Replicating a European Call with a One-Period Stock and Cash Portfolio
Summary
The document presents a one-period binomial example for pricing a European call and constructing a replicating portfolio. With the underlying ending in one of two states, the option's state-contingent payoffs determine the stock holding that, together with the short option position, removes uncertainty in the combined portfolio. The example connects this hedge to risk-neutral valuation: when a portfolio exactly replicates the option payoff at maturity, its current value must equal the discounted risk-neutral expectation of that payoff.
The question arises from confusing the value of the replicating portfolio with the value of a zero-cost arbitrage strategy. A replicating portfolio need not have zero initial value; its cost is the option price under no-arbitrage. The worked figures illustrate this equality, while the discussion raises rather than resolves the user's confusion. The setup is a simplified single-period model with specified states and a constant risk-free rate, so it does not address continuous rebalancing, transaction costs, or real-market parameter estimation.
Key ideas
- A stock and cash position can replicate an option when their terminal payoffs match in every modeled state.
- The hedge ratio is determined by the change in option payoff relative to the change in the underlying price.
- No-arbitrage pricing equates the cost of replication with the discounted risk-neutral expected payoff.
- A replicating portfolio can have nonzero initial value; replication does not mean the strategy starts at zero cost.
- The example assumes a one-period binomial setting and a constant risk-free rate.
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# Replicating portfolio
# Replicating portfolio
I have a doubt about the replicating portfolio methodology.
Example - Consider an European Call with $K=21$ and underlying with current price $S_0=20$. We assume that, at the maturity, the underlying can be $22$ with probability $p$ or $18$ with probability $1-p$. Let $r=0,12$ be the risk-free rate.
The option price is $f=e^{-rT}[pf_u+(1-p)f_d]=0,633$ for $p:=\frac{e^{rT}-d}{u-d}$. So the seller has adopted a long position on $\Delta=\frac{f_u-f_d}{S_0u-S_0d}=0,25$ stocks in the face of short position on the option. It follow a risk-free portfolio because we have the same payoff irrespective of the future state of world.
In alternative we can say that, for the absence of arbitrages and the $\mathbb{Q}$-martingality guaranteed by the First Theorem of APT, the present value of portfolio is
$V_0(\Theta)=\mathbb{E}^{Q}[e^{-\int_0^Tr_Sds}V_T(\Theta)|\Im_0]\overset{\operatorname{for}r\operatorname{constant}}{\overbrace{=}}e^{-rT}\mathbb{E}^{Q}[V_T(\Theta)]=4,367$
which will coincide to the current value of portfolio less than the price of no-arbitrage for the option, that is
$4,367=20\cdot 0,25-f\Rightarrow f=0,633$.
Now my doubt.
Why does the Risk-Neutral Evaluation Theorem say that $V_0(\Theta)=e^{-rT}\mathbb{E}^{\mathbb{Q}}[\varphi(S_T)]$?
I know that $\varphi(S_0)=e^{-rT}\mathbb{E}^{\mathbb{Q}}[p\varphi(S_T)^++(1-p)\varphi(S_T)^-]$ and that $V_0(\Theta)-\varphi(S_0)=e^{-rT}\mathbb{E}^{\mathbb{Q}}[V_T(\Theta)]$, but if $\Theta$ is not an arbitrage we have $V_0(\Theta)\neq 0$. I thought that if the portfolio is replicating we have $\varphi(S_T)=V_T(\Theta)$, so maybe I can write that $V_0(\Theta)=e^{-rT}\mathbb{E}^{\mathbb{Q}}[\overset{=0}{\overbrace{\varphi(S_T)+V_T(\Theta)}}]=0$ but I'm not sure.
Could you please clarify me these formulas? Thanks you all in advance.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.