problem solving in financial mathematics

a companion blog for a blog on option pricing models

Pricing a call option, a practice problem

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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.

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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{ }

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\copyright \ 2015 \text{ by Dan Ma}

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