Query Regarding Properties of Stock Options

[JNSamara]

I have a query regarding the effect on call and put option prices with changes in volatility.

In Hull Chapter 8 pg 208 they have two graphs, one for call and put, showing the option prices for different volatility values. The graphs are drawn assuming the options are at the money.

The put option price tends to 0 as the volatility tends to zero. This seems to make sense to me, as without any volatility the option with stay at the money and the option will be worthless....hence 0 price.

However the same thing is not shown on the call option graph. The graph shows that as the volatility tends to 0 the call option tends to some positive value.

I am a bit confused. I dont see why anyone would pay a premium to purchase a call option if it is certain to remain at the money.

I do understand from put-call parity that the put and call prices cannot simultaneously be equal to zero when they are at the money as this would result in arbitrage...but if there is no volatility i dont see how either of them can have any value at all.

Probably a random point but it is bugging me my lecturer wasnt able to help me.

 

[didster]

Hazarding a guess and thinking aloud (rather than detailed mathematics) but here goes:

Underlying
If there is no volatility, we can think about this as no risk - the future is known.
The Equity should then increase in value in line with the risk free rate.

Put
If you hold a Put which is at the money, you have an option to sell tomorrow the stock at today's price. (At the money means strike price = current price.) You know tomorrow's price is more, so may as well toss the option and sell at tomorrow's price.

Value of put option = 0;

Call
With a call at the money, you have an option to sell tomorrow the stock at today's price. But tomorrow's price is more than today's (having increased with the risk free rate). So you're more or less holding a guarantee (no risk) that you can buy something at less than the market price, with the guarantee being worth the value of the discount in price.

Value of call = some positive value (which I think = 1 - e{-rT})

 

[JNSamara]

Thanks a lot Didster.... I finally get it, been bugging me for ages actually.

My confusion came in with the definition of volatility. I think I thought about it as the underlying asset having no variance, which in effect means it does not move.

However I found a definition of volatility which is the variance of the rate of return....and as you correctly stated, a 0 volatility just means the return is constant, it doesnt mean the asset doesnt move.

Thanks again. Much prefer that type of answer to a detailed mathematical answer