I was wondering if there were analytical formulas to compute delta or gamma for perpetual American options?

  • $\begingroup$ In the Black-Scholes model, there is an analytical formula for the price of an perpetual American options. You could differentiate that with respect to the stock price in order to get Delta and Gamma? Do you have a different model in mind? $\endgroup$ – Kevin Jul 17 '19 at 19:00
  • $\begingroup$ @KeSchn No. That is the model I have in mind. However, does it even make sense to have a Delta or Gamma for Perpetual American Options? $\endgroup$ – Mutating Algorithm Jul 17 '19 at 19:12
  • $\begingroup$ They do make sense in order to hedge your risk (whether the BS model is the model which gives you accurate hedging statistics is a different question). But you can still can try to construct an Delta neutral portfolio, even if the option is American. You gotta be careful I suppose around dividends with a higher likelihood of options being exercised. Note that you do not have Theta since the option never expires. Indeed, your BS pricing PDE reduces to an one-dimensional ODE. $\endgroup$ – Kevin Jul 17 '19 at 19:21

The Black-Scholes differential equation is a second-order PDE in two dimensions and reads as \begin{align*} \frac{\partial f}{\partial t} + rx\frac{\partial f}{\partial x} + \frac{1}{2}\sigma^2 x^2 \frac{\partial^2 f}{\partial x^2}-rf&=0, \\ \Theta+rx\Delta+ \frac{1}{2}\sigma^2 x^2 \Gamma-rf&= 0, \end{align*} assuming that $f\in \mathcal{C}^{1,2}([0,T]\times\mathbb{R})$. With the right boundaries conditions, $f(t,S_t)$ is then the value of a European, path-independent claim.

In the case of perpetual options whose price $f=f(S_t)$ does not depend on time, $\Theta$ vanishes, which reduces the pricing problem to a second-order PDE in one dimension (i.e. an ODE) \begin{align*} rx\frac{\mathrm{d} f}{\mathrm{d} x} + \frac{1}{2}\sigma^2 x^2 \frac{\mathrm{d}^2 f}{\mathrm{d} x^2}-rf&=0. \end{align*} Such an ODE can be solved by guessing $f(x)=x^n$. Then, \begin{align*} nrx^n + \frac{1}{2}\sigma^2 n(n-1) x^n-rx^n&=0, \end{align*} which, after dividing by $x^n$ yields a quadratic equation in $n$ with solutions $n_1=1$ and $n_2=-\frac{2r}{\sigma^2}<0$. The general solution is then given by $$f(x) = A x^{n_1} + B x^{n_2}.$$

Let’s focus on a put option with strike price $K$. Since there is no depence on $t$, the optimal exercise condition $s$ is a constant and we get three cases

  1. $x<s$: The option ought to be exercised and thus, $f(x) = K-x$,
  2. $x=s$: Smooth pasting condition: $\frac{\mathrm{d}f}{\mathrm{d}x}\bigg|_{x=s}=-1$ and
  3. $x>s$: the price is given by the ODE above with boundary condition $\lim\limits_{x\to\infty}f(x)=0$.

Condition 3) implies that $A=0$ yielding $f(x)=Bx^{n_2}$. \begin{align*} 1) &\implies Bx^{n_2}=K-x \implies Bs^{n_2}=K-s \\ 2) &\implies Bn_2s^{n_2-1} = -1 \implies Bn_2s^{n_2} = -s \end{align*}

Both equations are satisfied if $s=\frac{Kn_2}{n_2-1}=\frac{2rK}{\sigma^2+2r}$. We furthermore obtain $B=\frac{\sigma^2}{2r}\left( \frac{2rK}{\sigma^2+2r}\right)^{1+\frac{2r}{\sigma^2}}$. Thus, finally, \begin{align*} P(S_t) = \begin{cases} K-S_t & \text{if } S_t<s, \\ BS_t^{n_2} & \text{if } S_t\geq s. \end{cases} \end{align*}

Delta and Gamma of your put option are then \begin{align*} \Delta &= \begin{cases} -1 & \text{if } S_t<s, \\ Bn_2S_t^{n_2-1} & \text{if } S_t\geq s. \end{cases}\\ \Gamma &= \begin{cases} 0 & \text{if } S_t<s, \\ Bn_2(n_2-1)S_t^{n_2-2} & \text{if } S_t\geq s. \end{cases} \end{align*}

Please note the relationship between Gamma and Delta in the case $S_t\geq s$ as in the ODE at the top. Clearly, your option does not have a Theta whereas Vega & Rho can be obtained by the product rule. The price formula also gives you an exercise rule and tells you when you ought to exercise your option. You may also want to read this stellar post.


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