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{\Large \textbf{Administrative Arrangements}}\\

\begin{itemize}
\item Course delivery using in-person lectures and problems classes.
\item Two lectures per week, one problem class every other week (ish).
\item Assessment is by exam (80\%, January 2024) and assignments (20\%), with four assignments (5\% each) 
in total.
\item The course page can be found on canvas at\\
\textit{https://canvas.ncl.ac.uk}
%\item Lectures on Mondays at 9:00 (ARMB 2.98) and Thursdays at 15:00 (HERB LT1).
%\item Problems classes/drop-ins are typically on Tuesdays, 17:00 (ARMB 2.98). Week 1 is a lecture! 
%\item Computer class: Herschel full cluster, Thursday 9:00. Week 10.
%\item Assessment is by exam (90\%), answers to set questions (5\%) and class test (5\%).
%\item Hand in work by 4pm, usually on Mondays, weeks 5 and 11 (2 assignments).
%\item Announcements will be made to your ncl.ac.uk account - please check regularly.
%\item Please bookmark the link to the webpage of the course, which is\\ 
%\textit{www.mas.ncl.ac.uk/$\sim$nag48/teaching/MAS3904/}\\ 
%This is also available via Blackboard.  
\end{itemize}

{\Large \textbf{Books}}\\

\begin{itemize}
\item J. Hull: Options, Futures and Other Derivatives (Prentice-Hall, 2003)
\item S. Ross: An Elementary Introduction to Mathematical Finance (CUP, 2003)
\item M. Capinski, T. Zastawniak: Mathematics for Finance (Springer, 2003)
\item S. Shreve: Stochastic Calculus for Finance 1 (Springer, 2004)
\item J. Franke, W. Hardle, C. Hafner: Statistics of Financial Markets (Springer, 2004)
\end{itemize}

\newpage

\noindent
{\large {\bf \Large Important dates}}

\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Timetable week & Teaching week & w/c & Notes \\
\hline
4 & 1 & 25/09/23 & \\
5 & 2 & 02/10/23 & \\
6 & 3 & 09/10/23 & \\
7 & 4 & 16/10/23 & Ass. 1 due by 4pm on Oct. 20th\\
8 & 5 & 23/10/23 & \\
9 & 6 & 30/10/23 & Ass. 2 due by 4pm on Nov. 3rd\\
\hline
10 & -- & 06/11/23 & Enrichment week (no teaching)\\
\hline
11 & 7 & 13/11/23 & \\
12 & 8 & 20/11/23 &  Ass. 3 due by 4pm on Nov. 24th\\
13 & 9 & 27/11/23 & \\
14 & 10 & 04/12/23 & Ass. 4 due by 4pm on Dec. 8th\\
15 & 11 & 11/12/23 & \\
\hline
-- & -- & 25/12/23 & Xmas!\\
\hline
\end{tabular}
\end{center}
\caption{Schedule}\label{tab:tabadmin}
\end{table}

%Dates for your diary are as follows:
%\begin{itemize}
%\item Thursday October 3 (week 1), assignment 1 given out
%\item Monday October 28 (week 5), hand in assignment 1 (by 4pm)
%\item Monday November 4 (week 6), mid-semester test, 9:00 (50 mins)
%\item Thursday November 14 (week 7), assignment 2 given out
%\item Thursday December 5 (week 10), computer practical, 9:00 Herschel cluster
%\item Monday December 9 (week 11), hand in assignment 2
%\item Monday January 6 (week 12), revision lecture, 9:00 ARMB 2.98
%\item Thursday January 9 (week 12), revision lecture, 15:00 Herschel LT1
%\end{itemize}

%\noindent
%{\large {\bf \Large Notes}}

%\begin{itemize}
%\item A full week-by-week schedule can be found on our course page.
%\item Work should be handed in at the general office, as is usual practice. 
%\item The mid-semester test will be split between ARMB 2.98 and DENT LTC. 
%The test itself will only cover material from the first \emph{two} 
%chapters. Students may bring one sheet of A4 with them.
%\item Assignment 2 will contain a small computing element, hence the practical 
%in week 10.
%\end{itemize}

\noindent
{\large {\bf \Large Late work policy}}
\medskip

For normal written coursework, a deadline extension of up to 7 days can be requested (by means of submitting a PEC form); work submitted within 7 days of the deadline without good reason will be marked for reduced credit (following the University sliding scale). You should note that no work can be accepted more than 7 days after the original deadline; where work cannot be submitted by this time, the PEC Committee may agree instead to 'discount' or 'exempt' the work (although this would not be routine). For any time-limited assessments (e.g. tests open for 24 hours or less, including NUMBAS tests), rescheduling can be requested (by means of submitting a PEC form). Late work cannot be accepted for NUMBAS assessment; however, a deadline extension can be requested (by means of submitting a PEC form).

For details of the policy (including procedures in the event of illness etc.) please consult the Mathematics, Statistics \& Physics Community pages on Canvas, under: Assessment Information, Late Work and Missed Assessments. 





\newpage

{\Large \textbf{Course outline in brief}}\\
\medskip

The course comprises five topics:
\begin{enumerate}
\item Risk-free and risky assets
\begin{itemize}
\item Interest, compounding of interest
\item Options of European and American type
\end{itemize}
\item Continuous time models of stock price / cash position 
\begin{itemize}
\item Log-Normal distribution
\item Brownian motion, Geometric Brownian motion
\item Black-Scholes pricing
\end{itemize}
\item Estimating Volatility
\begin{itemize}
\item Using historic data
\item Implied volatility
\end{itemize}
\item Exotic options and Monte Carlo simulation
\begin{itemize}
\item Lookback, barrier and Asian options
\item Pricing using simulations from the model
\end{itemize}
\item Introduction to It\^o calculus
\begin{itemize}
\item It\^o integral
\item Stochastic differential equations (SDEs)
\item Models of interest rate
\end{itemize}
\end{enumerate}

\newpage


{\large \textbf{Disclaimer}}\\

These notes may cover material that is not covered in 
the lectures. They may also omit some material that is 
covered. If you find any typos please let me know!

