Chapter 15: The 5-step Method in Discrete Time

Question 1

Equivalent measures

Answer 1

Two measures, P and Q, which apply to the same sigma-algebra F are said to be equivalent if for any event E in F:
P(E) > 0 if and only if Q(E) > 0

where P(E) and Q(E) are the probabilities of E under P and Q respectively.

Question 2

Summary of the 5 step method

Answer 2

• Establish the equivalent martingale measure Q
• Propose a fair price, Vt, for a derivative and its discounted value Et = Vt exp{-rt}
• Use the martingale representation theorem to construct a hedging strategy
• Show that this hedging strategy replicates the derivative payoff at time n.
• Therefore Vt is the fair value of the derivative at time n.

Question 3

Previsibility

Answer 3

ϕt is previsible if it is known based on information up to, but not including time t.

Question 4

Self-financing

Answer 4

A portfolio (ϕt, ψt) is self-financing if ϕt, ψt are previsible and dVt = ϕt dSt + ψt dBt

i.e. the required change in the value of the portfolio
… over each instant of time
… is equal to the pure instantaneous investment gain.

Question 5

“Complete” investment market

Answer 5

An investment market is complete if, for every derivative in that market, there exists a replicating strategy for that derivative.