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Part One mathematical and financial foundations



and

dS2 = a2 (S1 , S2 , t) dt + b2 (S1 , S2 , t) dX2 .

Note that I have two Brownian increments dX1 and dX2 . We can think of these as being

Normally distributed with variance dt, but they are correlated. The correlation between these

two random variables I will call ρ. This can also be a function of S1 , S2 and t but must satisfy

−1 ≤ ρ ≤ 1.

The ‘rules of thumb’ can readily be imagined:

dX12 = dt,



dX22 = dt and dX1 dX2 = ρ dt.



Itˆo’s lemma becomes

dV =



∂V

∂V

∂V

∂ 2V

∂ 2V

∂ 2V

dt +

dS1 +

dS2 + 12 b12 2 dt + ρb1 b2

dt + 12 b22 2 dt.

∂t

∂S1

∂S2

∂S1 ∂S2

∂S1

∂S2



(4.9)



4.14 SOME PERTINENT EXAMPLES

In this section I am going to introduce a few common random walks and talk about

their properties.

Remember that a stochastic differential equation model for variable S is something

of the form

dS =



dt +



dX.



The bit in front of the dt is deterministic and the bit in front of the dX tells

us how much randomness there is. Modeling is very much about choosing functions to go

where the underlining is; it is about choosing the functional form for the deterministic part and

the functional form for the amount of randomness. We will now look at some examples.

4.14.1



Brownian Motion with Drift



The first example is like the simple Brownian motion but with a drift:

dS = µ dt + σ dX.

A realization of this is shown in Figure 4.5. The point to note about this realization is that

S has gone negative, near the start. This random walk would therefore not be a good model

for many financial quantities, such as interest rates or equity prices. This stochastic differential

equation can be integrated exactly to get

S(t) = S(0) + µt + σ (X(t) − X(0)).



elementary stochastic calculus Chapter 4



0.6

0.5

0.4



S



0.3

0.2

0.1

0

0



0.2



0.4



0.6



0.8



1



−0.1

Time



Figure 4.5 A realization of dS = µ dt + σ dX.



4.14.2



The Lognormal Random Walk



My second example is similar to the above but the drift and randomness scale with S:

dS = µS dt + σ S dX.



(4.10)



A realization of this is shown in Figure 4.6. If S starts out positive it can never go negative;

the closer that S gets to zero the smaller the increments dS. For this reason I have had to start

the simulation with a non-zero value for S. This property of this random walk is clearly seen

if we examine the function F (S) = log S using Itˆo’s lemma. From Itˆo we have

1

d 2F

dF

dS + 12 σ 2 S 2 2 dt = (µS dt + σ S dX) − 12 σ 2 dt = µ − 12 σ 2 dt + σ dX.

dS

dS

S

This shows us that log S can range between minus and plus infinity but cannot reach these

limits in a finite time, therefore S cannot reach zero or infinity in a finite time.

How does the time series in Figure 4.6 which was generated on a spreadsheet using random

returns compare qualitatively with the time series in Figure 4.7 which is the real series for

Glaxo–Wellcome?

The integral form of this stochastic differential equation follows simply from the stochastic

differential equation for log S:

dF =



1 2

)t+σ (X(t)−X(0))



S(t) = S(0)e(µ− 2 σ



.



The stochastic differential equation (4.10) will be particularly important in the modeling of

many asset classes. And if we have some function V (S, t) then from Itˆo it follows that



dV =



∂V

∂V

∂ 2V

dt +

dS + 12 σ 2 S 2 2 dt.

∂t

∂S

∂S



(4.11)



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Part One mathematical and financial foundations



140



120



100



80



S



86



60



40



20



0

0



0.2



0.4



0.6



0.8



1



Time



Figure 4.6 A realization of dS = µS dt + σ S dX.



4.14.3



A Mean-reverting Random Walk



The third example is

dS = (ν − µS) dt + σ dX.

A realization of this is shown in Figure 4.8.

This random walk is an example of a mean-reverting random walk. If S is large, greater

than ν/µ, the negative coefficient in front of dt means that S will move down on average; if S

is small, less than ν/µ, it rises on average. There is still no incentive for S to stay positive in

this random walk. With r instead of S this random walk is the Vasicek model for the short-term

interest rate.

