Hello,
Being a beginner of lyx (no knowledge of latex), I have this lyx-generated
latex source (lyxout.txt attached)
When doing view / PDF (PDFLatex), latex run 1, I get a few "Missing ...
inserted"
and these errors:
>>>>>LaTeX Error: \begin{tabular} on input line 90 ended by \end{eqnarray*}.
... & = & d_{1}-\sigma\sqrt{T}\end{eqnarray*}
Your command was ignored.
Type I <command> <return> to replace it with another command,
or <return> to continue without it.
>>>>>Too many }'s.
\end{tabular}
\begin{tabular}{|c|c|c||c|}
You've closed more groups than you opened.
Such booboos are generally harmless, so keep going.
>>>>LaTeX Error: \begin{document} ended by \end{tabular}.
\end{tabular}
\begin{tabular}{|c|c|c||c|}
Your command was ignored.
Type I <command> <return> to replace it with another command,
or <return> to continue without it.
No pdf is generated. The lyx doc contains tables, and a lot of math formulae.
Any ideas?
Rds,
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\makeatletter
[EMAIL PROTECTED]"C:/BlackScholes/\string"/}}
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\documentclass[english]{article}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}
\usepackage{esint}
\makeatletter
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\newenvironment{lyxlist}[1]
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\usepackage{babel}
\makeatother
\begin{document}
\title{Black Scholes Merton}
\maketitle
The standard Black-Scholes framework is extended to cope with time-dependency
of the risk-free rate, dividend and volatility parameters.
In the following,
\begin{lyxlist}{00.00.0000}
\item [{$V$}] is the value of the option being priced,
\item [{$c$}] the price of a European call
\item [{$p$}] the price of a European put
\item [{$X$}] the option's strike
\item [{$T$}] the option's time to expiration date
\item [{$S$}] the price of the contract underlying the option
\item [{$\sigma$}] the standard deviation of the continuously compounded
log returns, per sqrt of annum, can be constant or time-dependent
$\sigma(t)$
\item [{$D$}] the present value of the dividends during the life of the
option, discounted at the risk-free rate, in case the underlying contract
pays discrete dividends
\item [{$q$}] the continuously compounded dividend yield, can be constant
or time-dependent $q(t)$
\item [{$r$}] the continuously compounded risk-free rate, can be constant
or time-dependent $r(t)$
\item [{$N(.)$}] the cumulative standard normal distribution function
\item [{$n(.)$}] the standard normal distribution density function
\end{lyxlist}
\section{Partial differential equation}
$V$ value of the option
\[
\frac{\partial V}{\partial
t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial
S^{2}}+\left[r-q\right]S\frac{\partial V}{\partial S}=rV\]
or its time-dependent equivalent\[
\frac{\partial V}{\partial
t}+\frac{1}{2}\sigma^{2}(t)S^{2}\frac{\partial^{2}V}{\partial
S^{2}}+\left[r(t)-q(t)\right]S\frac{\partial V}{\partial S}=r(t)V\]
In general, usual greek formulae remain valid with these replacements
$rT$ by $\int_{0}^{T}r(\tau)d\tau$
$qT$ by $\int_{0}^{T}q(\tau)d\tau$
$\sigma^{2}T$ by $\int_{0}^{T}\sigma^{2}(\tau)d\tau$
see Wilmott
\section{European call and put (upfront premium)}
