Affine LIBOR models driven by real-valued affine processes

ABSTRACT

The class of affine LIBOR models is appealing since it satisfies three central requirements of interest rate modeling. It is arbitrage-free, interest rates are nonnegative, and caplet and swaption prices can be calculated analytically. In order to guarantee nonnegative interest rates affine LIBOR models are driven by nonnegative affine processes, a restriction that makes it hard to produce volatility smiles. We modify the affine LIBOR models in such a way that real-valued affine processes can be used without destroying the nonnegativity of interest rates. Numerical examples show that in this class of models, pronounced volatility smiles are possible.

Keywords:

• LIBOR market models
• forward price models
• affine processes
• volatility smile

Mathematics Subject Classification:

• 91G30

Notes

1 can be described as the (convex) set where the extended moment generating function of X_t is defined for all times t ⩽ T and all starting values x ∈ E. By Lemma 4.2 in Keller-Ressel and Mayerhofer^[11], the set is in fact equal to the seemingly smaller set

2 One can extend bond price processes to [0, T] by setting for t > T_k, so that P( ·, T_k)/P( ·, T) is a martingale on [0, T] if and only if it is a martingale on [0, T_k]. Economically, this can be interpreted as immediately investing the payoff of a zero coupon bond into the longest-running zero coupon bond.

3 Since x₀ is fixed, any dependence of probability measures on the starting value of the Markov process X will be suppressed from now on.

4 This is similar for most affine processes but is best visible for Lévy processes.

5 The smile example of Keller-Ressel et al.^[13], figure 9.2, using an Ornstein-Uhlenbeck process seems to be numerically incorrect for strikes smaller than 0.4. With the mentioned initial yield curve the underlying interest rate is always larger than the strike, which corresponds to a zero implied volatility, destroying the displayed smile.

6 Actually, caplet prices coincide with prices of swaptions with only one underlying period. The difference between those two derivatives is the payoff time.

7 As in the floorlet case, if κ₁ = κ₂, then the forward swap rate is always larger than the strike. Note that S_{α, β}(t) can also be written as S_{α, β}(t) = ∑^β_{k = α + 1}w_k(t)F^k(t) with w_k > 0 (see, e.g., Brigo and Mercurio^[2]). It follows that if forward interest rates are bounded from below by positive constants, the same will be true for forward swap rates. This bound is an average of the corresponding forward interest rate bounds and therefore of the same order of magnitude, which for a meaningful model will be small.