# problem solving in financial mathematics

## a companion blog for a blog on option pricing models # Basic practice problem set 2 – one-period binomial pricing model

This practice problem reinforces the concept of binomial option pricing model discussed in the following two posts

found in this companion blog. The practice problems are basic and straightforward problem to price an option using a binomial tree based on a volatility factor $\sigma$.

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Practice Problems

Practice Problem 1
A 6-month call option on a stock is modeled as a binomial tree. Given the following information:

• The initial stock price is $65. • The strike price is$70.
• $r=$ 0.05.
• $\delta=$ 0.
• $\sigma=$ 0.3.

find the replicating portfolio and the price of this 6-month call option.

Practice Problem 2
This problem uses the same information as in Problem 1, except that the stock pays annual continuous dividends at the rate of $\delta=0.02$.

Practice Problem 3
A 6-month put option on a stock is modeled as a binomial tree. Given the following information:

• The initial stock price is $65. • The strike price is$60.
• $r=$ 0.05.
• $\delta=$ 0.02.
• $\sigma=$ 0.3.

use risk-neutral probabilities to price this 6-month put option. $\text{ }$
_____________________________________________________________________________________ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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Answers

1. Replicating portfolio: $\Delta=$ 0.43508237 shares (long), $B=$ -$22.8747876 (borrowing). $\text{ }$ Call option price = $\Delta S + B$ =$5.405566446 $\text{ }$

2. Replicating portfolio: $\Delta=$ 0.406303779 shares (long), $B=$ -$21.36173125 (borrowing). $\text{ }$ Call option price = $\Delta S + B$ =$5.048014419 $\text{ }$

• $\text{ }$ $p^*=$ 0.447164974 $1-p^*=$ 0.552835026 $uS=$ =$81.57471137 (up stock price) $dS=$ =$53.37034398 (down stock price)

Put option price = $p^*$ (0) + $(1-p^*)$ (6.629656019) = \$3.574614265 $\text{ }$

_____________________________________________________________________________________ $\copyright \ 2015 \text{ by Dan Ma}$

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