\newpage

\section*{Revision}
The majority of the course will use probability results for continuous 
random variables.

\subsection*{Probability Density Functions}
Let $X$ be a  random variable (r.v.). We say that  $X$ is  {\it continuous} 
if the probability of any fixed value $x$ is 0, i.e. $Pr(X=x)=0$, while   the probability that 
$X$ takes one of the values in some interval $[a,b]$, or $(a,b)$,  may be positive.
We describe the probability law of such a variable  
in terms of its {\it distribution function} (d.f.) \ $F(x)=Pr(X\leq x)$, \ $-\infty < x < \infty$. 
As we know, $F$  is always right-continuous and monotone, increasing (in fact, non-decreasing) from value $0$ at $-\infty$ to value $1$ at $\infty.$
\par We assume  more,  that the d.f. $F$ obeys a {\it density function}, say $f$. This means that 
 
$$F(x)=\int_{-\infty}^{x} f(u)du; \ F'(x)=f(x) \ \mbox{ for all } x \in (-\infty, \infty).$$

\noindent 
Recall that $f$ is a non-negative function and the total integral of $f$ is equal to 1. 

\par With $F$ or $f$ at hand, the probability that $X$ takes a value in a given interval $[a,b]$, \ $a<b$, 
is 
$$Pr(a < X\leq b)=F(b)-F(a)=\int_a^b f(x)dx.$$


The {\it mean value} (expectation) of a `nice' function $g(X)$ of the r.v. $X$ is defined by

$$
E[g(X)]=\int g(x)f(x)dx \ \mbox{ assuming it's finite }.
$$

\subsection*{Transformations of Random Variables}
Consider a random variable $X$ with p.d.f. $f_{X}(x)$. The p.d.f. of an arbitrary differentiable invertible 
transformation $Y=g(X)$ can be deduced as
\[
f_{Y}(y)=f_{X}(g^{-1}(y))\left|\frac{d}{dy}g^{-1}(y)\right|\,.
\]
Note that the term $\left|\frac{d}{dy}g^{-1}(y)\right|$ is known as the ``Jacobian'' of the transformation.

\newpage
\subsection*{Normal Random Variables}
\begin{itemize}
\item We say that the r.v. $X$ has a normal distribution   
with parameters $\mu$ and $\sigma^2$,   $X  \ \sim  \ {\textrm{N}}(\mu, \sigma^2)$, 
if $X$ 
has the density function 

$$f(x)=\frac{1}{ \sqrt{2\pi}\sigma }\exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right], \ -\infty < x < \infty; \ 
\mu \ \mbox{ is any real}, \ \sigma > 0.$$

\noindent
This density is also called normal
or Gaussian. The expectation and the variance of  $X$ are
$$E(X)=\mu,\,\,\,\,{\rm Var}(X)= \sigma^2.$$ 
The graph of $f(x), \ -\infty < x< \infty$, is the familiar bell-shaped curve, symmetric about the  axis  $x= \mu$.

\item A normal r.v. is called {\it standard} if $E(X)=0$ and ${\rm Var}(X)=1.$ 
In this case we use the notation $Z \ \sim {\textrm{N}}(0,1)$. 
That is to say, the density  of $Z$ is 
$$\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}, \ -\infty < x < \infty.$$
The graph of $\phi$ is symmetric about the $y$-coordinate axis. 
The corresponding d.f. is 
$$\Phi (x)= \int_{-\infty}^x \frac{1}{\sqrt{2\pi}}e^{-y^2/2}dy, \ -\infty < x < \infty$$
It is is called the standard normal (or Gaussian) distribution function.
\end{itemize}

\subsection*{Useful Properties of Normal Random Variables}
\begin{itemize}
\item \emph{Linear Combinations:} If the r.v. $X$ is normal, then so is $aX+b$, where $a,b$ are constants.
If $X$ has mean $\mu$ and variance $\sigma^2$, then $Z=(X-\mu)/\sigma$ is standard normal.
(Can you check this?) 
 This fact enables us 
to express probabilities related to $X$ in terms of $\Phi$.

\item \emph{Independence of random variables:} We say that the r.v.s $X_1,\ldots,X_n$ are {\it independent} if for 
arbitrary intervals $I_1,\ldots,I_n$, 
where $I_j=[a_j,b_j]$,  closed or open, 
we have
$$P[X_1\in I_1,\ldots,X_n\in I_n]=P[X_1\in I_1]\cdot\ldots \cdot P[X_n\in I_n].$$

If independent r.v.s $X_j$ are normal with parameters $\mu_j,\sigma_j^2,$ $j=1,\ldots,n$, then the sum 
$X_1+\ldots +X_n$ is normal as well, with mean $\mu_1+\ldots +\mu_n$ and variance
$\sigma^2_1+\ldots +\sigma^2_n$.

In general, we have $E[X+Y]=E[X]+E[Y]$ for arbitrary r.v.s $X$ and $Y$. 
In contrast to this, the equality ${\rm Var}[X+Y]= 
{\rm Var}[X]+ {\rm Var}[Y]$ holds only if $X$ and $Y$ are uncorrelated (e.g. independent).

\end{itemize}

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