Engineering BGM (Chapman and Hall/CRC Financial Mathematics Series)

The drift term appearing in the measure change 4. From this equation 4. Start by considering how changes in vj t arise. So the uj t a Tj term varies little in contrast to uj t K t, Tj , but is much larger, suggesting it should be immediately isolated. Hence the following approximate SDE for the stochastic part 4. H Reiterating in part, some terminology for parts of formula 4. From the definition 4.

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The first example is coterminal swaptions, the second is a set of swaps with a strictly increasing total tenor structure. Swaprate Dynamics 37 Example 4. Now make the swaprates for these three forward swaps lognormal as follows: Chapter 5 Properties of Measures The previous chapters show that BGM, swaprate models and market models in general are in many ways plays on measures; just find the right measures to make the variables of interest lognormal! First note from Geman et al [43] that in general measure changes are governed by: The two most practical measures for pricing either by simulation or timeslicer are the terminal measure Pn , for which the zero coupon bond B t, Tn is numeraire, and the spot Libor measure P0 , for which the pseudo-bank account consisting of rolled up zeros an analogue of the HJM bank account is numeraire.

Using the terminal measure for simulation can lead to blowouts in the sample standard deviation, see Example But the terminal measure is technically much easier to use than Spot Libor in timeslicers, see Section The result is to blow out the sample standard deviation, which can only be reduced by making further simulations, which increases computation time.

The forward value factor is an inherent deficiency with the terminal measure that makes the author prefer to work with the spot Libor measure for simulation. Chapter 6 Historical Correlation and Volatility If there are no market instruments implying future correlation, backward looking analysis of historical data is the only reasonable source of information. Hence the topic of this chapter is estimating historical correlation within a shifted BGM framework.

That said, increasingly in some markets like EUR, there is implied correlation information available in the form of options on differences in short-term and long-term swap rates CMS spread options , and in the future that will probably become the dominant source of correlation information. Once determined, correlation can either be input directly into the model becoming part of the parameter set, or it can be used as a desirable target for optimization routines that bestfit cap and swaption prices.

To get quarterly readings, one might use any of the standard curves readily available in banks, designed to fit relevant data and produce discount functions for all maturities. Apart from smoothness problems, when a day-by-day filmshow of forward curves is run always a good test of routines producing periodic curves or surfaces , one often observes flapping of the long end of the forward curve, where data is relatively sparse and errors compound.

Our convention will be that absolute maturities use capitals and the variables in which they appear, like K t, T , are also absolute, while relative maturities use small letters and the variables in which they appear, like K t, x , are relative. To statistically analyze historical correlation, the underlying models describing the day-to-day movement of yield curves must necessarily be stationary. For that reason the first row and column of R are often discarded as irrelevant. That raises the possibility that correlation can be estimated in one of the Gaussian, or flat or shifted BGM frameworks, and reasonably used in the others.

Various possible ways include: The point is that in swap-world the first 5 principal components are stable sometimes for years , capture most around But what is really relevant about 6. Moreover, that holds in general; when relative maturities are substituted for absolute maturities in variables like forwards, zero coupon bonds or swaprates, their SDEs change only in their drift.

But the actual form of the drift is unimportant because it makes no contribution in the quadratic variation estimators we will use. For similar reasons the measure underlying the SDE, like P0 in 6. Historical Correlation and Volatility 49 A similar estimator to 6. For a given set of data, the shifted covariance estimator 6. That supports the notion of one correlation fits all in shifted BGM. A sensible approach, however, is to use a Gaussian estimator for large shifts, and a lognormal one for small shifts.

Chapter 7 Calibration Techniques Most banks continuously monitor interest rate market data and after stripping, interpolating, and massaging it, present the processed data as market objects in the programming sense ready to be used in pricing algorithms. Typically the object will incorporate yieldcurve and volatility information for a given currency and particular sector like Treasuries or swaps or municipals in USD.

