Calibration of Markov-Functional Models - Modelling of Forward Libor and Swap Rates

The most important feature of Markov-functional models is the fact that their calibration to market prices of plain vanilla derivatives is relatively easy to perform. For convenience, we shall focus here on the calibration of the Markov-functional model of fixed-maturity forward swap rates. The case of forward Libor rates can be dealt with in an analogous way. A more extensive discussion of this issue can be found in Hunt et al. (1997).

First, we assume that the forward swap rate for the date T_n−1 follows a lognormal martin-gale under the corresponding forward measurePTn. More specifically, we postulate that the proces

˜κ(·, Tn−1) = κ(·, Tn−1, 1) satisfies

d˜κ(t, T_n−1) = ˜κ(t, T_n−1)ν(t, T_n−1) dW_t, (93) where W is a Brownian motion underPTnand ν(·, Tn−1) is a strictly positive deterministic function.

If we take the process

M_t= Z _t

0 ν(u, T_n−1) dW_u as the driving Markov process for our model, then clearly

κ(T_n−1, T_n−1) = ˜κ(0, T_n−1) e^M^Tn−1^{− 12} R_Tn−1

0 ν²(u,Tn−1) du

(94) and

B(T_n−1, T_n, M_T_n−1) =

1 + δ_nκ(0, T˜ _n−1) e^M^Tn−1^{− 12} R_Tn−1

0 ν²(u,Tn−1) du₋₁

. (95)

Suppose that we are given (digital) swaptions prices for all strikes κ > 0 and all expiration dates T₀, . . . , T_n−1. Our goal is to find the joint probability law of (˜κ(T₀, T₀), . . . , ˜κ(T_n−1, T_n−1)) under PTn. This can be achieved by deriving the functional dependence of each rate ˜κ(T_j, T_j) on the underlying Markov process; more specifically, we search for the function h_j : R+ → R+ such that

27Similarly as in the case of a plain-vanilla fixed-for-floating swap, in a constant maturity swap the fixed and floating payments occur at regularly spaced dates. The amounts of floating payments are based not on a Libor rate, but on some other swap rate, however.

κ(T_j, T_j) = h_j(M_T_j). To this end, we assume that for any j = 0, . . . , n− 1 there exists a strictly increasing function h_j such that this holds (in view of (94), this statement is valid for j = n− 1).

By the definition of the probability measurePTn, for i = j + 1, . . . , n

so that the right-hand side in the formula above is a function of M_T_j. Consequently, for

G_T_j(n− j) =

where g_j :R → R is a measurable function with strictly positive values. The right-hand side in (96) can be evaluated using the transition p.d.f. p_M(t, m; u, x) of the Markov process M, provided that the functional form of B(T_i, T_n, M_T_i) is known for every i = j + 1, . . . , n. To put it more explicitly

We work back iteratively from the last relevant date T_n−1. In the first step, i.e., when j = n− 2, the functional form of B(T_n−1, T_n, M_T_n−1) given by (95). Assume now that the functional forms of Our next goal is to show how to find the function h_j, under the assumption that the functional forms of bonds prices B(T_i, T_n, M_T_i) are known for every i = j +1, . . . , n. To this end, we assume that we are given all market prices of digital swaptions with expiration date T_j and any strictly positive strike level κ. We find it convenient to represent the price at time 0 of the j^th digital swaption, with strike κ and expiration date T_j, in the following way²⁸

DS^j₀(κ) = B(0, T_n)E_P_TnG_T_j(n− j)

28By definition, the j^th digital swaption, with unit notional principal, pays the amount δi at timeTi for i = j + 1, . . . , n whenever inequality ˜κ(Tj, Tj)> κ holds.

or equivalently,

DS^j₀(κ) = B(0, T_n)E_P_Tn g_j(M_T_j) I_{M

Tj>h⁻¹_j (κ)}

Finally, if we denote by f_M(x) = p_M(0, 0; T_j, x) the p.d.f. of M_T_j under PTn then DS^j₀(κ) = B(0, T_n)

Rg_j(x) I_{x>˜h_j_(κ)}f_M(x) dx, (99) where we write ˆh_j = h⁻¹_j . It is natural to assume that the function²⁹ DS^j₀ : R₊ → R₊ is strictly decreasing as a function of the strike level κ, with

DS^j₀(0) = Xn i=j+1

δ_iB(0, T_i) = G₀(n− j)

and DS^j₀(+∞) = 0. Since

EP_Tn g_j(M_T_j)

= G₀(n− j)B⁻¹(0, T_n)

it can be deduced from (99) that ˆh_j(0) =−∞. On the other hand, condition DS^j₀(+∞) = 0 implies that ˆh_j(+∞) = +∞. Finally, the function ˆhj implicitly defined through equality (99) is strictly increasing, so that it admits an inverse function h_j with desired properties. To wit, for h_j = ˆh⁻¹_j we have: h_j:R → R+is strictly increasing, with h_j(−∞) = 0 and hj(+∞) = +∞. This shows that the procedure above leads to a reasonable specification of the functional form ˜κ(T_j, T_j) = h_j(M_T_j).

