Lyceum 1: The Different Types Of Mathematics Seen In
Quantitative Finance
Aristotle
The real-world subject of quantitative finance uses tools from many branches of mathematics. And financial modeling can be approached in a variety of different ways. For some strange reason the advocates of different branches of mathematics get quite emotional when discussing the merits and demerits of their methodologies and those of their ‘opponents.’ Is this a territorial thing, what are the pros and cons of martingales and differential equation, what is all this fuss and will it end in tears before bedtime? Always both informative and objective, Team Wilmott will give you its unbiased analysis of the different camps. This is the first in a series of tutorials on the mathematics of finance.
Financial Modeling
Here’s a list the various approaches to modeling and a selection of useful tools.
The distinction between a ‘modeling approach’ and a ‘tool’ will start to become clear.
Modeling Approaches
Probabilistic
Deterministic
Discrete: difference equations
Continuous: differential equations
Useful Tools
Simulations
Approximations
Asymptotic analysis
Series solutions
Discretization methods

Green’s functions
While these are not exactly arbitrary lists, they are certainly open to some criticism or addition. Let’s first take a look at the Modeling Approaches.
Probabilistic:
One of the main assumptions about the financial markets, at least as far as quantitative finance goes, is that asset prices are random. We tend to think of describing financial variables as following some random path, with parameters describing the growth of the asset and its degree of randomness. We effectively model the asset path via a specified rate of growth, on average, and its deviation from that average. This approach to modeling has had the greatest impact over the last 30 years, leading to the explosive growth of the derivatives markets.
Deterministic:
The idea behind this approach is that our model will tell us everything about the future. Given enough data, and a big enough brain, we can write down some equations or an algorithm for predicting the future. Interestingly, the subject of dynamical systems and chaos fall into this category. And, as you know, chaotic systems show such sensitivity to initial conditions that predictability is in practice impossible. This is the ‘butterfly effect,’ that a butterfly flapping its wings in Brazil will ‘cause’ rainfall over Manchester. (Like what doesn’t!) A topic popular in the early 1990s, this has not lived up to its promises in the financial world.
Discrete/Continuous:
Whether probabilistic or deterministic the eventual model you write down can be discrete or continuous. Discrete means that asset prices and/or time can only be incremented in finite chunks, whether a dollar or a cent, a year or a day.
Continuous means that no such lower increment exists. For reasons that Team
Wilmott has never understood, the mathematics of continuous processes is often easier than that of discrete ones. But then when it comes to number crunching you have to anyway turn a continuous model into a discrete one.
In discrete models we end up with difference equations. An example of this is the binomial model for asset pricing. Time progresses in finite amounts, the timestep. In continuous models we end up with differential equations. The equivalent of the binomial model in discrete space is the Black-Scholes model, which has continuous asset price and continuous time. Whether binomial or
Black-Scholes both of these models come from the probabilistic assumptions about the financial world.
Now let’s take a look at some of the tools available.

Simulations:
If the financial world is random then we can experiment with the future by running simulations. For example, an asset price may be represented by its average growth and its risk, so let’s simulate what could happen in the future to this random asset. If we were to take such an approach we would want to run many, many simulations. There’d be little point in running just the one, we’d like to see a range of possible future scenarios.
Simulations can also be used for non-probabilistic problems. Just because of the similarities between mathematical equations a model derived in a deterministic framework may have a probabilistic interpretation.
Discretization methods:
The complement to simulation methods, there are many types of these. The best known of these are the finite-difference methods which are discretizations of continuous models such as Black-Scholes. Depending on the problem you are solving, and unless it’s very simple, you will probably go down the simulation or finite-difference routes for your number crunching.
Approximations:
In modeling we aim to come up with a solution representing something meaningful and useful, such as an option price. Unless the model is really simple, we may not be able to solve it easily. This is where approximations come in. A complicated model may have approximate solutions. And these approximate solutions might be good enough for our purposes.
Asymptotic analysis:
This is an incredibly useful technique, used in most branches of applicable mathematics, but almost unknown in finance. The idea is simple, find approximate solutions to a complicated problem by exploiting parameters or variables that are either large or small, or special in some way. For example, there are simple approximations for vanilla option values close to expiry.
Green’s functions:
This is a very special technique that only works in certain situations. The idea is that solutions to some difficult problems can be built up from solutions to special solutions of a similar problem. That’s not a very helpful definition, but we’ll be seeing enough of this later on.All of the above ideas will be explained in detail over the coming months. One of the aims of this educational series is to make the mathematics of finance as straightforward as possible. We will be using simple examples, understandable analogies, and lots of downloadable spreadsheets to achieve this.
Lyceum 2: Binomial Model I

Aristotle
Quick test
Before reading about the binomial method in detail, we’d like you to answer the following question. It is one day before the expiry of a call option struck at
100. The stock is currently valued at 100. You are reliably informed that there is a 60% chance of the stock rising to 101 and a 40% chance of it falling to
99. You are in a world with zero interest rate. What value would you give to the call option?
Make a note of your answer, we’ll come back to this later.
Branches
In the above question we had a stock price that was to take one of two values a day later. Using more general notation we have a situation like that shown in the figure below.
The stock price is currently S , can rise to uS or fall to vS . The probability of the rise is p . In our example S = 100, u = 1.01, v = 0.99 and p = 0.6.
Now let’s assume that we hold a call option on this asset that is going to expire tomorrow. This option has a strike of 100.
Holding just the stock or the option is risky:
If the asset rises we have 101, a profit of 1.
If it falls we have 99, a loss of 1
If the asset rises to 101 we get a payoff of 1.
If it falls to 99 we get no payoff, the asset expires out of the money
Here’s a trick, it’s called hedging. Let’s sell short a quantity 1/2 of the underlying asset so that now we have a portfolio consisting of a long option position and a

short stock position.
If the asset rises to 101 we have a portfolio worth
Max(101-100,0) - 0.5 x 101 = 1 – 0.5 x 101 = -99/2.
If the asset falls we have
Max(99-100,0) - 0.5 x 99 = 0 – 0.5 x 99 = -99/2.
So, whether the asset rises or falls, the portfolio takes the same value, -99/2.
If this portfolio takes the same value whatever happens to the underlying asset then it is a riskless portfolio. If it is riskless we can justifiably discount this guaranteed cashflow to the present, one day before expiry. In the simple case where the interest rate is zero we can say that the value of the portfolio the day before expiry must also be –99/2.
But what is this portfolio made up of today? The option and a short stock position. If we call the option value V, then we have
V – 0.5 x 100 = -99/2.
Thus the option value is 0.5.
The answer to our opening question is therefore 0.5.
If you answered 0.6 to the question you were probably looking at the expected payoff and discounting that. The expected payoff can be calculated from
0.6 x 1 + 0.4 x 0 = 0.6.
Do not despair if this was your answer.
This is a question that we often ask. First we check that people know about the binomial model, the answer is invariably yes. Then we hit them with the pricing question. If they truly understand the binomial concept they should get the answer 0.5. However, from our experiences 95% of people get the answer wrong, saying 0.6 instead.
The most important point to take away from the binomial model is the important idea that the real probability of the stock rising is irrelevant in the pricing of options.
The price of an option is dictated not by the probability of the underlying rising or falling but by the process of risk elimination. What this boils down to is that the

option value depends not on the direction of the stock, that it is rising or falling generally, but on its amount of randomness, here measured by u and v, on the timestep, and on the risk-free interest rate.
We would like to stress the point that discounting at the risk-free rate is strictly only permitted for risk-free cashflows!
Generalization
In order to generalize this option pricing methodology there are several key steps.
1. Pick u and v . The choice of these is governed by the volatility of the underlying asset and the timestep.
2. Set up the hedged portfolio. At this stage you won’t know how much of the underlying asset to hedge with, so call the quantity held short

