Issues with horizontal alignment regarding 3 equations

Question

This is my preamble :

\documentclass[12pt,thmsa]{article}
\usepackage{natbib}
\usepackage{amsmath,amsfonts,amssymb,amsthm}
\usepackage{breqn}
\usepackage{rotating}
\usepackage{threeparttable}
\usepackage{booktabs}
\usepackage[para,multiple]{footmisc}
\usepackage{mathtools}

\begin{document}

First equation issue

\begin{flalign}
    {w_1} &= \left\{N_1(-d_+(S_{dt},dt)+N_2(d_+(S_{dt},dt)),-d_+(S_{2dt},dt);-\rho_{12})+N_3(d_+(S_{dt},dt),d_+(S_{2dt},2dt),-d_+(S_{3dt},3dt);\rho_{12},-\rho_{13},-\rho_{23})...\right\},
    \end{flalign}


\begin{flalign}
{w_2} &= \left\{N_1(-d_-(S_{dt},dt)+N_2(d_-(S_{dt},dt)),-d_-(S_{2dt},dt);-\rho_{12})+N_3(d_-(S_{dt},dt),d_-(S_{2dt},2dt),-d_-(S_{3dt},3dt);\rho_{12},-\rho_{13},-\rho_{23})+...\right\},
\end{flalign}

This looks like:

My second equation troubling me:

\begin{align}
&\varepsilon {B_t} \left\{ {1 - {e^{ - \delta (T - t)}}N( - {d_ + }(\varepsilon {B_t},K,T - t)) - \delta \int\limits_t^T {{e^{ - \delta (s-t)}}N( - {d_ + }(\varepsilon {B_t},{B_s},s - t))\;ds} } \right\} \\
&= K \left\{ {1 - {e^{ - r(T - t)}}N( - {d_ - }(\varepsilon {B_t},K,T - t)) - r\int\limits_t^T {{e^{ - \delta (s-t)}}N( - {d_ - }(\varepsilon {B_t},{B_s},s - t))} {\kern 1pt} \;ds} \right\}.
\end{align}

This looks like this:

My third equation:

\begin{align}
P_3 = &Xe^{-rT/3}N_1[-d_-(B_{T/3},T/3)]-S_{T/3}N_1[-d_+(B_{T/3},T/3)]\nonumber\\
&+Xe^{-2rT/3}N_2[d_-(B_{T/3},T/3), -d_-(B_{2T/3},2T/3); -1/\sqrt{2}]\nonumber\\
&-S_{T/3, 2T/3}N_2[d_+(B_{T/3}, T/3),-d_+(B_{2T/3},2T/3);-1/\sqrt{2}]\nonumber\\
&+Xe^{-rT}N_3[d_+(B_{T/3},T/3), -d_+(B_{2T/3},2T/3), -d_+(X,T);1/\sqrt{2},-1/\sqrt{3},-\sqrt{2/3}]\nonumber\\
&-S_{T/3, 2T/3}N_3[d_-(B_{T/3},T/3),d_-(B_{2T/3},2T/3),-d_-(X,T);1/\sqrt{2},-1/\sqrt{3},-\sqrt{2/3}],
\end{align}

This looks like it:

Answer 1

I can propose this, based on aligned or multlined environments, and the medium-sized formulae from nccmath. Note you don't have to load amsmath if you load mathtools:

\documentclass[12pt,thmsa]{article}
\usepackage{natbib}
\usepackage{amsfonts,amssymb,amsthm}
%\usepackage{breqn}
\usepackage{rotating}
\usepackage{threeparttable}
\usepackage{booktabs}
\usepackage[para,multiple]{footmisc}
\usepackage{mathtools, nccmath}
\usepackage[showframe]{geometry}

