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Appendix A All the Math You Need. . . and No More (An Executive Summary)

Appendix A All the Math You Need. . . and No More (An Executive Summary)

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Appendix A all the math you need. . . and no more



Obviously, if you can get away with just the adding and subtracting, and occasional other

easy bits of arithmetic, well that’s just fine and dandy. And that’s what the simple option pricing

model, the binomial model, is all about. However, we don’t do an awful lot of that in this book,

because it is rather limiting. You can’t build much of a skyscraper with just a toy hammer and

a bit of string. And I do want us to build skyscrapers in this book, as it were.

Then there’s the abstract stuff; which is fantastic. The problem is that it is abstract probability

theory. Sometimes we have problems that aren’t probabilistic, then what? Probability theory

isn’t much use for that, is it? Sticking with the buildings analogy, I think of martingale theory

not as a tool but as a material. Steel is a material; it’s brilliant, you can build all sorts of things

with it, ships, bridges, etc. But it’s not that great for houses, and while it’s useful as the skeleton

of skyscrapers you will need other material to pad out that skeleton.

Differential calculus, now that’s not a material, that’s a box of tools. With the right tools

and some imagination you can build anything. Calculus doesn’t care whether a problem is

deterministic or probabilistic or something completely different. Calculus is just about how

things change or evolve, in time, space or with stock price. And that’s mostly what we do in

this book. Another advantage of focusing on the tools rather than the materials is that we don’t

have to limit ourselves in our modeling. Getting back to finance, many models at the cutting

edge of finance research are non linear. Calculus has no problems with nonlinearity, whereas

martingales do. If you are concentrating on the probabilistic models it seriously hampers your

scope for creativity. After all, outside of finance most models are non linear.

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 equations; what is all this fuss

about and will it end in tears before bedtime?

Here’s a list of 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.

A.2.1 Modeling Approaches

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.

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.



all the math you need. . . and no more Appendix A



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. The mathematics of continuous processes is often easier than that of discrete

ones. But then when it comes to number crunching you have to turn a continuous model into

a discrete one anyway.

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 time step. 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.

A.2.2 The Tools



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 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.



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Appendix A all the math you need. . . and no more



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.



A.3



e



The first bit of math you need to know about is e.

e is

• a number, 2.7183. . .

• a function when written ex ; this function is a.k.a. exp(x)

The function ex is just the number 2.7183 . . . raised to the power x; e2 is just 2.7183 . . .2 =

7.3891 . . . , e1 is 2.7183 . . . and e0 = 1. What about non-integer powers?

The function ex can be written as an infinite series





e =1+x +

x



1 2

2x



+



1 3

6x



+ ··· =

i=0



xi

.

i!



This gets around the non-integer power problem.

A plot of ex as a function of x is shown in Figure A.1.

The function ex has the special property that the slope or gradient of the function is also ex .

Plot this slope as a function of x and for ex you get the same curve again. It follows that the

slope of the slope is also ex , etc. etc.

14

12

10

8

6



exp(x)



4

2



−2



Figure A.1



−1.5



−1



The function ex .



−0.5



0

0



0.5

x



1



1.5



2



2.5



3



all the math you need. . . and no more Appendix A



2



1

log(x)



−0.5



0

0



0.5



1



1.5



2



2.5



3



x

−1



−2



−3



−4



Figure A.2 The function log x.



A.4



log



Take the plot of ex in Figure A.1 and rotate it about a 45◦ line to get Figure A.2. This new

function is ln x, the Naperian logarithm of x. The relationship between ln and e is

eln x = x or



ln(ex ) = x.



So, in a sense, they are inverses of each other.

The function ln x is also often denoted by log x, as in this book. Sometimes log x refers to

the function with the properties

10log x = x and log(10x ) = x.

This function would be called ‘logarithm base ten.’ The most useful logarithm has base e =

2.7183 . . . because of the properties of the gradient of ex .

The slope of the log x function is x −1 .

From Figure A.2 you can see that there don’t appear to be any values for log x for negative

x. The function can be defined for these but you’d need to know about complex numbers,

something we won’t be requiring here.



A.5



DIFFERENTIATION AND TAYLOR SERIES



I’ve introduced the idea of a gradient or slope in the sections above. If we have a function

denoted by f (x), then we denote the gradient of this function at the point x by

df

.

dx



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Appendix A all the math you need. . . and no more



Mathematically the slope is defined as

df

f (x + δx) − f (x)

= lim

.

δx→0

dx

δx

The action of finding the gradient is also called ‘differentiating’ and the slope can also be called

the ‘derivative’ of the function. This use of ‘derivative’ shouldn’t be confused with the use

meaning an option contract.

The slope can also be differentiated, resulting in a second derivative of the function f (x).

This is denoted by

d 2f

.

dx 2

We can take this differentiation to higher and higher orders.

