Why Linear Volatility Interpolation Can Create Calendar Arbitrage
Summary
The document explains a risk in interpolating implied volatility linearly across option expiries. The key no-arbitrage condition is that total implied variance, defined as maturity times implied volatility squared at a given strike, should not decrease with maturity. Implied volatility itself may decline across expiries in an arbitrage-free market, so a straight-line interpolation of volatility does not automatically preserve the condition on total variance.
The response writes interpolated volatility as a linear function of time and observes that its slope can be negative. Substituting that function into total variance and differentiating with respect to maturity produces a polynomial expression that can become negative, violating the calendar-arbitrage condition. A second answer characterizes the issue as negative forward variance and refers to a calculus-based argument. The discussion establishes a possible failure mode, but does not provide a numerical example or recommend a specific alternative interpolation method.
Key ideas
- Total implied variance at a fixed strike should be nondecreasing with expiry to avoid calendar arbitrage.
- Implied volatility can fall with maturity even when the surface is arbitrage-free.
- Linear interpolation of implied volatility can make the maturity derivative of total variance negative.
- Negative forward variance is the interpretation of this failure in forward-volatility terms.
- The discussion identifies a risk but does not prescribe a replacement interpolation method.
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Full text
# Why linear interpolation not appropriate for volatility surface construction?
# Why linear interpolation not appropriate for volatility surface construction?
We know linear interpolation is not appropriate for constructing a surface, but why?
In the book, "Foreign Exchange Option Pricing: A Practitioners Guide", the author writes:
> native linear interpolation with regard to time can lead to unrealistic forward volatility dynamics... this implies a negative forward variance between ...
I am not sure I understand the reasoning. Why does linear interpolation imply negative forward volatility ? Can anyone provide a better explanation? Is there any other reason that the simple linear interpolation should not be used?
## Answer by AFK (score 12, accepted)
https://quant.stackexchange.com/a/20780
Note that total implied variance defined as $$ V(T,K) = T\Sigma(T,K)^2 $$ should be an increasing function of $T$. Otherwise you have a calendar arbitrage (sell the call with shorter expiry and buy the cheap longer one).
If you interpolate linearly your implied volatility is $$ \Sigma(T,K) = w\Sigma(T_i,K) + (1-w)\Sigma(T_{i+1},K) $$ with weight $w = \frac{T_{i+1}-T}{T_{i+1}-T_i}$. This can also be written as $$ \Sigma(T,K) = \Sigma(T_i,K) + s(T-T_i) $$ with slope $s = (\Sigma(T_{i+1},K)-\Sigma(T_{i},K))/(T_{i+1}-T_i)$. Note that $s$ can be negative, i.e. $\Sigma(T_{i+1},K) < \Sigma(T_{i},K)$ even in an arbitrage-free situation: $V(T_{i+1},K) \ge V(T_{i},K)$.
Now all you have to do is check for calendar arbitrage: $$ \partial_T V(T,K) \geq 0 $$ A simple computation will show you that the lhs is a 2nd order polynomial in $T$ and that it can turn negative.
## Answer by Thomas Maloney (score 3)
https://quant.stackexchange.com/a/20758
It implies negative forward variance. I have the book, and went through the section following your quote. In math terms, he is making a proof by contradiction. He first assumes that you can interpolate Iinearly, and comes to the conclusion that it is not a good assumption. The argument does involve some calculus. I don't think I have a better explanation, so let me know if you have any questions about his argument.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.