How Strike and Time Affect Black–Scholes Option Prices
Summary
The document asks whether Black–Scholes prices move monotonically with strike and time, after noting that a call price rises with spot, volatility, and the interest rate when other inputs are held fixed. The response uses arbitrage reasoning to explain the strike relationship: a call with a higher strike should not cost more than one with a lower strike, while puts move in the opposite direction. The option’s price derivative with respect to a parameter can also establish the direction of change.
For time, the response discusses theta rather than giving a full proof of price monotonicity. For a European call with nonnegative interest rates and zero dividend yield, theta is described as negative; put theta can have either sign, though it is commonly negative for traded options. These claims depend on the stated assumptions and do not imply that every option’s value falls as maturity increases under all market conditions. The document offers intuition and sign guidance, not a complete derivation.
Key ideas
- A European call’s price decreases as its strike rises, consistent with avoiding an arbitrage opportunity.
- A European put’s price moves in the opposite direction with strike.
- For a call with nonnegative rates and no dividend yield, the response states that theta is negative.
- Put theta can be positive or negative, depending on conditions.
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Full text
# Increasing or decreasing BS-formula respect their parameters # Increasing or decreasing BS-formula respect their parameters The Black-Scholes formula depends of many parameters, is easy to note that it is increasing respect to the parameter $S_t$, $\sigma$ and $r$, it means to fix a all parameters and vary only one. Is possible to say the same for the strike price $K$ or the time $t$, I think BS-formula is decreasing respect to these parameters, it is easy to see for $K$ at time T because the formula gives $(S_T-K)^+$ but not sure. If anyone has an explicit proof of this I really appreciate it. Thank you!. ## Answer by Rylan (score 1) https://quant.stackexchange.com/a/76670 In both cases, for European options, it's a matter of determining whether the first derivative of the call price w.r.t. the parameter is always positive or negative. That said, intuition and good financial arguments are always helpful and can often be more enlightening than using the math. In the case of the strike price, I like your argument -- to make it very slightly more formal, you can say that if Call A had a higher strike than Call B, but also cost more, there would be an arbitrage opportunity. (Again, you can inspect the derivative to see also.) Of course, using the same argument, we can see that the opposite holds for puts. For theta, I'll link this answer as well as this link giving the formulas for theta. Particularly, we can see by inspection that for non-negative interest rates (and for 0 dividend yield), theta for a call is always negative (in this case, in the formula we are adding two terms that are only ever negative), this is not the case for puts. Put theta can be positive or negative even with positive interest rates, but in practical terms, theta will be negative for most traded options.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.