Deriving State-Dependent Volatility in a Symmetric Price Tree

Summary

The note derives the conditional volatility of a stock in a one-step binomial model where its price moves up or down by one unit with equal probability. With the stated definition, the conditional mean of the next price is the current price, while the conditional second moment exceeds the squared mean by one. Taking the square root gives a standard deviation of one price unit, and dividing by the current price yields volatility equal to the reciprocal of the current price.

This result means that the model’s percentage volatility decreases as the stock price rises, even though the absolute size of each possible move stays fixed. The conclusion depends on the specific additive, fixed-size price-step assumption and the one-period conditional volatility definition. It should not be read as a general empirical law for stocks or as a model with constant percentage volatility.

Key ideas

• Equal-probability moves of plus or minus one leave the conditional expected price unchanged.
• The conditional variance of the next price is one squared price unit.
• Under the stated definition, volatility is the reciprocal of the current price.
• The inverse relationship follows from fixed absolute moves, rather than fixed percentage moves.

Tags

# Vol binomial tree


# Vol binomial tree












Suppose that we have a stock $X_t$ valued at 100 euros per share. At each time step the price can go up or down 1 euro with prob $1/2$. Assuming that interest rates are $0$ and the volatility of the asset at time $t$ is defined as

$$vol(X_t)=\frac{\sqrt{\mathbb{E}[X_{t+1}^2\mid X_t]-\mathbb{E}[X_{t+1}\mid X_t]^2}}{X_t}$$ can we derive a closed formula for the volatility depending on $X_t$?

Does $vol$ increases when the price goes up?

## Answer by Bob Jansen (score 2)

https://quant.stackexchange.com/a/72112

The first term under the square is \begin{align}\frac{(X_t + 1)^2 + (X_t - 1)^2}{2} &= \frac{X_t^2 + 2X_t + 1 + X_t^2 - 2X_t + 1}{2} \\ &=X_t^2 + 1, \end{align} the second term is $X_t^2$. So $$\mathrm{vol}(X_t) = \frac{1}{X_t}.$$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.