13 Itˆo in higher dimensions
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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.