Embed Size px x x x x Here we use the one factor LGM model to price the standard IR exotic deals: callable swaps includingamortizing swaps , callable inverse floaters, callable super-floaters, callable range notes, autocaps and revolvers, and rangenotes. We lay out the complete pricing of these deals: how to represent the deals, how to select the calibration instruments,how to determine the appropriate calibration strategies and algorithms, the dierent deal evaluation algorithms for each dealtype, and the usage of adjusters to obtain the best possible prices and hedges. We then extend this analysis to the world of bonds. In this paper we specify how to price and hedge common exotic interest rate deals: callable swaps including amortizing swaps , callable inverse floaters, callable capped- floaters and super-floaters, callablerange notes, autocaps, revolvers, and captions. In Part II, we work out best practices for using the one factor LGM mode for pricing and hedging: Weintroduce the LGM model, we determine how to select eective calibration instruments, we derive ecientcalibration strategies and algorithms, we then discuss evaluation methodologies and determine ecient algo-rithms for evaluating the prices of exotic instruments under the calibrated model.

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We then extend this analysis to the world of bonds. Key words. LGM, interest rate models, calibration, exotic options.

Part head: Introduction. Here we introduce our notation and the mathematics of swaps and vanilla options. Throughout these sections we use best practices in choosing the calibration instru- ments, calibrating the model, evaluating the deal, and using internal adjustors to obtain risks to the proper instrumentsand clean up the prices.

In Part III, we extend the model to include a second factor for credit. The extended LGM model is used to price bond, focussing on bonds with embedded options. Discount factors, zeros, and FRAs. T end. Instantaneous rate for date T as seen at date t. Clearly, if one agrees at date t. Alternatively, we can re-phrase this as. This type of single payment deal is equivalent to a FRA forward rate agreement.

Clearly the fair amount is. We can always get their values by stripping. Fixed leg. On any given day t , these payments have the value. The mechanics of the swap market is covered in Appendix A. There we explain how to correctly construct date sequences and compute coverages. Floating leg. Here r j is generally.

At any date t , the forward fair or true rate r so that the value of the interest payment exactly equals the theoretical value:. Basis spread curves are obtained by stripping basis swaps. One can show that forward basis spreads are Martingales in the appropriate forward measures. That is, one assumes that the gamma of the forward spread is inconsequential. This is true regardless of the model used.

Handling the basis spread. Basis spreads are a nuisance. They are large enough that one. That is,. We will use this approach throughout. Swap rate and level. We can re-write the swap values in terms of the swap rate and level as.

A swaption is a European option on a swap. If one exercises on this date, one obtains the receiver swap. To introduce this model, suppose.

It is just the sum of a bunch zero coupon bonds, and hence is. There exists a probability measure in which the value of all tradeable instruments including the swaption divided by the numeraire is a Martingale. V opt. Moreover, the swap rate. So the. By the Martingale representation model, then, we conclude that. At this point we know that. Fundamental theory can take us no further.

The value of the payer swaption is obtained by reversing R f i x and R 0 in the above formulas:. V mkt. This value of the volatility is known as the implied volatility.

Modeling this rate as log normal. The value of the caplet is obtained by reversing R f i x and R 0 in the above formulas:. Following the above line of reasoning shows that the value of these digitals is. Part head: Pricing exotics via LGM. The LGM model. A modern interest rate model consists of three parts: a numeraire, a set of. The one factor.

LGM model has a single state variable, X. This is the evolution under the risk neutral measure induced by the numeraire, which will be named shortly. Clearly X t is Gaussian with the transition density.

We choose the numeraire to be. The last part of the model is the Martingale valuation formula. Suppose at time t the economy is in. The LGM model can be written most simply in terms of the reduced prices. In terms of the reduced prices V t, x , the LGM model is. Zero coupon bonds and the forward curves Let us go a bit further before summarizing. The reduced value of a zero coupon bond is.

Substituting for the numeraire and carrying out the integration yields the reduced zero coupon price. So eqs. The amount x of the shift is.

As always, model parameters have to be set a priori during the calibration procedure by combining both theoretical reasoning guessing and calibration of vanilla instruments. Aside: Connection to the Hull White model. Under the Hull White model, deals are valued. This is why calibrating directly on the Hull-White model instead of the LGM formulation of the model is often an inherently unstable procedure. Model invariances. First, all market prices remain unchanged if we change the model parameters by:.

To prove this, note that if we make the above transformation and then transform the internal variables x and X by. Second, all market prices remain unchanged if. To prove this, note that if we make the above transformation, and then transform the internal variables x and X by. This makes the last term in 4. Summary of the LGM model. The complete LGM model can be summarized as. These equations are the only facts about the model we need to price any security.

This model auto- matically reproduces the discount curve D T. Later we will present the calibration and pricing steps in exquisite detail. Calibration and hedging. Model calibration is the most critical step in pricing. It determines. To see this, suppose we. It invariably contains unknown mathematical parameters which are set by calibration. To calibrate, one. The calibrated model is now used to price the exotic deal. The only step in this procedure which uses market information is the calibration step.

Consider what happens at the nightly mark-to-market. The model is calibrated and deal is priced as above. Next the vega risks are calculated by bumping the vols in the volatility matrix cube one-by-one. An exotic deal only has vega risks to the n vanilla instruments used in calibration. This means that in the normal course of events, an exotic deal will be hedged by a linear combination of the vanilla instruments used during calibration.

If the span of the vanilla instruments provide a good representation of the exotic, then the hedges should exhibit rock solid stability, with the day-to-day amounts of the hedges changing only as much as necessary to account for the actual changes in the market place.

If the vanilla instruments do not provide a good representation of the exotic, then the hedges may exhibit instabilities, with day to day amounts of the hedges changing substantially even for relatively minor market changes. This is also known as applying an external adjuster.

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