Calendar Arbitrage Constraints at Fixed Absolute Strikes
Summary
The document asks how calendar no-arbitrage constraints change when options have the same absolute strike across expiries rather than the same forward moneyness. For equal forward moneyness, it gives the normalized call-price inequality that the longer-dated option must satisfy. With nonzero interest rates or proportional dividends, equal strikes generally do not preserve that moneyness relationship, so the simple inequality does not directly apply.
The response points to the calendar-spread condition associated with the local volatility formula, involving the call's time and strike sensitivities and the rate and dividend inputs. It says that without a simplifying fixed-moneyness assumption, calendar and strike arbitrage constraints must be considered together. A small set of call prices illustrates that monotonicity and convexity at one maturity alone do not yield an equally neat rule for prices at another maturity. The discussion is qualitative and does not derive a closed-form fixed-strike bound or work through the illustrative grid numerically.
Key ideas
- Equal forward moneyness across maturities gives a simple normalized call-price calendar condition.
- Equal absolute strikes do not generally imply equal forward moneyness when rates or dividends are nonzero.
- The general calendar condition involves time and strike derivatives of call prices.
- Fixed-strike grids require calendar and strike arbitrage constraints to be assessed together.
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Full text
# Calendar Arbitrage Constraint Using Fixed Strikes
# Calendar Arbitrage Constraint Using Fixed Strikes
Gatheral & Jacquier (2024) derive the calendar no-arbitrage condition for two call options $C(K_1,T_1)$ and $C(K_2,T_2)$ with different maturities ($T_2>T_1$) but the same forward-moneyness (i.e. $\frac{K_2}{F_2}=\frac{K_1}{F_1}$), given as:
$ \frac{C_2}{K_2}\geq\frac{C_1}{K_1} $
I'm having difficulty deriving an equivalent constraint if, rather than assuming the strikes share a forward-moneyness, we instead assume the strikes are equivalent (i.e. $K_2=K_1$). Obviously, in the realm of zero interest rates and dividends, equivalent strikes are also equivalent in forward-moneyness, so Gatheral's condition above holds. But, what about if rates and dividends are non-zero? For ease, we can assume a proportional, continuous dividend.
Any thoughts would be helpful!
## Answer by Andrea (score 1)
https://quant.stackexchange.com/a/82052
The general calendar spread condition is just the numerator of the local volatility formula
$C'_T+(r-q) \, K C'_K + q \, C \ge 0$
Unless you can simplify it (fixed forward moneyness), you will have to tackle calendar and strike arbitrage at the same time.
Now, working on a fixed-absolute-strike rectangular grid is equivalent to a non rectangular moneyness-strike grid.
Example, look at this situation (with $r=q=0$):
- $C(T_1, K=90)=30$
- $C(T_1, K=100)=24$
- $C(T_1, K=110)=20$
(all good, decreasing and convex).
Question for you is: what can you tell me about $C(T_2, K=95)$ and $C(T_2, K=105)$? (for a $T_2 > T_1$)
Plot them! You will see there are some constraints, but they will not be as pretty as the rectangular case.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.