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So in this chapter we assume volatility information is available in the form of interpolated swaption and stripped caplet implied volatilities entered in a quarterly implied volatility matrix with exercise times descending row-by-row and tenors moving left to right along the columns. If data is available awayfrom-the-money it is entered into identical but separate volatility matrices referenced either by absolute strike or relative delta strike, the whole making up an implied volatility cube.

Some entries in the cube are direct liquid market quotes which a good calibration would return exactly. Others are interpolated between the liquid quotes in some fashion, which can potentially introduce arbitrage opportunities the author is unaware of any criteria for ensuring an arbitrage free interpolation, other than it be generated by an arbitrage free interest rate model first fitted to the liquid data. In shifted BGM the shift and volatility are orthogonal in the sense that the shift a T can be found independently, as in Section Such functions return the correct instantaneous correlation, can exactly fit caps and a diagonal of coterminal swaptions allowing vegas with respect to them to be computed , can be made reasonably homogeneous stationary , and in addition can be made to approximately fit a selection of other swaptions boosting confidence in allround pricing.

Another approach, which essentially depends on the power of optimisers like the NAG [81] or IMSL [64] ones, is to bestfit a selection of liquid caps and swaptions in a generic calibration. Usually the motive is to generate indicative prices to help increase comfort levels with a shortrate model that is actually used for marking-to-market, risk management and hedging. The BGM model incorporates more volatility and correlation information than the shortrate model, whose comparative advantage is being Markov in a few variables and so fast. Recall that swaption, but not caplet, implied volatilities depend weakly on correlation.

Using semi-definite programming and a Pessler type volatility function, we also show how correlation can be freed up to participate in calibration while staying close to a historical target and permit an exact fit to the whole matrix. In the caplet formula 3. To fit a swaption skew go through the same steps as for caplets using 3.

Observe that the skew can only be fitted to a selection of data such as the caplets, or a subset of swaptions like a particular as a set of coterminal swaptions. At that point the shift a T is fully determined and cannot be adjusted to fit other data. So to fit larger sets of volatility data, clearly some sort of averaging of shifts over an appropriate set of caplets and swaptions will be required. The calibration fails when roots are imaginary, for example, when implied volatilities fall steeply, but with normal market data that is unlikely. Select M nodes T1 , T2 , It is straightforward but messy, to get quadratic expressions in the a0 ,a1 , Adjust the current mean value variables in equations 7.

In the equations 7. First the skew was fitted to return caplet volatilities at the strikes banding the accrual, and then this algorithm fitted caplet volatilities at one of those strikes and also coterminal swaption volatilities at-the-money. The idea in the following algorithm is to vary dummy swaption volatilities in the 1-factor routine of Section So similarly to Section Then return to Step-2 and iterate.

Caplet and swaption implied volatilities are then easily computed from 4. An implementation of the algorithm viewed by the author had the X matrix represented by cells on an Excel spreadsheet with the cell colour changing with size of the corresponding component; the resulting filmshow as X converged to the desired degree of bestfit and smoothness was both entertaining and educational.

Step-2 From X compute the implied volatilities of the target caplets and swaptions using the process described above. Step-4 Return to Step-2 and iterate until the desired fit and degree of smoothness are obtained. Using homogenous by layer type volatility functions like ours, swaption zetas that is, implied volatility squared multiplied by time to maturity, see 4. In addressing that task our basic assumptions which seem to have become fairly accepted wisdom are: Some 3 to 5 factors and no more, are needed to adequately explain movement of the simple forward curves.

We make a linear combination of the E i,k the target for the for the covariance Calibration Techniques 73 X k in the k th layer. In contrast, direct interpolation on stochastic variables turns out to be inaccurate and unsatisfactory. Note that none of the methods described in this chapter are arbitragefree, though in practice they work fairly accurately. Moreover, they are also inconsistent in that Section The author suggests consulting Schlogl [] for a more exacting analysis. The interpolation scheme 8. Chapter 9 Simulation Two simulation methods are described in this chapter.

