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58 Mathematics for Finance

Figure 3.4 Three-step binomial tree of stock prices

Exercise 3.13

FinddanduifS(1) can take two values, $87 or $76, and the top possible

value ofS(2) is $92.

Exercise 3.14

Suppose that the risk-free rate under continuous compounding is 14%,

the time stepτis one month,S(0) = 22 dollars andd=− 0 .01. Find the

bounds on the middle value ofS(2) consistent with Condition 3.2.

Exercise 3.15

Suppose that $32, $28 andxare the possible values ofS(2). Findx,

assuming that stock prices follow a binomial tree. Can you complete the

tree? Can this be done uniquely?

Exercise 3.16

Suppose that stock prices follow a binomial tree, the possible values of

S(2) being $121, $110 and $100. FinduanddwhenS(0) = 100 dollars.

Do the same whenS(0) = 104 dollars.

3.2.1 Risk-Neutral Probability

While the future value of stock can never be known with certainty, it is possible
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