Mark Joshi, The concepts and practice of mathematical finance exercise 3.6

Question

This is an exercise from Mark Joshi's book (exercise 3.6):

"A stock is worth 100. Each month its value increases or decreases by precisely 10. The riskless bond is worth $e^{r t}$ at time $t$ years with $r$ equal to 5% Price a four-month European put option struck at 110. Do the American case to."

Unfortunately, I struggle to show without computation that the answers are 13.06 and 13.38 for the European put and American one.

Could anybody help me to break down the calculation manually, without utilizing any computer programs please ?

Answer 1

Hints: Your binomial tree is $$ \begin{matrix} & & & & 140\\ & & & 130 & \\ & & 120 & & 120\\ & 110 & & 110\\ 100 & & 100 & & 100\\ & 90 & & 90\\ & & 80 & &80\\ & & & 70 \\ & & & & 60 \end{matrix} $$ In the first step you should determine the probabilities that the stock goes up, resp. down, such that the expected change in one month is equal to that of the riskless bond over one month.

To calculate \begin{align} P_{Euro}&=e^{-rT}\mathbb E\big[\big(K-S_T\big)^+\big]\\[2mm] \end{align} you take each final value of the Stock that gives a positive payoff. To get the probability for that payoff you count the number of ways the stock gets there.

We will discuss the American case when you achieved Joshi's result of $P_{Euro}\approx 13.06\,.$