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.