, delta.
3. Choose
 such that the portfolio values at expiry are the same, whether the asset moves up or down.
4. Discount this portfolio value to the present and calculate the current option value.
We will see some more of the details in another Lyceum meeting.
Lyceum 3: Binomial Model II
Aristotle
In an earlier Lyceum we saw some of the basic ideas behind the binomial pricing model. We’ll go into those ideas in more depth now.
No numbers, just symbols
The three constants u and v are chosen to give the binomial model the same characteristics as the asset we are modeling. That means we want to capture the volatility of the underlying asset, in particular the volatility over the next timestep, dt . Although we won’t be needing it, we’ll also carry around the probability p for a while.
We choose

and
We have introduced two new parameters here:

the drift of the asset and
 the volatility. We have chosen these parameters in such a way that:
The expected return on the asset is
 dt
The standard deviation of returns is
 dt
1/2
An equation for the value of an option
Suppose, for the moment, that we are one timestep away from expiry. Construct a portfolio consisting of one option and a short position in a quantity
 of the underlying. This portfolio has value where the value of the option V is to be determined.
One timestep later, at expiry, the portfolio takes one of two values, depending on whether the asset rises or falls. These two values are
V
+ is the option value if the asset rises and V
the value if it falls.
The values of both of these expressions are known if we know

. And
 is under our control.
Having the freedom to choose
 we can make the value of this portfolio the same whether the asset rises or falls. This is ensured if we make

Then the new portfolio value is
Since the value of the portfolio has been guaranteed, we can say that its value must coincide with the value of the original portfolio plus any interest earned at the risk-free rate, r . Thus
Rearranging we get where
The right-hand side of the equation for V is just like an expectation; it’s the sum of probabilities multiplied by outcomes.
We see that the probability of a rise or fall is irrelevant as far as option pricing is concerned. But what if we interpret p' as a probability? Then we could say that the option price is the present value of an expectation. But not the real expectation.
Let’s compare the expression for p ' with the expression for the actual probability p . The two expressions differ in that where one has the interest rate r the other has the drift

, but are otherwise the same.
We call p ' the riskneutral probability. It’s like the real probability, but the real probability if the drift rate were r instead of

.
Observe that the riskfree interest plays two roles in option valuation. It’s used once for discounting to give present value, and it's used as the drift rate in the asset price random walk.

Risk-neutral pricing
Interpreting p ' as a probability, the option pricing equation is the statement that
“the option value at any time is the present value of the riskneutral expected value at any later time.”
You can think of an option value as being the present value of an expectation, only it’s not the real expectation.
The above model and analysis has introduced many key concepts.

There is only on e price for an option, and that doesn’t depend on which direction you think the asset price is going, only on its volatility.

That option price can be interpreted as an expectation. This is why we can actually value options by using simulation/Monte Carlo methodology.

There is a simple algorithm for calculating an option price a timestep before expiry. As we’ll see, this can be extended to value an option at any time before expiry.
Lyceum 4: Binomial Model III
Aristotle
The Binomial Tree
The binomial model allows the stock to move up or down a prescribed amount over the next timestep. If the stock starts out with value S then it will take either the value uS or vS after the next timestep. We can extend the random walk to the next timestep.
We can extend the random walk to the next timestep. After two timesteps the asset will be at either u 2 S , if there were two up moves, uvS , if an up was followed by a down or vice versa, or v
2
S , if there were two consecutive down moves. Imagine extending this random walk out all the way until expiry. The result is the binomial tree.

Observe how the tree bends due to the geometric nature of the asset growth.
Valuing back down the tree
We certainly know V
+ and V
at expiry, time T , because we know the option value as a function of the asset then, this is the payoff function. If we know the value of the option at expiry we can find the option value at the time T - dt for all values of S on the tree. But knowing these values means that we can find the option values one step further back in time.

The continuous-time limit

dt
Lyceum 5: What Is A Differential Equation?
Aristotle
You’ve seen the Black-Scholes equation, either in its Greek form

or in its differential form
Whichever form you’ve seen or prefer, this is just an example of a differential equation. To be precise it’s an example of a partial differential equation. ‘Partial’ means that the dependent variable V (so called because it ‘depends on’ other variables) is a function of more than one independent variable, here S , the asset price and t , the time. There are many, many different types of differential equation, but the ones you are most likely to encounter in the financial world are usually similar in form to the Black-Scholes equation.
These equations are called ‘differential equations’ because they involve differences between quantities. That’s what the curly d’s are all about, the s. A term such as means the difference in V values divided by difference in t values. But that’s not how we think of it. Instead think of it as the slope of the function V in the t direction. But what does ‘slope’ mean in this context?
The option value is a function of two variables, asset price S and time t . If it helps, think of V as being the height of a hill with the two variables being distances in a northerly and westerly directions. We’re going to be looking at the slope of this mountain in each of the two directions, these will be sensitivities of the option price to changes in the asset and in time. These slopes or gradients are what you experience in your car when you see a sign such as ‘1-in-10 gradient.’ That is precisely the same as a slope of 0.1.
Similarly means the slope in the S direction. But what about?

This can also be written as
This can be interpreted as the slope in the S direction of…the slope in the S direction! The slope of the slope.
The Black-Scholes equation is a relationship between the height of the hill and the slopes, and slopes of slopes, in various directions. Readers of The
Hitchhikers Guide to the Galaxy will recall the character who designed the fjords of Norway. Well, imagine another character who designed a Welsh hillside to satisfy the Black-Scholes equation!
The Black-Scholes partial differential equation is a parabolic equation, meaning that it has a second derivative with respect to one variable, S , and a first derivative with respect to the other, t . Equations of this form are also known as heat or diffusion equations.
The equation, in its simplest form, goes back to almost the beginning of the 19th century. Diffusion equations have been successfully used to model
 diffusion of one material within another, smoke particles in air
 flow of heat from one part of an object to another
 chemical reactions, such as the Belousov-Zhabotinsky reaction which exhibits fascinating wave structure
 electrical activity in the membranes of living organisms, the
Hodgkin--Huxley model
 dispersion of populations, individuals move both randomly and to avoid overcrowding
 pursuit and evasion in predator-prey systems
 pattern formation in animal coats, the formation of zebra stripes

 dispersion of pollutants in a running stream
In most of these cases the resulting equations are more complicated than the
Black-Scholes equation.
The Black-Scholes equation can be accurately interpreted as a reaction-convection-diffusion equation. The basic diffusion equation is a balance of a first-order t derivative and a second-order S derivative. If these were the only terms in the Black-Scholes equation it would still exhibit the smoothing-out effect, that any discontinuities in the payoff would be instantly diffused away.
The only difference between these terms and the terms as they appear in the basic heat or diffusion equation, is that the diffusion coefficient is a function of one of the variables S . Thus we really have diffusion in a non-homogeneous medium. The greater the amount of diffusion, or here the volatility, the faster the diffusion.
The first-order S -derivative term can be thought of as a convection term. If this equation represented some physical system, such as the diffusion of smoke particles in the atmosphere, then the convective term would be due to a breeze, blowing the smoke in a preferred direction.
The final term, rV , is a reaction or absorption term. Think of this as the passive-smoking effect, atmospheric smoke is absorbed by the lungs.
Putting these terms together and we get a reaction-convection-diffusion equation. An almost identical equation would be arrived at for the dispersion of pollutant along a flowing river with absorption by the sand. In this, the dispersion is the diffusion, the flow is the convection, and the absorption is the reaction.
Lyceum 6: Discretization
Aristotle
In this Tutorial we are going to look at discretization. How can we represent the value of an option that depends on an underlying asset, price S , and time, t , in a computer program, and how can we determine the values of the key Greeks, delta, gamma and theta?