Mean-reverting models are used for modeling a random variable that ‘isn’t going anywhere.’

That’s why they are often used for interest rates; Figure 4.9 shows the yield on a Japanese

Government Bond.

Let’s take a look at the Vasicek model for the spot interest rate r

dr = (ν − γ r) dt + σ dX

where γ is the reversion rate and ν/γ is the mean rate.

By setting W = r − ν, W is a solution of

dW = −γ W dt + σ dX.



elementary stochastic calculus Chapter 4



Figure 4.7 Glaxo–Wellcome share price (volume below). Source: Bloomberg L.P.



This random walk for W is an Ornstein–Uhlenbeck process. An analytic solution for this

equation exists, and we shall derive it now.

Introduce the integrating factor I = eγ t . Write

d (I W ) = I dW + W dI = eγ t (−γ W dt + σ dX) + γ W eγ t dt

= σ eγ t dX.

Integrating over [0, t] gives

t



I W = constant + σ



eγ s dX(s),



0



so that

t



W = W (0) e−γ t + σ



eγ (s−t) dX(s).



0



By using integration by parts we can simplify (4.12).

t

0



t



eγ (s−t) dX(s) = X − γ

0



eγ (s−t) X(s) ds.



(4.12)



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Part One mathematical and financial foundations



1.2



1



0.8



S



88



0.6



0.4



0.2



0

0



0.2



0.4



0.6



0.8



1



Time



Figure 4.8 A realization of dS = (ν − µS)dt + σ dX.



And we can write (4.12) as

t



W (t) = W (0) e−γ t + σ X(t) − γ



eγ (s−t) X(s) ds .



0



Hence

t



r = ν + W = ν + (r(0) − ν) exp (−γ t) + σ X(t) − γ



eγ (s−t) X(s) ds .



0



4.14.4



And Another Mean-reverting Random Walk



The final example is similar to the third, and I will again write it in terms of r, but I am going

to adjust the random term slightly:

dr = (ν − µr) dt + σ r 1/2 dX.

Now if r ever gets close to zero the randomness decreases, perhaps this will stop r from

going negative? Let’s play around with this example for a while. And we’ll see Itˆo in practice.

Write F = r 1/2 . What stochastic differential equation does F satisfy? Since

d 2F

dF

= 12 r −1/2 and

= − 14 r −3/2

dr

dr 2



elementary stochastic calculus Chapter 4



Figure 4.9 Time series of the yield on a JGB. Source: Bloomberg L.P.



we have

dF =



4ν − σ 2

− 12 µF dt + 12 σ dX.

8F



I have just turned the original stochastic differential equation with a variable coefficient in front

of the random term into a stochastic differential equation with a constant random term. In so

doing I have made the drift term nastier. In particular, the drift is now singular at F = r = 0.

Something special is happening at r = 0.

Instead of examining F (r) = r 1/2 , can I find a function F (r) such that its stochastic differential equation has a zero drift term? For this I will need

(ν − µr)



dF

d 2F

+ 12 σ 2 r 2 = 0.

dr

dr



This is easily integrated once to give

dF

2

2

= Ar −2ν/σ e2µr/σ

dr

for any constant A. I won’t take this any further but just make one observation. If

2ν

≥1

σ2



(4.13)



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Part One mathematical and financial foundations



we cannot integrate (4.13) at r = 0. This makes the origin non-attainable. In other words, if

the parameter ν is sufficiently large it forces the random walk to stay away from zero.

This particular stochastic differential equation for r will be important later on, it is the Cox,

Ingersoll & Ross model for the short-term interest rate.

These are just four of the many random walks we will be seeing.



4.15 SUMMARY

This chapter introduced the most important tool of the trade, Itˆo’s lemma. Itˆo’s lemma allows

us to manipulate functions of a random variable. If we think of S as the value of an asset for

which we have a stochastic differential equation, a ‘model,’ then we can handle functions of

the asset, and ultimately value contracts such as options.

If we use Itˆo as a tool we do not need to know why or how it works, only how to use it.

Essentially all we require to use the lemma successfully is a rule of thumb, as explained in the

text. Unless we are using Itˆo in highly unusual situations, then we are unlikely to make any

errors.