\subsection{Price}
\begin{tabular}{|c|c|c||c|}
\hline
const $\sigma$ & Discrete dividends $D$ & Continuous dividend yield $q$ &
time-dependent $q$\tabularnewline
\hline
\hline
$r$ & \begin{eqnarray*}
c & = & (S-D)N(d_{1})-Xe^{-rT}N(d_{2})\\
p & = & Xe^{-rT}N(-d_{2})-(S-D)N(-d_{1})\\
d_{1} & = & \frac{\ln\frac{S-D}{X}+rT+\frac{1}{2}\sigma^{2}T}{\sigma\sqrt{T}}\\
d_{2} & = & d_{1}-\sigma\sqrt{T}\end{eqnarray*}
& \begin{eqnarray*}
c & = & Se^{-qT}N(d_{1})-Xe^{-rT}N(d_{2})\\
p & = & Xe^{-rT}N(-d_{2})-Se^{-qT}N(-d_{1})\\
d_{1} & = &
\frac{\ln\frac{S}{X}+\left(r-q\right)T+\frac{1}{2}\sigma^{2}T}{\sigma\sqrt{T}}\\
d_{2} & = & d_{1}-\sigma\sqrt{T}\end{eqnarray*}
& \begin{eqnarray*}
c & = & Se^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})-Xe^{-rT}N(d_{2})\\
p & = & Xe^{-rT}N(-d_{2})-Se^{-\int_{0}^{T}q(\tau)d\tau}N(-d_{1})\\
d_{1} & = &
\frac{\ln\frac{S}{X}+rT-\int_{0}^{T}q(\tau)d\tau+\frac{1}{2}\sigma^{2}T}{\sigma\sqrt{T}}\\
d_{2} & = & d_{1}-\sigma\sqrt{T}\end{eqnarray*}
\tabularnewline
\hline
$r(t)$ & \begin{eqnarray*}
c & = & (S-D)N(d_{1})-Xe^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})\\
p & = & Xe^{-\int_{0}^{T}r(\tau)d\tau}N(-d_{2})-(S-D)N(-d_{1})\\
d_{1} & = &
\frac{\ln\frac{S-D}{X}+\int_{0}^{T}r(\tau)d\tau+\frac{1}{2}\sigma^{2}T}{\sigma\sqrt{T}}\\
d_{2} & = & d_{1}-\sigma\sqrt{T}\end{eqnarray*}
& \begin{eqnarray*}
c & = & Se^{-qT}N(d_{1})-Xe^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})\\
p & = & Xe^{-\int_{0}^{T}r(\tau)d\tau}N(-d_{2})-Se^{-qT}N(-d_{1})\\
d_{1} & = &
\frac{\ln\frac{S}{X}+\int_{0}^{T}r(\tau)d\tau-qT+\frac{1}{2}\sigma^{2}T}{\sigma\sqrt{T}}\\
d_{2} & = & d_{1}-\sigma\sqrt{T}\end{eqnarray*}
& \begin{eqnarray*}
c & = &
Se^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})-Xe^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})\\
p & = &
Xe^{-\int_{0}^{T}r(\tau)d\tau}N(-d_{2})-Se^{-\int_{0}^{T}q(\tau)d\tau}N(-d_{1})\\
d_{1} & = &
\frac{\ln\frac{S}{X}+\int_{0}^{T}\left(r(\tau)-q(\tau)\right)d\tau+\frac{1}{2}\sigma^{2}T}{\sigma\sqrt{T}}\\
d_{2} & = & d_{1}-\sigma\sqrt{T}\end{eqnarray*}
\tabularnewline
\hline
\end{tabular}\begin{tabular}{|c|c|c||c|}
\hline
$\sigma(t)$ & Discrete dividends $D$ & Continuous dividend yield $q$ &
time-dependent $q$\tabularnewline
\hline
\hline
$r$ & \begin{eqnarray*}
c & = & (S-D)N(d_{1})-Xe^{-rT}N(d_{2})\\
p & = & Xe^{-rT}N(-d_{2})-(S-D)N(-d_{1})\\
d_{1} & = &
\frac{\ln\frac{S-D}{X}+rT+\frac{1}{2}\int_{0}^{T}\sigma^{2}(\tau)d\tau}{\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\\
d_{2} & = & d_{1}-\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}\end{eqnarray*}
& \begin{eqnarray*}
c & = & Se^{-qT}N(d_{1})-Xe^{-rT}N(d_{2})\\
p & = & Xe^{-rT}N(-d_{2})-Se^{-qT}N(-d_{1})\\
d_{1} & = &
\frac{\ln\frac{S}{X}+\left(r-q\right)T+\frac{1}{2}\int_{0}^{T}\sigma^{2}(\tau)d\tau}{\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\\
d_{2} & = & d_{1}-\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}\end{eqnarray*}
& \begin{eqnarray*}
c & = & Se^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})-Xe^{-rT}N(d_{2})\\
p & = & Xe^{-rT}N(-d_{2})-Se^{-\int_{0}^{T}q(\tau)d\tau}N(-d_{1})\\
d_{1} & = &
\frac{\ln\frac{S}{X}+rT-\int_{0}^{T}q(\tau)d\tau+\frac{1}{2}\int_{0}^{T}\sigma^{2}(\tau)d\tau}{\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\\
