How Interest Rates Affect Black–Scholes Put Prices Across Expiries

Summary

The document explains why a European put can become cheaper as time to expiry increases in a Black–Scholes example. With the stock and strike equal, volatility and the risk-free rate held constant, the reported put price falls as expiry extends from one year to much longer horizons. The response attributes this behavior to the interest-rate component of the model: a higher rate raises the call's value while lowering the put's value.

Put-call parity offers the intuition. A long call combined with a short put creates synthetic long stock exposure, whose financing cost is linked to the risk-free rate. The example illustrates that longer maturity alone does not guarantee a higher put premium. It is a brief conceptual explanation, however, and does not derive the pricing formula or examine how dividends, changing rates, volatility assumptions, or American exercise rights affect the result.

Key ideas

• In the stated Black–Scholes example, the put price declines as expiry lengthens.
• The response identifies the risk-free rate component as the main explanation.
• Put-call parity links a call and put to synthetic long stock exposure.
• A longer expiry does not by itself ensure a more valuable put.

Tags

# Why does black scholes model give lower prices for puts with further time to expiry?


# Why does black scholes model give lower prices for puts with further time to expiry?












Consider BS-model with parameters: Stock = 100, Strike = 100, Texp = 1 year, Vol = 13%, Rf Rate = 3%. For these parameters the BS put price is 3.76. Then consider the same parameters but with Texp = 20 years. The new BS put price is 3.29. Consider now Texp = 100 years. The put price is even lower still at 0.09.

What gives? Does it make sense that longer expiry options can be valued less?

## Answer by JoshK (score 3)

https://quant.stackexchange.com/a/57905

It's the interest rate component. That is more meaningful in the formula. Note that the call becomes more expensive.

Think about it this way. You could buy the call and sell the put instead of being long the stock. This gives you a synthetic long position. You need to pay the market the cost of borrow (r). That makes the calls more expensive and the puts cheaper.

In a twisted way you could say that the market expects the price of the stock to rise by the risk free rate.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.