How to find the price variance of an infinitely expanding Binomial Tree?

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.

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.

• 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. – Kermittfrog Jan 11 at 11:19
• No, I am looking for the Variance of the Stock Price actually. – Anirban Saha Jan 11 at 11:30
• The $U$ you used here is the u times (growth multiplier) if I am not mistaken? – Anirban Saha Jan 11 at 11:31
• 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)? – Kermittfrog Jan 11 at 12:21
• @Kermittfrog I have rephrased the question. Thank you for your suggestion – 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 $$N$$.

• I still have the feeling that this is not what you are looking for, though. – Kermittfrog Jan 12 at 8:00
• 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. – Anirban Saha Jan 12 at 11:07