Engineering BGM (Chapman & Hall Crc Financial Mathematics Series)

Glasserman type methods [44], [45] avoid bias from drifts by discretizing the SDEs of positive continuous time martingales, and produce accurate results for time steps of the order of a couple of weeks. Big-step methods [90] use predictor-corrector techniques to approximate drift and volatility, and step in intervals of years between decision times like Bermudan exercises. Discretization of the Z t, Tj and V t, Tj as martingales can be rigorously enforced and decreases bias by avoiding drift unlike, for example, simulating K t, Tj under Pn.

Both Z t, Tj and V t, Tj should be strictly positive zeros, numeraires and shifted forwards are strictly positive , which can be ensured by integrating a lognormal SDE with exponential increments. Under the terminal measure Pn From 2. In either case, to simulate H t, Tj from H 0, Tj in one step that is, by big-step simulation we need to separately produce expressions for the drift and volatility terms in 9. Simulation of subsequent steps then follows the same pattern, because the model is Markov in the H t, Tj.

Hence the following approach similar to that of Pelsser et al [90] using fairly robust approximations in various places. XTL -node on timeslice-TL. Depending on node density 4 to 6 figures of accuracy are possible. For example, if the timeslices are the resets of a Bermudan swaption, node values at timesliceT will be the continuation value for some XT and the intrinsic value for other XT. Once nodes on say timeslice-TL are distributed to cover 5 to 6 standard deviations of Mn TL , values of H TL , Tj can be assigned to those nodes using the drift approximation 9.

Step-2 Assign nodes to timeslice-TL to represent the Gaussian random variable Mn TL , and then work out via H TL , Tj the discounted intrinsic value i[Mn TL ] of relevant instruments on each node Mn TL , for example, the underlying swap in the case of a Bermudan values would also be initially attached to the furthest, that is, the very first timeslice in this same fashion. Step-3 With the aid of the formulae in Section But here, from 9.

Engineering BGM

In particular, this path dependency prevents nodes being loaded with either initial or intrinsic values. Approximating hk s by its initial value hk 0 works, but leads to inaccuracies at distant timeslices and in out-of-the-money options. For P0 that involves the path dependent term Distributing nodes where unconditional densities are highest for example, as in the Gauss-Hermite integration routine leads to inaccuracies in splining due to big gaps where the density is low. Whichever method is used an accuracy of five to six decimal places is an appropriate target to aim for.

Whether or not we can expect to use just one lookup table for the whole valuation process, depends largely on whether or not the conditional variances change with timeslice. Now recalibrate using the separable multi-factor method of Section Chapter 11 Pathwise Deltas The most important risk measures are deltas sensitivity to movements of the underlying yield curve and vegas sensitivity to changes in the implied volatilities of the instruments to which the model is calibrated ; in this chapter we show how to compute pathwise deltas along the lines of Glasserman and Zhao [44].

One reasonably accurate method of computing D C0 is to bump and grind: More accurate is the pathwise method: So use the low variance property of hj t , see 3. Partial derivatives of zeros and swaps From 9. The next lemma, which follows from That provides a way of checking, see [44], the accuracy of the pathwise delta method. Because a knock-in together with a knock-out is the same as the underlying option, barrier options will be generally cheaper than their vanilla equivalents. Hence their attraction to customers who are willing to take a view in exchange for lower premiums.

An illustration is the following: As with all other barrier options, for example, FX barriers, there are sixteen flavours corresponding to combinations of cap or floor, up or down, in or out, and barrier less or greater than strike.

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Chapter 12 Bermudans The owner of a payer receiver Bermudan swaption has the right to exercise into an underlying payer receiver swap at some subset of its fixed side reset dates. A good source of revenue for banks is an arrangement popular with investors, in which they receive higher than usual fixed interest in exchange for giving the bank the right to cancel the deal if it does not suit. At the centre of the structure is a callable swap, which is a payer swap that can be cancelled at any of its reset dates.