Figure 1: The finite-difference grid
In the world of finite differences we use the grid or mesh shown in Figure 1. This grid usually has equal time steps, the time between nodes, and equal asset steps.
Let’s introduce some notation. The time step will be dt and the asset step dS , both of which are constant. Thus the grid is made up of the points at asset values S = i
* dS and times t = T - k * dt where i and k are integers. Notice how we’ve changed the direction of time, as k increases so real time decreases. This is because when we solve the Black-Scholes equation we must work backwards in time from known option values at expiry. We will write the option value at each of these grid points as V(i, k) .
Suppose that we know the option value at each of the grid points, can we use this information to find the sensitivities of the option value with respect to S and t?
Approximating theta
Our approximation for the first time-derivative of V is simply
Theta(i, k) = ( V(i, k) – V(i, k+1) ) / dt
It uses the option value at the two points marked in Figure 2. The error in this approximation is O( dt ).

Figure 2: An approximation to the theta.
Approximating delta
Let’s examine a cross section of our grid at one of the time steps.
Figure 3: Three approximations to the delta
In Figure 3 is shown this cross-section. The figure shows three things: the function we are approximating (the option curve), the values of the function at the grid points (the dots) and three possible approximations to the first derivative (the three straight lines). These are called a forward difference, a backward difference and a central difference, respectively. One of these approximations is better than the others, and it is obvious from the diagram which it is. The central difference has an error of O( dS * dS ). Our approximation for the first S derivative of V is simply
Delta(i, k) = ( V(i + 1, k) – V(i - 1, k) ) / dS
Approximating gamma
The gamma of an option is the second derivative of the option with respect to the

underlying. The natural approximation for this is
Gamma(i, k) = _
( V(i + 1, k) – 2 * V(i, k) + V(i - 1, k) ) / dS / dS
Exercise
Write a VB program that takes the function V = Log(S) * (T – t) and numerically calculates the delta, gamma and theta at different asset values and times.
Lyceum 7: Solving Black-Scholes By Finite Differences
Aristotle
Discretizing BS
Last week we saw how to set up a grid and how to discretize option prices to get the greeks. In this Tutorial we’ll see how to use all of this to create a very simple finite-difference algorithm for pricing vanilla options. The first step is to understand the equation we are solving. To start with, this is going to be the basic Black-Scholes equation, but we can still easily solve more complicated problems. The Black-Scholes equation, written in terms of the greeks, is
The notation here is the standard. This equation is more usually written as a partial differential equation, but that’s not strictly necessary, especially since we are trying to keep the jargon to a minimum. To recap. from last week, we can write the greeks in terms of the option values at the various nodes as
Delta(i, k) = (V(i + 1, k) – V(i - 1, k)) / dS / 2
Gamma(i, k) = (V(i + 1, k) – 2 * V(i, k) + V(i - 1, k)) / dS
/ dS
Theta(i, k) = (V(i, k) – V(i, k + 1)) / dt
The delta and gamma are going to appear in the algorithm in these forms but the theta will become
V(i, k + 1) = V(i, k) – dt * Theta(i, k)
Suppose that we know V(i, k) for all i we can then find Delta(i, k) and Gamma(i, k) and theta (from Black-Scholes) and finally find V(i, k + 1). In other words, we can work backwards from expiry to find the option value today. We’ll see this happening in the following code.

Our first explicit finite-difference code
Below is the Visual Basic code for pricing a European call option. The notation is obvious except for NoAssetSteps. This is the number of steps the asset price grid is divided up into. The more steps, the more accurate the code, but the slower it will run. Since we won’t be storing the deltas and gammas we don’t need to make them arrays, they are only used on a temporary basis.
Function OptionValue(Asset As Double, Strike As Double, _
Expiry As Double, _
Volatility As Double, Intrate As Double, _
NoAssetSteps As Integer)
Dim VOld(0 To 100) As Double '(a)
Dim VNew(0 To 100) As Double '(a)
Dim S(0 To 100) As Double dS = 2 * Strike / NoAssetSteps '(b)
NearestGridPt = Int(Asset / dS) '(c) dummy = (Asset - NearestGridPt * dS) / dS '(c)
Timestep = dS * dS / Volatility / Volatility / _
(4 * Strike * Strike) '(d)
NoTimesteps = Int(Expiry / Timestep) + 1 '(d)
Timestep = Expiry / NoTimesteps '(d)
For i = 0 To NoAssetSteps
S(i) = i * dS
VOld(i) = Application.Max(S(i) - Strike, 0) '(e)
Next i
For j = 1 To NoTimesteps '(f)
For i = 1 To NoAssetSteps - 1
Delta = (VOld(i + 1) - VOld(i - 1)) / (2 * dS) '(g)
Gamma = (VOld(i + 1) - _
2 * VOld(i) + VOld(i - 1)) / (dS * dS) '(g)
VNew(i) = VOld(i) + Timestep * _
(0.5 * Volatility * Volatility * _
S(i) * S(i) * Gamma + Intrate * S(i) * Delta - _
Intrate * VOld(i)) '(h)
Next i
VNew(0) = 0 '(i)
VNew(NoAssetSteps) = 2 * VNew(NoAssetSteps - 1) _
- VNew(NoAssetSteps - 2) '(i)
For i = 0 To NoAssetSteps
VOld(i) = VNew(i) '(j)
Next i
Next j '(f)
OptionValue = (1 - dummy) * VOld(NearestGridPt) + _
dummy * VOld(NearestGridPt + 1)

End Function
The code may be cut and pasted directly into a module in VBA.
Notes:
(a) We only need two arrays for the option value. We update the old values at step (j)
(b) Setting up the asset step size
(c) The current asset value might not lie on a grid point so later we’ll have to do some interpolating
(d) The timestep is chosen to make sure that the method is stable and that both today and expiry lie on grid points
(e) This is where the payoff is set up
(f) The timestepping loop starts at expiry and works back to the present
(g) Here’s where we calculate delta and gamma
(h) This line combines the definition of theta with the Black-Scholes formula
(i) We know the value of a call when the asset is zero, it is also zero. We know that the gamma is zero for very large asset price
(j) Here’s the updating
Please email aristotle@wilmott.com
if you need clarification of any of this.
Exercise: This code is written just to price a call option. How would you modify it to price a put option?
Next time out we will extend the finite-difference method and examine how to price American options.
Lyceum 8: Simple Extensions of the Finite-difference Method
Aristotle
Dividends
Last time out we saw how to solve the Black-Scholes equation using the explicit finite- difference method. This time we’ll see how to modify the last installment’s code to include a dividend yield on the stock and to price American options. When the underlying asset is a currency with a known foreign interest rate or is a stock with dividends then the BS equation becomes.
Here D is the dividend yield on the stock or the foreign interest rate.
(Approximating discretely paid dividends by a continuously paid dividend yield may or may not matter.)

