Module Details

The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
Title Stochastic Modelling in Insurance and Finance
Code MATH375
Coordinator Dr B Gashi
Mathematical Sciences
Bujar.Gashi@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2016-17 Level 6 FHEQ Second Semester 15

Aims

  1. Be able to understand the stochastic modelling for different actuarial and financial problem.

  2. Develop the necessary skills to construct asset liabilities models and to value financial derivatives.

  3. Prepare the students to sit for the exams of CT8 subject of the Institute of Actuaries.


Learning Outcomes

Understand the continuous time log-normal model of security prices, auto-regressive model of security prices and other economic variables (e.g. Wilkie model). Compare them with alternative models by discussing advantages and disadvantages. Understand the concepts of standard Brownian motion, Ito integral, mean-reverting process and their basic properties. Derive solutions of stochastic differential equations for geometric Brownian motion and Ornstein-Uhlenbeck processes.

Acquire the ability to compare the real-world measure versus risk-neutral measure.  Derive, in concrete examples, the risk-neutral measure for binomial lattices (used in valuing options). Understand the concepts of risk-neutral pricing and equivalent martingale measure.  Price and hedge simple derivative contracts using the martingale approach.

Be aware of the first and second partial derivative (Greeks) of an option price. Price zero-coupon bonds and interest–rate derivatives for a general one-factor diffusion model for the risk-free rate of interest via both risk-neutral and state-price deflator approach. Understand the limitations of the one-factor models.

Understand the Merton model and the concepts of credit event and recovery rate. Model credit risk via structural models, reduced from models or intensity-based models.

Understand the two-state model for the credit ratings with constant transition intensity and its generalizations: Jarrow-Lando-Turnbull model.


Syllabus

(a)    Measures of risk
Variance of return, downside sime-variance of return, shortfall probabilities, Value at Risk (VaR), Tail VaR; perform calculation using risk measures and explain how the distribution of returns and the thickness of tails will influence the assessment of risk.

 (b)   Stochastic modelling of the behaviour of the security prices
Continuous time log-normal model of security prices, auto-regressive models of security prices and other economic variables (e.g. Wilkie model): discussion, simple calculations involving the models, comparison with alternative models, advantages and disadvantages.

(c)    Ito Formula: theory and applications
Standard Brownian motion, Ito integral, mean-reverting processes: definition and basic properties. Ito’s formula: statement and application in simple problems. Stochastic differential equations for geometric Brownian motion and Ornstein-Uhlenbeck process: derivation of solutions.
(d)   Opt ions pricing, valuation and hedging
Arbitrage, complete markets and factors influencing the option prices. Specific results: valuation of a forward contract; upper and lower bounds for European and American call and put options.
(e)   Martingale measures and derivative pricing model
Complete market, risk-neutral pricing and equivalent martingale measure, price and hedge simple derivative contracts using the martingale approach. Black–Scholes partial differential equation. Pricing via state-price deflators: apply in simple models as binomial model and Black-Scholes model and demonstrate equivalence to risk-neutral pricing. Partial derivatives (Greeks) of an option price: first and second derivative.
 
(f)      Models for the term-structure of interest rates
Models for the term structure interest rates: desirable characteristics; Vasicek, Cox-Ingersoll-Ross and Hull-White models. Pricing zero-coupon bonds and interest–rate derivatives for a general one-factor diffusion model for the risk-free rate of interest: risk-neutral approach versus state-price deflator approach.
 
(g)    Models for credit risk
Credit event, recovery rate, Merton model. Approaches to modelling credit risk: structural models, reduced form models, intensity-based models. Two-state model for credit ratings: with constant transition intensity, generalized to the Jarrow-Lando-Turnbull model, generalized to incorporate stochastic transition intensity.

Recommended Texts

Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.
Explanation of Reading List:

Key text:

Hull, J.C. (2000): Options, Futures, and Other Derivatives, 4th Ed. Prentice Hall

Recommended text:

[1] Karatzas I. and Shreve S.E. (1991): Brownian motion and stochastic calculus, Springer

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH101; MATH102; MATH103; MATH162; MATH262; MATH264; MATH263  

Co-requisite modules:

 

Modules for which this module is a pre-requisite:

 

Programme(s) (including Year of Study) to which this module is available on a required basis:

 

Programme(s) (including Year of Study) to which this module is available on an optional basis:

Programme:NG31 Year: 3

Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Unseen Written Exam  2.5 hours  Second Semester  100  Yes  Standard UoL penalty applies  Assessment 1 Notes (applying to all assessments) All 5 questions carry equal weight. 
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes