Approximating Option Delta from a Binomial Price Tree
Summary
The document explains how option delta can be approximated from the two successor-node option values in a binomial tree. Delta is the derivative of option value with respect to the underlying price, so the relevant Taylor approximation is in the underlying price, not in time. A finite difference between option values at the up and down prices estimates this slope.
The answer gives the approximation as the difference between those values divided by the difference between the corresponding underlying prices. In a tree, the two prices are available at the next time step, where the branches have separated; using those option values avoids constructing additional trees just to obtain multiple prices at the initial time. This is a local approximation, and the document does not discuss its error or performance under different tree specifications.
Key ideas
- Delta measures how option value changes with the underlying price.
- A finite difference across the up and down branches approximates the price derivative.
- The Taylor expansion is with respect to the underlying price, rather than time.
- Next-step tree values provide distinct underlying prices for estimating hedge ratios.
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Full text
# Delta of an option derived from the binomial model
# Delta of an option derived from the binomial model
I have the following function $V=V(S,t)$, $V^- = V(vS,t+\delta t)$, $V^+ = V(uS, t +\delta t)$. The book proceeds to explain that if we use Taylor series expansion on the above we will confirm that $\frac{\partial V}{\partial S} \sim \Delta$ by substituting the results into $\Delta = \frac{V^+ - V^-}{S(u-v)}$.
I get the following results for the expansion: $V^- = V^- (vS,t)+ \frac{\partial V^-}{\partial t} \delta t$
$V^+ = V^+ (uS,t)+\frac{\partial V^+}{\partial t} \delta t$
Not sure how I proceed, subbing right now does not yield any sensible result. Not sure where we get partial derivative of $V$ with respect to $S$ either.. Is that from expanding $V(S,t)$, in that case, how do expand it. Basically, how do I arrive at the result that shows that partial derivative of $V$ with respect to $S$ is delta?
## Answer by Gordon (score 2, accepted)
https://quant.stackexchange.com/a/21719
This is not the Taylor expansion with respect to $t$, instead, it is the Taylor expansion with respect to $S$. Moreover, the prices at time $t+\delta t$ is used for approximation. That is, \begin{align*} \frac{\partial V}{\partial S}\big|_t &\approx \frac{V(uS, t) - V(vS, t)}{uS - vS}\\ &\approx \frac{V(uS, t+\delta t) - V(vS, t+\delta t)}{S(u-v)}\\ &= \frac{V^+-V^-}{S(u-v)}. \end{align*} The reason is that, at time $t=0$, there is only a single price available, and, to compute the delta and gamma respective hedge ratios, other trees are needed. For computational efficiency, we use the prices at time $0+\delta t$, where multiple prices are available.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.