Early Exercise
American options may be exercised at any time, not just expiry. If we believe that arbitrage opportunities shouldn’t exist then we have to make sure that the theoretical option value is at least the payoff. Mathematically, we want where ‘Payoff’ is the payoff function. We also want the delta to be continuous, we don’t want any jumps in the hedge ratio. This is incredibly easy to implement in the explicit finitedifference method, as you’ll see below.
Our second explicit finite-difference code
Below is the Visual Basic code for pricing an American call option on a dividend-paying stock. The notation is the same as last week, with the addition of
Divvie as the dividend yield on the underlying, US determines whether the option is American (US = "Y") or not, and the Payoff(i).
Function USOptionValue(Asset As Double, Strike As Double, _
Expiry As Double, Volatility As Double, _
Divvie as Double, Intrate As Double, _
US as String, NoAssetSteps As Integer) '
(a)
Dim VOld(0 To 100) As Double
Dim VNew(0 To 100) As Double
Dim S(0 To 100) As Double
Dim Payoff(0 to 100) As Double ' (b) dS = 2 * Strike / NoAssetSteps
NearestGridPt = Int(Asset / dS) dummy = (Asset - NearestGridPt * dS) / dS
Timestep = dS * dS / Volatility / Volatility / _
(4 * Strike * Strike)
NoTimesteps = Int(Expiry / Timestep) + 1

Timestep = Expiry / NoTimesteps
For i = 0 To NoAssetSteps
S(i) = i * dS
VOld(i) = Application.Max(S(i) - Strike, 0)
Payoff(I) = Vold(i) ' (c)
Next i
For j = 1 To NoTimesteps
For i = 1 To NoAssetSteps
Delta = (VOld(i + 1) - VOld(i - 1)) / (2 * dS)
Gamma = (VOld(i + 1) - 2 * VOld(i) + VOld(i - 1)) / (dS * dS)
VNew(i) = VOld(i) + Timestep * (0.5 * Volatility * Volatility
* _
S(i) * S(i) * Gamma + _
(Intrate - Divvie) * S(i) * Delta - Intrate * VOld(i)) ' (d)
Next i
VNew(0) = 0
VNew(NoAssetSteps) = 2 * VNew(NoAssetSteps - 1) _
- VNew(NoAssetSteps - 2)
For i = 0 To NoAssetSteps
VOld(i) = VNew(i)
Next i
If US = "Y" then
For i = 0 To NoAssetSteps
VOld(i) = Application.Max(Vold(i), Payoff(i)) ' (e)
Next i
Endif
Next j
USOptionValue = (1 - dummy) * VOld(NearestGridPt) + _
dummy * VOld(NearestGridPt + 1)

End Function
The code may be cut and pasted directly into a VBA module.
(a) The function call now asks for dividend information and whether the option is
American or not
(b) We need to remember the payoff
(c) Setting up the payoff
(d) This now includes the dividend yield
(e) This is the line that tests whether the option should be exercised
Exercise: Change the above code to value Bermudan options, options for which exercise is permitted on or between specified dates.
Lyceum 9: Out Barrier Options and the Finite-difference Method
Aristotle
This time out we look at the pricing of our first exotic option, an up-and-out call.
Knockout options have the same pay-off as vanilla options unless the underlying asset has reached some prescribed level, the barrier or trigger, prior to expiry. If the barrier is triggered then typically the option expires worthless. Financially and mathematically this means that the option value must be zero when the asset value is the barrier level.
In the code below we have had to include Barrier , the position of the barrier
(above the Strike ). We’ve also made the top of the asset-price array coincide with the position of the barrier.
Function OutBarrierOptionValue(Asset As Double, Strike As
Double, _
Expiry As Double, Volatility As Double, _
Divvie as Double, Intrate As Double, _
Barrier as Double, NoAssetSteps As
Integer)‘(a)
Dim VOld(0 To 100) As Double
Dim VNew(0 To 100) As Double
Dim S(0 To 100) As Double dS = Barrier / NoAssetSteps ‘(b)
NearestGridPt = Int(Asset / dS)

dummy = (Asset - NearestGridPt * dS) / dS
Timestep = dS * dS / Volatility / Volatility / _
(Barrier * Barrier) ‘(c)
NoTimesteps = Int(Expiry / Timestep) + 1
Timestep = Expiry / NoTimesteps
For i = 0 To NoAssetSteps
S(i) = i * dS
VOld(i) = Application.Max(S(i) - Strike, 0)
Next i
For j = 1 To NoTimesteps
For i = 1 To NoAssetSteps - 1
Delta = (VOld(i + 1) - VOld(i - 1)) / (2 * dS)
Gamma = (VOld(i + 1) - _
2 * VOld(i) + VOld(i - 1)) / (dS * dS)
VNew(i) = VOld(i) + Timestep * _
(0.5 * Volatility * Volatility * _
S(i) * S(i) * Gamma + _
(Intrate – Divvie) * S(i) * Delta - _
Intrate * VOld(i))
Next i
VNew(0) = 0
VNew(NoAssetSteps) = 0 ‘(d)
For i = 0 To NoAssetSteps
VOld(i) = VNew(i)
Next i
Next j
OutBarrierOptionValue = (1 - dummy) * VOld(NearestGridPt) +
_ dummy * VOld(NearestGridPt + 1)
End Function
Notes:
(a) The function call now asks for barrier information
(b) The asset array only needs to go as far as the barrier
(c) The timestep has been chosen to ensure stability of the explicit method
(d) This is the barrier boundary condition
Exercise:
Change the above code to value
1. an up-and-out call option if there is a rebate at the time of knockout
2. a down-and-out call option
© Wilmott Associates 2001

Lyceum 10: Classifying Exotic Derivatives
Aristotle
Below is the termsheet for a Perfect Trader or Passport option. How would you price this contract and how would you hedge it?
USD/DEM ‘Perfect Trader’ Option
Notional Amount
Option Maturity
USD 50,000,000
Three months from Trade Date
Allowed Position Long or short up to Notional
Amount
Transaction
Frequency
Up to two times daily
Settlement Amount Max(0,sum total in DEM of the gains + losses on each of the trades)
Upfront Premium 3.35% of Notional Amount
This contract is really just an option on a trading account; the trader can buy or sell
DEM twice a day, up to a prearranged limit, and keep the profits made over three months. A net loss is written off. Does this mean that you can apply a
Black-Scholes formula to get the price and hedge ratio? Not really, this contract has many features that make it one of the most complex of current products.
Before we begin the process of pricing, hedging and general risk management we should try to classify a contract by characteristics that make it similar or dissimilar to more familiar contracts.
Here's how we could classify this contract:
Time Dependence
Cashflow
Decisions
Dimension
Path Dependence
No
No
Yes
Weak
3

