Replicating a European Put with an American Put, Stock, and Bonds

Summary

The note explains a passage in a paper on characterizations of American put options. Its central point is that the hedge described is part of a strategy that replicates an investor’s European option position using an American option, the underlying stock, and bonds. As the underlying price changes, the positions in the option and bond are adjusted so the portfolio remains self-financing and matches the value of the replicated holding.

In that setup, “investor” refers to the holder of the replicating portfolio, who must rebalance to maintain the replication; it does not mean every ordinary buyer of a put must dynamically hedge. The answer suggests “arbitrageur” might be clearer terminology. The note provides a conceptual interpretation rather than deriving the paper’s theorem, and the respondent acknowledges not having worked through the paper completely.

Key ideas

• The hedge is used to replicate a European option position with an American option, stock, and bonds.
• Changing underlying prices require rebalancing the option and bond holdings.
• A self-financing replicating portfolio must maintain the value of the position it represents.
• The described investor is acting as a hedger or arbitrageur, rather than simply buying a put.

Tags

# Hedging behind the decomposition of american put options


# Hedging behind the decomposition of american put options












Now I'm reading a paper:"alternative characterizations of american put options" , the authors are Carr,Jarrow,Myneni

http://www.math.nyu.edu/research/carrp/papers/pdf/amerput7.pdf

After theorem 1 (in page 4),the author said :

I don't quite understand why the "investor" should hedge the put option when the stock price is below the boundary. I think only the "writer" of the option should hedge,but not the "investor".What's the meaning of this paragraph?

## Answer by Bob Jansen (score 1, accepted)

https://quant.stackexchange.com/a/17154

That seems to be a nice paper but I haven't worked through it completely yet.

As I understand it, the goal is to replicate the holding (by an investor) of an European option using an American option, stock and bonds in a self-financing manner. As the value of the underlying changes this requires rebalancing of the option and the bond, i.e. hedging. Since the portfolio is self-financing the value has to be equal. Thus, once you have such a portfolio the value of the American option is the only unknown and it can be derived from the known prices.

Surely, an investor in the traditional sense will just buy the European or American put and not set up a dynamic hedge. Nowadays the authors might have written arbitrageur instead.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.