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I am a beginner to financial mathematics, and my lecturer asked me to ponder about how volatility may affect the value of an option (as a function of spot price). For example, if an option had a (daily) volatility of 1%, how would the option value look like say 5 days before expiry? So far, the only quantity I know which can affect the value time-wise is interest rate - an extra factor of $\text{e}^{-rt}$ needs to be applied, and the step-function becomes slightly curved. I haven't seen any resources which describe how volatility comes into play. Will greatly appreciate some explanation about this! (I have seen an upvoted question about volatility and binary options, but it’s quite out of my depth!)

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  • $\begingroup$ This answer here might be helpful: it shows a diagram which has option pay-off as a function of Volatility. $\endgroup$ Nov 7 '20 at 6:01
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The option payoff diagram for European Option will be exactly the same.

The intuitive reason that option value changes with volatility is that it changes the probability of winning the jackpot.

Think about it, options provide you downside protection. So, for example, when the stock is volatile, it has a higher probability of getting to a high price. If you held a call option, you got the jackpot. If the stock dropped, you don't care how much it drop below the strike price. It kind of creates an asymmetry of risk and benefit.

Of course, there is no free lunch in derivative markets, it makes option with high underlying volatility worth more and priced higher.

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Have you reviewed the Black Scholes formulae? For a given spot price, volatility is back-calculated based on options premium that investors are willing to pay.

Given constant volatility as you suggest, the option value will decay as it gets close to the expiration. In other words, an option at a strike price equal to spot price, will have an intrinsic value 5 days from expiration (this is the option premium that an investor is willing to bet on the price of the underlying asset at expiration). Simply put, this premium is dependent on 1)volatility and 2)the time left to expiration. At expiration, the time remaining goes to zero, so the intrinsic value also becomes zero ie an option at strike price =spot price will have zero value. HTH

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  • $\begingroup$ Hi, thanks for your answer; is this decay exponential as well? Does it follow a similar form to e^-rt? The BS formula isn’t working too intuitively for me (I see the normal CDFs involved but this doesn’t seem like a time evolution) $\endgroup$
    – user107224
    Nov 7 '20 at 2:11
  • $\begingroup$ No it is a $\sqrt T$ type of thing, not exponential. $\endgroup$
    – noob2
    Nov 7 '20 at 11:20

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