d_{2} & = & d_{1}-\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}\end{eqnarray*}
\tabularnewline
\hline
$r(t)$ & \begin{eqnarray*}
c & = & (S-D)N(d_{1})-Xe^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})\\
p & = & Xe^{-\int_{0}^{T}r(\tau)d\tau}N(-d_{2})-(S-D)N(-d_{1})\\
d_{1} & = &
\frac{\ln\frac{S-D}{X}+\int_{0}^{T}r(\tau)d\tau+\frac{1}{2}\int_{0}^{T}\sigma^{2}(\tau)d\tau}{\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\\
d_{2} & = & d_{1}-\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}\end{eqnarray*}
& \begin{eqnarray*}
c & = & Se^{-qT}N(d_{1})-Xe^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})\\
p & = & Xe^{-\int_{0}^{T}r(\tau)d\tau}N(-d_{2})-Se^{-qT}N(-d_{1})\\
d_{1} & = &
\frac{\ln\frac{S}{X}+\int_{0}^{T}r(\tau)d\tau-qT+\frac{1}{2}\int_{0}^{T}\sigma^{2}(\tau)d\tau}{\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\\
d_{2} & = & d_{1}-\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}\end{eqnarray*}
& \begin{eqnarray*}
c & = &
Se^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})-Xe^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})\\
p & = &
Xe^{-\int_{0}^{T}r(\tau)d\tau}N(-d_{2})-Se^{-\int_{0}^{T}q(\tau)d\tau}N(-d_{1})\\
d_{1} & = &
\frac{\ln\frac{S}{X}+\int_{0}^{T}\left(r(\tau)-q(\tau)\right)d\tau+\frac{1}{2}\int_{0}^{T}\sigma^{2}(\tau)d\tau}{\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\\
d_{2} & = & d_{1}-\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}\end{eqnarray*}
\tabularnewline
\hline
\end{tabular}
\subsection{Delta}
\begin{tabular}{|c|c||c|}
\hline
Discrete dividends $D$ & Continuous dividend yield $q$ & time-dependent
$q$\tabularnewline
\hline
\hline
\begin{eqnarray*}
\Delta_{c} & = & N(d_{1})\\
\Delta_{p} & = & N(d_{1})-1\end{eqnarray*}
& \begin{eqnarray*}
\Delta_{c} & = & e^{-qT}N(d_{1})\\
\Delta_{p} & = & e^{-qT}\left[N(d_{1})-1\right]\end{eqnarray*}
& \begin{eqnarray*}
\Delta_{c} & = & e^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})\\
\Delta_{p} & = &
e^{-\int_{0}^{T}q(\tau)d\tau}\left[N(d_{1})-1\right]\end{eqnarray*}
\tabularnewline
\end{tabular}
\subsection{Gamma}
\begin{tabular}{|c|c|c||c|}
\hline
& Discrete dividends $D$ & Continuous dividend yield $q$ & time-dependent
$q$\tabularnewline
\hline
\hline
$\sigma$ & \begin{eqnarray*}
\Gamma_{c}=\Gamma_{p} & = & \frac{n(d_{1})}{(S-D)\sigma\sqrt{T}}\end{eqnarray*}
& \begin{eqnarray*}
\Gamma_{c}=\Gamma_{p} & = &
e^{-qT}\frac{n(d_{1})}{S\sigma\sqrt{T}}\end{eqnarray*}
& \begin{eqnarray*}
\Gamma_{c}=\Gamma_{p} & = &
e^{-\int_{0}^{T}q(\tau)d\tau}\frac{n(d_{1})}{S\sigma\sqrt{T}}\end{eqnarray*}
\tabularnewline
\hline
$\sigma(t)$ & \begin{eqnarray*}
\Gamma_{c}=\Gamma_{p} & = &
\frac{n(d_{1})}{(S-D)\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\end{eqnarray*}
& \begin{eqnarray*}
\Gamma_{c}=\Gamma_{p} & = &
e^{-qT}\frac{n(d_{1})}{S\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\end{eqnarray*}
& \begin{eqnarray*}
\Gamma_{c}=\Gamma_{p} & = &
e^{-\int_{0}^{T}q(\tau)d\tau}\frac{n(d_{1})}{S\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\end{eqnarray*}
\tabularnewline
\end{tabular}
\subsection{Vega}
Only defined when the volatility is constant.
