Option Replication and Completeness When Arbitrage Is Possible

Summary

This note examines whether a call option can be replicated in a one-period binomial market that admits arbitrage. It proposes a bond-and-stock portfolio and gives stock-price outcomes and a range of strikes and hedge positions for which the calculated initial call price is positive. The central question is whether such a portfolio establishes completeness when arbitrage is present.

The example is useful for distinguishing replication from arbitrage-free pricing. A positive value and matching terminal payoffs may describe a replicating portfolio, but arbitrage-free assumptions are needed for the usual unique, economically consistent price and fundamental theorem linking completeness to equivalent martingale measures. The note poses this distinction rather than resolving it, and its stated calculations are not a general result. Any conclusion depends on checking the portfolio's payoffs in both states and the market assumptions; an in-the-money call alone does not demonstrate that an investor always gains.

Key ideas

• A one-period stock-and-bond portfolio can be assessed by matching its payoffs to a call in each state.
• A positive replication cost does not by itself show that a market is arbitrage-free.
• The standard link between completeness and unique no-arbitrage pricing relies on no-arbitrage assumptions.
• The numerical example raises a question but does not establish a general condition for replication.

Tags

# Binomial Model - completeness in presence of arbitrage


# Binomial Model - completeness in presence of arbitrage












Consider a uniperiodal binomial model where I buy one bond of value $B_0$ and rate $r=0.1$, and $h$ stocks with price $S_0=5$. The value of the portfolio at time $t=0$ is

$$ V_0 = B_0 + hS_0, $$

that should be equal to the price $p$ of a call option with strike price $K$, given that the binomial model is complete. If I select the price of the stock at time $t=1$ to be $S_1=6$ or $8$, the market is not arbitrage-free, but I could determine that for $0<K<5.5$ there exists a price $p>0$, corresponding to $-0.1<h<-0.1+0.2K$, and $B_0=6$. In other words, the call option is replicable (meaning that $p>0$, right?) if $K<5.5$ in a market where arbitrage is possible.

Physically it seems acceptable for me. In a market where I always gain money, I just need to buy a call option always in-the-money. Is it reasonable?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.