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Risk neutral probability
1. 3.2.1 Risk โ Neutral Probability
While the future value of stock can never be known with certainty, it is posible to
work out expected stock prices within the binomial tree model. It is then natural to compare
these expected prices and risk-free investments. This simple idea will lead us towards
powerful and surprising applications in the theory of derivative securities (for example,
options, forwards, futures), to be studied in later chapters.
To begin with, we shall work out the dynamics of expected stock prices ๐ธ(๐(๐)). For
๐ = 1
๐ธ(๐(1)) = ๐๐(0)(1 + ๐ข) + (1 โ ๐)๐(0)(1 + ๐) = ๐(0) (1 + ๐ธ(๐พ(1)))
Where
๐ธ(๐พ(1)) = ๐๐ข + (1 โ ๐)๐
Is the expected one-step return. This extends to any ๐ as follows.
Proposition 3.4
The expected stock prices for ๐ = 0,1,2, โฆ are given by
๐ธ(๐(๐)) = ๐(0) (1 + ๐ธ(๐พ(1)))
๐
Proof
Since the one-step returns ๐พ(1), ๐พ(2), โฆ are independent, so are the random variabels
1 + ๐พ(1), 1 + ๐พ(2), โฆ it follows that
๐ธ(๐(๐)) = ๐ธ (๐(0)(1 + ๐พ(1))(1 + ๐พ(2)) โฆ (1 + ๐พ(๐)))
= ๐(0)๐ธ(1 + ๐พ(1))๐ธ(1 + ๐พ(2)) โฆ ๐ธ(1 + ๐พ(๐))
= ๐(0) (1 + ๐ธ(๐พ(1))) (1 + ๐ธ(๐พ(2))) โฆ (1 + ๐ธ(๐พ(๐))).
Because the ๐พ(๐) are identically distributed, they all have the same expectation,
๐ธ(๐พ(1)) = ๐ธ(๐พ(2)) = โฏ = ๐ธ(๐พ(๐)),
2. Which proves the formula for ๐ธ(๐(๐)).
If the amount ๐(0) were to be invested risk-free at time 0, it would grow to
๐(0)(1 + ๐) ๐
after ๐ steps. Clearly, to compare ๐ธ(๐(๐)) and ๐(0)(1 + ๐) ๐
we only need to
compare ๐ธ(๐พ(1) and ๐.
An investment in stock always involves an element of risk, simply because the price
๐(๐) is unknown in advance. A typical risk-averse investor will require-that ๐ธ(๐พ(1)) > ๐,
arguing that he or she should be rewarded with a higher expected return as a compensation
for risk. The reverse situation when ๐ธ(๐พ(1)) < ๐ may nevertheless be attractive to some
investors if the risky return is high with small non-zero probability and low with large
probability. (A typical example is a lottery, where the expected return is negative). An
investor of this kind can be called a risk-seeker. We shall return to this topic in chapter 5,
where a pricise definition of risk will be developed. The border case of a market in which
๐ธ(๐พ(1)) = ๐ is referred to as risk-neutral.
It proves convenient to introduce a special symbol ๐โ for the probabilityas well as ๐ธโ
for the corresponding expectation satisfying the condition
๐ธโ(๐พ(1)) = ๐โ ๐ข + (1 โ ๐โ)๐ = ๐
For risk-neutrality, which implies that
๐โ =
๐ โ ๐
๐ข โ ๐
We shall call ๐โ the risk-neutral probability and ๐ธโ the risk-neutral expectation. It is
important to understand that ๐โ is an abstract mathematical object, which may or may not
be equal to the actual market probability ๐. Only in a risk-neutral market do we have ๐ =
๐โ. Even though the risk-neutral probability ๐โ may have no relation to the actual
probability ๐, it turns out that for the purpose of valuation of derivate securities the relevant
probability is ๐โ, rather then ๐. This application of the risk-neutral probability, which is of
great practical importance, will be discussed in detail in chapter 8.
Exercise 3.17
3. Let ๐ข =
2
10
๐๐๐ ๐ =
1
10
Investigate the properties of ๐โ as a function of ๐.
Exercise 3.18
Show that ๐ < ๐ < ๐ข if and only if 0 < ๐โ < 1.
Condition (3.4) implies that
๐โ(๐ข โ ๐) + (1 โ ๐โ)(๐ โ ๐) = 0.
Geometrically, this means that the pair (๐โ, 1 โ ๐โ) regarded as a vector on the plan
๐ 2
is orthogonal to the vector with coordinates (๐ข โ ๐, ๐ โ ๐), which represents the possible
one-step gains (or losses) of an investor holding a single share of stock, the purchase of
which was financed by a cash loan attracting interest at a rate ๐, see Figure 3.5. the line
joining the point (1,0) and (0,1) consists of all points with coordinates (๐, 1 โ ๐), where
0 < ๐ < 1. One of these points corresponds to the actual market probability and one to
the risk-neutral probability.
Another interpretation of condition (3.4) for the risk-neutral probability is
illustrated in figure 3.6. if masses ๐โ and 1 โ ๐โ are attached at the points with coordinates
๐ข and ๐ on the real axis, then the centre of mass will be at ๐.