\begin{tabular}{|c|c||c|}
\hline
Discrete dividends $D$ & Continuous dividend yield $q$ & time-dependent
$q$\tabularnewline
\hline
\hline
\begin{eqnarray*}
\upsilon_{c}=\upsilon_{p} & = & (S-D)n(d_{1})\sqrt{T}\end{eqnarray*}
& \begin{eqnarray*}
\upsilon_{c}=\upsilon_{p} & = & Se^{-qT}n(d_{1})\sqrt{T}\end{eqnarray*}
& \begin{eqnarray*}
\upsilon_{c}=\upsilon_{p} & = &
Se^{-\int_{0}^{T}q(\tau)d\tau}n(d_{1})\sqrt{T}\end{eqnarray*}
\tabularnewline
\end{tabular}
\subsection{Theta}
We will demonstrate for the most general case:
\begin{eqnarray*}
c & = &
Se^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})-Xe^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})\\
p & = &
Xe^{-\int_{0}^{T}r(\tau)d\tau}N(-d_{2})-Se^{-\int_{0}^{T}q(\tau)d\tau}N(-d_{1})\\
d_{1} & = &
\frac{\ln\frac{S}{X}+\int_{0}^{T}\left(r(\tau)-q(\tau)\right)d\tau}{\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}+\frac{1}{2}\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}\\
d_{2} & = &
\frac{\ln\frac{S}{X}+\int_{0}^{T}\left(r(\tau)-q(\tau)\right)d\tau}{\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}-\frac{1}{2}\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}\end{eqnarray*}
\[
\Theta_{c}=-\frac{\partial c}{\partial T}=-S\frac{\partial}{\partial
T}\left[e^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})\right]+X\frac{\partial}{\partial
T}\left[e^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})\right]\]
\[
=-S\left[-q(T)e^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})+e^{-\int_{0}^{T}q(\tau)d\tau}n(d_{1})\frac{\partial
d_{1}}{\partial
T}\right]+X\left[-r(T)e^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})+e^{-\int_{0}^{T}r(\tau)d\tau}n(d_{2})\frac{\partial
d_{2}}{\partial T}\right]\]
\[
=Sq(T)e^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})-Xr(T)e^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})-Se^{-\int_{0}^{T}q(\tau)d\tau}n(d_{1})\frac{\partial
d_{1}}{\partial T}+Xe^{-\int_{0}^{T}r(\tau)d\tau}n(d_{2})\frac{\partial
d_{2}}{\partial T}\]
\[
\frac{\partial d_{1}}{\partial
T}=\frac{\left[r(T)-q(T)\right]\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}-\left[\ln\frac{S}{X}+\int_{0}^{T}\left(r(\tau)-q(\tau)\right)d\tau\right]\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}}{\int_{0}^{T}\sigma^{2}(\tau)d\tau}+\frac{1}{2}\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\]
\[
\frac{\partial d_{2}}{\partial
T}=\frac{\left[r(T)-q(T)\right]\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}-\left[\ln\frac{S}{X}+\int_{0}^{T}\left(r(\tau)-q(\tau)\right)d\tau\right]\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}}{\int_{0}^{T}\sigma^{2}(\tau)d\tau}-\frac{1}{2}\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\]
\[
n(d_{1})=\frac{1}{\sqrt{2\pi}}e^{-\frac{d_{1}^{2}}{2}}=\frac{1}{\sqrt{2\pi}}\exp\left[-\frac{\left[\ln\frac{S}{X}+\int_{0}^{T}\left(r(\tau)-q(\tau)\right)d\tau\right]^{2}}{2\int_{0}^{T}\sigma^{2}(\tau)d\tau}\right]\exp\left[-\frac{1}{8}\int_{0}^{T}\sigma^{2}(\tau)d\tau\right]\exp\left[-\frac{1}{2}\left[\ln\frac{S}{X}+\int_{0}^{T}\left(r(\tau)-q(\tau)\right)d\tau\right]\right]\]