When we read this table we see that the contract has no dependence on time, each day is like every other, there are no special dates to be aware of. There are also no interim cash flows, the holder of the option pays for it upfront and then receives his return at expiry. There are decisions to be made, when to trade in the underlying asset in this case. This tells me to watch out for some optimization procedure.
The commonest form of optimality is seen in American options. When to exercise an American option is an optimization problem. In the trader option the optimality is much more sophisticated. There is some dependence on the history of the asset price, but nothing complicated. In that sense the contract could be worse, it could have an Asian feature for example! There are three dimensions, which means that computationally it will probably take a moderate amount of time to price and hedge, and we will not be expecting any nice formulae. Chances are, we are looking at using a numerical scheme.
Now let's step back and look at these characteristics in turn, what they all mean and why they make contracts exotic.
History in the making
The next most important exotic feature is path dependency. Many contracts have values that depend on the history of the asset price path and not just a value at expiry. Path-dependent contracts come in two forms, weakly path dependent and strongly path dependent. The weakly-path dependent contracts include the basic barrier options, knock-ins and knock-outs. They depend on the price history insofar as the contract will have a different value depending on whether or not the barrier level has been triggered. The strongly path-dependent contracts include Asians, depending on some average of the underlying over its past, and lookbacks, depending on extreme values attained by the underlying. There are many more such contracts, including Parisians which are barrier options with value depending on how long the underlying has been beyond the barrier. For some of these exotics the fastest and most accurate way to price them is by simple extensions of the Black-Scholes partial differential equation and finite-difference methods.
Basket case
Monte Carlo simulations are currently the only realistic way of valuing high-dimensional products depending on many underlying assets, the 'basket

options.' A typical basket option gives the holder the right to purchase the highest valued, say, out of a basket of 20 stocks for a fixed strike. The high dimensionality rules out the use of finite-difference methods for the pricing. Popular at the moment are the quasi
Monte Carlo methods that are part way between random and uniform that are more efficient than the basic Monte Carlo techniques. Path dependency usually also introduces extra dimensions, and hence can make contracts harder to value.
Decisions, decisions,…
One of the most exotic of current exotics is the 'pass-port,' 'perfect trader' or simply, 'trader' option, shown at the start. Instead of being a derivative of some underlying asset it is a derivative of a trading account in some underlying. In other words, a trader can speculate in some asset (on paper), keeping all the profit but not having to take any of the loss. Of course, such a contract is very valuable and the trader must pay a significant premium for it. This contract is highly complex. Each day the trader has to decide how much of the underlying to buy or sell. This aspect of decision making adds to the complexity of this instrument.
Hedging and risk management
Exotic options can often be difficult to price theoretically, but equally often a trader will have a good idea, based on experience and instinct, of what they are actually worth. What he does not necessarily know is how to hedge away the risk. There are two standard forms of hedging, delta or dynamic hedging and static hedging. If you have a theoretical pricing model then the sensitivity of the price to the underlying asset tells you the delta and thus how many of the underlying asset to sell to maintain a riskless position. This is correct provided that the model is correct. Any errors in the model will lead to unhedged risk.
© Wilmott Associates 2001
Lyceum 11: Monte Carlo Simulation For Pricing
Aristotle
The random walk of assets
One of the foundations of quantitative finance theory is the random walk for asset prices. Modeling financial instruments as random walks has been one of the great success stories of finance and economics. The random walk theory leads on to the modern portfolio theory of Markowitz, the continuous-time asset

allocation models of Merton and the Black-Scholes theory of option pricing. And, of course, the ideas have also been applied to the world of fixed income. The models for the random evolution of stock prices, or interest rates, require knowledge of just two quantities, the expected return on the asset and the asset' s volatility. Figure 1 shows several simulations of the path of an asset.
Figure 1: Some simulations of an asset path. Each path has the same expected return and volatility, but each path is different.
Real and risk -neutral worlds
Because the theory for the behavior of assets is founded upon randomness and probability, it is natural for simulations to play a key role in the application of the theories to practice. These simulation methods are usually called Monte Carlo, for obvious reasons. Phelim Boyle is one of the famous names in the use of
Monte Carlo methods to price derivatives. He showed how to find the fair value of an option by simulating many, many possible asset price paths. It' s an interesting but subtle point that the model for assets assumes random price paths yet the Black-Scholes theory uses dynamic delta hedging to eliminate all randomness and hence risk from an option portfolio. A result of this is that when you price an option you don' t actually simulate what the asset might do in practice but what it would do in an artificial world. This world is called the risk-neutral world. The difference between the real world and the risk-neutral world is in the value of the asset' s expected return. So, when you come to simulate asset price paths you have to ensure that the expected return on the asset is set to the risk-free rate. In words, the theoretical value of an option is ' the expected value of the present value of the payoff under a risk-neutral random walk'. Whether you price by Monte Carlo simulation, analytical solution of the Black-Scholes equation or solve the Black-Scholes equation numerically, you should get the same answer.

Advantages of MC
The main advantage of Monte Carlo simulations for pricing derivatives is in the ease of programming. It' s a technique that is very simple to program, and can even be done on a spreadsheet. It is straightforward to price quite exotic contracts with fancy path dependency. You can even price many path-dependent contracts simultaneously for relatively little extra computational effort. The disadvantages are in speed of computation and in the pricing of derivatives with in-built decisions, such as American options or the trader options we discussed last week. As far as speed is concerned, the finite difference solution of a Black-Scholes-type equation will be faster than a Monte
Carlo simulation as long as there are four or fewer underlying assets. But, with five or more underlyings, such as you get in basket options, you' d be better off using Monte Carlo simulations. This is not true if you are pricing American options. At present Monte Carlo simulations have not been very successfully applied to pricing American options, at least not without significant programming effort. So, if you find yourself with an American-style basket option you are in trouble!
Quasi MC
Very occasionally you will find yourself having to price a basket option with many, many underlyings. If the contract is European, independent of price paths, on lognormal assets (asset movements scale with asset price) with constant volatilities and constant correlations (some big ' ifs' !) then there is a nice formula for the option' s fair value. This formula takes the form of an expectation, or a multi-dimensional integral. This expectation can be estimated using Monte Carlo simulations. However, there' s a faster technique, called Quasi-Monte Carlo
(QMC), that is even better. Instead of choosing random numbers in the simulation you choose carefully distributed but completely deterministic numbers. This technique has been around since the 1960s but has been used in finance for less than a decade. The beauty of the various Monte Carlo methods is that the computational time is virtually independent of the number of dimensions, the number of underlying assets. The basic Monte Carlo method gives an accuracy of order 1/N 1/2 where N is the number of function evaluations/simulations. So for an extra decimal place in accuracy you need 100 times as many points, and the program will take 100 times as long to run.
However, if you can use one of the QMC methods for pricing then you get a much better order of accuracy of 1/N. The extra decimal place accuracy will only take ten times as long to compute.
References
Black, F & Scholes, M 1973 The pricing of options and corporate liabilities.

