Narrow Results

We show that for any finite group $G$, there exist infinitely many simple anticommutative algebras with multiplicative bases $(𝔄,\mathcal{ℬ})$ such that the group $\mathrm{\text{Aut}}((𝔄,\mathcal{ℬ}))$ is isomorphic to $G$.

A magma is just a set $A$ with a single binary operation

Let $(A,〈\cdot ,\dots ,\cdot 〉)$ be an $n$-algebra of arbitrary dimension and over an arbitrary base field $𝔽$. A basis ${ℬ}={\{e_i\}}_{i\in I}$ of $A$ is said to be multiplicative if for any $i_1,\dots ,i_n\in I$, we have either $〈e_{i_1},\dots ,e_{i_n}〉=0$ or $0\ne 〈e_{i_1},\dots ,e_{i_n}〉\in 𝔽e_j$ for some (unique) $j\in I$. If $n=2$, we are dealing with algebras admitting a multiplicative basis while if $n=3$ we are speaking about triple systems with multiplicative bases. We show that if $A$ admits a multiplicative basis then it decomposes as the orthogonal direct sum $A={\oplus }_{\alpha }{I}_{\alpha },$ of well-described ideals admitting each one a multiplicative basis. Also, the minimality of $A$ is characterized in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is by means of the family of its minimal ideals.

The study on the multiple-curve interest rate models becomes increasingly active since the 2007 credit crunch, for which one curve, typically the OIS curve, is used for discounting purpose, while the LIBOR curves (associated with various market tenors) are used for projecting the future cash flows. In this work, we extend the standard LIBOR market model to accommodate such multiple-curve setting by means of a multiplicative basis. The multiplicative basis is modeled as an exponential function of multi-factor square-root processes. Under the multiplicative basis setup, the OIS forward rates are correlated with the implied (additive) LIBOR-OIS spreads. We then derive closed-form pricing formulas for caplet, swaption, and interest rate futures in the multiplicative basis framework. In particular, we show that the valuation of caplet and swaption can be easily computed by a proper integral of real-valued functions, which facilitates the calibration of our model. Finally, we discuss a slight modification of our model to allow for negative interest rates.

In Zhong (2018), LIBOR market model with multiplicative basis, International Journal of Financial Engineering, 5(2), we proposed a LIBOR market model with multiplicative basis, namely, the LMM-MB model, to model the joint evolution of the LIBOR rates and the OIS forward rates. This model leads to tractable pricing formulas for the standard interest rate derivatives such as the (vanilla) caplet, swaption and futures. In this paper, we study the pricing of some non-standard interest rate derivatives under the LMM-MB model, specifically the in-arrears (IA) cap and the ratchet cap. Similar to the vanilla caplet, we show that the pricing of the IA caplet can be readily computed by a proper integral of real-valued functions. We then derive an analytical approximation for the ratchet cap. In the case of non-zero spread, the ratchet cap can be approximated by using a two-dimensional fast Fourier transform method. In the case of zero spread, the ratchet cap can be computed from a proper integral of a single variable function. Numerical results reveal a good match of our close-form formulas with the Monte Carlo simulation method.