Binomial Tree Variance and Its Lognormal Limit
Summary
The note considers how to describe future asset-price variance in a binomial model and connects the discrete tree to the continuous-time lognormal model. At a finite future step, the price distribution is determined by the number of up moves, which has a binomial probability. The response expresses the expected price and variance by summing across those possible move counts, with the expected price set by the risk-free growth assumption.
As the time step becomes infinitesimal, the binomial model converges to a lognormal distribution whose parameters depend on the initial price, drift, and volatility. This convergence concerns refining the time grid for a fixed horizon, not taking the horizon to infinity while keeping the number of steps fixed. The note provides a framework for finite-time variance and a limiting distribution, but does not derive a separate infinite-horizon variance or assess the perpetual American put formula in the question.
Key ideas
- At a finite tree step, the price distribution follows probabilities over the possible counts of up moves.
- Expected price under the stated construction grows at the continuously compounded risk-free rate.
- Finite-time price variance can be computed by weighting squared deviations across the tree’s possible prices.
- The lognormal limit follows as the time step approaches zero, not merely as the horizon grows without bound.
- The response does not provide an infinite-horizon variance derivation or validate the option-pricing formula.
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Full text
# How to find the price variance of an infinitely expanding Binomial Tree?
# 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.
## Answer by Kermittfrog (score 1, accepted)
https://quant.stackexchange.com/a/60469
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 $N$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.