\[
n(d_{2})=\frac{1}{\sqrt{2\pi}}e^{-\frac{d_{2}^{2}}{2}}=\frac{1}{\sqrt{2\pi}}\exp\left[-\frac{\left[\ln\frac{S}{X}+\int_{0}^{T}\left(r(\tau)-q(\tau)\right)d\tau\right]^{2}}{2\int_{0}^{T}\sigma^{2}(\tau)d\tau}\right]\exp\left[-\frac{1}{8}\int_{0}^{T}\sigma^{2}(\tau)d\tau\right]\exp\frac{1}{2}\left[\ln\frac{S}{X}+\int_{0}^{T}\left(r(\tau)-q(\tau)\right)d\tau\right]\]
\[
=n(d_{1})\exp\left[\ln\frac{S}{X}+\int_{0}^{T}\left(r(\tau)-q(\tau)\right)d\tau\right]=n(d_{1})\frac{S}{X}e^{^{\int_{0}^{T}r(\tau)d\tau}}e^{^{-\int_{0}^{T}q(\tau)d\tau}}\]
\[
\Theta_{c}=Sq(T)e^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})-Xr(T)e^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})\]
\[
-Se^{-\int_{0}^{T}q(\tau)d\tau}n(d_{1})\frac{\partial d_{1}}{\partial
T}+Xe^{-\int_{0}^{T}r(\tau)d\tau}n(d_{2})\frac{\partial d_{2}}{\partial T}\]
\[
=Sq(T)e^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})-Xr(T)e^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})\]
\[
-Se^{-\int_{0}^{T}q(\tau)d\tau}n(d_{1})\frac{\partial d_{1}}{\partial
T}+Xe^{-\int_{0}^{T}r(\tau)d\tau}n(d_{1})\frac{S}{X}e^{^{\int_{0}^{T}r(\tau)d\tau}}e^{^{-\int_{0}^{T}q(\tau)d\tau}}\left[\frac{\partial
d_{1}}{\partial
T}-\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\right]\]
\[
=Sq(T)e^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})-Xr(T)e^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})\]
\[
-Se^{-\int_{0}^{T}q(\tau)d\tau}n(d_{1})\frac{\partial d_{1}}{\partial
T}+n(d_{1})Se^{^{-\int_{0}^{T}q(\tau)d\tau}}\left[\frac{\partial
d_{1}}{\partial
T}-\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\right]\]
\[
\Theta_{c}=Sq(T)e^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})-Xr(T)e^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})-Se^{-\int_{0}^{T}q(\tau)d\tau}n(d_{1})\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\]
Similarily, the theta for a european put is
\[
\Theta_{p}=-Sq(T)e^{-\int_{0}^{T}q(\tau)d\tau}N(-d_{1})+Xr(T)e^{-\int_{0}^{T}r(\tau)d\tau}N(-d_{2})-Se^{-\int_{0}^{T}q(\tau)d\tau}n(d_{1})\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\]
\subsection{Rho}
Only defined when the riskfree rate is constant.
\begin{tabular}{|c}
\hline
\begin{eqnarray*}
\rho_{c} & = & XTe^{-rT}N(d_{2})\\
\rho_{p} & = & -XTe^{-rT}N(-d_{2})\end{eqnarray*}
\tabularnewline
\end{tabular}
\subsection{DDiv}
Only defined when the dividend yield is a constant.
\begin{tabular}{|c}
\hline
\begin{eqnarray*}
\left.\frac{\partial}{\partial q}\right\rfloor _{c} & = & -STe^{-qT}N(d_{1})\\
\left.\frac{\partial}{\partial q}\right\rfloor _{p} & = &
STe^{-qT}N(-d_{1})\end{eqnarray*}
\tabularnewline
\end{tabular}
\section{European call put (futures-style margining)}
Note that all exchange-listed options where the premium is handled
with futures-style margining are american. However, we can consider
the European case. Let $c$and $p$ be the upfront premium options,
and $C$ and $P$ the equivalent margined options.
\subsection{Price}
$C=e^{rT}c$ or $C=e^{\int_{0}^{T}r(\tau)d\tau}c$ , the same for
puts.
