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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

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    $\begingroup$ We need the total portfolio to be risk free for the argument to work, any correlated asset would leave risk (unless the asset is 100% correlated) so your obvious choice is to use the underlying. Also complete markets are ones where all contingent claims can be replicated - but the replicating portfolios are dynamic delta hedges in the underlying. It's not that you can use any old asset. $\endgroup$ Commented Jul 25 at 12:51
  • $\begingroup$ I've seen different "derivations" of the idea. Some, like I assume you too, want to construct a risk free portfolio using some combination of the derivative and the underlying. In this case, it becomes quite intuitive that you need an underlying that would perfectly offset the derivative, i.e. perfect correlation (shorted). I am more familiar with the idea of setting up a portfolio that replicates the option payoff in all scenarios, using a combination of a stock and a risk free asset. In this case, I guess it's the same argument: ... $\endgroup$ Commented Jul 25 at 13:19
  • $\begingroup$ Agreed. Either your goal is to obtain a derivative portfolio with locally vanishing risk or dynamically replicating a derivative whose payoff is linked to a specific underlying... you kind of have to trade the underlying. Note that if you cannot, or that there remains some risk (e.g. jump diffusion of stochastic volatility), the model is theoretically incomplete and there is no unique pricing measure. Similarly if you chose to use bonds rather than the risk free money market account, this would simply yield pricing equations under a different numeraire (yet equivalent martingale measure). $\endgroup$ Commented Jul 27 at 18:17

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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)

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