Replicating a Two-Asset State-Contingent Option

Summary

The document presents a finite-state option replication problem with two stocks, each of which can take one of two future values. A traded option pays in the state where both stocks finish high, and the goal is to price and replicate a separate claim that pays when the first stock is high while the second is low. Interest rates are assumed to be zero, and both stocks start at the same stated value.

The suggested method enumerates the four joint states and combines cash, both stocks, and the traded option. Set the portfolio payoff equal to the target claim’s payoff in each state, then solve the resulting four equations for the four holdings. The answer asserts that this produces a unique solution, but it leaves the system unsolved and gives no resulting price or portfolio. The replication depends on the stated instruments spanning the target payoffs.

Key ideas

• Two binary-valued assets create four possible joint future states.
• A replicating portfolio can combine cash, each stock, and a traded contingent claim.
• Match the portfolio payoff to the target option payoff in every state to form a linear system.
• The document outlines the replication method but does not calculate the solution.

Tags

# Pricing options with two assets


# Pricing options with two assets












I'm studying for a test and am stuck on this practice question:

With interest rates equal to 0, two different stocks $S_1$ and $S_2$, both valued at \$1 today, can be worth \$2 or \$0.50 at some point in the future. If the option that pays \$1 when both $S_1 = S_2 = \$2$ is traded in the market and is worth \$0.125, calculate the price and replicating portfolio of the option that pays \$1 when $S_1 = \$2$ but $S_2 = \$0.5$. You may leave your answer in matricial form.

## Answer by gt6989b (score 0, accepted)

https://quant.stackexchange.com/a/10365

Hint The future world has 4 states: $(0.5,0.5), (2,0.5), (0.5,2), (2,2)$. You have 4 instruments - cash, each stock, and an option they are both \$2 which is traded. Take $x,y,z,w$ of each and match the portfolio to the price of the option in each market state.

You get 4 equations and 4 unknowns, solve, and supposedly you get a unique solution, which immediately yields the replicating portfolio.

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