There is a certain asymmetry in (European) call/put prices, even with negligible interest and dividend rates. For example, assume current price $100, interest rate 0.01%, dividend 0, volatility 25%, expiration 3 months. According to BS model, 110 Call is $1.68 while 90 Put is $1.31. Why are they different? Cost of carry is negligible, ΔS/S is the same in both cases. I suspect it has to do with σ2t term and possibly lognormality of returns, but looking for more intuitive explanation ("explain to me like I am 6").
DISCLAIMER: Any explanation by me WILL sound like I am 6 years old :->)
Here's a passage from Natenberg's book : Option Volatility and Pricing Strategies:
The lognormal distribution of prices assumed in the Black-Scholes model helps explain why options with higher exercise prices seem to carry more value than options with lower exercise prices. For example, suppose a certain commodity is at exactly 100. If there are no interest considerations and we assume a normal distribution of possible commodity prices, then the 105 call and the 95 put, being equally far out of the money, should have identical values. In fact, under the assumptions of the Black-Scholes model the 105 call will always have a greater theoretical value. The lognormal distribution allows for greater upside price movement than downside price movement in the underlying commodity. Consequently the 105 call will have a greater possibility of price appreciation than the 95 put. Of course, if one were to assume a normal distribution of prices, then both options would indeed have the same value.
The premium/price of a put option should be different than that of a call option.
Relative put and call prices differ by the riskless rate of interest. So when interest rates rise, a call's price rises and a put's price decreases. This is explained by the put-call parity.
“When European options are at the money and the stock pays no dividends call prices exceed put by the riskless rate of interest.”
So when the interest rate is small and unchanged but volatility rises, the difference in premiums is expanded under the Black- Scholes model. Similarly if time to expiry changes the effect also causes a rise in premium prices.
Just a note if both assets are equivalent in every way then the call may be overpriced or the put is underpriced