FURTHER READING

• Neftci (1996) is the only readable book on stochastic calculus for beginners. It does not

assume any knowledge about anything. It takes the reader very slowly through the basics

as applied to finance.

• Once you have got beyond the basics, move on to Øksendal (1992) and Schuss (1980).



CHAPTER 5



the Black–Scholes

model

In this Chapter. . .

•

•

•

•



5.1



the foundations of derivatives theory: delta hedging and no arbitrage

the derivation of the Black–Scholes partial differential equation

the assumptions that go into the Black–Scholes equation

how to modify the equation for commodity and currency options

INTRODUCTION



This is, without doubt, the most important chapter in the book. In it I describe and explain

the basic building blocks of derivatives theory. These building blocks are delta hedging and no

arbitrage. They form a moderately sturdy foundation to the subject and have performed well

since 1973 when the ideas became public.

In this chapter I begin with the stochastic differential equation model for equities and exploit

the correlation between this asset and an option on this asset to make a perfectly risk-free

portfolio. I then appeal to no arbitrage to equate returns on all risk-free portfolios to the

risk-free interest rate, the so called ‘no free lunch’ argument.

The arguments are trivially modified to incorporate dividends on the underlying and also to

price commodity and currency options and options on futures.

This chapter is quite theoretical, yet all of the ideas contained here are regularly used in

practice. Even though all of the assumptions can be shown to be wrong to a greater or lesser

extent, the Black–Scholes model is profoundly important both in theory and in practice.



5.2



A VERY SPECIAL PORTFOLIO



In Chapter 2 I described some of the characteristics of options and options markets. I introduced

the idea of call and put options, amongst others. The value of a call option is clearly going

to be a function of various parameters in the contract, such as the strike price E and the time

to expiry T − t, where T is the date of expiry and t is the current time. The value will also



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Part One mathematical and financial foundations



depend on properties of the asset itself, such as its price, its drift and its volatility, as well as

the risk-free rate of interest.1 We can write the option value as

V (S, t; σ , µ; E, T ; r).

Notice that the semicolons separate different types of variables and parameters:

•

•

•

•



S and t are variables;

σ and µ are parameters associated with the asset price;

E and T are parameters associated with the details of the particular contract;

r is a parameter associated with the currency in which the asset is quoted.



I’m not going to carry all the parameters around, except when it is important. For the moment

I’ll just use V (S, t) to denote the option value as a function of its variables.

One simple observation is that a call option will rise in value if the underlying asset rises, and

will fall if the asset falls. This is clear since a call has a larger payoff the greater the value of

the underlying at expiry. This is an example of correlation between two financial instruments,

in this case the correlation is positive. A put and the underlying have a negative correlation.

We can exploit these correlations to construct a very special portfolio.

Use to denote the value of a portfolio of one long option position and a short position in

some quantity , delta, of the underlying:

= V (S, t) −



S.



(5.1)



The first term on the right is the option and the second term is the short asset position. Notice

the minus sign in front of the second term. The quantity

will for the moment be some

constant quantity of our choosing. We will assume that the underlying follows a lognormal

random walk

dS = µS dt + σ S dX.

It is natural to ask how the value of the portfolio changes from time t to t + dt. The change in

the portfolio value is due partly to the change in the option value and partly to the change in

the underlying:

d



= dV −



dS.



Notice that has not changed during the time step; we have not anticipated the change in S.

From Itˆo we have

dV =



∂V

∂ 2V

∂V

dt +

dS + 12 σ 2 S 2 2 dt.

∂t

∂S

∂S



Thus the portfolio changes by

d

1



=



∂V

∂ 2V

∂V

dt +

dS + 12 σ 2 S 2 2 dt −

∂t

∂S

∂S



Actually, I’m lying. One of these parameters does not affect the option value.



dS.



(5.2)



the Black–Scholes model Chapter 5



5.3



ELIMINATION OF RISK: DELTA HEDGING



The right-hand side of (5.2) contains two types of terms, the deterministic and the random.

The deterministic terms are those with the dt, and the random terms are those with the dS.

Pretending for the moment that we know V and its derivatives then we know everything about

the right-hand side of (5.2) except for the value of dS. And this quantity we can never know in

advance.

