# problem solving in financial mathematics

## a companion blog for a blog on option pricing models # Pricing a call option, a practice problem

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

in a companion blog.

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Practice Problems

Suppose the stock of a certain company is currently selling for $50 per share. The price per share at the end of one year is expected to increase to$60 (20% increase) or to decrease to $40 (20% decrease). The stock pays no dividends during the next year. The annual risk-free interest rate is 5%. Assume all the options discussed here are European options. 1. A call option on the stock specifies an exercise price (strike price) of$55 and is set to expire at the end of one year. What is the fair price of this call option? Use the one-period binomial option model. What is the portfolio replicating the payoff of this call option?
2. Consider a put option on the same underlying stock and with the same strike price and the same time to expiration as the put option in #1. Use the put-call parity to derive the price of this put option.
3. Price the put option described in #2 using the binomial option model. $\text{ }$
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Answers

1. Replicating portfolio: $\Delta=$ 0.25 shares, $B=$ -$9.512294245 (borrowing). $\text{ }$ Call option price = $\Delta S + B$ =$2.9877 $\text{ }$
2. Put option premium = $5.305324103 3. $\text{ }$ 4. Replicating portfolio: $\Delta=-0.75$ shares (short), $B=$$42.8053241 (lending). $\text{ }$
Put option price = $\Delta S + B$ = \$5.3053241 $\text{ }$

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

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