Financial Maths

Coursework 2

Mathematical Finance

Group 27

Q1. Hedging in Complete and Incomplete market
Solution:
Complete market Suppose we have m states. A complete market A is one with the marketed subspace Span(A.1,A.2, ⋯, A.n) includes all possible payoffs over the m states, i.e., if it contains all possible m-dimensional vectors.

Incomplete market Suppose we have m states. An incomplete market corresponds to a market with fewer linear independent assets than states, i.e. Rank (A) < m.

Implications:
Complete market Let A be payoff matrix and b the payoff we want to hedge.
1. For hedging in Complete Markets with no redundant assets, the perfect hedge is.
2. For hedging in Complete Markets with redundant assets, there can always be found m linear independent basis assets in A to form A1 such that A1 is invertible, and the perfect hedging portfolio for any focus asset b is given by.
3. For hedging in Complete Markets with no redundant assets, state price vector is. The focus asset price.

Incomplete market
1. For hedging in Incomplete Markets with no redundant assets, the best hedge is.
2. For hedging in Incomplete Markets with redundant assets, similar approach as in Complete Markets. ( Ales Cerny, Mathematical Techniques in Finance, 2009)

Q2. Hedging Problem
Solution:
a.
Returns matrix of basis assets is

The payoff the digital put option is
Price of return S = 1, and

Therefore,

Since

The hedge which minimizes the expected squared hedging error is

By Matlab code: =[1.3*sqrt(0.3),1.05*sqrt(0.3);1.1*sqrt(0.5),1.05*sqrt(0.5);0.8*sqrt(0.2),1.05*sqrt(0.2)]
=[0;0;sqrt(0.2)]
=inv('*)*'*

We get: x = -2.0000
2.2857
Therefore, we should short