Quants work long hours. Some work almost as hard as nurses. It is surely too much to ask that they also solve an equation numerically!

Example You need to value a fixed-income contract and so you have to choose a model. Do you (a) analyse historical fixed-income data in order to develop an accurate model, which is then solved numerically, and finally back-tested using a decade's worth of past trades to test for robustness, or (b) use Professor X's model because the formula are simple and, quite frankly, you don't know any numerical analysis, or (c) do whatever everyone else is doing? Typically people will go for (c), partly for reasons already discussed, which amounts to (b).

Example You are an aeronautical engineer designing a new airplane. Boy, those Navier-Stokes equations are hard! How do you solve non-linear equations? Let's simplify things, after all you made a paper plane as a child, so let's just scale things up. The plane is built, a big engine is put on the front, it's filled with hundreds of passengers, and it starts its journey along the runway. You turn your back, without a thought for what happens next, and start on your next project.

One of those examples is fortunately not real. Unfortunately, the other is.

Quants love closed-form solutions. The reasons are

1. Pricing is faster

2. Calibration is easier

3. You don't have to solve numerically.

Popular examples of closed-form solutions/models are, in equity derivatives, the Heston stochastic volatility model (Heston, 1993), and in fixed income, Vasicek (1977),^{[1]} Hull & White (1990), etc.

Although the above reasons for choosing models with closed-form solutions may be true they are not important criteria per se in the scheme of things. Indeed, there are a couple of obvious downsides to restricting yourself to such models. Just because you want an easy life doesn't make the model work. By choosing ease of solution or calibration over accuracy of the model you may be making calibration less stable.

Models with closed-form solutions have several roles in applied mathematics and quantitative finance. Closed-form solutions are

• useful for preliminary insight

• good for testing your numerical scheme before going on to solve the real problem

• for examining second-year undergraduate mathematicians.

People who only know about the 'popular' models do not make good quants, good risk managers, or good traders. They don't even make good researchers.

Lesson9: Valuation is Not Linear

You want to buy an apple, so you pop into Waitrose. An apple will cost you 50p. Then you remember you've got friends coming around that night and these friends really adore apples. Maybe you should buy some more? How much will 100 apples cost?

I've started asking this question when I lecture. And although it is clearly a trick question, the first answer I get is always £50.

Anyone who thinks that is the right answer will make a fantastic, classically trained, old-fashioned, and eventually loss-making, and finally unemployed, quant. As anyone who has ever shopped should know, the answer is 'less than £50.' Economies of scale.

Here's a quote from a well-known book (my emphasis): 'The change of numeraire technique probably seems mysterious. Even though one may agree that it works after following the steps in the chapter, there is probably a lingering question about why it works. Fundamentally it works because valuation is linear.... The linearity is manifested in the statement that the value of a cash flow is the sum across states of the world of the state prices multiplied by the size of the cash flow in each state.... After enough practice with it, it will seem as natural as other computational tricks one might have learned.'

Note it doesn't say that linearity is an assumption, it is casually taken as a fact. Valuation is apparently linear. Now there's someone who has never bought more than a single apple!

Example The same author may be on a sliding royalty scale so that the more books he sells the bigger his percentage. How can nonlinearity be a feature of something as simple as buying apples or book royalties yet not be seen in supposedly more complex financial structured products? (Maybe he is selling so few books that the nonlinearity has not kicked in!)

Example A bank makes a million dollars profit on CDOs. Fantastic! Let's trade 10 times as much! They make $10 million profit. The bank next door hears about this and decides it wants a piece of the action. They trade the same size. Between the two banks they make $18 million profit. Where'd the $2 million go? Competition between them brings the price and profit margin down. To make up the shortfall, and because of simple greed, they increase the size of the trades. Word spreads and more banks join in. Profit margins are squeezed. Total profit stops rising even though the positions are getting bigger and bigger. And then the inevitable happens, the errors in the models exceed the profit margin (margin for error), and between them the banks lose billions. 'Fundamentally it works because valuation is linear.' Oh dear!

This is not really the place for me to explain all the non-linear quantitative finance models in existence. There are many, and I will give some references below. But how many do you know? I suspect you'll struggle to name a single one. Researchers, modellers, banks, software companies, everyone has a lot invested in this highly unrealistic assumption of linearity, and yet many of the best models I've seen are non-linear.^{[2]}

To appreciate the importance of nonlinearity you have to understand that there is a difference between the value of a portfolio of contracts and the sum of the values of the individual contracts.^{[3]} In pseudo math, if the problem has been set up properly, you will get

