One-Step Binomial Replication and Mispriced Option Arbitrage
Summary
The document discusses the one-step binomial model used to price a call by constructing a portfolio of stock and option positions. It explains that the example’s portfolio is long 0.25 shares and short one call, and that reversing those positions gives the trade used when the call is priced below its no-arbitrage value. The example calculates the reversed portfolio’s initial value and shows equal stated profits across the up and down scenarios.
The answer also distinguishes the scenario payoff from interest earned by lending the portfolio’s value at the risk-free rate. It points out a possible mismatch in the original question: the payoff shown implies a strike of $21 rather than the assumed $20. The option’s maturity is not specified, so the example cannot establish whether the scenario profit exceeds the lending return. The answer briefly describes alternative hedged combinations but does not derive them in detail.
Key ideas
- A one-step binomial hedge combines stock and an option to create a riskless portfolio.
- The example’s portfolio is long 0.25 shares and short one call.
- Shorting that portfolio reverses both positions, producing a long call and short stock position.
- The answer calculates equal profits in the up and down scenarios for its stated inputs.
- The option maturity is needed to compare those profits with risk-free interest earned.
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Full text
# If an option is undervalued, how does shorting a portfolio generate profit?
# If an option is undervalued, how does shorting a portfolio generate profit?
I am reading Hull's Options book. He introduces a one-step binomial model and a no-arbitrage argument, using the example shown in the picture below:
> Consider a portfolio consisting of a long position in $\Delta$ shares of the stock and a short position in one call option (he does not explicitly give the strike, but he clearly assumes that it is \$20).
Question 1: Why do we short one call option? Why do we not long a call or short a put?
After some calculation, $\Delta = 0.25$, and if we assume risk-free rate is 4%, then the no-arbitrage price of the option is 0.545.
> If the value of the option were more than 0.545, the portfolio would cost less than 4.455 to set up and would earn more than the risk-free rate. If the value of the option were less than 0.545, shorting the portfolio would provide a way of borrowing money at less than the risk-free rate.
Question 2: could you explain how we make a profit if the price of the option is less than 0.545? When it is less than 0.545 and we short the portfolio, do we gain \$20 $\Delta$ from the share? What about the short position of the call?
## Answer by KaiSqDist (score 1, accepted)
https://quant.stackexchange.com/a/78796
Question 1: Why do we short one call option? Why do we not long a call or short a put?
You could do the other combinations, but then you would have to:
- Short Put > Short Stock
- Long Call > Short Stock
- Long Put > Long Stock
To delta hedge the portfolio and think about the individual results in terms of the riskless rate like in your second quoted paragraph.
Question 2: could you explain how we make a profit if the price of the option is less than 0.545? When it is less than 0.545 and we short the portfolio, do we gain $20 Δ from the share? What about the short position of the call?
The "portfolio" you are talking about is a short single call and long 0.25 units of the stock. Therefore, if you short the portfolio, it becomes long a single call and short 0.25 units of the stock. If the option is less than 0.545 (say 0.5 instead of 0.545), this portfolio is worth:
\begin{equation} \Pi = 0.25*20 - 0.5 = 4.5 \\ Scenario\:U\:Profit = 4.5 - 0.25*(22-20) + 1 - 4.5 = 0.5 \\ Scenario\:D\:Profit = 4.5 - 0.25*(18-20) + 0 - 4.5 = 0.5 \end{equation}
Therefore, if you lend a portfolio at the riskless rate,
\begin{equation} Interest\:Earned = \Pi*\exp(r*\tau) - 4.5 = 4.5*exp(0.04*\tau) - 4.5 \end{equation}
I would bet that the $\tau$ of the option is such that $Interest\:Earned$ is less than \$0.5. From,
\begin{equation} min[Scenario\:U\:Profit, Scenario\:D\:Profit] = \\\$0.5 > Interest\:Earned \end{equation}
What is the $\tau$ of the option please? You need to specify.
Also, the strike of the option is not \$20, it seems to be \$21 as the payoff in $Scenario\:U$ is \$1 instead of \$2.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.