How Implied Volatility and Maturity Affect Option Prices
Summary
This discussion examines why longer-dated options can cost more even when implied volatility appears to fall with maturity. It cautions that implied volatility term structures vary: they may rise, fall, or fluctuate, including at the money. In the Black–Scholes framework, the price depends in part on volatility scaled by the square root of time, so a declining volatility level does not by itself imply a lower option value as maturity increases.
The answers also offer explanations involving the underlying’s chance of reaching relevant prices over a longer horizon and the discounting of future cash flows. The main practical lesson is to compare option prices with the full volatility term structure and time dependence, rather than infer price direction from implied volatility alone. These are short forum responses rather than a complete derivation; the bond analogy is simplified, and the discussion does not establish a general pricing rule for every option or market condition.
Key ideas
- Implied volatility can rise, fall, or vary irregularly across maturities.
- A fall in implied volatility does not guarantee a lower price for a longer-dated option.
- In Black–Scholes, volatility scaled by the square root of time is relevant to option value.
- Longer horizons and discounting can both affect maturity comparisons.
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Full text
# Questions on the relationship between option price and maturity # Questions on the relationship between option price and maturity From the plot of volatility surface, as maturity goes up, the implied volatility will decrease. Dose it mean that options with the same strike have higher value when maturity is larger. If so why price of large maturity option is often higher, which can contribute to calendar spread? ## Answer by james42 (score 1) https://quant.stackexchange.com/a/18012 I'll try to make a guess. Maybe it's due to the properties of the underlying, since we know that GBM will go to almost every point with a sufficient amount of time. The higher the time, the higher the probability of hitting a given price, the higher the value of an option. ## Answer by SmallChess (score 0) https://quant.stackexchange.com/a/18013 An option is hedged by holding some stocks and risk-free bonds. ``` Option Price = a * stock + (1-a) * risk-free-bonds ``` where a is the position of the stock needed for hedging in the self-finance portfolio. Therefore, the option price depends on price of the zero coupon bond. Now, if we have two options, everything the same but time-to-maturity, let's say T1, T2, T2 > T1. The T2 option needs to be hedged with a T2 zero-coupon-bond. However, the T2 bond costs less than the T1 bond because everyone prefers a bond that pays back the money earlier. Therefore, the price of the option must go up for the T2 option. The relationship can be explained by how Black-Scholes is derived in the first-place (delta continuous hedging). ## Answer by Ulysses (score 0) https://quant.stackexchange.com/a/18042 First of all, IV does not always decrease with maturity. Even with focus on the ATM IV volatility, the term structure can be very different in different circumstances: it can go up, down or be rather bumpy. Nevertheless, in the BS formula the option price is a function of $\sigma(\tau)\sqrt \tau$, so even if $\sigma(\tau)\downarrow$ it may still happen that $\sigma(\tau)\sqrt \tau \uparrow $, otherwise clearly you'll get a negative calendar spread which would be quickly arbitraged by other market participants.
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