Take a look at Figure A.3. In particular, note the two dots marked on the bold curve. The

bold curve is the function f (x). The dot on the left is at the point x on the horizontal axis and

the function value is f (x), the distance up the vertical axis. The dot to the right of this is at

x + δx with function value f (x + δx). What can we say about the vertical distance between

the two dots in terms of the horizontal distance?

Start with a trivial example. If the distance δx is zero then the vertical distance is also zero.

Now consider a very small but non-zero δx.

The straight line tangential to the bold curve f (x) at the point x is shown in the figure. This

line has slope df/dx evaluated at x. Notice that the right-hand hollow dot is almost on this

bold line. This suggests that a good approximation to the value f (x + δx) is

f (x + δx) ≈ f (x) + δx



df

(x).

dx

'the curve'



f(x + dx)

quadratic

approximation

to the

curve



f(x)



linear

approximation

to the

curve



x



Figure A.3



A schematic diagram of Taylor series.



x + dx



all the math you need. . . and no more Appendix A



This is a linear relationship between f (x + δx) − f (x) and δx. This makes sense since on

rearranging we get

df

f (x + δx) − f (x)



dx

δx

which as δx goes to zero becomes our earlier definition of the gradient.

But the right-hand hollow dot is not exactly on the straight line. It is slightly above it.

Perhaps a quadratic relationship between f (x + δx) − f (x) and δx would be a more accurate

approximation. This is indeed true (provided δx is small enough) and we can write

f (x + δx) ≈ f (x) + δx



df

d 2f

(x) + 12 δx 2 2 (x).

dx

dx



This approximation, shown on the figure as the grey dot, is more accurate. One can take this

approximation to cubic, quartic, . . . The Taylor series representation of f (x + δx) is the infinite

sum





f (x + δx) = f (x) +

i=1



1 i d if

(x).

δx

i!

dx i



Taylor series is incredibly useful in derivatives theory, where the function that we are interested in, instead of being f , is V , the value of an option. The independent variable is no longer

x but is S, the price of the underlying asset. From day to day the asset price changes by a

small, random amount. This asset price change is just δS (instead of δx). The first derivative

of the option value with respect to the asset is known as the delta, and the second derivative is

the gamma.

The value of an option is not only a function of the asset price S but also the time t: V (S, t).

This brings us into the world of partial differentiation.

Think of the function V (S, t) as a surface with coordinates S and t on a horizontal plane.

The partial derivative of V (S, t) with respect to S is written

∂V

∂S

and is defined as

∂V

V (S + δS, t) − V (S, t)

= lim

.

δS→0

∂S

δS

Note that in this V is only ever evaluated at time t. This is like measuring the gradient of

the function V (S, t) in the S direction along a constant value of t.1 The partial derivative of

V (S, t) with respect to t is similarly defined as

∂V

V (S, t + δt) − V (S, t)

= lim

.

δt→0

∂t

δt

Higher-order derivatives are defined in the obvious manner.

1



Or think of it as the slope of a hill going North. The time derivative would be the slope going West.



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Appendix A all the math you need. . . and no more



The Taylor series expansion of the value of an option is then

V (S + δS, t + δt) ≈ V (S, t) + δt



∂V

∂V

∂ 2V

+ δS

+ 12 δS 2 2 + · · · .

∂t

∂S

∂S



This series goes on for ever, but I’ve only written down the largest and most important terms,

those which are required for the Black–Scholes analysis.



A.6



EXPECTATION AND VARIANCE



Much of the modeling in finance uses ideas from probability theory. Again you don’t need to

know that much to understand most of the theory.

The first important idea is that of expectation or mean. If you roll a die there is an equal, 16 ,

probability of each number coming up. What is the expected number or the average number if

you roll the die many times. The answer is





1

6



+2×



1

6



+3×



1

6



+4×



1

6



+5×



1

6



+6×



1

6



= 3 12 .



Here we just multiply each of the possible numbers that could turn up by the probability of

each, and sum. Although 3 12 is the expected value, it cannot, of course, be thrown since only

integers are possible.

Generally, if we have a random variable X (the number thrown, say) which can take any of

the values xi (1, 2, 3, 4, 5, 6 in our example) for i = 1, . . . , N each of which has a probability

P (X = xi ) (in the example, 16 ) then the expected value is

N



xi P (X = xi ).



E[X] =

i=1



Expectations have the following properties:

E[cX] = cE[X]

and

E[X + Y ] = E[X] + E[Y ].

If the outcome of two random events X and Y have no impact on each other they are said

to be independent. If X and Y are independent we have

E[XY ] = E[X]E[Y ].

Expectations are important in finance because we often want to know what we can expect

to make from an investment on average.