These random terms are the risk in our portfolio. Is there any way to reduce or even eliminate

this risk? This can be done in theory (and almost in practice) by carefully choosing . The

random terms in (5.2) are

∂V

−

∂S



dS.



If we choose

=



∂V

∂S



(5.3)



then the randomness is reduced to zero.

Any reduction in randomness is generally termed hedging, whether that randomness is due to

fluctuations in the stock market or the outcome of a horse race. The perfect elimination of risk,

by exploiting correlation between two instruments (in this case an option and its underlying) is

generally called delta hedging.

Delta hedging is an example of a dynamic hedging strategy. From one time step to the next

the quantity ∂V

∂S changes, since it is, like V , a function of the ever-changing variables S and t.

This means that the perfect hedge must be continually rebalanced. In later chapters we will see

examples of static hedging, where a hedging position is not changed as the variables evolve.

Delta hedging was effectively first described by Thorp & Kassouf (1967). (We will see more

of Thorp when we look at casino Blackjack as an investment in Chapter 17.)



5.4



NO ARBITRAGE



After choosing the quantity

by the amount



as suggested above, we hold a portfolio whose value changes



d



=



∂V

∂ 2V

+ 12 σ 2 S 2 2

∂t

∂S



dt.



(5.4)



This change is completely riskless. If we have a completely risk-free change d in the portfolio

value then it must be the same as the growth we would get if we put the equivalent amount

of cash in a risk-free interest-bearing account:

d



=r



dt.



(5.5)



This is an example of the no arbitrage principle.

To see why this should be so, consider in turn what might happen if the return on the portfolio

were, first, greater and, second, less than the risk-free rate. If we were guaranteed to get a return

of greater than r from the delta-hedged portfolio then what we could do is borrow from the



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Part One mathematical and financial foundations



bank, paying interest at the rate r, invest in the risk-free option/stock portfolio and make a

profit. If, on the other hand, the return were less than the risk-free rate we should go short the

option, delta hedge it, and invest the cash in the bank. Either way, we make a riskless profit

in excess of the risk-free rate of interest. At this point we say that, all things being equal, the

action of investors buying and selling to exploit the arbitrage opportunity will cause the market

price of the option to move in the direction that eliminates the arbitrage.



5.5 THE BLACK–SCHOLES EQUATION

Substituting (5.1), (5.3) and (5.4) into (5.5) we find that

∂ 2V

∂V

+ 12 σ 2 S 2 2

∂t

∂S



dt = r V − S



∂V

∂S



dt.



On dividing by dt and rearranging we get

∂V

∂V

∂ 2V

+ 12 σ 2 S 2 2 + rS

− rV = 0.

∂t

∂S

∂S



(5.6)



This is the Black–Scholes equation. The equation was first

written down in 1969, but a few years passed, with Fischer

Black and Myron Scholes justifying the model, before it was published. The derivation of the

equation was finally published in 1973, although the call and put formulae had been published

a year earlier.2

The Black–Scholes equation is a linear parabolic partial differential equation. In fact,

almost all partial differential equations in finance are of a similar form. They are almost always

linear, meaning that if you have two solutions of the equation then the sum of these is itself

also a solution. Or at least they tended to be linear until recently. In Part Five I will show

you some examples of recent models which lead to non-linear equations. Financial equations

are also usually parabolic, meaning that they are related to the heat or diffusion equation of

mechanics. One of the good things about this is that such equations are relatively easy to solve

numerically.

The Black–Scholes equation contains all the obvious variables and parameters such as the

underlying, time, and volatility, but there is no mention of the drift rate µ. Why is this? Any

dependence on the drift dropped out at the same time as we eliminated the dS component of

the portfolio. The economic argument for this is that since we can perfectly hedge the option

with the underlying we should not be rewarded for taking unnecessary risk; only the risk-free

rate of return is in the equation. This means that if you and I agree on the volatility of an

asset we will agree on the value of its derivatives even if we have differing estimates of the

drift.

Another way of looking at the hedging argument is to ask what happens if we hold a portfolio

consisting of just the stock, in a quantity , and cash. If

is the partial derivative of some

option value then such a portfolio will yield an amount at expiry that is simply that option’s

2



The pricing formulae were being used even earlier by Ed Thorp to make money.