\subsection{Delta, Gamma, Vega, ddiv}
All these are derivatives w.r.t. a parameter that is neither $r$
nor $T$. So:
\[
\Delta_{C,P}=e^{\int_{0}^{T}r(\tau)d\tau}\Delta_{c,p}\]
\[
\Gamma_{C,P}=e^{\int_{0}^{T}r(\tau)d\tau}\Gamma_{c,p}\]
\[
\upsilon_{C,P}=e^{\int_{0}^{T}r(\tau)d\tau}\upsilon_{c,p}\]
\[
\left.\frac{\partial}{\partial q}\right\rfloor
_{C,P}=e^{\int_{0}^{T}r(\tau)d\tau}\left.\frac{\partial}{\partial
q}\right\rfloor _{c,p}\]
\subsection{Theta\[
\theta_{C,P}=-\frac{\partial C,P}{\partial
T}=-\frac{\partial\left[e^{\int_{0}^{T}r(\tau)d\tau}c,p\right]}{\partial T}\]
\[
\theta_{C,P}=-r(T)e^{\int_{0}^{T}r(\tau)d\tau}c,p+e^{\int_{0}^{T}r(\tau)d\tau}\theta_{c,p}\]
\[
\theta_{C,P}=-e^{\int_{0}^{T}r(\tau)d\tau}\left[r(T)c,p-\theta_{c,p}\right]\]
\[
\theta_{C}=-e^{\int_{0}^{T}r(\tau)d\tau}\left[r(T)\left[Se^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})-Xe^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})\right]-Sq(T)e^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})+Xr(T)e^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})+Se^{-\int_{0}^{T}q(\tau)d\tau}n(d_{1})\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\right]\]
\[
=-e^{\int_{0}^{T}r(\tau)d\tau}\left[Sr(T)e^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})-Xr(T)e^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})-Sq(T)e^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})+Xr(T)e^{-\int_{0}^{T}r(\tau)d\tau}N(d_{2})+Se^{-\int_{0}^{T}q(\tau)d\tau}n(d_{1})\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\right]\]
\[
=-e^{\int_{0}^{T}r(\tau)d\tau}\left[S\left[r(T)-q(T)\right]e^{-\int_{0}^{T}q(\tau)d\tau}N(d_{1})+Se^{-\int_{0}^{T}q(\tau)d\tau}n(d_{1})\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\right]\]
\[
\theta_{C}=-S\left[r(T)-q(T)\right]e^{\int_{0}^{T}(r(\tau)-q(\tau))d\tau}N(d_{1})-Se^{\int_{0}^{T}(r(\tau)-q(\tau))d\tau}n(d_{1})\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\]
\[
\theta_{P}=-e^{\int_{0}^{T}r(\tau)d\tau}\left[r(T)\left[Xe^{-\int_{0}^{T}r(\tau)d\tau}N(-d_{2})-Se^{-\int_{0}^{T}q(\tau)d\tau}N(-d_{1})\right]+Sq(T)e^{-\int_{0}^{T}q(\tau)d\tau}N(-d_{1})-Xr(T)e^{-\int_{0}^{T}r(\tau)d\tau}N(-d_{2})+Se^{-\int_{0}^{T}q(\tau)d\tau}n(d_{1})\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\right]\]
\[
=-e^{\int_{0}^{T}r(\tau)d\tau}\left[Xr(T)e^{-\int_{0}^{T}r(\tau)d\tau}N(-d_{2})-Sr(T)e^{-\int_{0}^{T}q(\tau)d\tau}N(-d_{1})+Sq(T)e^{-\int_{0}^{T}q(\tau)d\tau}N(-d_{1})-Xr(T)e^{-\int_{0}^{T}r(\tau)d\tau}N(-d_{2})+Se^{-\int_{0}^{T}q(\tau)d\tau}n(d_{1})\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\right]\]
\[
=e^{\int_{0}^{T}r(\tau)d\tau}\left[S\left[r(T)-q(T)\right]e^{-\int_{0}^{T}q(\tau)d\tau}N(-d_{1})-Se^{-\int_{0}^{T}q(\tau)d\tau}n(d_{1})\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\right]\]
\[
\theta_{P}=S\left[r(T)-q(T)\right]e^{\int_{0}^{T}(r(\tau)-q(\tau))d\tau}N(d_{1})-Se^{\int_{0}^{T}(r(\tau)-q(\tau))d\tau}n(d_{1})\frac{\sigma^{2}(T)}{2\sqrt{\int_{0}^{T}\sigma^{2}(\tau)d\tau}}\]
}
\subsection{Rho}
Only defined when the riskfree rate is constant.
\begin{tabular}{|c}
\hline
\begin{eqnarray*}
\rho_{C}=\frac{\partial C}{\partial r}=\frac{\partial}{\partial
r}\left[e^{rT}c\right]= &
re^{rT}\left[Se^{-qT}N(d_{1})-Xe^{-rT}N(d_{2})\right]+e^{rT}XTe^{-rT}N(d_{2})=
& rSe^{(r-q)T}N(d_{1})+(T-r)XN(d_{2})\\
\rho_{P}=\frac{\partial P}{\partial r}=\frac{\partial}{\partial
r}\left[e^{rT}p\right]= &
re^{rT}\left[Xe^{-rT}N(-d_{2})-Se^{-qT}N(-d_{1})\right]-e^{rT}XTe^{-rT}N(-d_{2})=
& (r-T)XN(-d_{2})-rSe^{(r-q)T}N(-d_{1})\end{eqnarray*}
\tabularnewline
\end{tabular}
\end{document}
