Introduce the stochastic modelling for different actuarial and financial problem.

Help students to develop the necessary skills to construct asset liabilities models and to value financial derivatives, in continuous time.

Prepare the students to sit for the exams of CM2 subject of the Institute and Faculty of Actuaries.

(LO1) 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.

(LO2) 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.

(LO3) 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.

(LO4) 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.

(LO5) Understand the two-state model for the credit ratings with constant transition intensity and its generalizations: Jarrow-Lando-Turnbull model.

(S1) Problem solving skills

(S2) Numeracy

(a) Stochastic modelling of the behaviour of the security prices:

Continuous time log-normal model of security prices, the distribution functions for the accumulated amount of a singre premium and for the present value of a sum due at a given specified future time provided (1+i) is log-normal distributed.

(b ) 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.

(c) Options 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.

(d) Martingale measures and derivative pricing model:

Complete mar ket, 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 Black-Scholes model and demonstrate equivalence to risk-neutral pricing. Partial derivatives (Greeks) of an option price: first and second derivative.

(e) 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.

(f) 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: w ith constant transition intensity, generalized to the Jarrow-Lando-Turnbull model, generalized to incorporate stochastic transition intensity.