\begin{document}
\begin{align}
    w_1 &= \begin{multlined}[t]\Bigl\{N_1(-d_+(S_{dt},dt)+N_2(d_+(S_{dt},dt)),-d_+(S_{2dt},dt);-\rho_{12})\\[-1ex]
    +N_3(d_+(S_{dt},dt),d_+(S_{2dt},2dt),-d_+(S_{3dt},3dt);\rho_{12},-\rho_{13},-\rho_{23})...\Bigr\},\end{multlined}\\
{w_2} &= \begin{multlined}[t]\Bigl\{N_1(-d_-(S_{dt},dt)+N_2(d_-(S_{dt},dt)),-d_-(S_{2dt},dt);-\rho_{12})\\[-1ex]
+ N_3\bigl(d_-(S_{dt},dt),d_-(S_{2dt},2dt),-d_-(S_{3dt},3dt);\rho_{12},-\rho_{13},-\rho_{23}\bigr)+...\Bigr\}\end{multlined}
\end{align}
\vspace{1cm}
\begin{align}
\varepsilon {B_t}&\medmath{\Biggl\{ 1 - {e^{ - \delta (T - t)}}N\bigl( - {d_ + }(\varepsilon {B_t},K,T - t)\bigr) - \delta \int\limits_t^T e^{ - \delta (s-t)} N\bigl( -d_{+}(\varepsilon B_t, B_s, s - t)\bigr)\;ds \Biggr\}} \\
= K &\medmath{\Biggl\{ 1 - e^{ - r(T - t)}N\bigl( -d_{-}(\varepsilon B_t, K, T - t)\bigr) - r\int\limits_t^T e^{ - \delta (s-t)}N\bigl(-d_{-} (\varepsilon B_t, B_s, s - t)\bigr) \;ds \Biggr\}}.
\end{align}
\vspace{1cm}
\begin{equation}
\begin{aligned}
P_3 &= {} Xe^{-rT/3}N_1\Bigl[-d_-\Bigl(B_{\tfrac{T}{3}},\mfrac{T}{3}\Bigr)\Bigr]-S_{T/3}N_1\Bigl[-d_+\Bigl(B_{\tfrac{T}{3}},\mfrac{T}{3}\Bigr)\Bigr] \\
&+Xe^{-2rT/3}N_2\Bigl[d_-\Bigl(B_{\tfrac{T}{3}},\mfrac{T}{3}\Bigr), -d_-\Bigl(B_{\tfrac{2T}{3}},\mfrac{2T}{3}\Bigr); -\mfrac{1}{\sqrt{2}}\Bigr] \\
&-S_{\tfrac{T}{3}, \tfrac{2T}{3}} N_2\Bigl[d_+\Bigl(B_{\tfrac{T}{3}},\mfrac{T}{3}\Bigr),-d_+ \Bigl(B_{\tfrac{2T}{3}},\mfrac{2T}{3}\Bigr);-\mfrac{1}{\sqrt{2}}\Bigr] \\
&+Xe^{-rT}N_3\Biggl[d_+\Bigl(B_{\tfrac{T}{3}},\mfrac{T}{3}\Bigr), -d_+\Bigl(B_{\tfrac{2T}{3}},\mfrac{2T}{3}\Bigr), -d_+(X,T);\mfrac{1}{\sqrt{2}},-\mfrac{1}{\sqrt{3}},-\sqrt{\mfrac{2}{3}}\Biggr] \\
&-S_{\tfrac{T}{3}, \tfrac{2T}{3}}N_3\Biggl[d_-\Bigl(B_{\tfrac{T}{3}}, \mfrac{T}{3}\Bigr),d_-(B_{2T/3}, 2T/3),-d_-(X,T);\mfrac{1}{\sqrt{2}},-\mfrac{1}{\sqrt{3}},-\sqrt{\mfrac{2}{3}}
\Biggr],
\end{aligned}
\end{equation}

\end{document}

Answer 2

Your equations are a bit too long. I tried and divided them in many rows. It's the best I could do, but please consider defining some longer pieces of the formulas as constants/letters, otherwise it's difficult to keep track and typeset it.