Journal of Political Economy 81 637- 59
Boyle, P 1977 Options: a Monte Carlo approach. Journal of Financial Economics
4 323-338
Markowitz , H 1959 Portfolio Selection: efficient diversification of investment.
John Wiley
© Wilmott Associates 2001
Lyceum 12: Credit Risk Modelling
Aristotle

The modeling of credit risk is in a terrible state. In a perfect world, we'd name those responsible for getting this subject into such a mess. However, fear of lawsuits prevents us…but those of you who know the literature will also know whom we're talking about!
Let's review how the current crop of models originated, then we'll knock them down, and finally, we hope, make some suggestions for better lines of attack. First there was the Black-Scholes model for equity options. There are many, many assumptions underlying this model, all of which are demonstrably wrong. Such is the nature of financial modeling. As well as requiring rather sweeping assumptions, the model depends on the elegant concepts of delta hedging and no arbitrage. Again, in practice delta hedging is far from perfect…and all of us are making a living from arbitrage, in a sense.
Nevertheless, the BS model is jolly good. The model can be easily adapted to cope with many new exotic derivatives, and many of the assumptions can be dropped without too much effort. Perhaps the success of this equity derivatives model went to the heads of academics working in the fixed-income world. The fixed-income world is far more complicated than the equity world. It's more complicated because traders think of different parts of the yield curve as different products; in the equity world there's usually only the one underlying for most contracts. This brings us up against that nasty creature 'correlation.' In the simplest theoretical interest rate models, which mimic almost exactly the equity model, there is only one source of randomness. A consequence of this is that you should be able to hedge one part of the yield curve with any other, blatantly not possible in practice. Introducing more and more random factors only makes the modeling and pricing more time consuming without necessarily making matters any better.
Anyway, the parlous state of interest rate modeling is another story,
We're not likely to get as much support for trashing all interest rate models as we will get for trashing all credit risk models. (Speaking of which, how did credit models evolve from interest rate models? This is a real no brainer! The original interest rate models used the variable r as the short-term interest rate. One class of credit risk models replaces this r with p , where p means probability of default.
The math now carries over unchanged. Why is this not such a good idea?) Let's see some of the details of credit risk modeling. There are two popular approaches

to credit risk modeling, firm valuation and reduced form.
The firm-valuation models were the first attempts at coming to grips with modeling credit risk. Model the value of the company issuing the risky instruments, taking into account their assets and liabilities. Ideally, this involves an examination of balance sheets to get an idea of the success or otherwise of the company. The value of the company is typically treated as a stochastic or random quantity. When this value falls to some prescribed level this is seen as the trigger for default.
There's a lot to be said in favor of these models but the main downside is the difficulty in estimating parameters. Because of these difficulties with these models there have grown up a much simpler class of models.
These are the models that we mentioned at the top of this article. They assume that default is a completely exogenous and random event, often having nothing to do with the health of the bond issuer. These models are popular because the mathematics is identical to that for interest rate models. The parameters in the model are impossible to measure…you don't know the likelihood of default until it is too late. What do you do when you can't measure parameters? Typically you assume that the market knows the parameters and so you 'fit' or 'calibrate' your model so that it outputs theoretical prices that match real prices.
Whichever approach you take, the resulting model in effect gives a price to a credit product that is the present value of the expected payoff. This is justifiable in a couple of situations, but otherwise not. The two situations are when you can hedge away risk. Or when you don't care about risk. First of all, if you can hedge away risk to get a completely risk-free portfolio then you can argue that the resulting portfolio should earn the risk-free interest rate. But can you hedge credit risk? No, almost never. You can't guarantee that a 'credit event' such as default will occur simultaneously in several products. And you need such correlation for effective risk

elimination.
Second, do you care about risk? This depends. Think of the payoff for the simplest conceivable risky bond, one that pays off the principal at maturity unless thereis some credit event in the meantime. This has a very extreme payoff profile, being discrete and bimodal. The risk and return are not readily interpreted in terms of expected returns and standard deviations. If you can invest in lots and lots of such bonds, all of which are uncorrelated, then your return will tend to a Normal distribution. In the limit as you invest a fixed amount in an infinite number of such bonds your return becomes the expected return. Nice idea, but think of what happened to Long Term Capital Management. Their apparently uncorrelated risky instruments suddenly became correlated, leading to enormous losses.
The two points which worry us are:

Hedging credit risk is virtually impossible

The payoffs for credit instruments can be very extreme, bimodal for example
Our view is that pricing via simple expectations is not adequate. Nor are most forms of mean-variance analysis. We believe that credit risk is one the best uses for the economic theories of 'utility.' Simply speaking, utility theory is a way of assigning a non monetary value to a random outcome. Think of utility as being a measure of 'happiness.' A person's (or company's) 'utility function' gives an idea of the relative worth of different outcomes: two billion dollars won't make you twice as happy as one billion! Utility is a very subjective idea…if used for pricing financial instruments you will find that different investors will price risky instruments differently.
We didn't used to like utility theory. We thought of it as being an artificial construct of economists. Now we're converted. Interesting results follow from using this theory: different people value contracts differently; economies of scale appear naturally; there are often optimal static hedges that can be put in place to improve a contract's value.
Market Model Assumptions Problems
Equity One-factor Brownian No frictions, continuous Volatility,
(lognormal) random trading, delta hedging walk volatility, volatility (and discrete hedging)
Fixed One-, two-, three-, n - No frictions, continuous Correlation, correlation,

income factor random walk trading, delta hedging correlation (as well as of any product with vol. and discrete one, two, three, n other hedging!) products
Credit One-factor for:
Probability of default Hedging!
Value of the firm Known evolution of company’s value
No way, José! Very, very rarely can you hedge default
Very, very difficult to estimate parameters
Please send your thoughts on this subject to aristotle@wilmott.com
.
© Wilmott Associates 2001
Lyceum 13: No Arbitrage and Risk Neutrality in Horse Racing I
Aristotle
Previously in the Lyceum…
A few Lyceums ago (what is the plural of Lyceum?) we saw how the absence of arbitrage opportunities led to the idea of risk-neutral pricing. The value of an option can be interpreted as the present value of the expected payoff, with the expectation being with respect to the risk-neutral asset price path. In this context risk-neutral just means that the asset price increases with a growth rate that is the same as the risk-free interest rate. In other words, what we really believe that the asset price is going to do in the future (in terms of its growth rate) is irrelevant. We don't even need to know the growth rate of an asset to price its options, only its volatility.
Something related happens in the world of sports betting.
Setting the odds in a sporting game
When it's a horse race, football or baseball game the odds are set not to reflect the real probabilities of a horse or a team winning but to reflect the betting than has occurred. Depending on how the betting goes, the odds will be set so that the House/Bookie cannot lose. For example, in a soccer match between
England and Germany the Germans are more likely to win, but the patriotic
English will bet more heavily on England (presumably). The odds given by the

bookies will reflect this betting and make it look like England is more likely to win.
Of course, in Germany the situation is reversed. The best bet would be on
Germany, but placed in England, and one on England placed in Germany!
In practice bookies in one country would lay off their bets on bookies in other countries so all bookies have roughly the same odds. Otherwise there would be straightforward arbitrage opportunities.
In practice it's unlikely for there to be a sure-fire bet (unless the bookie has made a mistake, the race is fixed, or you can find two or more bookies that aren't directly or indirectly laying off their bets on each other).
But you can win, on average. By exploiting the difference between the real probability of a horse winning and the odds you can get. (There are differences between real odds and what you get paid in all casino games, but it's only in
Blackjack that this can be exploited.)
The Mathematics
Suppose that there are N horses in a race, with an amount Wi bet on the ith horse. The odds set by the bookie are q i
: 1. This means that if you bet 1 on horse i you will lose the 1 if the horse loses, but will take home q i
+ 1 if the horse wins, your original 1 plus a further q i
. How does the bookie set the odds to ensure he never loses?
The total takings before the race is
If horse j wins the bookie has to pay out
All that the bookie has to do is to ensure that

or equivalently
Nothing too complicated.
But see how the odds have been chosen to reflect the betting. Nowhere was there any mention of the likelihood of horse j winning!
Arbitrage
Suppose the bookie made an error when setting the odds. How could you determine whether there was an arbitrage opportunity? (Don't forget that only positive bets are allowed, there's no going short here!)
Let's introduce some more notation. The wi are the bets that you place. (We can forget about the wagers made by everyone else, the W s.) Let's assume that your total wager is 1, so that
(1)
The amount you win is
(2) if horse j is the winner.
Can you find a w i
for all i
such that they add up to one, are all positive and that expression (2) is positive for all j ? If you can there is an arbitrage opportunity.
More next time…
Please send your thoughts on this subject to aristotle@wilmott.com.

