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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!

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3 Answers 3

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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.

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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)

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Your answer is referring to Delta portfolio weight, but that is not yet introduced here. The question is only on Delta as sensitivity of Option Price to Underlying Price. –  emcor Jul 20 at 23:12
    
@emcor The question was about why should one rewrite delta from being a ratio to being an equality between 2 portfolios in 2 states of the world. I think the easiest way to understand it is to look at the delta hedging priniciple. –  Matt B. Jul 21 at 6:40

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.

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The question is based on the Binomial Model where S can only go up or down, so Delta holds exactly for these possible changes (not just approximately for small changes). –  emcor Jul 20 at 22:24

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