The expectation or mean is also known as the first moment of the distribution of the random

variable X. It can be thought of as being a typical value for X. The scatter of values around

the mean can be measured by the second moment or the variance:

Var(X) = E (X − E[X])2 .



all the math you need. . . and no more Appendix A



Variances have the following property:

Var(cX) = c2 Var(X).

When X and Y are independent

Var(X + Y ) = Var(X) + Var(Y ).

The standard deviation is the square root of the variance and is perhaps more useful as a

measure of dispersion since it has the same units as the variable X:

Standard deviation(X) =



Var(X).



If the standard deviation is small then values of X are concentrated around the mean, E[X].

If the standard deviation is large then values of X are more widely scattered.

Standard deviations are important in finance because they are often used as a measure of

risk in an investment. The higher the standard deviation of investment returns the greater the

dispersion of the returns and the greater the risk.



A.7



ANOTHER LOOK AT BLACK–SCHOLES



Now that we understand about differentiation we can take another look at the Black–Scholes

equation:

∂V

∂V

∂ 2V

+ 12 σ 2 S 2 2 + rS

− rV = 0.

∂t

∂S

∂S

The option value V (S, t) depends on (or ‘is a function of’) the asset price S and the time t.

The first derivative of the option value with respect to time is called the theta:

=



∂V

.

∂t



Notice that this is a partial derivative and so theta is the gradient of the option value in the

direction of changing time, asset price fixed. It measures the rate of change of the option value

with time if the asset price doesn’t move, hence the other name ‘time decay.’

The first derivative of the option value with respect to the asset price is called the delta:

=



∂V

.

∂S



This is the slope in the S direction with time fixed. Asset prices change very rapidly and so

the dominant change in the option value from moment to moment is the delta multiplied by

the change in the asset price. This is just a simple application of Taylor series; the difference

between the option price at time t when the asset is at S and a later time t + δt when the asset

price is S + δS is given by

V (S + δS, t + δt) − V (S, t) =



δS + · · · .



The · · · are terms which are, generally speaking, smaller than the leading term. They are still

important, as we’ll see in a moment.



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Appendix A all the math you need. . . and no more



Because the change in option value and the change in asset price are so closely linked we

can see that holding a quantity

of the underlying asset short we can eliminate, to leading

order, fluctuations in our net portfolio value. This is the basis of delta hedging.

The second derivative of the option value with respect to the asset price is called the gamma:

=



∂ 2V

.

∂S 2



This is also just the S derivative of the delta. If the asset changes by an amount δS then the

delta changes by an amount δS. Thus the gamma is a measure of how much one might have

to rehedge, and gives a measure of the amount of transaction costs from delta hedging.

Now we can interpret all the terms in the Black–Scholes equation, but what does the equation

itself mean?

Written in terms of the greeks, the Black–Scholes equation is

+ 12 σ 2 S 2 + rS



− rV = 0.



Reordering this we have

= rV − rS



− 12 σ 2 S 2



= r(V − S ) − 12 σ 2 S 2 .



When we have a delta hedged position we hold the option with value V and are short

the underlying asset. Thus our portfolio value is at any time



of



V −S .

So we can write the Black–Scholes equation in words as

Time decay = (interest received on cash equivalent of portfolio value) − 12 σ 2 S 2 gamma.

The option value grows by an equivalent of interest that would have been received by a

riskless pure cash position. But the delta hedged option is not a cash position. That’s where the

final, gamma, term comes in.

Ignoring the interest on the cash equivalent, the theta and gamma terms add up to zero. Of

course, you can’t ignore this interest unless the portfolio has zero value or rates are zero.

The delta hedge is only accurate to leading order. If one is hedging with finite time intervals

between rehedges then there is inevitably a little bit of randomness that we can’t hedge away.

We can see this if we go to higher order in the Taylor series expansion of V (S + δS, t + δt):

V (S + δS, t + δt) − V (S, t) =



δS +



δt +



1

2



δS 2 . . . .



The term is predictable if we know the time δt between hedges (and it has already appeared in

the Black–Scholes equation). But the term is multiplied by the random δS 2 . We can’t hedge

this away perfectly. It is, in practice, the source of hedging errors. However, if we rehedge

sufficiently frequently (i.e. δt is very small) then the combined effect of the gamma terms is

via an average of the δS 2 . And this average is σ 2 S 2 δt. Why is it the average that matters? It’s

like betting on the toss of a biased coin. If you have an advantage then you can exploit it by

betting a small amount but very, very often. In the long run you will certainly win. (In terms

of standard deviations, as the time between hedges decreases so does the standard deviation of

the hedging error accumulated over the life of the option.)



all the math you need. . . and no more Appendix A



We can now see that the gamma term in the Black–Scholes equation is to balance the higherorder fluctuations in the option value. Naturally, it therefore depends on the magnitude of these

fluctuations, the volatility of the underlying asset.



A.8



SUMMARY



That wasn’t hard, was it?



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