Roughly speaking, these sorts of deals are set up as follows. The investor deposits funds with the bank which are safely invested and generate floating Libor back to the bank. The bank then sets up with its exotic derivatives desk a callable swap in which the exotics desk receives that floating Libor and pays a fixed coupon. But because the swap is cancellable, that is equivalent to the exotics desk getting a free receiver Bermudan, which is worth money.

That free money is then in part used to increase the fixed coupon in the underlying payer swap, which goes back to the investor as enhanced yield. So the investor takes a view and bets that interest rates will remain steady; if he is right, he receives enhanced coupon for the full duration of the deposit, if he is wrong, he gets his deposit back early and must find an alternative investment at probably lower rates. For example, a callable constant maturity spread swap might, instead of the fixed coupon, pay a constant coupon plus a positive multiple of the 2-year constant maturity swaprate less the year constant maturity swaprate.

Receiver Bermudans are also often used to partially hedge fixed coupon mortgaged backed securities MBS ; if rates drop and mortgagees refinance at a lower coupon, banks may still want to receive the original higher coupon. Bermudans are now so common as to be almost vanilla products; some traders calibrate to them!

Nevertheless, because they are the archetypical callable product, their pricing exhibits many of the techniques needed to price more esoteric callable exotics; hence this chapter. They are also very stable instruments, relatively easy to price and hedge, and not overly sensitive to correlation or the swaption volatility smile.

Note also that any exercise decisions determined by inequalities can be based on either non-discounted or discounted values; because numeraires are positive the results must be the same. For simplicity, we will illustrate his method via an example in Section Using the recurrence K k It remains to identify or approximate the continuation values. So the conditional expectation that is the continuation value will be a function of those variables evaluated at T i.

Some robust heuristic approximation will be needed, because it is not practical to regress on a large number of basis functions. Reasonable results can be obtained by regressing on just the intrinsic values. The constant 1 is included to add useful stability. Step-4 Repeat Step-3 backwards through the resets T 7 , T 6 , The choice of regression variables is partly an art and partly a science, and generally they should be tailored to the callable instrument being valued and involve variables that in a linear combination can mimic its potential behavior.

Thus for vanilla Bermudans useful regression variables might include: He has yet to implement the time-consuming upper bound method described in Section He owns the option, has perhaps paid less than the optimal price for it, and can decide to exercise it according to the optimal routine he has used to compute his bid, or indeed at any other time of his choosing.

A series of papers have tackled the upper bound problem starting with essentially equivalent independent approaches by Rogers [] and Haugh and Kogan [51], and an alternative approach by Jamshidian [67]. Subsequently Andersen and Broadie [8] added operational depth, which was refined by Joshi [69], [70], [71]. Substituting M t for h t in Upper bounds can now be computed by judicious choices of martingales h t to insert in Andersen and Broadie used a martingale defined as follows.

Spare money can be invested in the numeraire resulting in a self-financing strategy that must be a martingale after discounting. On the other hand, variances tend to be lower cutting the number of paths needed. The critical simulation equations are Computing B0 and D B0 is then simply a matter of averaging individual contributions over all trajectories. The corresponding changes in value of the exotic option that we wish to hedge, then yield the required hedge parameters.

For clarity of exposition, we first derive the required BGM shift and volatility perturbations for coterminal swaptions and then consider the case when calibration is to a smaller miscellaneous set of liquid instruments. Vega and Shift Hedging The time convention Condition Bump and grind methods for computing vegas involve a recalibration that must try to mimic these perturbations in some way; that is not easy, and leads to inaccuracies when the functions are overly distorted.

The matrices u and A in Practically speaking, with quarterly nodes N will be over and m much less. In the Gaussian HJM framework the link is possible with deterministic volatilities for domestic and foreign instantaneous forwards, and also the FX rate; that is, deterministic volatilities are totally compatible with lognormal models for the prices of domestic and foreign bonds and the spot and forward FX rates.

