May 1, 2003Rolf PoulsenAMS, IMF, KUrolf@math.ku.dk

Topics in Continuous-Time Finance

Suggestions for Final Projects

blah blah blah

American Options

Theoretical angle: Is American option pricing rigorously justified? Specifi-cally, why is it that

πA(t) = sup

τ stopping time

EQt (e−r(τ−t)Pay-off(τ)) ?

Can this be supported by an “any other price would give arbitrage”-statement?It can. In continuous time it’s tricky, but a discrete word (a binomial modelfor instance) it’s doable. Work your way trough Section 4.3 in Pliska (1997)(the book can be found in the Actuarial library).

PDE angle: Solve the PDE for the price numerically. Does naive Crank-Nicolson loose an order, meaning that it’s no faster then the binomial method?What happens when an iterative technique like “successive over-relaxation” isused on the linear complementary formulation? Work/implement your waythrough Section 4.5 in Seydel (2002).

(Something that has not been mentioned at the lectures) Simulation angle:Seemingly, it is not feasible to price American options by simulation. Butthings are not what they seem. Work trough to first sections of Longstaff &Schwartz (2001). Implement their method for the base-case American put.Compare to the binomial model. The ambitious student(s, most likely) willlook at the proofs.

PDEs & “Not Plain Vanilla” Options

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Asian OptionsSection in 6.2 Seydel (2002). Jump conditions. Could take a look at Alziary,Dechamps & Koehl (1997).

2-D SpaceWhatever do you do with a 2-D space variable (“2 stocks”)? Explicit method:Doable (with more things to keep track of) but usual order and especiallystability problems. Implicit methods: Nasty coefficient matrices. Solution:Alternating direction implicit method.Accessible introduction with finance use: Wilmott (1998)[Chapter 48] Test-case: Margrabe’s exchange option. Interesting case: Pay-off (S1 − S2 − K)+.“Numerical literature” introduction: Strikwerda (1989)[Section 7.3] or Mitchell& Griffiths (1980)[Section 2.12]. These are more demanding, but you get towork on easier test problems. (For instance solving Exercise 7.3.2 in Strikw-erda more than suffices for a project.)

A Passport OptionToo tricky . . .?

“Plain Vanilla” Options, But Not the Black/Scholes World

Stochastic Volatility ModelsRead Heston’s article. If you are good at complex analysis: Give the detailsof the inversion. If (like me) you’re not: Implement the formula. Verify theimplemantation through simulation. Possible the “smart/mixing” simulationapproach from Romano & Touzi/Lewis.Plot model smiles. Find some option data and compare.

Jump-DiffusionHmmm, I don’t know. Work your way through Merton’s original article?

Local Volatility ModelsGet to appreciate Dupire’s forward equation; demonstrate that it works inthe Black/Scholes-case. Implement for other volatility functions. Dig up op-tion data. Try to back out the volatility function. This reqires considerableregularization.

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Static HedgingConstruct the replicating portfolios by puts and calls. Compare (throughsimulation) static hedging to dynamic hedging (as in Project 1). Look at howDerman, Ergener & Kani (1995) do it.

Interest Rates

The CIR modelSee the questions elsewhere on the course homepage. There are (evidently) fartoo many questions for one project. Feel free to pick and choose.

Classic short rate models“Implementation-free” angle: Read the original article Vasicek (1977) andanswer the questions posted elsewhere on the course homepage.

Empirical angle: Get hold of time series data on observed ZC yield curves.(There are some Danish curves on the homepage.) Pick out “some yieldtime series’; estimate an Ornstein/Uhlenbeck/Vasicek model on it. Statis-tics/econometric questions: What are parameter estimates; what are theirstandard deviations; how stable are the estimates; how well-specified is themodel?Finance questions: How well are yield curves explained by the model; possi-ble after risk-premia estimation & inclusion? How well are yield volatilitiesexplained? Bjork (1998)[Chapter 17] explains how to calibrate the Vasicekmodel “in theory”. Implement that. How do these calibrated models behave“out of sample”? Recipe: Rogers & Stummer (2000); read & repeat (on DKdata for instance).

Embedded Interest Rate OptionsOption 1: Bermudan option approach to callable mortgage-backed bonds.Jørgensen, Miltersen & Sørensen (1999): Read and repeat. Requires the nu-merical solution of PDE, but real data are not particularly needed.

Option 2: Work through the pricing algorithm for options on coupon bonds inone-factor/Vasicek models. Implement it. Test it via i) simulation ii) solutionsfor ZCB options. Compare observed prices of embedded Bermudan options(in “realkreditobligationer”) to theoretical prices of European options.

Option 3 Find some model where you can price cap-contracts. Work throughthe last part of the “Mathematical Finance 2001”-exam. Get some data &

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determine the spreads. Take a look at my “Capped ARMs”-paper (on thehomepage).LIBOR market models What are they all about?

A Gaussian Two-factor Model Work your way through the first pages of Sec-tion 4.2 in Brigo & Mercurio (2001).

References

Alziary, B., Dechamps, J.-P. & Koehl, P.-F. (1997), ‘A p.d.e. approach toAsian options: Analytical and numerical evidence’, Journal of Banking

and Finance 21, 613–640.

Bjork, T. (1998), Arbitrage Theory in Continuous Time, Oxford.

Brigo, D. & Mercurio, F. (2001), Interest Rate Models Theory and Practice,Springer.

Derman, E., Ergener, D. & Kani, I. (1995), ‘Static Options Replication’, Jour-

nal of Derivatives 2, 78–95.

Jørgensen, P. L., Miltersen, K. R. & Sørensen, C. (1999), A Comparison ofCall Strategies for Callable Annuity Mortgages. Working paper, OdenseUniversity.

Longstaff, F. A. & Schwartz, E. (2001), ‘Valuing american options by sim-ulation: A simple least-squares approach’, Review of Financial Studies

14, 113–147.

Mitchell, A. R. & Griffiths, D. F. (1980), The Finite Difference Method in

Partial Differential Equations, Wiley.

Pliska, S. (1997), Introduction to Mathematical Finance, 1. edn, Blackwell.

Rogers, L. C. G. & Stummer, W. (2000), ‘Consistent fitting of one-factor mod-els to interest rate data’, Insurance: Mathematics & Economics 27, 45–63.

Seydel, R. (2002), Tools for Computational Finance, Springer.

Strikwerda, J. (1989), Finite Difference Schemes and Partial Differential

Equations, Wadsworth & Brooks/Cole.

Vasicek, O. (1977), ‘An Equilibrium Characterization of the Term Structure’,Journal of Financial Economics 5, 177–188.

Wilmott, P. (1998), Derivatives, Wiley.

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