# Sufficient conditions for no static arbitrage

In Carr and Madan (2005), the authors give sufficient conditions for a set of call prices to arise as integrals of a risk-neutral probability distribution (See Breeden and Litzenberger (1978)), and therefore be free of static arbitrage (via the Fundamental Theorem of Asset Pricing)

These conditions are:

In the case that we have a full range of call prices:

• $C(K)$ is monotically decreasing
• $C(K)$ is convex

Or if $C(K)$ is twice differentiable:

• $$C'(K) \leq 0 \tag1$$
• $$C''(K) \geq 0\tag2$$

Carr and Madan do not mention the following constraints, thought they may be implied (?):

• $$C(K) \geq 0\tag3$$
• $C(0)$ is equal to the discounted spot price $\tag 4$

Other authors do mention the constraints (1-4) together. For example Fengler and Hin (2012) call these the "standard representation of no-arbitrage constraints"

In Reiswich (2010), the author presents the following condition:

• $$\frac{\partial P}{\partial K} \geq 0\tag{5a}$$

Or equivalently, via Put-Call Parity:

• $$\frac{\partial C}{\partial K} \geq \frac{C(K) - e^{-r\tau}S}{K}\tag{5b}$$

Reiswich claims that (5) is stricter than what is implied by (1-4) (i.e. there are sets of call prices which satisfy (1-4) but not (5)). Is this really true? If so, how do we reconcile this with Carr and Madan's claim of sufficiency?

Edit: Alternately, if (5) must hold is a no-arbitrage setting, and if (1-4) are sufficient, then how do we derive (5) from (1-4)?

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I think that you are missing one key condition on the call prices that I would say is standard, namely that the call prices should be bounded below by an "intrinsic" value. Specifically, we would expect $C(K) \ge (S-e^{-rT}K)_+$, and this can easily be seen to yield a static arbitrage if violated. This condition (in a slightly different form) can be found for example in the paper of Davis and Hobson which is very relevant to this question.
So why is this enough? Let's suppose that we work in the classical risky asset case (the paper by Reiswich is in FX markets, so your equation (5b) has an interest rate which is the foreign rate, lets take this to be zero), so in fact we want to show $$\frac{\partial C}{\partial K} \ge \frac{C(K)-S}{K}.$$ But now think of the call price as a function of $K$, we know this is a convex function, which lies above the line $(S-e^{-rT}K)$, and if we take a tangent at $K$, this is a line with gradient $\frac{\partial C}{\partial K}$ passing through the point $(K,C(K))$. At zero, this line (by the convexity of the call price function) must lie below the call price curve, but the line passes the $y$-axis at the point $C(K)-K \frac{\partial C}{\partial K}$, and this must be less than $S$. Rearranging gives the inequality of Reiswich.