Oh, mind that the align environment is meant to align many equations at once with different numbers. What you want is probably split or multline

\documentclass[12pt,thmsa]{article}
\usepackage{natbib}
\usepackage{amsmath,amsfonts,amssymb,amsthm}
\usepackage{breqn}
\usepackage{rotating}
\usepackage{threeparttable}
\usepackage{booktabs}
\usepackage[para,multiple]{footmisc}
\usepackage{mathtools}

\begin{document}
\begin{multline}
 w_1 = \{N_1(-d_+(S_{dt},dt)+N_2(d_+(S_{dt},dt)),-d_+(S_{2dt},dt);-\rho_{12})\\
 +N_3(d_+(S_{dt},dt),d_+(S_{2dt},2dt),-d_+(S_{3dt},3dt);\rho_{12},-\rho_{13},-\rho_{23})\ldots\},
\end{multline}
\begin{multline}
{w_2} = \{N_1(-d_-(S_{dt},dt)+N_2(d_-(S_{dt},dt)),-d_-(S_{2dt},dt);-\rho_{12})\\
+N_3(d_-(S_{dt},dt),d_-(S_{2dt},2dt),-d_-(S_{3dt},3dt);\rho_{12},-\rho_{13},-\rho_{23})+\ldots\},
\end{multline}
\begin{equation}
\begin{split}
&\varepsilon {B_t} \biggl\{ 1 - {e^{ - \delta (T - t)}}N( - {d_ + }(\varepsilon {B_t},K,T - t)) \\ &{}-\delta \int\limits_t^T {{e^{ - \delta (s-t)}}N( - {d_ + }(\varepsilon {B_t},{B_s},s - t))\;ds}  \biggr\} \\
&= K \biggl\{ 1 - {e^{ - r(T - t)}}N( - {d_ - }(\varepsilon {B_t},K,T - t)) \\
&{}- r\int\limits_t^T {{e^{ - \delta (s-t)}}N( - {d_ - }(\varepsilon {B_t},{B_s},s - t))} {\kern 1pt} \;ds \biggr\}.
\end{split}
\end{equation}
\begin{equation}
\begin{split}
P_3 &= Xe^{-rT/3}N_1[-d_-(B_{T/3},T/3)]-S_{T/3}N_1[-d_+(B_{T/3},T/3)]\\
&{}+Xe^{-2rT/3}N_2[d_-(B_{T/3},T/3), -d_-(B_{2T/3},2T/3); -1/\sqrt{2}]\\
&{}-S_{T/3, 2T/3}N_2[d_+(B_{T/3}, T/3),-d_+(B_{2T/3},2T/3);-1/\sqrt{2}]\\
&{}+Xe^{-rT}N_3[d_+(B_{T/3},T/3), -d_+(B_{2T/3},2T/3),\\
&{} -d_+(X,T);1/\sqrt{2},-1/\sqrt{3},-\sqrt{2/3}]\\
&{}-S_{T/3, 2T/3}N_3[d_-(B_{T/3},T/3),\\
&d_-(B_{2T/3},2T/3),-d_-(X,T);1/\sqrt{2},-1/\sqrt{3},-\sqrt{2/3}],\\
\end{split}
\end{equation}
\end{document}

Answer 3

As Moriambar also says your equations are too long. I think you have the same problem in all the equation so I concentrated on the first. Here I first split after the first main - but even then the second row was too wide. I have tried to get the contents of paranthesis to align using \phantom. Still, it is not really readable, so if it is possible to find shorter expressions it would help.

\begin{flalign}
  {w_1} &= \Bigl\{N_1\Bigl(-d_+(S_{dt},dt)+N_2(d_+(S_{dt},dt)),-d_+(S_{2dt},dt);-\rho_{12}\Bigr)\nonumber\\
  &\phantom{{}={}\Bigl\{}
  +N_3\Bigl(d_+(S_{dt},dt),d_+(S_{2dt},2dt),\nonumber\\
  &\phantom{{}={}\Bigl\{{}+N_3\Bigl(}
  -d_+(S_{3dt},3dt);\rho_{12},-\rho_{13},-\rho_{23}\Bigr)\ldots\Bigr\},
\end{flalign}

In cases like this I think you should not use \left and \right. Instead use e.g. \Bigl and \Bigr defined in amsmath since they don't need to be in pair on the same row. There are many other sizes of them. Second, the alignment & should be before the = tio get spacing right. You can use \phantom to start on other points. Normally I try to have the equation number after the last row, which means supressing the others using \nonumber. (There is some other way to do it aslo but I forgot it right now).