For the reader’s convenience, we shall recapitulate the main steps of the calibration proce-dure. In the first step, we numerically find the function h_n−2 which expresses ˜κ(T_n−2, T_n−2) in terms of M_T_n−2. To this end, we need first to evaluate the function g_n−2 using formula (97) with B(T_n, T_n, x) = 1 and B(T_n−1, T_n, x) given by (95)

In the second step, we first determine B(T_n−2, T_n, x) using relationship (98), that is, B⁻¹(T_n−2, T_n, x) = 1 + h_n−2(x)g_n−2(x).

Then, we find g_n−3using (97), and subsequently we determine the rate ˜κ(T_n−3, T_n−3), or rather the corresponding function h_n−3.

Continuing this procedure, we end up with the following representation of the finite family of swap rates

(˜κ(T₀, T₀), . . . , ˜κ(T_n−1, T_n−1)

= g₀(M_T₀), . . . , g_n−1(M_T_n−1) .

This representation uniquely specifies the probability law of the considered family of swap rates under the terminal forward measurePTn.

Remarks. In view of (93), the price at time t≤ Tn−1 of the (n− 1)^th digital swaption equals DSⁿ⁻¹_t (κ) = δ_nB(t, T_n)PTn{˜κ(Tn−1, T_n−1) > κ | Ft}

that is,

DSⁿ⁻¹_t (κ) = δ_nB(t, T_n)N ˜h₂(t, T_n−1)

(100) where N denotes the standard Gaussian cumulative distribution function, and the coefficient ˜h₂ is given in the formulation of Proposition 3.2. Needless to say that formula (100) is not valid in the present setup, even for t = 0, for any digital swaption with maturity T₀, . . . , T_n−2. Moreover, it is clear that assumption (93) is not necessary; we need only assume that the functional form of the swap rate ˜κ(T_n−1, T_n−1) with respect to some underlying Markov process M is explicitly known (and is a monotone function of M_T_n−1).

29Recall that the functionDS^j₀represents the observed market prices of digital swaptions. Therefore, the foregoing assumptions about the behaviour of this function are indeed quite natural.

Bibliography

Andersen, L., and Andreasen, J. (1997) Volatility skews and extensions of the Libor market model.

Working paper, General Re Financial Products.

Brace, A. (1996) Dual swap and swaption formulae in the normal and lognormal models. Working paper, University of New South Wales.

Brace, A., G¸atarek, D., and Musiela, M. (1997) The market model of interest rate dynamics.

Mathematical Finance 7, 127–154.

Brace, A., Musiela, M., and Schl¨ogl, E. (1998) A simulation algorithm based on measure relation-ships in the lognormal market model. Working paper, University of New South Wales.

Brace, A., and Womersley, R.S. (2000) Exact fit to the swaption volatility matrix using semidefinite programming. Working paper.

B¨uhler, W., and K¨asler, J. (1989) Konsistente Anleihenpreise und Optionen auf Anleihen. Working paper, University of Dortmund.

Cox, J., and Ross, S. (1976) The valuation of options for alternative stochastic processes. Journal of Financial Economics 3, 145–166.

D¨oberlein, F., and Schweizer, M. (1998) On term structure models generated by semimartingales.

Working paper, Technische Universit¨at Berlin.

D¨oberlein, F., Schweizer, M., and Stricker, C. (1999) Implied savings accounts are unique. Forth-coming in Finance and Stochastics.

Dun, T., Schl¨ogl, E., and Barton, G. (2000) Simulated swaption delta-hedging in the lognormal forward LIBOR model. Working paper, University of Sydney and University of Technology, Sydney.

Flesaker, B. (1993) Arbitrage free pricing of interest rate futures and forward contracts. Journal of Futures Markets 13, 77–91.

Flesaker, B., and Hughston, L. (1996a) Positive interest. Risk 9(1), 46–49.

Flesaker, B., and Hughston, L. (1996b) Positive interest: foreign exchange. In: Vasicek and Beyond, L.Hughston, ed. Risk Publications, London, pp. 351–367.

Flesaker, B., and Hughston, L. (1997) Dynamic models of yield curve evolution. In: Mathemat-ics of Derivative Securities, M.A.H.Dempster, S.R.Pliska, eds. Cambridge University Press, Cambridge, pp. 294–314.

Geman, H., El Karoui, N., and Rochet, J.C. (1995) Changes of numeraire, changes of probability measures and pricing of options. Journal of Applied Probability 32, 443–458.

Glasserman, P., and Kou, S.G. (1999) The term structure of simple forward rates with jump risk.

Working paper, Columbia University.

Glasserman, P., and Zhao, X. (2000) Arbitrage-free discretization of lognormal forward Libor and swap rate model. Finance and Stochastics 4, 35–68.

Goldys, B. (1997) A note on pricing interest rate derivatives when Libor rates are lognormal.

Finance and Stochastics 1, 345–352.

