CONVEXITY CONUNDRUMS: PRICING CMS SWAPS, CAPS, AND FLOORS. PATRICK S. HAGAN GORILLA SCIENCE 11 PALISADE PLAZA EDGEWATER, NJ. Slope function corresponds to ′( ) in Hagan’s Convexity Conundrums paper. Linear TSR models only differ in their specification of the slope. CMS paid at arbitrary time under Hagan’s model.  P. Hagan. Convexity conundrums: Pricing CMS swaps, cpas, and floors. Wilmott.
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We re in hot competition with another bank over a deal. As the deal evolves our trading team starts getting pushed around the market and it dawns on us that the other bank s pricing is better than ours at least for this class of deals.
We could fix this problem by inventing a universal method for achieving the best possible prices for all deal types. That topic will be covered in a future column next to the column on Elvis sightings.
Here we focus on a single class conundrumw deals the constant maturity swaps caps and floors.
We develop a framework that leads to the standard methodology for pricing these deals and then use this framework to systematically improve the pricing. Let us start by agreeing on basic notation. We also use cvg t st t end dcb. So if interest accrues at rate R then cvg t st t end dcb r is the interest accruing in the interval t st to t end. These dates are usually quarterly. If the CMS leg is set-in-advance this is donvexity then R j is the rate for a standard swap that begins at t j and ends N years later.
Although set-in-advance is the market standard it is not uncommon for contracts to specify CMS legs set-in-arrears.
CMS caps and floors are constructed in an almost identical fashion. Any margin payments m can also be valued easily. So all we need do is value a single payment of the three types paid at t p. To express this rate mathematically let s s In return for making these payments the payer receives the floating leg payments.
Neglecting convvexity basis spread the floating convexihy is worth paid at the start date s 0 minus paid at the end date s n. These are clearly freely tradeable instruments so we can choose the level as our numeraire.
The usual theorems then guarantee that haan exists a probability measure such that the value V t of any freely tradeable deal divided by the numeraire is a Martingale. It is helpful to examine the valuation of a plain vanilla swaption. Consider a standard European option on the reference swap. Since the Martingale formula 2.
One completes the pricing by integrating to calculate the expected value. Martingale so it s average value is today s value: The last term is the convexity correction.
There are two steps in evaluating the convexity correction. uagan
One obvious model is to allow only parallel shifts of the yield curve. The second step is to evaluate this expected value. For deals set-in-advance if the CMS leg dates t 0 t We can carry out the second step by replicating the payoff in 2.
This formula replicates the value of the CMS caplet in terms of European swaptions at different strikes x. At this point some pricing systems break the integral up into 0bp or so buckets and re-write the convexity correction as the sum of European swaptions centered in each bucket. These swaptions are then consolidated with the other European swaptions in the vanilla book and priced in the vanilla pricing system. This replication method is the most accurate method of evaluating CMS legs.
It also has the advantage of automatically making the CMS pricing and hedging consistent with the desk s handling of the rest of its vanilla book.
However this method is opaque and compute intensive. After briefly considering CMS floorlets and CMS swaplets we develop simpler approximate formulas for the convexity correction as an alternative to the replication method.
Convexity Conundrums: Pricing CMS Swaps, Caps, and Floors*
Thus the CMS floolets can also be priced through replication with vanilla receivers. Here the first term in 2. The other terms represent the convexity correction written in terms of vanilla payer and receiver swaptions.
These too can be evaluated by replication. It should be noted that CMS caplets and floorlets satisfy call-put parity. Therefore the value of this combination must be equal at all earlier times as well: Substituting this into our formula for a CMS caplet 2. This can be partially mitigated by using the correct volatilities. Accordingly call-put parity should be used to evaluate in-the-money caplets and floorlets as a CMS swap payment plus an out-of-the-money floorlet or caplet.
These formulas are adequate for many purposes. When finer pricing is required one can systematically improve these formulas by using the more sophisticated models for G developed in the Appendix and by adding the quadratic and higher order terms in the expansion 3. These payoffs emphasize away-from-the-money rates more than standard swaptions so the convexity corrections can be quite sensitive to the market s skew and smile.
Alternatively when the very highest accuracy is needed replication can be used to obtain near perfect results. Models of the yield curve A. Standard model The standard method for computing convexity corrections uses bond math approximations: In a similar spirit the zero coupon bond for the pay date t p is approximated as Z t s 0 Z t; t p A. Parallel shifts This model takes into account the initial yield curve shape which can be significant in steep yield curve environments.
Non-parallel shifts We can allow non-parallel shifts by approximating Z t; s j Z t; s 0 D s j D s 0 e [h s j h s0 ]x A. HAGAN determines the level in terms of the swap rate. We follow the standard if bad practice of referring to both the physical instrument and its value as the numeraire. W 44 Wilmott magazine. Cash-settled swaptions How wrong are we? How wrong are we? No-arbitrage conditions for cash-settled swaptions Fabio Mercurio Financial Engineering Banca IMI, Milan Abstract In this note, we derive no-arbitrage conditions that must be satisfied by the pricing function.
Interest rate Derivatives There is a wide variety of interest rate options available. The most widely offered are interest rate caps and floors. Increasingly we also see swaptions offered. Implementation details for the pricing of European swaptions in different frameworks are presented. Introduction This note describes the pricing. Enter all the candidate and examination details.
Convexity Conundrums: Pricing CMS Swaps, Caps, and Floors* – PDF
Stapleton 2 and Marti G. The analysis of interest rates over time is complicated because rates are different for different maturities. Interest rate for borrowing money for the next 5 years is ambiguous, because. Hedging Illiquid FX Options: Posthuma 2 and S. Guaranteed Annuity Options B. July Document Revision Number: The Black Scholes Model In Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing.
Options and beyond Jerome. Interest rate risk, page 1 Maturity and interest-rate risk Suppose you buy one of these three bonds, convexiyy selling at a yield to maturity of 8 percent. The Fixed Income Benchmark 1.
Models and their uses 3. Spot rates and their properties 4. Introduction to swaps Steven C. Brown Texas-Austin and Donald. Learning Curve An introduction to the use of the Bloomberg system in swaps analysis Received: To review the basics of the time value of money. This involves reviewing discounting guaranteed future cash flows at annual, semiannual and continuously. Black-Scholes Equation is derived using two methods: Equity-index-linked swaps Equivalent to portfolios of forward contracts calling for the exchange of cash flows based on two different investment rates: Interest Rate Volatility I.
Published in Journal of Investment Management, Vol. Here we present the standard methodology for pricing accrual. While it is true that short-term rates are more volatile than long-term rates, the longer duration of the longer-term bonds makes their prices and their. My email address is on my website Lecture given 19th February. Definitions Ameriprise Workshop Overview Definitions The Black model has been the standard model for European options on currency, interest rates, and stock indices with it s main drawback being.
Fixed Income ortfolio Management Interest rate sensitivity, duration, and convexity assive bond portfolio management Active bond portfolio management Interest rate swaps 1 Interest rate sensitivity, duration.
Trading Strategies of Vanilla. Estimate the cash flows coupons and return of principal 2. The principal features of floating rate bonds can be summarised simply: Introduction In early s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there.