Chapter 21, Problem 1PS

Summary Introduction

Call option:

It is an agreement where the buyer is entitled the right to buy a stock at a pre-specified price within a pre-specified period. The stock on which the call option is provided is called the underlying asset.

Put option:

It is an agreement where the buyer is entitled the right to sell a stock at a pre-specified price within a pre-specified period. The stock on which the put option is provided is called the underlying asset.

Put-call parity: The put call parity provides a relationship among the stock price, strike price, call price and the put price. According to this theory, the difference between the price of a call option and put option on the same asset underlined with the same strike price and expiry date equals to the difference between the stock price and the present value of the strike price.

  $\begin{array}{l}\mathrm{Pr}iceofcall-\mathrm{Pr}iceofput=Stockprice-PVofStrikeprice\\ \mathrm{Pr}iceofcall-\mathrm{Pr}iceofput=Stockprice-Strikeprice\times \left(e^{-riskfreerate\times timeto\exp ire}\right)\end{array}$

To determine:

The impact of volatility on the option values and to provide a numerical example using put call parity relationship to support it.

Answer to Problem 1PS

More volatility on the option value, higher the value of the call option and put option. It was verified by the numerical example.

Explanation of Solution

The value of call option increases when the stock price increases while the value of put option increases when the stock price decreases. So, the value of call option and put option depends on the movement of the stock price or the underlying asset.

Volatility refers to the fluctuation in the level of market price of the underlying asset. So, volatility may cause the stock price to increase or decrease. So, with volatility, it is likely that stock price increases or decreases with time, which causes the value of call and put option to increase. Longer the time period to expire the option, greater is the volatility.

Let's take a numerical example to verify it. Consider two cases where volatility or expiration period is changed and other factors remain the same and then calculate the value of call option.

Case 1 Consider a stock with price $68.73. The value of put option is $2.50 and risk free rate is 8%. Both put and call option on this stock has a strike price is $75. For both, the expiration period is 1 year. Using the put call parity, the value of call option is calculated.

Given:

Stock price=$68.73

Strike price = $75

Price of put option= $2.50

Risk free rate=0.08

Time to expire= 1 year

Calculation:

Using put call parity equation,

  $\begin{array}{c}\mathrm{Pr}iceofcall-\mathrm{Pr}iceofput=Stockprice-Strikeprice\times \left(e^{-riskfreerate\times timeto\exp ire}\right)\\ \mathrm{Pr}iceofcall-\$2.50=\$68.73-\$75\times \left(e^{-0.08\times 1}\right)\\ \mathrm{Pr}iceofcall=\$3\end{array}$

Here, the price / value of call option is $3.

Case 2 Consider a stock with price $68.73. The value of put option is $2.50 and risk free rate is 8%. Both put and call option on this stock has a strike price is $75. For both, the expiration period is 5 year. Using the put call parity, the value of call option is calculated.

Given:

Stock price=$68.73

Strike price = $75

Price of put option= $2.50

Risk free rate=0.08

Time to expire= 5 year

Calculation:

Using put call parity equation,

  $\begin{array}{c}\mathrm{Pr}iceofcall-\mathrm{Pr}iceofput=Stockprice-Strikeprice\times \left(e^{-riskfreerate\times timeto\exp ire}\right)\\ \mathrm{Pr}iceofcall-\$2.50=\$68.73-\$75\times \left(e^{-0.08\times 5}\right)\\ \mathrm{Pr}iceofcall=\$20.96\end{array}$

Here, the price / value of call option is $20.96.

Conclusion

Longer the expiration period, greater is the volatility and higher the value of call option.