$ (resp. $2r\leq-\sigma^{2}$) and $\delta=0$, the price of a cash-or-nothing, perpetual American binary undeniability (resp. put) with strike $K>x$ (resp. $K < x$) is $x/K$. It is sufficient to consider the specimen of the call; the put is handled similarly. Since $a>0$, $$e^{a\xi-\left|a\right|b}=e^{a\left(\xi-b\right)}=e^{-a\sigma}=x/K.$$ Below, we graph the value of an American binary put with $K = 100$, $r = 0.04$, $σ = 0.2$, and $T = 1$. GNU Octave/MATLAB lawmaking to generate a graph similar to the whilom is given below. K = 100.; x = K:1:K*2; r = 0.04; sigma = 0.2; delta = 0.01; T = 1.; a = 1. / sigma * log (K ./ x); xi = (r - delta) / sigma - sigma / 2.; b = sqrt (xi * xi + 2. * r); v = 0.5 * exp (a * (xi - b)) .* ( ... 1 + sign (a) .* erf ((b * T - a) / sqrt (2 * T)) ... + exp (2 * a * b) ... .* (1 - sign (a) .* erf ((b * T + a) / sqrt (2 * T))) ... ); plot ([0 1 x], [1 1 v], 'linewidth', 2); turning ([K/2 K*2 0 1+2^(-5)]); xlabel ('Initial stock price (x)'); ylabel ('American binary put value (P)'); First passage times For real numbers $a$ and $\xi$ let $$\tau_{a\xi}=\inf\left\{ t\geq0:\xi t+W_{t}=a\right\} $$ be the random variable respective to the first time the Brownian motion with skid $\xi$ reaches level $a$. The pursuit is well-known (see, e.g., [1]): The density of $\tau_{a\xi}$ is $$f_{\tau_{a\xi}}\left(t\right)=\frac{\left|a\right|}{\sqrt{2\pi t^{3}}}\exp\left(-\frac{\left(a-\xi t\right)^{2}}{2t}\right).$$ In deriving the above, it is sufficient to consider the specimen of $a\geq0$ and to use the reflection principle to derive the density for $a < 0$. A lengthy computation yields an expression for the Laplace transform of the distribution of the provisionary first passage time of Brownian motion to a particular level, which is of separate interest: Let $c$ be real, $b=\sqrt{\xi^{2}+2c}$ and $0 < T < \infty$. If $\xi^{2}+2c\geq0$, \begin{align*} E\left[e^{-c\tau_{a\xi}} \mathbf{1}_{\tau_{a\xi} \leq T} \right] & =\int_{0}^{T}e^{-ct}f_{\tau_{a\xi}}\left(t\right)dt\\ & =\frac{1}{2}e^{a\left(\xi-b\right)}\left\{ 1+\text{sgn}\left(a\right)\text{erf}\left(\frac{bT-a}{\sqrt{2T}}\right)+e^{2ab}\left[1-\text{sgn}\left(a\right)\text{erf}\left(\frac{bT+a}{\sqrt{2T}}\right)\right]\right\} . \end{align*} Let $c$ be real and $b=\sqrt{\xi^{2}+2c}$. If $\xi^{2}+2c\geq0$, $E\left[e^{-c\tau_{a\xi}}\right]=e^{a\xi-\left|a\right|b}.$ To victorious at the results of the previous section, it suffices to note that the price of an American binary put (resp. call) with strike $K < x$ (resp. $K > x$) and time-to-expiry $T$ is exactly $$E\left[e^{-r\tau^{\star}} \mathbf{1}_{\tau^{\star} \leq T} \right]$$ where $$\tau^{\star}=\inf\left\{ t\geq0:S_{t}=K\right\} =\inf\left\{ t\geq0:\left(\frac{r-\delta}{\sigma}-\frac{\sigma}{2}\right)t+W_{t}=\frac{1}{\sigma}\log\frac{K}{x}\right\}.$$ The results now follow immediately from the fact that $\tau^{\star}=\tau_{a\xi}$ with $a$ and $\xi$ specified as in the previous section. Bibliography Shreve, Steven E. Stochastic calculus for finance II: Continuous-time models. Vol. 11. Springer Science & Business Media, 2004. Mathematical finance Parsiad Azimzadeh AboutPhD (University of Waterloo), MMath (University of Waterloo), BSc (Simon Fraser University)Latest postsAn introduction to regular Markov chainsmlinterp: Fast wrong-headed dimension linear interpolation in C++Optimal stopping III: a comparison principleOptimal stopping II: a dynamic programming equationOptimal stopping I: a dynamic programming principleGNU Octave financial 0.5.0 releasedMonte Carlo simulations in GNU Octave financial packageIntroductory group theoryClosed-form expressions for perpetual and finite-maturity American binary optionsFast Fourier Transform with examples in GNU Octave/MATLABPagesHomeSelected publicationsWelcomeTagsMarkov villenage (1)Optimal stopping (3)GNU Octave (2)Notes (2)Mathematical finance (1) RSS | Design: HTML5 UP