# 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

1. 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

2. price the option using Black-Scholes with this pseudo-interest-rate.

Is there something I'm missing or is it even mathematically correct?

Greetings

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.

• Got it. Basically, this means that you just need the interest rate for time T, no matter what the kind the term structure of interest rates is. Futhermore, it seems even wrong trying to calculate the pseudo-term-structure stated in my question, since you invest in a zero coupon bond and not in a normal coupon bond. – Simon Jan 16 '19 at 15:20
• Exactly. Where the real term structure of coupon bonds (or for the interbank interest rate market deposits and swaps) is used is to bootstrap a zero coupon rate term structure that is then interpolated for the option maturity $T$. – Antoine Conze Jan 16 '19 at 15:27

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 ?

• In this question, I don't refer to the term structure of volatility, but to the term structure of interest rates (not sigma, but r). By "investing in this term structure" I mean buying a bond (where its cashflows are discretely discounted by the specific spot rates of the normal term structure at each time until maturity) vs. buying a bond (where its cashflows are discretely discounted by a constant interest rate, since the term structure is flat) – Simon Jan 16 '19 at 9:34