Finding the Initial Replicating Portfolio in a CRR Model

Summary

The discussion clarifies two meanings of an option’s initial hedging investment in a Cox-Ross-Rubinstein binomial model. The portfolio’s initial value is the option’s fair price, expressed as the stock position times the initial stock price plus the bond position. If the desired quantities are the stock and bond holdings themselves, that value equation alone is insufficient to determine both positions.

The answer supplies the missing conditions by matching the portfolio’s value at the next time step in both possible states. The up-state equation uses the increased stock price and grown bond value; the down-state equation uses the decreased stock price and grown bond value. Solving these two equations gives the replicating holdings, which should also satisfy the initial-value equation. The explanation is limited to a one-step hedge setup within the stated binomial model; it does not work through numerical holdings or discuss how to extend the calculation across the full tree.

Key ideas

• The initial value of a replicating portfolio equals the fair value of the option.
• The initial-value equation alone cannot determine both stock and bond holdings.
• Match the portfolio value to the option value in both possible next-step states.
• The two state-contingent equations determine the initial stock and bond positions.
• The resulting holdings must reproduce the option’s initial price.

Tags

# In a CRR model, find the Initial investment of the hedging strategy


# In a CRR model, find the Initial investment of the hedging strategy












> Given a Cox-Ross-Rubinstein model with $T=10$, $u=1.1$, $d=0.9$, $r=0.02$, $S_0=100$ and a European call option with Strike $K=220$, find the initial investment of the hedging strategy.

I know how to compute the fair price at $t=0$ for the European call, say it is $x$, and I know that for the initial investment $(\alpha_0,\beta_0)$ of the Hedging strategy where $\alpha_0$ is the weight on the stock and $\beta_0$ on the bond, we have $x=100\alpha_0+\beta_0$. But there needs to be another equation such that I can solve it.

## Answer by Rylan (score 0, accepted)

https://quant.stackexchange.com/a/76629

I personally think the phrase "initial investment" is a bit ambiguous, and I would have assumed it meant "[currency] value of the hedging portfolio", which you correctly note is equal to the fair value of the option.

Another possibility, which it seems like is the one you're looking for, is finding $(\alpha_0, \beta_0)$ themselves. In finding the price of the option at time 0, you are effectively finding these also. In particular, you can explicitly solve: $$100u\alpha_0 + (1 + r) \beta_0 = \text{Value of the option at t=1 if the stock goes up}$$ $$100d\alpha_0 + (1 + r) \beta_0 = \text{Value of the option at t=1 if the stock goes down}$$

And note that there should be no inconsistency with the equation you have provided, $100 \alpha_0 + \beta_0 = x$.

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