# Replicating portfolio and risk-neutral pricing for interest rate options

For equity options, the pricing of options depends on the existence of a replicating portfolio, so you can price the option as the constituents of that replicating portfolio. However, I am not seeing how the same analysis can be applied to value interest rate options. Does the concept of replication apply to interest rate derivatives? If so, what would a replicating portfolio look like?

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It's made of interest rate instruments, of course, one per dimension of your SDE. Money market accounts, swaps and zero coupon bonds are common choices. – Brian B Dec 13 '12 at 1:44
Interest rate models often model a continuous curve based on a finite number of unobservable factors. So it is not that straightforward to go from the SDE to the maturity and notional of the swaps you should buy/sell. – AFK Jan 30 '15 at 22:17

The concept of replication is indeed applied to IR products, after all they are also hedged in practice.

However, in the equity world we start with the replicating portfolio and then arrive to the pricing formula. In contrast, for IR products we employ a convenient numeraire which helps us to arrive at the pricing formula directly (in a non-constructive and arguably less intuitive way). Furthermore, in the IR world the market instruments themselves are many and interlinked (cash rates, FRAs, swaps, basis swaps, etc.); in equities you have a single well defined underlying.

To see the replicating portfolio one has to just look at the hedging ratios with respect to all relevant market observables. A swaption would be exposed to two points of the swap curve, and this exposure can be reconstructed by a collection of swaps and other instruments: i.e. the replicating portfolio.

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For forward libor models, one can hedge interest rate options by using bonds. (Note that forward libor is a tradeable security under the forward measure). See http://www.columbia.edu/~mh2078/market_models.pdf

For affine yield models (like Vasicek or CIR models) the inverse problem is the most useful. Given an interest rate process, I can compute a theoretical bond price. From observable market prices of bonds I can then compute the parameters of my interest rate model (by minimizing the L2 norm, for example). Given these (risk neutral!) parameters I can then price any instrument that depends on that interest rate. But, I cannot necessarily hedge these instruments.

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Let $0 \leq T < U$. Consider a European call on a U-Bond (Zero-coupon bond maturing at time U) with time of maturity $T$.

What you do is that you hedge the call option with the aid of the U-Bond and the T-Bond. I could go in to more details on how to do this in particular models, but I would basically just write the same things as in this book:

Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit http://www.amazon.com/Interest-Rate-Models-Practice-Inflation/dp/3540221492

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As Brian B states above the short answer includes Money market accounts, swaps and zero coupon bonds among other instruments. Lets say we have an interest rate derivative that we need to value via replication. Now if we think of what we mean by a replicating portfolio its clear that the main ingredient needed is to match the pay structure\payout of the interest rate derivative with existing market instruments -- thus the instruments needed depend on the payout of the derivative.

For example, if we have an interest rate swap (IRS), we see that the cash flow (and hence payout) of the instrument is the same as if we had a chain (sequence) of forward rate agreements (FRA's). Similarly we can look at modeling the IRS with bonds of an appropriate tenor. There is no single definitive answer, because depending on the economics of the IRS, we may not have FRA's that match the payout, or we may not have bonds with correct maturities, we may have the bonds, but they may not be liquid enough to be valued etc. This notion is known as the completeness/incompleteness of the market.

Hence given a interest rate derivative, usually it would be the case that the existing market instruments are interest related products, but the actual components that make up the replicating portfolio depend on the pay out structure of the derivative, and the ability to be able to obtain market prices for the assets in the replicating portfolio. In fact, if these two conditions are met, then it does not matter which portfolio of underlying instruments you use.

However if these conditions are not met, then you usually have to make do with what is available or modify/proxy instruments to obtain a valuation -- in practice, this is most often the case.

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@ezbentley oops sorry I answered your question for interest rate derivatives in general! I did not see that you wanted a discussion on options in particular -- the gist of the answer remains the same though. – Don Shanil Jun 3 '14 at 1:32