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Basically, I'd really like to give a natural interpretation of

\begin{equation} \Delta(t) =\frac{\partial O}{\partial S} \end{equation}

as prescribed by the Black-Scholes model for European derivatives, where O(t) is the option price and S(t) is the price of the underlying. Δ(t) is supposed to be the fraction (hence a number between 0 and 1) of the asset (whose price is S(t)) bought by the writer during the interval between option subscription and expiry date.

  1. Who guarantees that $\displaystyle\frac{\partial O}{\partial S}\in(0,1$ ?

  2. As a writer, the criterion to buy more or fewer assets (tuning Δ(t)) should be driven only by the asset trend S(t), and the strike price X. Why take ∂O/∂S? What does it tell concretely?

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2 Answers 2

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1 ) Delta (for European options) is the mathematical derivative of the option price with respect to the price of the underlying (that's what the formula you wrote is doing). In mathematics, Derivatives are a fundamental tool of calculus. A derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). In the case of delta, it measures the rate of change of the theoretical option value with respect to changes in the underlying asset's price.

Since a European call option gives the right to purchase the underlying at some predetermined price at some time in the future. there are only two possible extreme cases in between which the value of delta will fall:

  • Delta = 0: the price of the derivative does not change if the underlying changes: this is the case for far out the money options - e.g. the right to buy something in 2 days for the price of 1000 that costs only 1 at the moment (it is worthless). This corresponds to the very left hand side of the graphic below.
  • Delta = 1: the price of the derivative changes one for one with the price of the underlying e.g. the right to buy something in 2 days for the price of 1 that costs 1000 at the moment (it is certain that the right will be used, hence the movement is identical to owning the underlying outright). The very right hand side of the graphic below.

Geometrically, it is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. In terms of option pricing, you can plot the price of the option with respect to different values of the underlying, while keeping all else constant. The delta is the slope at any point you look at (at any given value of spot). It helps to look at this in a simple graph (done in Julia, code can be found here, how to make it interactive can be seen here, albeit for theta).

enter image description here

Side remark, that is very much simplified, because in reality you can have Spot, Forward, Spot premium adjusted, forward premium adjusted Delta for example. Non Vanilla options are anyhow different and delta can be very different from 0-1.

2 ) The asset trend has nothing to do with option pricing. Delta hedging is done so that the portfolio value remains unchanged when small changes occur in the value of the underlying security. You do not want to bet or guess the direction of an underlying. That is gambling. Naturally, this hedge is only short lived and works only in the near proximity to the current market conditions (a linear approximation to a curve can only do so much) . That's why market makers frequently need to rebalance if they want to hedge their exposure.

Edit: I know you ment the trend in the underlying security but as I wrote, this trend is entrirely unimportant for pricing options. In a nutshell, option pricing is entirely based on "no arbitrage" and "replication / hedging" arguments within a risk neutral framework. As a result, the derivative price does not depend on the drift / trend of the underlying.

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  • By "asset trend" I mean the trend of the underlying security.
    – ric.san
    Apr 13, 2023 at 11:14
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Δ(t) is supposed to be the fraction (hence a number between 0 and 1) of the asset (whose price is S(t)) bought by the writer during the interval between option subscription and expiry date.

Not by defenition... delta is the sensitivity of the option price to the price of the underlying. Meaning, if the price of the underlying rises by $1, the price of the option rises by Δ (roughly, as delta is not linear).

So what's the connection with the percent to buy for delta-hedging? The point of delta hedging is to eliminate exposure to the underlying. So If I have a call option for 100 shares that has an exposure of Δ to the underlying, then if I short -Δ*100 shares, that cancels out my exposure from the option. So delta can be thought of as the amount of the underlying that you need to offset the exposure, but that's not the definition.

If the value of the underlying changes, then my delta also changes, and I have to adjust how much of the underlying I own (or am short) in order to stay delta-hedged.

Who guarantees that Δ is between zero and one

For vanilla options, the black-scholes model mathematically guarantees that the value of Δ is between zero and one. You could have leveraged or other exotics options (like spread options) that have deltas higher than 1.

As a writer, the criterion to buy more or fewer assets (tuning Δ(t)) should be driven only by the asset trend S(t), and the strike price X.

You don't "tune" Δ if you're delta-hedging - you buy more or less of the underlying in order to offset the Δ of the option that's given to you. It's only driven by the trend in the sense that if the underlying goes up, my delta goes up (i.e. goes more negative if I'm short), and I must buy more of the underlying to stay hedged. It's not really based on "trend" so much as the mathematics of delta.

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