# An equation for European options

So, any European type option we can characterize with a payoff function $P(S)$ where $S$ is a price of an underlying at the maturity.

Let us consider some model $M$ such that within this model $V(S,\tau,P)$ is a price of an option with a payoff $P$ at time to maturity $\tau$ and an asset's price $S$ at time $\tau$. Using only non-arbitrage principles we obtain $$V(S,\tau,P_1+P_2) = V(S,\tau,P_1)+V(S,\tau,P_2).$$ With this formula we get one symmetry for the equation on the price of the option which holds regardless of the model $M$. Usually all equations are linear in payoff since they are linear themselves which comes from the fact that this equation are obtained using infinitesimal generators of the stochastic processes for the price.

Are there any other non-arbitrage principles which can make some additional restrictions on the equations for the price of the European option? I thought about using some facts on $V(S,\tau,P_1(P_2))$ and options on options.

Edited: the non-arbitrage argument for the linearity is the following. Suppose for some $S,\tau$ we have $V(S,\tau,P_1+P_2) > V(S,\tau,P_1)+V(S,\tau,P_2)$. Then we can short $V(...,P_1+P_2)$ and long $V(...,P_1)$ and $V(...,P_2)$ - so at the maturity we have nothing to pay, but at the current time the difference $$V(S,\tau,P_1+P_2) - V(S,\tau,P_1)-V(S,\tau,P_2)$$ is our profit.

• I think that in some case you can consider some non linearity over the price of the sum of pay-off. For example, super replication prices are not always the sum of super replication prices of individual payoffs. Regards. – TheBridge Apr 5 '11 at 7:01
• I edited my answer with a non-arbitrage argument for the linearity. I imply perfect liquidity of the market and trade costs negligible with respect to the difference in prices. – Ilya Apr 5 '11 at 8:06
• Thank's for completing your hypothesis, by the way I think that this subject is close to one approach to risk neutral valuation via so called "pricing rules" which is essentially a functionnal analysis of the subject. Some references to this include some papers by L.C.G. Rogers, and maybe also something by N.Touzi but I am not completly sure about those references. I can check them more thouroughly if you are interested. Regards – TheBridge Apr 6 '11 at 21:23

I think you provided the two that must be met. Pricing must be linear.$$V(\ldots, 2P) = 2 \cdot V(\ldots, P)$$ And pricing must meet the law of iterated value. Where $\tau \in (t, T)$ $$V_t(\ldots, \tau, V_{\tau}(\ldots, T, P)) = V_t(\ldots, T, P)$$ These two laws must be met for any cash flow to prevent arbitrage.
• @Gortaur -- Good call. That should be clearer. If you know that you will have a better valuation $V_{\tau}$ at time $\tau$, then make that valuation now at time $t$. – Richard Herron Apr 5 '11 at 16:52