22 No arbitrage in the binomial, Black–Scholes and ‘other’ worlds
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Part One mathematical and ﬁnancial foundations
101
100
100
99
Figure 15.35
The trinomial tree.
1 − ∆101
V − ∆100
0 − ∆100
0 − ∆99
Figure 15.36
Is hedging possible?
is only possible in the two special cases, the Black–Scholes continuous time/continuous asset
world, and the binomial world. And in the far more complex ‘real’ world, delta hedging is not
possible.3
15.23 SUMMARY
In this chapter I described the basics of the binomial model, deriving pricing equations and
algorithms for both European- and American-style exercise. The method can be extended in
many ways, to incorporate dividends, to allow Bermudan exercise, to value path-dependent
contracts and to price contracts depending on other stochastic variables such as interest rates.
I have not gone into the method in any detail for the simple reason that the binomial method
is just a simple version of an explicit ﬁnite-difference scheme. As such it will be discussed
3 Is it good for the popular models to have such an unrealistic property? These models are at least a good starting point.
I ﬁnd it fascinating that the two most popular modeling ‘worlds,’ the Black–Scholes and the binomial, are the only two
in which hedging is possible. I think this says more about mathematical modelers than it does about the nature of the
ﬁnancial markets.
the binomial model Chapter 15
in depth in Part Six. Finite-difference methods have an obvious advantage over the binomial
method; they are far more ﬂexible.
FURTHER READING
• The original binomial concept is due to Cox, Ross & Rubinstein (1979).
• Almost every book on options describes the binomial method in more depth than I do. One
of the best is Hull (2005) who also describes its use in the ﬁxed-income world.
APPENDIX: ANOTHER PARAMETERIZATION
The three constants u, v and p are chosen to give the binomial walk the same drift and
standard deviation as that given by the stochastic differential equation (3.7). Having only these
two equations for the three parameters gives us one degree of freedom in this choice. This
degree of freedom is often used to give the random walk the further property that after an up
and a down movement (or a down followed by an up) the asset returns to its starting value,
S.4 This gives us the requirement that
v(uS) = u(vS) = S
i.e.
uv = 1.
(15.4)
Our starting point, the lognormal random walk,
dS = µS dt + σ S dX
has the solution, found in Section 4.14.2,
S(t) = S(0)e
√
1
µ− 2 σ 2 t+σ φ t
where φ is a standardized Normal random variable.
For the binomial random walk to have the correct drift over a time period of δt we need
puS + (1 − p)vS = SE e
√
1
µ− 2 σ 2 δt+σ φ δt
= Se µ δt ,
i.e.
pu + (1 − p)v = e µ δt .
Rearranging this equation we get
p=
e µδt − v
.
u−v
(15.5)
4 Other choices are possible. For example, sometimes the probability of an up move is set equal to the probability of a
down move i.e. p = 1/2.
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Then for the binomial random walk to have the correct variance we need (details omitted)
pu2 + (1 − p)v 2 = e (2µ+σ
2 )δt
.
(15.6)
Equations (15.4), (15.5) and (15.6) can be solved to give
u=
1
2
e −µ δt + e (µ+σ
2 )δt
+
1
2
e −µ δt + e (µ+σ 2 )δt
2
− 4.
Approximations that are good enough for most purposes are
u ≈ 1 + σ δt 1/2 + 12 σ 2 δt,
v ≈ 1 − σ δt 1/2 + 12 σ 2 δt
and
p≈
µ − 12 σ 2 δt 1/2
1
+
.
2
2σ
Of course, if this is being used for pricing options, you must replace the µ with r everywhere.
CHAPTER 16
how accurate is the
Normal approximation?
In this Chapter. . .
•
•
•
why the Normal distribution is so popular
how fat the tails really are
what dropping the Normal assumption entails
16.1
INTRODUCTION
Without a shadow of a doubt the assumption of Normally distributed returns is one of the most
important assumptions in quantitative ﬁnance. It allows us to make enormous advances because
it comes with a lot of relatively easy-to-use analytical tools. Yet the Normal distribution has
often been criticized for being unrealistic in its description of large events: stock crashes. The
Normal distribution vastly underestimates their probability. In this short chapter we look at why
the distribution is popular, see some simple statistics associated with tail events and look at
alternatives.
16.2
WHY WE LIKE THE NORMAL
DISTRIBUTION: THE CENTRAL
LIMIT THEOREM
The Central Limit Theorem (CLT) states: ‘Given a distribution
with a mean m and variance s 2 , the sampling distribution of
the mean approaches a Normal distribution with a mean of m
and a variance of s 2 /N as N increases.’
By ‘sampling distribution of the mean’ is meant
1
N
N
Xi
i=1
where the Xi are all drawn from the initial distribution.
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Part One mathematical and ﬁnancial foundations
In layman’s terms, if you add up lots of random numbers all drawn from the same ‘building
block’ distribution then you get a Normal distribution. And this works for any building-block
distributions (except for some ‘small print’ which we’ll see in a moment). This explains why
the Normal distribution is important in practice; it occurs whenever a distribution comes from
adding up lots of random numbers. Perhaps stock price daily returns should be Normal since
you ‘add up’ thousands of returns during each day.
And since the Normal distribution only has the two parameters, the mean and the variance,
it follows that the skew and kurtosis etc. of the building-block distribution don’t much matter
to the ﬁnal distribution.
The ‘small print’ are the conditions under which the Central Limit Theorem is valid. These
conditions are:
• The building-block distributions must be identical (you aren’t allowed to draw from different
distributions each time).
• Each draw from the building-block distribution must be independent from other draws.
• The mean and standard deviation of the building-block distribution must both be ﬁnite.
There are generalizations of the CLT in which these conditions are weakened, but we won’t
go into those here.
16.3 NORMAL VERSUS LOGNORMAL
I often ask new students what distribution is assumed by the Black–Scholes model for the asset
return. The answer (before I have taught them ‘properly’) is usually equally likely to be either
Normal or lognormal. But then I get the same answers when I ask them what is the distribution
assumed for the asset return.
You will know that the simple assumption for returns is that they are Normal and that,
provided the parameters drift and volatility are constant, the resulting distribution for the asset
is lognormal.
Here is a quick way of demonstrating and explaining lognormality that relies only on the
Central Limit Theorem.
Start with a stock price with value S0 . Add a random return R1 to this to get the stock price,
S1 , at the next time step:
S1 = S0 (1 + R1 ).
After the second time step, and a random return of R2 , the stock price is
S2 = S0 (1 + R1 )(1 + R2 ).
After N time steps we have
N
SN = S0
(1 + Ri ).
i=1
What is the distribution for the stock price SN ?
(16.1)