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22 No arbitrage in the binomial, Black–Scholes and ‘other’ worlds

22 No arbitrage in the binomial, Black–Scholes and ‘other’ worlds

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292



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.



293



294



Part One mathematical and financial foundations



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





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.



296



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)



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