Pricing a European Put with a Binomial Stock Tree

Summary

The note outlines how to price a European put in a four-month binomial stock model where the share price changes by a fixed amount each month. The stock starts at 100, moves up or down by 10 per step, and the riskless bond compounds at a 5% annual rate. The strike is 110. The suggested approach first finds the risk-neutral probabilities by requiring the expected one-month stock change to match the return implied by the bond.

At maturity, identify the terminal stock prices below the strike, calculate each put payoff, and weight it by the probability of reaching that price. The tree’s path counts supply the probabilities of each terminal outcome, and discounting the expected payoff gives the European value. The answer treats the American option only as a later exercise: it asks the learner to establish the European result before extending the calculation. It supplies hints and a tree, not the full calculations, and the stated numerical answer is from the question rather than independently derived in the response.

Key ideas

• Set the one-step risk-neutral probabilities by matching expected stock growth to the riskless return.
• Use the binomial tree to enumerate terminal stock prices and count the paths to each price.
• For a put, only terminal prices below the strike produce a positive payoff.
• Discount the risk-neutral expected payoff to obtain the European put value.
• The response gives a calculation outline but leaves the American valuation for a later step.

Tags

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


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












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 by Kurt G. (score 1)

https://quant.stackexchange.com/a/78670

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 $P_{Euro}\approx 13.06\,.$

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