European Option Pricing with a Nonflat Interest Rate Curve
Summary
The document addresses whether a flat rate must be constructed from a normal interest-rate term structure to use Black–Scholes for a European option. The answer explains that the relevant discounting rate for an option expiring at time T is the zero-coupon rate for that maturity. Under the T-forward measure, the zero-coupon bond serves as numeraire, and the model’s volatility is interpreted as the volatility of the underlying forward price. It also notes that Merton’s early option-pricing work allowed nonconstant interest rates.
A separate response emphasizes that a European option is determined by the underlying’s terminal distribution at expiry, so an expiry-specific volatility slice can be used in this framing. It contrasts American options, where early exercise makes prices dependent on conditions through time. The discussion is conceptual and does not derive a pricing formula or resolve the question’s proposed present-value equivalence calculation; the original meaning of investing in the term structure is left unclear.
Key ideas
- European option discounting uses the zero-coupon rate for the option’s maturity.
- Under the maturity-specific forward measure, the relevant volatility is that of the underlying forward price.
- A European option’s price in the described framework depends on the terminal distribution at expiry.
- American options can depend on market conditions over time because they may be exercised early.
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# Arbitrage-free calculation of flat term structure out of normal term structure for e.g. pricing european options # Arbitrage-free calculation of flat term structure out of normal term structure for e.g. pricing european options since e.g. the Black-Scholes model requires a constant interest rate (flat term structure) but the real world often has normal term structure, I was wondering if it is mathematically correct to - numerically calculate the interest rate r at which an investment in this pseudo-term-structure has the same present-value as investing in the present normal term structure - price the option using Black-Scholes with this pseudo-interest-rate. Is there something I'm missing or is it even mathematically correct? Greetings ## Answer by Antoine Conze (score 1, accepted) https://quant.stackexchange.com/a/43537 Black-Scholes does not really require a constant interest rate. For a european option with maturity $T$ the only rate involved is the zero coupon rate for maturity $T$. The theory behind this comes from working under the $T$-forward measure (the risk neutral measure associated with the zero coupon bond as numeraire). The only subtelty is that the model volatility represents the volatility of the underlying forward price. In fact Merton's paper "Theory of rational option pricing", written around the same time as the BS paper (this is why people sometimes refer to the BS model as being the Black-Scholes-Merton model), did not assume that the interest rate is constant. But Merton's paper was published a few months after Black & Scholes paper so the idea that the rate should be constant stuck. ## Answer by Ezy (score 0) https://quant.stackexchange.com/a/43516 European options prices do not depend on the implied volatility term structure. To price a european option all that is needed is the terminal distribution of the spot at the expiry time $T$. So in that sense it is mathematically consistent to price european options using the implied vol slice at expiry $T$ even if the surface displays term structure. On the other hand this is not true for american options due to the possibility of early exercise which creates a dependency on the local volatility of the spot at various times. That being said it is unclear what you mean by « investing in this term structure » could you please clarify ?
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