Replicating a State-Contingent Option Payoff with Stock and Bonds
Summary
The document explains how to price a derivative by matching its payoffs across possible future states. In a three-state example, a stock, a risk-free bond, and a traded option provide three payoff patterns. The unknown option is replicated by solving one linear equation per state, then applying the law of one price: its value must equal the cost of the replicating holdings.
The worked example gives a portfolio of one share, a short bond position, and a short position in the traded option, with a resulting price of $0.48. This illustrates payoff-matrix replication in a finite-state model. The result depends on the stated asset prices, interest rate, and traded-option price, and assumes the listed instruments span the target payoff and can be traded without frictions. The example's calculation should be checked carefully: its stated portfolio holdings do not appear to reproduce the target payoff in all three states as written.
Key ideas
- Replication prices a derivative by matching its payoff in every modeled state.
- A payoff matrix turns the replication problem into a system of linear equations.
- The law of one price equates the derivative's value with the cost of its replicating portfolio.
- The method requires enough traded instruments to span the target payoff and relies on consistent inputs.
Tags
Full text
# How to replicate this option?
# How to replicate this option?
I have a question I am not sure how to approach: Suppose interest rates is 50%, a stock worth \$1 today can be worth \$2, \$1, \$0.5 next year.
If the option that pays \$1 only when S = \$2 is traded in the market and is worth \$0.125, calculate the price and replicating portfolio of the option that pays \$0.5 when S = \$1.
It has something to do with pay-off matrix but I don't know how to apply it?
## Answer by SRKX (score 5, accepted)
https://quant.stackexchange.com/a/10264
If I understand well, you have a market with 3 states: up, flat or down.
You have 3 instruments:
- The stock
- The risk-free rate (50%)
- The option
If you can create a portfolio today with these 3 instruments that can replicate de payoff of the option you have to price, then the law of one price tells you that the price of the option should be the price of this portfolio.
So, you have to solve a system of 3 equations with 3 unknown:
Assume you will hold $a$ stock, $b$ risk-free bond, $c$ option, and using fraction notation for 0.5, 1.5, the system is:
Up State: $a \cdot 2 + b \cdot \frac{3}{2} + c \cdot 1=0$
Flat State: $a \cdot 1 + b \cdot \frac{3}{2} + c \cdot 0=\frac{1}{2}$
Down State: $a \cdot \frac{1}{2} + b \cdot \frac{3}{2} + c \cdot 0=0$
From the third equation, you get $a=-3b$.
Substituting into the second, you get $-3b+\frac{3}{2}b = \frac{1}{2}$ which yield $b=-\frac{1}{3}$ and hence $a=-3 \cdot \left( - \frac{1}{3} \right)=1$.
Finally, substituting in the first equation, you get $1 \cdot 2 + \left( - \frac{1}{3} \right) \cdot \frac{3}{2} + 1 \cdot c =0$.
This yields $c= - \frac{3}{2}$.
So the portfolio holding 1 stock, selling $\frac{1}{3}$ risk-free bond and selling $\frac{3}{2}$ options replicates perfectly the payoff of the option you have to price. The price of this option is hence $1 \cdot 1\$ - \frac{1}{3} \cdot 1\$ - \frac{3}{2} \cdot 0.125\$=0.48\$$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.