For tenors > 100 years (i.e. 150 years), my Black-Scholes model is telling me the price of a call option will be equal to its current stock price. Can anyone intuitively explain this?
Thank you
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Sign up to join this communityFor tenors > 100 years (i.e. 150 years), my Black-Scholes model is telling me the price of a call option will be equal to its current stock price. Can anyone intuitively explain this?
Thank you
You can think in terms of the Black-Scholes stock price dynamics:
Assuming a positive drift such as the conditions mentionned by LocalVolatility in its comment are satisfied, you can see that the underlying spot price will tend to infinity in an infinite amount of time.
Therefore, the spot price will be sufficiently large ($S>>K$) to assume that your payoff $(S-K)^+$ is roughly equal to the underlying spot.
You then see that the only way to replicate this call is to buy the underlying, which has the same payoff as your call when the time to maturity goes to infinity.
By no arbitrage, they must therefore have the same price: hence your call worthing the spot price $S$.
A European call price respectively admits $\max(0, P(0,T)(F(0,T)-K))$ and $P(0,T) F(0,T)$ for lower and upper bounds, where $P(0,T) = e^{-rT}$ is the price of the zero coupon bond expiring at $T$ and $F(0,T)$ the equity forward price.
Only if $\lim_{T \to \infty} F(0,T) \gg K$ will you have the result you mention since in that case both bounds coincide to $P(0,T) F(0,T)$. If the stock does not pay dividends $P(0,T)F(0,T) = S_0$.