Replicating a One-Period Call Option with Stock and a Bond
Summary
The document explains how to price a call option in a one-period binomial model by constructing a portfolio of the underlying stock and a bond that matches the option’s payoff in both possible states. The stock starts at one, moves up or down by five percent, the strike is one, and the interest rate is zero. The call therefore pays five cents in the up state and nothing in the down state.
Solving the two payoff equations gives a position of one-half share and a short position of 0.475 bonds. The initial cost of that replicating portfolio is 0.025, which is the option premium under the model’s no-arbitrage pricing logic. The explanation offers a direct check on the algebra in the original question. Its conclusion depends on the stated single-period, two-state setup and zero interest rate; it does not address transaction costs or more complex price dynamics.
Key ideas
- A one-period call’s payoffs are determined separately for each possible terminal stock price.
- The replicating portfolio combines the underlying stock with a risk-free bond.
- Matching the payoff in both states gives one-half share and a short position of 0.475 bonds.
- With a zero interest rate, the replicating portfolio costs 0.025 initially, so that is the model call price.
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# Option Pricing Formula
# Option Pricing Formula
I am really upset about an exercise, because i don't get it. Maybe someone can find my error?
We are talking about a single period model. Therefore our Asset is denoted by $S(0)=1$, $U=0.05$, $D=-0.05$, $R=0$ and $K=1$. Question seems easy: How much is the premium for the option $C(0)$?
I know: $C(0)=x_{C}S(0)+y_{C}A(0)$
where $x_{C} = \frac{S(0)(1+U)-K}{S(0)(U-D)}$, $y_{C}=\frac{(1+D)(S(0)(1+U)-K)}{A(0)(U-D)(1+R)}$
from solving the linear system $\begin{Vmatrix} xS(0)(1+U)+yA(0)(1+R)=S(0)(1+U)-K\\ xS(0)(1+D)+yA(0)(1+R)=0 \end{Vmatrix}$.
Provided solution should be $C(0)=0.025$, but i have no idea how, because by simply substituting given values the solution isn't correct. Are there any errors before?
## Answer by Jan Stuller (score 6, accepted)
https://quant.stackexchange.com/a/54740
The Stock price after the single period can be 1.05 or 0.95. If the stock ends up at 1.05, the option pay-off is 0.05, if the stock price ends up at 0.95 the option pay-off is zero. We want to figure out the price of the option by replicating it with the underlying Stock and a Bond (rates are zero, so the bond price at time zero is 1 and after the single period it is also 1). We wanna solve the following set of equations ($x$ is the number of stocks and $y$ is the number of bonds you hold to replicate the option pay-off at maturity):
$$x*0.95 +y = 0$$
$$x*1.05 + y = 0.05$$
Subtrack the first equation from the second and you get $x*0.1 = 0.05$, therefore $x=0.5$. You can now sustitute this into the second equation to get:
$$0.525 + y = 0.05$$
This solves to $y=-0.475$, therefore at maturity, if you are long 0.5 units of the Stock and short 0.475 units of the Bond, you replicate the option pay-off in both states.
Rates are zero so the option price at initial time is just 0.5 times the stock price - 0.475 * the bond price = 0.025. That's your answer.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.