© Wilmott Associates 2001
Lyceum 14: No Arbitrage and Risk Neutrality in Horse Racing II
Aristotle
Previously in the Lyceum…
Let’s continue with the search for an arbitrage opportunity in a horse race.
The w i are the bets that you place. Let’s assume that your total wager is 1, so that
(1)
The amount you win is
(2) if horse j is the winner.
Can you find a w i
for all i
such that they add up to one, are all positive and that expression (2) is positive for all j ? If you can there is an arbitrage opportunity.
The requirement that (2) is positive can be written as
(3)
Can we find positive ws such that (1) and (3) hold? This is very easy to visualize, at least when there are two or three horses. Let’s look at the two-horse race.

In the figure above the axes represent the amount of the wager on each of the two horses. The red line shows the constraint (1). The wagers must lie on this line. The two dots, one black and the other orange, mark the point
( 4) in each of two situations. The black dot is the typical situation where there is no arbitrage opportunity and the orange dot does have an associated arbitrage opportunity. Let’s see the details.
To find an arbitrage opportunity we must find a pair ( w 1, w 2) lying on the red line such that each coordinate is greater than a certain quantity, depending on the qs . Plot the point (4) and draw a line vertically up, and another line horizontally to the right, as shown in the figure, emanating from the orange dot.
Does the quadrant defined by these two lines include any of the red line? If not, as would be the case with the black dot, then there is no arbitrage possible. If some of the red line is included then arbitrage is possible.

How best to profit from the opportunity?
There’s a simple test to see whether we are in a black dot or an orange dot situation. In general, if then there is no arbitrage. If the sum is less than one, there is an arbitrage.
You can benefit from the arbitrage by placing wagers w i such that they lie on the part of the red line encompassed by the quadrant. Which part of the red line, though, is up to you. By that I mean that you must make some statement about what you are trying to achieve or optimize in the arbitrage. One possibility is to look at the worst-case scenario and maximize the payback in that case.
Alternatively, specify real probabilities for each of the horses winning
In the next Lyceum we’ll see more about how to optimally place your bets.
Please send your thoughts on this subject to aristotle@wilmott.com
Lyceum 15: Optimality In Horse Racing
Aristotle
We saw recently how odds are established by bookies. We even saw how to spot arbitrage opportunities. In practice, of course, you could spend a lifetime looking for arbitrage opportunities that rarely occur in real life. In this Lyceum we are going to see if we can exploit the difference between the odds as set by the bookie and the odds that you estimate. Remember, the odds set by the bookie are really determined by the wagers placed, which are more to do with irrational sentiment
(“I’m going to bet on this horse ‘cos it’s got the same name as the pet rat I had when I was a child”) than with a cold-hearted estimation of the probabilities.
We need some more notation. Let’s use p i
as the probability of the i th horse winning the race. This is supposed to be the real probability, not the bookie’s probability. Obviously, the odds must sum to one:
.
If we wager w i
on the i th horse then we expect to make

. (1)
This is under the assumption that the total wager, the sum of all the w s , is one. An obvious goal is to make this quantity positive, we want to get a positive return on average . But there may be many ways to make this positive. How do we decide which way is best?
Another quantity we might want to look at is the standard deviation of winnings.
This is given by
. (2)
This measures the dispersion of winnings about the average, and is often interpreted as a measure of risk. If this were zero our profit or loss would be a sure thing.
Here’s an example.
Horse Bookie’s odds
Nijinsky 5
Red Rum
Oxo
Red Marauder
Gay Lad
Roquefort
Red Alligator
Shergar
2
2
2
2
6
1
1
Your Estimate of
Probability
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0.1
Wager
How should you bet? The following calculations are easily done on a spreadsheet .
Scenario 1: Maximize expected return.
Since you place no premium on reducing risk you should bet everything on the horse that maximizes
.
In this case, that is Red Rum. The expected return is 40% with a standard

deviation of 280%. A very risky bet!
Horse Bookie’s odds
Nijinsky
Red Rum
5
6
Oxo 1
Red Marauder 1
Gay Lad 2
Roquefort
Red Alligator
Shergar
2
2
2
Your Estimate of Wager
Probability
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0.1
0
0
0
0
1
0
0
0
Scenario 2: Minimize standard deviation
An interesting strategy. The solution is given below.
Horse Bookie’s odds
Nijinsky 5
Red Rum
Oxo
6
1
Red Marauder 1
Gay Lad 2
Roquefort
Red Alligator
Shergar
2
2
2
Your Estimate of
Probability
0.2
Wager
0.063062
0.2
0.1
0.1
0.1
0.1
0.1
0.1
0.054068
0.189203
0.189246
0.126108
0.126108
0.126108
0.126097
I say ‘interesting’ because this strategy results in zero standard deviation, an a return of
–62%. In other words, a guaranteed loss!
Scenario 3: Maximize return divided by standard deviation
A strategy that seeks to benefit from a positive expectation but with a smaller risk.
For mathematical reasons (the Central Limit Theorem) this is a natural strategy.
The solution is given below.
Horse Bookie’s odds
Nijinsky
Red Rum
5
6
Your Estimate of
Probability
Wager
0.2
0.2
0.459016
0.540984

Oxo 1
Red Marauder 1
Gay Lad
Roquefort
2
2
Red Alligator
Shergar
2
2
0.1
0.1
0.1
0.1
0.1
0.1
0
0
0
0
0
0
The expected return is now 31% with a standard deviation of 164%.
Please send your thoughts on this subject to aristotle@wilmott.com
.
© Wilmott Associates 2001
Lyceum 16: Another Look at Risk-neutral probabilities
Aristotle
We can hardly stress enough the importance of the concept of risk neutrality, in the form of risk-neutral pricing and risk-neutral probabilities. It is by no means the end of the subject of pricing and hedging, indeed in some respects it clouds ones view of more realistic possibilities, but a thorough understanding is nevertheless a sine qua non for quantitative finance.
We have seen the idea of a binomial tree, how it is built up and how it is used to price options. Let’s look at this in a slightly different way. I am going to give you the tree of asset prices and I want you to calculate option prices. OK?
Suppose we are one day from expiry, the stock stands at 100, tomorrow the stock could be at either 102 or 99. A call option, with strike 100, expires

tomorrow. Interest rates are zero. What is the price of the call?
We’ve seen how to set up a delta-hedged portfolio that is risk free, and thereby derived an algorithm for pricing the option. Recall that the option price does not depend on the real probability of the stock price rising or falling, but on a quantity called the risk-neutral probability, which we shall denote by p’
.
There is a shortcut to finding the p’
. Ask the question, what p’
makes the expected stock price tomorrow the same as the stock price today? The answer is given by the solution of
This is the required risk-neutral probability. This probability is then used in the option price calculation. Simply calculate the expected option price at expiry using p’
and, with zero interest rates, this is the option price today:
Before discussing this idea and building up an entire tree, let’s put interest rates back in. The idea is very similar.
Suppose that the interest rate applying at the present time is r , to find p’ we must solve the problem: What p’
makes the expected stock price tomorrow, a time dt away, the same as the forward stock price?
In other words,
The final step is to use this probability to calculate the expected option price at expiry and then discount that amount at the risk-free rate:

Below is a very simple binomial tree. What are the probabilities associated with each branch?
In the absence of interest rates, it’s obvious that all of the probabilities are 0.5.
What if I make the simple modification as below?
All that I’ve done is to swap around the two numbers at the bottom. But this has messed up the probabilities. There is no probability (i.e. a number from zero to

one) such that we can go from 97 to either 98 or 99. In financial terms, this tree permits arbitrage.
When it comes to setting up the tree for pricing derivatives there are two things we must ensure. First that there are no arbitrage opportunities, and second that the branching captures the volatility that we want to model. There are many ways to set up the tree to satisfy these requirements, and you should re-read the earlier Lyceum where this is discussed in detail.
Please send your thoughts on this subject to aristotle@wilmott.com
.
© Wilmott Associates Limited 2001
Lyceum 17: The Central Limit Theorem
Aristotle
Fundamental to the whole subject of finance is the simple concept known as
“The Central Limit Theorem.” Although not always acknowledged, this is how money is made. Contrary to the popular belief that quants slave away at sophisticated math models to guarantee every trade is profitable, sometimes they make money, sometimes they lose it but…on average they come out ahead. Let’s see how this works.
Distributions
A one-dollar bet on the toss of a fair coin gives you a 50% chance of making a dollar, for a head, say, and a 50% chance of losing one dollar, for a tail. We could represent this outcome by the simple bar chart shown below. The heights of the two bars are both 0.5 = 50%. This is an example, a rather simple example, of a probability density function.

Mean
The average amount you’ll make on this bet is zero. The average of +1 and –1 each having 50% probability of occurring is just zero. It’s a fair coin.
Standard Deviation
The standard deviation, measuring the spread of possible outcomes, is 1.
Adding Up the Random Numbers
Now suppose we play a little game where we toss the coin twice. Each time there is $1 riding on the outcome. How much money might we have at the end?
It will be $2, $0 or -$2. Unless one of the coin lands on an edge, payoffs of $1 and $1 aren’t possible.
And the probabilities of these outcomes? If the chance of getting one head is
50%, the probability of getting one head and then another is 50% of 50% or
25%. There’s the same chance of getting two tails, leaving 50% chance of a head and a tail or a tail and a head. Again, we can plot the probability density function.
The average is still zero. But the standard deviation is now the square root of 2.
Let’s toss the coin 32 times. The distribution looks like this.

The mean is still zero, but the standard deviation is now square root of 32.
We are getting close to a statement of the Central Limit Theorem.
As the number of tosses gets larger and larger, so the shape of the probability density function gets closer and closer to the famous bell-shaped curve of the
Normal or Gaussian distribution. The mean is always zero and the standard deviation is the square of the number of coin tosses.
If this limiting behavior only worked for the tossing of coins then it wouldn’t be that much use. The amazing thing is that the Normal distribution is the limiting distribution if you add up many random numbers from any basic building-block distribution …provided a few simple criteria are satisfied:
The mean of the individual distribution must be finite, and constant (it can’t vary from one toss to the next for example)
The standard deviation of the individual distribution must be finite and constant
(you mustn’t bet different amounts on each toss)
Each random number must be independent of previous ones
Now let’s see the result.
The Central Limit Theorem
Add up N independent, identically distributed random numbers, each draw having mean of m and standard deviation of s , then the sum will tend to a
Normal distribution as N tends to infinity with mean of Nm and standard deviation

Example
You cut a deck of playing cards. If you get a court card or Ace you lose one dollar. If you get a 2 to 10 inclusive you win one dollar. Your average is
Your variance is so that the standard deviation is 12/13.
Here’s your distribution after one cut.
After 10 cuts your profit distribution looks something like this.

Despite heavy odds in your favor you may still lose money.
After 10,000 cuts of the deck (a very heavy session!) the distribution of your winnings would look like this.
The mean is 10,000 x 5/13 and the standard deviation is 100 x 12/13. The mean is now enormously greater than the standard deviation. Although it is possible to lose money, the chances are microscopically small.
The Central Limit Theorem is important because it tells you what happens after a large number of bets or investments. It shows that the two most important factors in an investment are its mean and its standard deviation. The best investment is one that has the largest mean (and must definitely be positive!) and the smallest standard deviation.

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© Wilmott Publications Limited 2002
Lyceum 18: The Kelly Criterion
Aristotle
We like to draw comparisons between the worlds of finance and of gambling, you may have noticed. Our philosophy is that gambling is just a particular form of investing, one that is often technically simpler to understand than the world of stocks and shares, convertible bonds and options. If you can't cope with the mathematics of roulette, or the emotional rollercoaster of Blackjack, then you certainly shouldn't be working in a bank…except maybe shining shoes. We are now going to explore a very simple idea, known to all professional gamblers but still not known by everyone in finance.
Risk and Return
We saw in the last Lyceum piece how expectations and standard deviations are important when investing over and over again, in bet after bet on the toss of a coin for example. In the long run you'll make a tidy profit if your expectation is positive, but perhaps with some nasty drawdown early on. If you've got bottomless pockets then that drawdown is not going to worry you. More typically, however, a short run of bad luck may wipe you out and wipe out your chances of the long-run profit.
Let's address the issue of the 'long run' and see if there's any way we can get to our long-run profit by clever money management.
The Long Run
You've got $1,000 in your pocket and an opportunity to play some game of chance over and over again. Your analysis of the game, the possible outcomes, and your probability of winning each time, suggest that you've got an edge.
The probability density function for outcomes of a $1 bet is , and the positive edge means that the average payoff for that bet is 1+ , with > 0.
The standard deviation of the outcome is given by

Later on we'll be making an assumption about the relative size of and , but for the moment let's keep things as general as possible.
Bet a Fraction of Your Wealth Each Time
The thousand dollars won't last long if we bet it all in one go. Let's instead bet a fraction f . And we'll keep betting the same fraction f every game, whether we're up to $10,000 or down to $10.
After one bet we have
After two bets we have
The subscripts simply acknowledge the fact that the s are random, and different each game.
After the evening's session of N games we've got
(A product of N terms.)
We could take expectations of this expression, or we could ask the question
"What is the expected growth rate?" That is, if we compared the money we've got at the end of the evening with the exponential growth we'd get from putting the money in the bank for a while, what is the equivalent interest rate?
To calculate such a thing we need to take logarithms of this expression.
Now we can state our investment goal, we will invest a fraction f of our wealth each game in such a way as to maximize the expected return. This means maximize the following expression:
Since the s come from the same distribution each time, this amounts to

maximizing by the optimal choice of f .
Now we come to the fun bit.
Optimal Investment Fraction
Typically the bet is at best only going to give us a slight edge, so that the mean is going to be much smaller than the standard deviation . If that is the case we can approximate expression (1) by
This is what we want to maximize.
We can find the maximum by differentiating with respect to f . We want
So, the choice of fraction which maximizes our long-term growth rate is just
This is the famous Kelly criterion, beloved of all pro gamblers. The idea behind it and the result are important for money management in finance as well. Don't invest too much, it's too risky; don't invest too little, you have to wait forever to get some payback.
Indeed, some of the results from asset allocation models show remarkable similarity to this. Even the Sharpe ratio, the measure of performance for funds and traders etc. is related to this simple concept.
Exercise: Make a spreadsheet that simulates the betting process. Plot the P&L against time for various bet fractions f , including the Kelly fraction
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