How to optimize a series of equations whose outputs are a variable of the subsequent equatinos

External question — Quantitative Finance Stack Exchange Author: angryserver Original post: https://quant.stackexchange.com/questions/44716 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. 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}

External answer — Quantitative Finance Stack Exchange Author: Attack68 Original post: https://quant.stackexchange.com/a/44719 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. 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.