Replicating Option Payoffs with Tradable Assets in Complete Markets

# Replicating portfolio - must the underlying asset be used?


# Replicating portfolio - must the underlying asset be used?












I am reading through famous sources like Hull, Wilmott, et.c., where they construct the replicating portfolio in a one-step binomial model. This involves setting up a system of equations, where we have two unknowns, ∆ and B (shares of underlying stock, and risk-free money), corresponding to equations. The no-arbitrage principles allows us to equate the option and portfolio future states, assuming they produce identical cash flows.

What I kind of intuitively understand but would want a more formal explanation of is why we assume that ∆ must be the underlying asset as opposed to some other asset? Intuitively, it makes sense that we need some correlation between the stock and the option, we can't just have ANY random stock with no relation to the option. But is there some formal assumption that this system of equations makes use of?

I have read something about complete markets, where all derivatives can be replicated using other assets, but I haven't found a definite statement about it having to be the underlying.

So we assume the portfolio cash flows is identical to the option. And this must be a portfolio with the underlying stock. But "WHY"? Or is there some assumption that only these three securities exist in our market?

Thanks

## Answer by lehalle (score 2)

https://quant.stackexchange.com/a/83875

Let's take the example of a call option to give you easily the intuition: you are an investment bank, you sell it to a client at $t=0$. In 3 months your client will be able to call (from you) a share of the underlying stock at a strike of \$100 (she will do it if its spot price in the future $S_{3m}$ is greater than 100).

For you, the investment bank, the goal is to replicate the payoff. It means that you want to build a portfolio at $t=0$ (now) and adjust it (if possible in real time, provided you neglect transaction costs and market impact) such that in 3 months you will have exactly what you need:

- nothing if the spot price is below the strike

- one share to deliver, or anything you can convert at no cost in this share, if $S_{3mo}>100$.

If you can compute (or estimate) the expected cost of this replication process seen from $t=0$, this is the price (i.e. the replication cost) of the option, and you ask this premium upfront to your client when she signs the contract.

I guess the intuition is there: you can put whatever you want in your replicating portfolio as long as it can allow you to deliver (or to buy one share at $S_{3mo}$, whatever it will be) in 3 months.

The idea of a complete market is a way to say that you can find the tradable instruments to put in your replicating portfolio the enable this delivery at maturity.

(for more detail, have a look at C-A L and A Raboun, 2022. Financial Markets in Practice: From Post-Crisis Intermediation to FinTechs, authors explain in different way the principles of risk intermediation)