Interpreting Call Delta Hedging in a Binomial Model

# Interpretation of equation derived from the delta of a call European call option


# Interpretation of equation derived from the delta of a call European call option












I have started reading an introductory book called: A Course in Derivative Securities by Kerry Back. On page 12 they mention the following:

The delta of the call option is $\delta = (C_{u} - C_{d}) / (S_{u} - S_{d})$ and then they rewrite this to $\delta S_{u} - C_{u} = \delta S_{d} - C_{d}$, where $S_{u}$ is the stock price in the "up state", $S_{d}$ for the "down state" and $C_{u} = max(0, S_{u} - K)$, $K$ is the exercise price.

Now I am wondering, besides from the math, why is it, intuitively, that $\delta S_{u} - C_{u} = \delta S_{d} - C_{d}$ on day 0. Does this also hold on any other day? If so, could someone intuitively tell me why (I get the derivation though, but I lack the deeper understanding of why).

Thanks a lot!

## Answer by emcor (score 1, accepted)

https://quant.stackexchange.com/a/14041

Call Delta is generally defined as $$\Delta_C=\partial_S C=\frac{dC}{dS}=\frac{C_u-C_d}{S_u-S_d}$$, so it is the derivative or tangential change in $C$ from change in $S$, discretized in the Binomial Model.

As we know, this derivative goes symmetrically both ways, when $C_u$ goes up or $C_d$ down, so one can in general rewrite this equation:

$$\Delta S_u-C_u=\Delta S_d-C_d$$

As explained, this holds on every day just by the definition of $\Delta$, but it has no direct intuitive interpretation yet as such. You may say that for no-arbitrage, a symmetric structure is in general "better", but the result here is just a consequence of discretizing the derivative in $\Delta$. Later on, one may find that Delta also happens to be the stock-weight for a replicating portfolio.

## Answer by Matt B. (score 1)

https://quant.stackexchange.com/a/14044

Intuitively, holding $\delta$ stocks in your portfolio is going to make you money if the stock goes up (but you're going to lose on the option you've sold), and lose you money if the stock goes down (but you make on the option which becomes worthless).

The equality is the basis for the concept of $\delta$-neutrality, ie whatever happens, your portfolio value is unaffected, and is equal to today's value (discounted by the risk free rate)

## Answer by Kumar (score 0)

https://quant.stackexchange.com/a/14047

Its not the day that matters. Delta will remain the same on subsequent days if the price of the option and its underlying remain unchanged since the time the delta was calculated.

Delta is a linear approximation and measures only the slope, but options are non-linear instruments, they have a convexity. Delta holds only for small changes in underlying so any large change in underlying means that delta needs to be revalued.

Also, $\delta S_u - C_u = \delta S_d - C_d$ holds by definition of delta hedging and how delta is evaluated.