How to find the price variance of an asset in a Binomial Tree Model? Suppose the price of the Stock is $S_t$ at time $t$ and it has a probability of $p$ that will go up $u$ times to $u \cdot S_t$ and a probability $(1-p)$ that it will go down to $d \cdot S_t$ at time $t+1$. And this goes on indefinitely. enter image description here

I am trying to price a Perpetual American Put Option whose price is given by $ V_{t} = K \left[ \frac{K}{S_{t}} \left( 1 - \frac{2r}{ 2r+\sigma^2 } \right) \right] ^{2r/\sigma^2 }$, where $K$ is the Strike Price, $V_t$ is the Price of the option at time $t$. $r$ is the risk-free rate of interest and $\sigma^2$ is the price variance of this stock. Finding the price variance for a finite time-frame is straight forward, but any resources towards finding the same for the infinite time period would be helpful. Thank you in advance.

  • $\begingroup$ Hi Anirban, just to make sure: With volatility, do you mean the volatility parameter that you have to plug into the binomial tree method? Usually, this is a degree of freedom in your modelling approach. In the classical Cox-Ross-Rubinstein 1979 tree, you could find $\sigma$ from $U\equiv e^{\sigma\sqrt{\Delta t}}$. I have the feeling that this is not what you are looking for, though. $\endgroup$ – Kermittfrog Jan 11 at 11:19
  • $\begingroup$ No, I am looking for the Variance of the Stock Price actually. $\endgroup$ – Anirban Saha Jan 11 at 11:30
  • $\begingroup$ The $U$ you used here is the u times (growth multiplier) if I am not mistaken? $\endgroup$ – Anirban Saha Jan 11 at 11:31
  • $\begingroup$ Are you looking for the price variance or for the (logarithmic, annualized) return volatility of the stock? In either case, you can derive those given $p,u,d$ from your tree directly. Could you rephrase your question (in here)? $\endgroup$ – Kermittfrog Jan 11 at 12:21
  • $\begingroup$ @Kermittfrog I have rephrased the question. Thank you for your suggestion $\endgroup$ – Anirban Saha Jan 11 at 12:38

I hope that I have understood the gist of your question, if not I may try to adjust this answer.

Let the time step in a binomial tree be $\Delta t \equiv \frac{T}{N}$. For $N \to \infty$, the distribution of the stock price (of any specific point in time $t>t_0$ converges to the lognormal distribution with scale parameter $\log S_0 + (r-\frac{1}{2}\sigma^2)t$ and shape parameter $\sigma^2 t$, i.e.

$$S_t\sim \mathrm{LogNormal}(\log S_0 + (r-\frac{1}{2}\sigma^2)t,\, \sigma^2 t)$$

In a binomial tree, the asset price distribution at any future time step $t_k=k\Delta t$ is binomial:

$$ P(S_{t_k}=S_0\times U^l \times D^{k-l})=\binom{k}{l}p^l(1-p)^{k-l}\quad,l\in[0,\ldots,k]$$

Thus, the expected future stock price is

$$\mathrm{E}(S_{t_k})= S_0\sum_{l=0}^k\binom{k}{l}p^l(1-p)^{k-l}U^lD^{k-l}$$ By construction, this equals $S_0e^{r\times t_k }$, with $r$ the continuously compounded risk free rate. The future stock price variance can then be found as

$$\mathrm{Var}(S_{t_k})\equiv\mathrm{E}\left(\left(S_{t_k}-\mathrm{E}(S_{t_k})\right)^2\right)= \sum_{l=0}^{k}\binom{k}{l}p^l(1-p)^{k-l}S_0^2\left(U^{2l-k}-e^{r\times t_k}\right)^2$$

Note: Obviously, the binomial distribution will converge to the lognormal for $\Delta t \to 0$, not for $t_k\to \infty$ for any given $N$.

  • $\begingroup$ I still have the feeling that this is not what you are looking for, though. $\endgroup$ – Kermittfrog Jan 12 at 8:00
  • $\begingroup$ This is what the answer is to what I was looking for. I was hoping there would be a simplified expression instead of a summation but yes this is correct. $\endgroup$ – Anirban Saha Jan 12 at 11:07

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