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The basic question is, given $f(x) = y$ and $f(y) = z$, how can you find $x$ such that $z$ is at its maximum?


I can optimize each equation independently, but I do not know how to optimize when combining equations. A concrete example is as follows:

Imagine forex market that is made up of $x$ and $y$, where $x$ and $y$ are both currencies. Users can send in $x$ to receive $y$, and vice versa. The market structure is defined by \begin{equation} x * y = k \end{equation} where $k$ is a constant number, say $1$, and the product of $x$ and $y$ must always be equal to this number.

The price of $x$ or $y$ is simply $x / y$, such that $k$ always stays the same. If someone sends $x'$ of the currency as payment and receives $y'$ in return, the new equation for the market must be true.

\begin{equation} \dfrac{(x + x')}{(y - y')} = k \end{equation}

Given all this information, imagine you were to make a trade on two markets of this structure. How would you optimize your input, $x0'$, such that your output $x_1'$, is maximized, and $y_0'$ is equivalent on both trades?

\begin{equation} \dfrac{(x_0 + x_0')}{(y_0 - y_0')} = k_0 \;\;\; and \;\;\; \dfrac{(y_1 + y_0')}{(x_1 - x_1')} = k_1 \end{equation}

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  • $\begingroup$ I kind of see what you are asking, but my instinct is saying there is a really simple mathematical way of expressing this problem but the description and choice of symbols are obfuscating it.... $\endgroup$ – Attack68 Mar 21 at 19:22
  • $\begingroup$ @Attack68 I truly believe it is simple but cannot figure out how to attack it. I tried to simplify it with the first sentence. $\endgroup$ – quantfinancequest Mar 21 at 19:39
  • $\begingroup$ But then you are just optimising $z=f(f(x))$ or $f(g(x))=z$ if your 2 functions $f$ are not actually the same, so not sure where the difficulty arises (besides it not resulting in a convex or differentiable function) $\endgroup$ – Attack68 Mar 21 at 19:42
  • $\begingroup$ @Attack68 then how do you optimize $z=f(g(x))$? Is that simply a derivative of each function? $\endgroup$ – quantfinancequest Mar 21 at 19:47
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To optimize:

$$z = f(g(x))$$

using traditional calculus with chain rule:

$$ \frac{dz}{dx} = \frac{df}{dg} \frac{dg}{dx} $$

Set $\frac{dz}{dx} = 0$ and that will determine either minimum, maximum or saddle points.

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  • $\begingroup$ Thank you. Provided there is no other extremely-explicit answer, I will accpet this one. Thank you for your help. $\endgroup$ – quantfinancequest Mar 21 at 19:56

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