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.