Arbitrage Bounds on Call Prices Across Strikes and Maturities
Summary
The document explains how arbitrage arguments imply shape constraints on call option prices. For a fixed maturity, a call price must not rise as the strike increases. It must also be convex in strike: otherwise, a butterfly position can produce an upfront credit and a nonnegative payoff, positive in some states. The argument compares calls at neighboring strikes to show why violating convexity creates an arbitrage.
It also discusses maturity ordering for American calls. If a longer-dated call were cheaper than a shorter-dated call with the same strike, buying the longer option and selling the shorter one would create an initial credit; early exercise of the short option can be matched by exercising the longer option. The maturity argument depends on American exercise rights. The document does not fully distinguish strict from weak convexity conditions, and the arbitrage reasoning assumes prices and exercise rights can be acted on as described.
Key ideas
- A call price cannot increase when its strike increases without creating an arbitrage opportunity.
- Call prices must be convex across strikes, as violations can be traded using a butterfly spread.
- For American calls with the same strike, longer maturity should not be priced below shorter maturity under the argument presented.
- The maturity comparison relies on the ability to exercise American options before expiration.
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Full text
# Qualitative properties of call
# Qualitative properties of call
I have read somewhere that we can show by using arbitrage argument the following relationship for call option :
$$\frac{\partial{C_t(T,K)}}{\partial{K}}\leq0$$
$$\frac{\partial^2{C_t(T,K)}}{\partial{K^2}} \geq0$$
$$\frac{\partial{C_t(T,K)}}{\partial{T}} \geq0$$
where $C_t(T,K)$ is a call price at time t, for a strike K, with maturity T, obviously we can show these relation by taking the derivation for example with respect to the maturity for the last relation, or for showing the second relation (i.e. the butterfly spread relation, we just need to notice that the payoff is a convex function the strike and then the second derivative is always positive or null,but I am looking for arbitrage argument to show these relations. If someone has any idea to show them I would be grateful.
## Answer by M. Jeunesse (score 7)
https://quant.stackexchange.com/a/25476
$$\begin{array}{rcl} (1) & \partial_KC_t(T,K) & \leq 0 \\ (2) & \partial^2_KKC_t(T,K) & > 0 \\ (3) & \partial_T C_t(T,K) & \geq 0 \\ \end{array}$$
If $(1)$ doesnot hold, it exists $K_1<K_2$ such that $C_t(T,K_1)<C_t(T,K_2)$. Then as barrycarter said in his comment, you sell $C_t(T,K_2)$ and you buy $C_t(T,K_1)$, so your cash position is $C_t(T,K_2)-C_t(T,K_1)>0$, at maturity you receive $(S_T-K_1)^+-(S_T-K_2)^+\geq 0$. There is an arbitrage.
If $(2)$ doesnot hold, it exists $\epsilon>0$ and $K>\epsilon$ such that $C_t(T,K-\epsilon)+C_t(T,K+\epsilon)\leq 2 C_t(T,K)$. Then you buy $C_t(T,K-\epsilon)$ and $C_t(T,K+\epsilon)$ and you sell $2C_t(T,K)$, your cash position is $2 C_t(T,K) - C_t(T,K-\epsilon)+C_t(T,K+\epsilon)\geq 0$. at maturity you get $(S_T-(K+\epsilon))^++(S_T-(K-\epsilon))^+-2(S_T-K)^+\geq 0$ which is the butterfly spread you mention. Note that with non-null probability, this payoff is positive. There is again an arbitrage.
Assuming you talk about american call options. If $(3)$ doesnot hold, it exists $T_1<T_2$ such that $C_t(T_1,K)>C_t(T_2,K)$. Then you buy $C_t(T_2,K)$, you sell $C_t(T_1,K)$, your cash position is $C_t(T_1,K)-C_t(T_2,K)>0$. At any time $\tau\leq T_1$, the buyer of $C_t(T_1,K)$ can exercise its right, and then you owe him $(S_\tau-K)^+$, but since you buy an american option $C_t(T_2,K)$, you can also exercise your right at time $\tau$ and you net position. Again since you build a positive cash position leading to a non-negative positive (here equal to $0$), you build an arbitrage.
Note that this third relationship will hold with any american options. And note that the two first relationships will also work with american calls by adapting the $T$ to $\tau$ being the exercise time of the agent who buys call options.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.