But, as shown by Schlogl [], in cross-economy BGM some among the domestic and foreign interest rate volatilities, and FX forward volatilities must be stochastic. Nevertheless, as with swaption volatilities in domestic BGM, with appropriate choices it is possible to obtain approximations for stochastic volatilities in cross-economy BGM that are good enough to return by simulation fairly accurate values for the implied volatilities to which the model is calibrated. To set the scene, in the following Section Our notation for the foreign economy will be to superfix f to domestic variables to denote the equivalent foreign variable; for example, if PT and B t, T are respectively the T -forward measure and zero coupon in the domestic economy, then PfT and B f t, T are the equivalent in the foreign economy.

Evaluating the spot volatility parity Then, various possibilities for the parity relationship include: Dividing the model free spot parity relationship This equation holds for all t, so matching SDEs on each side must yield the required measure changes and volatility parity relationships between the two economies. ST1 t ST1 t Similarly to The volatility parity equation Then, substituting for the bond volatilities from 3. Practically useful possibilities reduce to one of: In that case, the parity equations Note that this includes the possibility, see Section In the next example, for instance, the best choice is clearly foreign rates and FX deterministic.

Simulation can now be done under the spot measure using virtually any scheme. For example, with the notation of Section To simulate the FX, proceed as follows. And, of course, each contract STj t dies at its maturity Tj. Using the relationship That can be easily constructed from the spot and domestic and foreign three month discount rate timeseries Cross-Economy BGM using At least three factors one for domestic rates, one for foreign rates and one for FX and more usually seven to nine factors are needed in a crosseconomy model to properly capture correlations between the forwards in both economies and the FX rates, all of which must be calibrated to interest rate and FX implied volatilities subject to the volatility parity relationship.

This is a complex structure with some data constraints: Reliable implied volatilities for FX forward options out to maturities of 5 to 10 years usually exist, and together with domestic and foreign interest rate implied volatilities constitute a full set of data to which to calibrate. Beyond 5 to 10 year maturities, where only interest rate volatilities are dependable, parity is used to define the FX forward volatilities.

But to do that, and produce a successful and believable calibration, we need some notion of how FX forward implied volatilities might behave at more distant maturities say beyond 10 years. That is, we will let the interest rate volatility data and a linear constraint determine the long term FX implied volatilities. We now generalize the Pedersen method of Section The historical correlation matrix R given by Term structures of shift a T and af T chosen to fit skews in the domestic and foreign economies as outlined in Section Step-2 From X compute the implied volatilities swpnDi X and swpnFi X of the target domestic and foreign caplets and swaptions, and implied volatilities ivolFXi X of the target FX forward options using the methods described above.

Step-3 Insert those values along with X and A, and the target implied volatilities targetDi , targetFi and targetFXi into the objective function Chapter 15 Inflation A major application of the cross-economy BGM model of Chapter is to inflation see Jarrow et al [68] for the original paper articulating cross-economy HJM to inflation , where the consumer price index CPI takes the place of the foreign currency exchange rate. As in the crosseconomy model, superfix f to nominal-world variables to denote real-world variables. Calibration involves stripping nominal and real curves, and identifying forward CPI volatility functions with correlations along with a satisfactory forward inflation curve.

Once that is done, the pricing and hedging of products is usually straightforward. Please accept our apologies for any inconvenience this may cause. Exclusive web offer for individuals. Add to Wish List. Toggle navigation Additional Book Information. Summary Also known as the Libor market model, the Brace-Gatarek-Musiela BGM model is becoming an industry standard for pricing interest rate derivatives.

Written by one of its developers, Engineering BGM builds progressively from simple to more sophisticated versions of the BGM model, offering a range of methods that can be programmed into production code to suit readers' requirements. Using this version, the author develops basic ideas about construction, change of measure, correlation, calibration, simulation, timeslicing, pricing, delta hedging, barriers, callable exotics Bermudans , and vega hedging.

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An appendix provides notation and an extensive array of formulae. The straightforward presentation of various BGM models in this handy book will help promote a robust, safe, and stable environment for calibrating, simulating, pricing, and hedging interest rate instruments. Request an e-inspection copy. Financial Modelling with Jump Processes. The Bookshelf application offers access: