# problem solving in financial mathematics

## Pricing a put 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 put option on the stock specifies an exercise price (strike price) of$45 and is set to expire at the end of one year. What is the fair price of this put option? Use the one-period binomial option model. What is the portfolio replicating the payoff of this put option?
2. Consider a call 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 call option.
3. Price the call option described in #2 using the binomial option model.

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1. Replicating portfolio: $\Delta=$ -0.25 shares (short), $B=$ $14.26844137 (lending). $\text{ }$ Put option price = $\Delta S + B$ =$1.768441368

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2. Call option premium = $8.963117265 3. $\text{ }$ 4. Replicating portfolio: $\Delta=0.75$ shares (long), $B=$ -$28.53688274 (borrowing).
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Call option price = $\Delta S + B$ = $8.96311726 $\text{ }$ _____________________________________________________________________________________ $\copyright \ 2015 \text{ by Dan Ma}$ Advertisements ## 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. _____________________________________________________________________________________ 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. $\text{ }$ _____________________________________________________________________________________ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ _____________________________________________________________________________________ Answers 1. Replicating portfolio: $\Delta=$ 0.25 shares, $B=$ -$9.512294245 (borrowing).
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Call option price = $\Delta S + B$ = $2.9877 $\text{ }$ 2. Put option premium =$5.305324103
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4. Replicating portfolio: $\Delta=-0.75$ shares (short), $B=$ $42.8053241 (lending). $\text{ }$ Put option price = $\Delta S + B$ =$5.3053241
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$\copyright \ 2015 \text{ by Dan Ma}$