Imagine we have just inherited a barrier option from someone who got fired earlier today. As with many barrier options it has a lot of volatility risk because of the way gamma changes sign. We value it at $10 using one of the non-linear models to be mentioned shortly. Not worth as much as I'd hoped. Because of all that volatility risk I m being conservative in my valuation. The volatility risk is because the contract has gamma. If I could reduce gamma then I will reduce volatility risk and the value should increase, there'll be less risk and so less reason to be conservative. I look around and I see an exchange-traded, vanilla option selling for $5. This option has caught my eye because it has similar gamma to the barrier, around the same asset value, etc. but of the opposite sign. If I buy this option it will reduce my volatility risk! I buy the option, it costs $5. I now value the portfolio consisting of the barrier and the vanilla, valued as one unit, and I get the answer... $16. In other words, the benefit to my portfolio of buying the vanilla has outweighed its cost. I seem to have made $1. This is 'static hedging.' We could try buying two of these options, thinking that buying one vanilla makes us $1 'out of thin air' so we can conjure up $2 by buying two options. Sadly, we find that the portfolio of barrier option plus two vanillas is actually only $20.5, not the hoped-for $22. Again, this is because of nonlinearity and what this tells us is that somewhere between buying no vanillas and buying two is the optimal static hedge.

There are many papers on static hedging of exotics (often barriers) with vanillas, and they all have in common that there is no financial benefit whatsoever to such hedging. They are ad hoc fixes to poor models. However, when the model is non-linear the benefit is clearly seen in terms of extra value.

Here is a partial list of the advantages to be found in some non-linear models.

• Perfect calibration: Actual and implied quantities are not confused. We don't have to believe that there is any useful information about parameters contained in the market prices of derivatives. Nonlinearity ensures that reasonable market prices of liquid instruments are matched by default. No more tying yourself in numerical knots to calibrate your unstable stochastic volatility model. Nonlinearity means that there are genuine model-based reasons for static hedging, a benefit of which is perfect calibration.

Calibration is automatic. And you can calibrate to both bid and ask prices, and to liquidity. How many of you calibrating Heston can make that claim?

• Speed: The models will be almost as fast to solve numerically as their equivalent linear models (sometimes faster because calibration happens by default).

• Easy to add complexity to the model: The modular form of the models means that it is easy to add complexity in terms of jumps, stochastic volatility, etc.

• Optimal static hedging: Hedging exotics with traded vanillas will increase 'value' because of the non-linearity. This can be optimized.

• Can be used by buy and sell sides: Traditionally the buy side uses one type of models and the sell side another. This is because the buy side is looking for arbitrage while the sell side are valuing exotics which their risk management insists are priced to be consistent with vanillas (i.e. the thought of accepting that there may be arbitrage is abhorrent!) By changing the 'target function' in certain optimization problems some of the models can be used by hedge funds looking for statistical arbitrage and by investment banks selling exotics that are calibrated to vanillas.

That list is just to whet your appetite. Even if you think everything else in this chapter is baloney, you must at least look up the following articles. That's assuming you are ready to move outside your comfort zone.

And now the reading list:

• Hoggard, Whalley & Wilmott (1994) is an early paper on a non-linear model, in this case a model for pricing in the presence of transaction costs. Also look at other papers on costs by Whalley et al.

• Avellaneda, Levy & Paras (1995) introduce the Uncertain Volatility Model (UVM). (See also Avellaneda & Paras, 1976 and Avellaneda & Buff, 1977.) This is a very simple and clever model, very compact in form compared with models such as Heston and considerably easier to solve and to calibrate (it's automatic). The trick is to price conservatively and look at worst-case scenarios. A follow-up paper showed just why nonlinearity is so important (optimal static hedging and value maximization).

• In Hua & Wilmott (1997, 1998, 1999, 2001) we look at modelling crashes in a worst-case scenario framework, and again a non-linear model results. This model and the UVM both permit perfect hedging and are complete-market models.^{[4]} In later articles we show how to strip the model down to its bare essentials into a wonderfully simple (and very popular with investors!) risk management technique called

CrashMetrics.

• In Ahn & Wilmott (2003, 2007, 2008) and several follow-up articles we introduce a model based on stochastic volatility, but with a twist. Instead of taking the common route of introducing a market price of volatility risk to derive an equation we work within a framework in which we calculate the mean option value and its standard deviation. Effectively we say that, let's be honest, you can't hedge volatility no matter what academics say, so let's just accept it as fact and get on with reducing risk as much as we can. As well as being no harder to implement than standard volatility models - actually it's easier because calibration is automatic - it can be used for both arbitrage and for valuation of exotics. It's about the nearest thing to an all-singing, all-dancing, model as you can find at the moment.

[1] To be fair to Vasicek I'm not sure he ever claimed he had a great model, his paper set out the general theory behind the spot-rate models, with what is now known as the Vasicek model just being an example.

[2] I have to declare an interest here, I and colleagues have developed some of these.

[3] What I call, to help people remember it, the 'Beatles effect.' The Fab Four being immeasurably more valuable as a group than as the sum of individuals ...Wings, Thomas the Tank Engine,...

[4] Complete market' is a common phrase and framework, but its meaning can be a bit ambiguous once you move outside classical, linear models. Here I say the models are complete in the same way that the Black-Scholes for pricing American options is complete.

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