MAS3904/MAS8904 Stochastic Financial Modelling Dr Andrew (Andy) Golightly a.golightly@ncl.ac.uk Semester 1, 2018/19 Administrative Arrangements Lectures on Tuesdays at 14:00 (PERCY G13) and Thursdays at 14:00 (HERB LT3). Problems classes/drop-ins are typically on Tuesdays, 17:00 (PERCY G13). Week 1 is a lecture! Computer class: Herschel building, Red/Blue cluster, Monday 11:00. Week 10. Assessment is by exam (90%), answers to set questions (5%) and class test (5%). Hand in work by 4pm, usually on Mondays, weeks 5 and 11 (2 assignments). Announcements will be made to your ncl.ac.uk account - please check regularly. Please bookmark the link to the webpage of the course, which is www.mas.ncl.ac.uk/ nag48/teaching/mas3904/ This is also available via Blackboard. Books J. Hull: Options, Futures and Other Derivatives (Prentice-Hall, 2003) S. Ross: An Elementary Introduction to Mathematical Finance (CUP, 2003) M. Capinski, T. Zastawniak: Mathematics for Finance (Springer, 2003) S. Shreve: Stochastic Calculus for Finance 1 (Springer, 2004) J. Franke, W. Hardle, C. Hafner: Statistics of Financial Markets (Springer, 2004) 1
Important dates Dates for your diary are as follows: Thursday October 4 (week 1), assignment 1 given out Monday October 29 (week 5), hand in assignment 1 (by 4pm) Thursday November 8 (week 6), mid-semester test, 14:00 (50 mins) Thursday November 15 (week 7), assignment 2 given out Monday December 3 (week 10), computer practical, 11:00 Herschel Red/Blue Monday December 10 (week 11), hand in assignment 2 Tuesday January 8 (week 12), revision lecture, 14:00 PERCY G13 Thursday January 10 (week 12), revision lecture, 14:00 Herschel LT3 Notes A full week-by-week schedule can be found on our course page. Work should be handed in at the general office, as is usual practice. The mid-semester test will be in HERB LT3. The test itself will only cover material from the first two chapters. Students may bring one sheet of A4 with them. Assignment 2 will contain a small computing element, hence the practical in week 10. Late work policy Please note: It is not possible to extend submission deadlines for coursework in this module and no late work can be accepted. The module contains one in-course test that cannot be rescheduled. For details of the policy (including procedures in the event of illness etc.) please look at the School web site: https://internal.ncl.ac.uk/maths/students/late-work/ www.mas.ncl.ac.uk/ nag48/teaching/mas3904/ 2
Course outline in brief The course comprises five topics: 1. Risk-free and risky assets Interest, compounding of interest Options of European and American type 2. Continuous time models of stock price / cash position Log-Normal distribution Brownian motion, Geometric Brownian motion Black-Scholes pricing 3. Estimating Volatility Using historic data Implied volatility 4. Exotic options and Monte Carlo simulation Lookback, barrier and Asian options Pricing using simulations from the model 5. Introduction to Itô calculus Itô integral Stochastic differential equations (SDEs) Models of interest rate www.mas.ncl.ac.uk/ nag48/teaching/mas3904/ 3
Revision The majority of the course will use probability results for continuous random variables. Probability Density Functions Let X be a random variable (r.v.). We say that X is 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 distribution function (d.f.) F(x) = Pr(X x), < x <. As we know, F is always right-continuous and monotone, increasing (in fact, non-decreasing) from value 0 at to value 1 at. We assume more, that the d.f. F obeys a density function, say f. This means that F(x) = x f(u)du; F (x) = f(x) for all x (, ). Recall that f is a non-negative function and the total integral of f is equal to 1. 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 b) = F(b) F(a) = b a f(x)dx. The mean value (expectation) of a nice function g(x) of the r.v. X is defined by E[g(X)] = g(x)f(x)dx assuming it s finite. 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)) d dy g 1 (y). Note that the term is known as the Jacobian of the transformation. d dy g 1 (y) www.mas.ncl.ac.uk/ nag48/teaching/mas3904/ 4
Normal Random Variables We say that the r.v. X has a normal distribution with parameters µ and σ 2, X N(µ,σ 2 ), if X has the density function f(x) = 1 2πσ exp [ (x µ)2 2σ 2 ], < x < ; µ is any real, σ > 0. This density is also called normal or Gaussian. The expectation and the variance of X are E(X) = µ, Var(X) = σ 2. The graph of f(x), < x <, is the familiar bell-shaped curve, symmetric about the axis x = µ. A normal r.v. is called standard if E(X) = 0 and Var(X) = 1. In this case we use the notation Z N(0,1). That is to say, the density of Z is φ(x) = 1 2π e x2 /2, < x <. The graph of φ is symmetric about the y-coordinate axis. The corresponding d.f. is Φ(x) = x 1 2π e y2 /2 dy, < x < It is is called the standard normal (or Gaussian) distribution function. Useful Properties of Normal Random Variables Linear Combinations: If the r.v. X is normal, then so is ax + b, where a,b are constants. If X has mean µ and variance σ 2, then Z = (X µ)/σ is standard normal. (Can you check this?) This fact enables us to express probabilities related to X in terms of Φ. Independence of random variables: We say that the r.v.s X 1,...,X n are independent if for arbitrary intervals I 1,...,I n, where I j = [a j,b j ], closed or open, we have P[X 1 I 1,...,X n I n ] = P[X 1 I 1 ]... P[X n I n ]. If independent r.v.s X j are normal with parameters µ j,σj 2, j = 1,...,n, then the sum X 1 +...+X n is normal as well, with mean µ 1 +...+µ n and variance σ1 2+...+σ2 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 Var[X +Y] = Var[X]+Var[Y] holds only if X and Y are uncorrelated (e.g. independent). www.mas.ncl.ac.uk/ nag48/teaching/mas3904/ 5