No-Arbitrage Linearity and Consistency for European Option Prices
Summary
The document considers what no-arbitrage principles constrain the pricing equation for European options. It models an option price as a function of the underlying price, time to maturity, and terminal payoff. A portfolio argument shows that the price of a combined payoff must equal the sum of the prices of its components: otherwise, taking opposite positions would produce an initial profit with no net maturity payoff.
The response adds two required properties: scaling a payoff by a constant scales its price by the same amount, and valuation must be consistent across intermediate dates, so valuing a future option price now agrees with valuing its ultimate payoff directly. These are presented as general cash-flow conditions for avoiding arbitrage. The text offers reasoning rather than a formal proof under specified market assumptions, and does not derive a particular option-pricing equation or address market frictions.
Key ideas
- No-arbitrage implies that the price of a sum of payoffs equals the sum of their prices.
- If payoff prices are not additive, opposing positions can create an arbitrage under the stated setup.
- Scaling a payoff by a constant must scale its price by the same constant.
- Valuation across intermediate dates must agree with direct valuation of the final payoff.
- The stated principles constrain pricing but do not specify a particular option model.
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# An equation for European options
# 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.
## Answer by Richard Herron (score 3, accepted)
https://quant.stackexchange.com/a/912
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.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.