Delta for put options

A trader sold a client one-month put options on 2,000 shares of an underlying equity. Options exercise price is 1,300EUR, option premium is 19.09 EUR and underlying equity trading at 1,340EUR per share. Determine whether the change in price of the put option will be greater for an increase or decrease in the price of the underlying equity. Answer: change in price of put option is greater for decrease in price of underlying than for an increase in price of underlying. For put options, the delta will underestimate the price effect of decreases in the underlying equity and will overestimate the price effect of increases in the underlying equity. This is due to the convex relationship between put option prices and the price of the underlying equity. This can be addressed by adjusting the put option price for the effect of gamma, which is analogous to the convexity adjustment of a bond’s price. 1. How does delta underestimate price effect of decreases in underlying and overestimate price effect of increases in underlying? 2. How is there a convex relationship between puts and underlying, when as stock price increases, payoff to a put goes up but by reducing amounts?

For put options, delta overestimates the impact of increases in the value of the underlying and underestimates the impact of decreases. This is because of the relationship between the prices of options and their underlying assets is convex.

Options entail convexity…both puts and calls (read up abt Jensen’s Inequality for options when free). Delta is like bond duration, a linear approx. Gamma is like convexity. If you understand bond duration & convexity, this wouldn’t be difficult.