Thanks for bearing with me through a dozen lessons. As a reward for your patience and tolerance I am going to give you a bonus lesson! And this bonus lesson is probably the easiest one for quants to implement immediately. The lesson is ... dump the binomial method!

I really like the binomial method. But only as a teaching aid. It is the easiest way to explain

1. hedging to eliminate risk

2. no arbitrage

3. risk neutrality.

I use it in the CQF to explain these important, and sometimes difficult to grasp, concepts.^{[1]} But once the CQFers have understood these concepts they are instructed never to use the binomial model again, on pain of having their CQFs withdrawn!

Ok, I exaggerate a little. The binomial model was the first of what are now known as finite-difference methods. It dates back to 1911 and was the creation of Lewis Fry Richardson, all-round mathematician, sociologist, and poet.

A lot of great work has been done on the development of these numerical methods in the last century. The binomial model is finite differences with one hand tied behind its back, hopping on one leg, while blindfolded. So when I refer to the 'binomial method here what I am really criticizing is people s tendency to stick with the simplest, most naive finite-difference method, without venturing into more sophisticated territory, and without reading up on the more recent numerical-methods literature.

Why is the binomial method so ubiquitous? Again, habit is partly to blame. But also all those finance professors who know bugger all about numerical methods but who can just about draw a tree structure, they are the ones responsible. Once an academic writes his lecture notes then he is never going to change them. It s too much effort. And so generations of students are led to believe that the binomial method is state of the art when it is actually prehistoric.

Summary

Question 1: What are the advantages of diversification among products, or even among mathematical models? Answer 1: No advantage to your pay whatsoever!

Question 2: If you add risk and curvature what do you get? Answer 2: Value!

Question 3: If you increase volatility what happens to the value of an option?

Answer 3: It depends on the option!

Question 4: If you use ten different volatility models to value an option and they all give you very similar values what can you say about volatility risk?

Answer 4: You may have a lot more than you think!

Question 5: One apple costs 50p, how much will 100 apples cost you?

Answer 5: Not £50!

How did you do in the quiz at the start? If you are new to quant finance you may have got some of the answers correct. If you have just come out of a Masters in Financial Engineering then you probably got most of them wrong. But if you're a quant or risk manager who likes to think for himself and is not happy with the classical 'results' of quantitative finance, then maybe you even got all of them right!

QF is interesting and challenging, not because the mathematics is complicated, it isn't, but because putting maths and trading and market imperfections and human nature together and trying to model all this, knowing all the while that it is probably futile, now that's fun!

References and Further Reading

Ahmad, R & Wilmott, P 2005 Which free lunch would you like today, Sir? Delta hedging, volatility arbitrage and optimal portfolios. Wilmott magazine November 64-79

Ahmad, R & Wilmott, P 2007 The market price of interest rate risk: Measuring and modelling fear and greed in the fixed-income markets. Wilmott magazine January

Ahn, H & Wilmott, P 2003 Stochastic volatility and mean-variance analysis. Wilmott magazine November 84-90

Ahn, H & Wilmott, P 2007 Jump diffusion, mean and variance: how to dynamically hedge, statically hedge and to price. Wilmott magazine May 96-109

Ahn, H & Wilmott, P 2008 Dynamic hedging is dead! Long live static hedging! Wilmott magazine January 80-87

Ahn, H & Wilmott, P 2009 A note on hedging: restricted but optimal delta hedging; mean, variance, jumps, stochastic volatility, and costs. In preparation

Avellaneda, M & Buff, R 1997 Combinatorial implications of nonlinear uncertain volatility models: the case of barrier options. Courant Institute, NYU

Avellaneda, M & Paras, A 1996 Managing the volatility risk of derivative securities: the Lagrangian volatility model. Applied Mathematical Finance 3 21-53

Avellaneda, M, Levy, A & Paras, A 1995 Pricing and hedging derivative securities in markets with uncertain volatilities. Applied Mathematical Finance 2 73-88

Derman, E & Kani, I 1994 Riding on a smile. Risk magazine 7 (2)

32-39 (February)

Dupire, B 1993 Pricing and hedging with smiles. Proc AFFI Conf, La Baule June 1993

Dupire, B 1994 Pricing with a smile. Risk magazine 7 (1) 18-20 (January)

Haug, EG 2007 Complete Guide to Option Pricing Formulas. McGraw-Hill

Heston, S 1993 A closed-form solution for options with stochastic volatility with application to bond and currency options. Review of Financial Studies 6 327-343

Ho, T & Lee, S 1986 Term structure movements and pricing interest rate contingent claims. Journal of Finance 42 1129-1142

Hoggard, T, Whalley, AE & Wilmott, P 1994 Hedging option portfolios in the presence of transaction costs. Advances in Futures and Options Research 7 21-35

Hua, P & Wilmott, P 1997 Crash courses. Risk magazine 10 (6) 64-67 (June)

Hua, P & Wilmott, P 1998 Value at risk and market crashes. Derivatives Week

Hua, P & Wilmott, P 1999 Extreme scenarios, worst cases, Crash-Metrics and Platinum Hedging. Risk Professional

Hua, P & Wilmott, P 2001 CrashMetrics. In New Directions in Mathematical Finance (eds Wilmott, P & Rasmussen, H)

Hull, JC & White, A 1990 Pricing interest rate derivative securities. Review of Financial Studies 3 573-592

Rubinstein, M 1994 Implied binomial trees. Journal of Finance 69

771-818

Schonbucher, PJ & Wilmott, P 1995 Hedging in illiquid markets: nonlinear effects. Proceedings of the 8th European Conference on Mathematics in Industry

Schoutens, W, Simons, E & Tistaert, J 2004 A perfect calibration! Now what? Wilmott magazine March 66-78

Vasicek, OA 1977 An equilibrium characterization of the term structure. Journal of Financial Economics 5 177-188

Whalley, AE & Wilmott, P 1993a Counting the costs. Risk magazine 6 (10) 59-66 (October)

Whalley, AE & Wilmott, P 1993b Option pricing with transaction costs. MFG Working Paper, Oxford

Whalley, AE & Wilmott, P 1994a Hedge with an edge. Risk magazine 7 (10) 82-85 (October)

Whalley, AE & Wilmott, P 1994b A comparison of hedging strategies. Proceedings of the 7th European Conference on Mathematics in Industry 427-434

Whalley, AE & Wilmott, P 1995 An asymptotic analysis of the Davis, Panas and Zariphopoulou model for option pricing with transaction costs. MFG Working Paper, Oxford University

Whalley, AE & Wilmott, P 1996 Key results in discrete hedging and transaction costs. In Frontiers in Derivatives (eds Konishi, A & Dattatreya, R) 183-196

Whalley, AE & Wilmott, P 1997 An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Mathematical Finance 7 307-324

Wilmott, P 2000 The use, misuse and abuse of mathematics in finance. Royal Society Science into the Next Millennium: Young scientists give their visions of the future. Philosophical Transactions 358 63-73

Wilmott, P 2002 Cliquet options and volatility models. Wilmott magazine December 2002

Wilmott, P 2006a Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd

Wilmott, P 2006b Frequently Asked Questions in Quantitative Finance. John Wiley & Sons Ltd

[1] It's also instructive to also take a quick look at the trinomial version, because then you see immediately how difficult it is to hedge in practice.

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