The course starts with a financial market in a continuous time framework, where the

Black/Scholes model is typically taken as the prime example. The "modern" martingale theory is

made use of in characterizing absence of arbitrage, in a fairly simple way, for continuous

diffusion, or Ito-processes.

- After the introduction of the theory of pricing and hedging derivatives in a complete model,

including American type derivatives, we move to forward and futures contracts, and then to term

structure models, including the models of Vasicek, Cox-Ingersoll-Ross. The Heath-Jarrow-Morton

framework is also given some coverage. If time permits, we extend the theory of options to

include stochastic interest rates, where we present the Amin-Jarrow (1992) solution for the pricing

of a European call option by the use of the forward measure.

- We then analyze optimal consumption and portfolio choice in a dynamic setting for the time and

state continuous models. As an example we show explicitly how to solve the celebrated Merton¿s

problem, when the consumer has constant relative risk aversion.

- The last part consists of dynamic equilibrium theory in a continuous time setting. Here we

present the Arrow-Debreu equilibrium, and the Lucas consumption based equilibrium model. It is

shown, in particular, how one may implement the Lucas-type equilibrium in an Arrow-Debreu

setting for a complete market, based on an idea of Roy Radner. As an illustration, we show how

the Black/Scholes formula may alternatively be derived in an equilibrium of this type.

Furthermore, we derive the Cox-Ingersoll-Ross term structure model as an equilibrium solution, as

it was originally published. We round off with the consumption based capital asset pricing model

(CCAPM). If time permits, we may also study incomplete models at an introductory level, and

perhaps also extend the model of the underlying uncertainty to Lévy-based processes containing

unpredictable jumps at random time points.

The course requires knowledge in modern microeconomics, some micro based macroeconomics,

intermediate and graduate finance, and modern probability theory. For the latter, it is an advantage

to have taken, or take in parallel, the course "An introduction to stochastic analysis with

applications", also given in the spring semester.

Topics

1. The Black/Scholes model

2. State prices and equivalent martingale measures

3. Term structure models

4. Derivative pricing

5. Portfolio and consumption choice

6. Equilibrium in continuous time models