Goldys, B., Musiela, M., and Sondermann, D. (1994) Lognormality of rates and term structure models. Working paper, University of New South Wales.

Heath, D., Jarrow, R., and Morton, A. (1992) Bond pricing and the term structure of interest rates:

A new methodology for contingent claim valuation. Econometrica 60, 77–105.

Hull, J.C., and White, A. (1999) Forward rate volatilities, swap rate volatilities, and the imple-mentation of the LIBOR market model. Working paper, University of Toronto.

Hunt, P.J., and Kennedy, J.E. (1997) On convexity corrections. Working paper, ABN-Amro Bank and University of Warwick.

Hunt, P.J., and Kennedy, J.E. (1998) Implied interest rate pricing model. Finance and Stochastics 2, 275–293.

Hunt, P.J., and Kennedy, J.E. (2000) Financial Derivatives in Theory and Practice. John Wiley

& Sons, Chichester.

Hunt, P.J., Kennedy, J.E., and Pelsser, A. (1997) Markov-functional interest rate models. Forth-coming in Finance and Stochastics.

Hunt, P.J., Kennedy, J.E., and Scott, E.M. (1996) Terminal swap-rate models. Working paper, ABN-Amro Bank and University of Warwick. Forthcoming in Finance and Stochastics.

Jamshidian, F. (1996) Pricing and hedging European swaptions with deterministic (lognormal) forward swap rate volatility. Working paper, Sakura Global Capital.

Jamshidian, F. (1997) Libor and swap market models and measures. Finance and Stochastics 1, 293–330.

Jamshidian, F. (1999) Libor market model with semimartingales. Working paper, Netanalytic Limited.

Jin, Y., and Glasserman, P. (1997) Equilibrium positive interest rates: a unified view. Forthcoming in Review of Financial Studies.

Lotz, C., and Schl¨ogl, L. (1999) Default risk in a market model. Working paper, University of Bonn.

Miltersen, K., Sandmann, K., and Sondermann, D. (1997) Closed form solutions for term structure derivatives with log-normal interest rates. Journal of Finance 52, 409–430.

Musiela, M. (1994) Nominal annual rates and lognormal volatility structure. Working paper, Uni-versity of New South Wales.

Musiela, M., and Rutkowski, M. (1997a) Martingale Methods in Financial Modelling. Springer, Berlin Heidelberg New York.

Musiela, M., and Rutkowski, M. (1997b) Continuous-time term structure models: Forward measure approach. Finance and Stochastics 1, 261–291.

Musiela, M., and Sawa, J. (1998) Interpolation and modelling term structure. Working paper, University of New South Wales.

Neuberger, A. (1990) Pricing swap options using the forward swap market. Working paper, London Business School.

Rady, S. (1997) Option pricing in the presence of natural boundaries and a quadratic diffusion term. Finance and Stochastics 1, 331–344.

Rady, S., and Sandmann, K. (1994) The direct approach to debt option pricing. Review of Futures Markets 13, 461–514.

Rutkowski, M. (1997) A note on the Flesaker-Hughston model of term structure of interest rates.

Applied Mathematical Finance 4, 151–163.

Rutkowski, M. (1998) Dynamics of spot, forward, and futures Libor rates. International Journal of Theoretical and Applied Finance 1, 425–445.

Rutkowski, M. (1999) Models of forward Libor and swap rates. Applied Mathematical Finance 6, 29–60.

Sandmann, K., and Sondermann, D. (1993) On the stability of lognormal interest rate models.

Working paper, University of Bonn.

Sandmann, K., and Sondermann, D. (1997) A note on the stability of lognormal interest rate models and the pricing of Eurodollar futures. Mathematical Finance 7, 119–125.

Sandmann, K., Sondermann, D., and Miltersen, K.R. (1995) Closed form term structure deriva-tives in a Heath-Jarrow-Morton model with log-normal annually compounded interest rates.

In: Proceedings of the Seventh Annual European Futures Research Symposium Bonn, 1994.

Chicago Board of Trade, pp. 145–165.

Schl¨ogl, E. (1999) A multicurrency extension of the lognormal interest rate market model. Working paper, University of Technology, Sydney.

Schmidt, W.M. (1996) Pricing irregular interest cash flows. Working paper, Deutsche Morgan Grenfell.

Schoenmakers, J., and Coffey, B. (1999) Libor rates models, related derivatives and model calibra-tion. Working paper.

Sidenius, J. (1997) Libor market models in practice. Working paper.

Uratani, T., and Utsunomiya, M. (1999) Lattice calculation for forward LIBOR model. Working paper, Hosei University.

Yasuoka, T. (1998) No arbitrage relation between a swaption and a cap/floor in the framework of Brace, Gatarek and Musiela. Working paper, Fuji Research Institute Corporation.

Yasuoka, T. (1999) Mathematical pseudo-completion of the BGM model. Working paper, Fuji Research Institute Corporation.

In document Modelling of Forward Libor and Swap Rates (Page 41-46)