Barrier options: Symmetry and Static Hedging
=========

Prove the Reflection Theorem from 
http://www.math.ku.dk/~rolf/static0.pdf

Applications:

(A.i)  Derive price formulas for barrier option prices in the 
       Black/Scholes model

(A.ii) Explain - and implement - the construction of static hegdes for barrier 
       options by matching "the adjusted pay-off function" 
 
(A.iii) Derive expressions for the distribution of Br. motion  
        and its running maximum (W_t, max_{u <= t} W_u)

(A.iv)  Prove the explicit expression for the distribution (and how to 
        sample from it) given by
http://www.math.ku.dk/~rolf/BeagleholeDybvigZhou.pdf
       fof the maximum of a Brownian bridge

(A.v)   Show how Brownian bridge simulation can enhance barrier option 
        price simulation; from Dybvig et al. above to Joshi and Leung
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=907386



Basket Options: Approximation and Application
==========

What is a basket option and why is there no closed-form solution for 
its price in the Black-Scoles model?

Approximations:
     - moment matching 
     - Milevsky & Posner
http://www.yorku.ca/milevsky/Papers/JD1998A.pdf 
     - any number of other suggestions 
Describe, compare, replicate numbers in original papers 

Application: Basically, how do you price this
http://www.math.ku.dk/~rolf/PlusValuta.pdf
Steps: 
     - What does the contract do? where is the basket option? 
     - Do does one build arbitrage-free currency models? Björk Ch. 16-17
     - Where on the www do find the necessary data (current and historical)
     to estimate a price?
     - Does the bank charge fairly for the basket option embedded in 
     PlusValuta?  




Pricing Options on Coupon Bonds
========

Why is there no closed-form solution even in Gaussian models (like Vasicek)? 

What can we do in 1-factor models? Jamshidian's trick 
http://www.math.ku.dk/~rolf/teaching/mfe04/mfe04.exam.pdf

Stochastic duration approximations:
http://www.math.ku.dk/~rolf/teaching/PhDcourse/munk.pdf 

Any number of other methods exist.

Implement (1-d, 2-d cases) and compare.