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## Tuesday, November 12, 2013

### replication - 1-step binomial model, evolving to real option

Now I feel the binomial tree model is a classic analytical tool for option pricing....

The 1-step binomial scenario is simple but non-trivial. Can be mind-bending. Usually we are given 5 numbers for Sd, S0, Su, Cd, Cu, and the problem is phrased like "Use some number of stock and bond to replicate the contract C" to get the same "payout outcomes" [1].

First, ignore the Cd, Cu values. The Sd, S0, Su 3 numbers alone imply RN probabilities of the up-move and down-move.

Next, using the RNP values we can replicate ANY contract including the given C contract.

The number of shares in the replication is actually the delta-hedge of the call.

[1] "Payout outcomes" mean the contract pays Cd dollars in the down-state and Cu dollars in the up-state.
------ That's the first knowledge pearl to internalize...------------

* 1-step binomial call option
- this option contract can be replicated with stocks + bonds. Rebalancing not necessary.
- RNP/MG is an alternative to replication

* 1-step 3-state call option
- can't replicate with stocks + ....
- RNP non-unique
(That's assuming the 3 outcomes don't accidentally line up.)

* 2-step 3-state call option, i.e. allowing rebalancing
- can replicate with stocks + bonds but needs rebalancing (self-financed, of course)
- RNP/MG is an alternative to replication

* fine-grained call options -- infinite steps, many states
- can replicate (terminal) payout with stocks + bonds, but needs dynamic delta-hedge (self-financed of course)
- * required number of stocks = delta