Question

Explain why both put and call options are worth more if the stock return standard deviation
is higher but put and call options are affected oppositely by the stock price.

please refer the text book FINANCIAL MANAGEMENT (theory and practice) 16E by BRIGHAM and EHRHARDT

Answer 1

Value of both put and call options increase with the increase in volatility.

Please see the image for Black Scholes model equations for option valuation

When standard deviation increases, d1 increases and d2 decreases. Correspondingly N(d1) increases and N(d2) decreases. Similarly as d1 increases N(-d1) decreases. Also, as d2 increases N(-d2) increases.

Hence with increase in standard deviation: N(d1) and N(-d2) increases; N(d2) and N(-d1) decreases.

Therefore both put and call price increases.

Buying a put option gives the buyer the right to sell the underlying at the strike price. Therefore if spot price goes up value of put option reduces. Whereas call option is the right to buy the underlying at strike price. Hence, value of call will go up when stock price increases.

This can also be understood from the BS model equation.

While call price is proportional to Spot price, put price decreases with spot price.