Oh Kang Kwon

Forward rate dependent Markovian transformations of the Heath-Jarrow-Morton term structure model

Abstract. In this paper, a class of forward rate dependent Markovian transformations of the Heath-Jarrow-Morton [16] term structure model are obtained by considering volatility processes that are solutions of linear ordinary differential equations. These transformations generalise the Markovian systems obtained by Carverhill [8], Ritchken and Sankarasubramanian [20], Bhar and Chiarella [1], and Inui and Kijima [18], and also generalise the bond price formulae obtained therein.

Finite Dimensional Affine Realisations of HJM Models in Terms of Forward Rates and Yields

Finite dimensional Markovian HJM term structure models provide ideal settings for the study of term structure dynamics and interest rate derivatives where the flexibility of the HJM framework and the tractability of Markovian models coexist. Consequently, these models became the focus of a series of papers including Carverhill (1994), Ritchken and Sankarasubramanian (1995), Bhar and Chiarella (1997), Inui and Kijima (1998), de Jong and Santa-Clara (1999), Björk and Svensson (2001) and Chiarella and Kwon (2001a). However, these models usually required the introduction of a large number of state variables which, at first sight, did not appear to have clear links to the market observed quantities, and the explicit realisations of the forward rate curve in terms of the state variables were unclear. In this paper, it is shown that the forward rate curves for these models are affine functions of the state variables, and conversely that the state variables in these models can be expressed as affine functions of a finite number of forward rates or yields. This property is useful, for example, in the estimation of model parameters. The paper also provides explicit formulae for the bond prices in terms of the state variables that generalise the formulae given in Inui and Kijima (1998), and applies the framework to obtain affine representations for a number of popular interest rate models.

Classes of Interest Rate Models Under the HJM Framework

Although the HJM term structure model is widely accepted as the mostgeneral, and perhaps the most consistent, framework under which to studyinterest rate derivatives, the earlier models of Vasicek,Cox–Ingersoll–Ross, Hull–White, andBlack–Karasinski remain popular among both academics andpractitioners. It is often stated that these models are special cases ofthe HJM framework, but the precise links have not been fully establishedin the literature. By beginning with certain forward rate volatilityprocesses, it is possible to obtain classes of interest models under theHJM framework that closely resemble the traditional models listed above.Further, greater insight into the dynamics of the interest rate processemerges as a result of natural links being established between the modelparameters and market observed variables.

A Complete Markovian Stochastic Volatility Model in the HJM Framework

This paper considers a stochastic volatility version of the Heath, Jarrow and and Morton (1992) term structure model. Market completeness is obtained by adapting the Hobson and Rogers (1998) complete stochastic volatility stock market model to the interest rate setting. Numerical simulation for a special case is used to compare the stochastic volatility model against the traditional Vasicek (1977) model.

Monte Carlo Market Greeks in the Displaced Diffusion LIBOR Market Model

The problem of developing sensitivities of exotic interest rates derivatives to the observed implied volatilities of caps and swaptions is considered. It is shown how to compute these from sensitivities to model volatilities in the displaced diffusion LIBOR market model. The example of a cancellable inverse floater is considered.

Optimal Betting Strategies for Simultaneous Games

In this paper we consider the problem of optimal betting on simultaneous games when the bookmaker accepts bets on the joint outcome of subsets of any combination of those events, called parlays, accumulators, or multibets. When the bookmakers take is a fixed proportion of the wager, multibetting on all n games replicates any other betting strategy and dominates separate (simultaneous) bets on individual games. When, more typically, the bookmaker quotes multiplicative payouts, and hence takes a higher percentage on multibets than bets on single games, the optimal betting strategy depends on the bettors utility function. We consider the special case of a Kelly (log utility) bettor, and the more general case of a λ-Kelly bettor who wagers a fixed fraction λ ≤ 1 of the full-Kelly bet. Our main result is that when the bookmaker offers multiplicative payouts, a Kelly or λ-Kelly bettor realizes the same monetary outcome by multibetting as would have been achieved by sequential betting on single games (were sequential bets possible). It follows, therefore, that even when games are sequential, a Kelly or λ-Kelly bettor should only bet sequentially when there is an expectation of the bookmakers odds becoming more favorable nearer to game time.

Least Squares Monte Carlo Credit Value Adjustment with Small and Unidirectional Bias

Credit value adjustment (CVA) and related charges have emerged as important risk factors following the Global Financial Crisis. These charges depend on uncertain future values of underlying products, and are usually computed by Monte Carlo simulation. For products that cannot be valued analytically at each simulation step, the standard market practice is to use the regression functions from least squares Monte Carlo method to approximate their values. However, these functions do not necessarily provide accurate approximations to product values over all simulated paths and can result in biases that are difficult to control. Motivated by a novel characterization of the CVA as the value of an option with an early exercise opportunity at a stochastic time, we provide an approximation for CVA and other credit charges that rely only on the sign of the regression functions. The values are determined, instead, by pathwise deflated cash flows. A comparison of CVA for Bermudan swaptions and cancellable swaps shows that the proposed approximation results in much smaller errors than the standard approach of using the regression function values.

A Class of Stochastic Volatility HJM Interest Rate Models

This paper considers a class of stochastic volatility HJM term structure models with explicit finite dimensional realisations. The resulting bond market is arbitrage free but incomplete resulting in a non-unique martingale measure. Nevertheless, the market price of risk is partially determined by the forward rate drift and volatility. Numerical simulation for bond and bond option prices are included to illustrate the effect of stochastic volatility on these prices.

Mean Reversion Level Extensions of Time‐Homogeneous Affine Term Structure Models

It is well‐known that time‐homogeneous affine term structure models are incompatible with most observed initial forward rate curves. For the Vasiček (1977) and Cox et al. (1985) models, time‐inhomogeneous extensions capable of fitting any given initial forward rate curve were introduced in Hull and White (1990), and similar extensions, for short rate models in general, were introduced in Björk and Hyll (2000), Brigo and Mercurio (2001), and Kwon (2004). In this paper, we introduce a general and systematic method for obtaining time‐inhomogeneous extensions of affine term structure models that are compatible with any observed initial forward rate curve. These extensions are minimal in the sense that the system of Riccati equations determining the bond prices remain essentially unchanged under the extension. Moreover, the extensions considered in Björk and Hyll (2000), Brigo and Mercurio (2001), and Kwon (2004), for time‐homogeneous affine term structure models, are all special cases of the extensions introduced in this paper.

Algorithm for Projecting Onto Simplicial Cones and Application to Portfolio Optimization

Many problems from diverse areas of research can be reformulated as the problem of computing the projection of a given vector in a Euclidean space onto the simplicial cone generated by a set of linearly independent vectors. For example, the well-known problem in finance of determining the minimum variance portfolio with no short sales constraint can be transformed to a problem of this type. Although the projection problem does not admit a solution in closed form, various numerical techniques have been proposed for its solution. This paper introduces an efficient new algorithm for the solution to this problem, and compares its performance to other algorithms proposed in the literature. It is shown that the algorithm is very efficient even for high dimensional problems, and orders of magnitude faster than methods introduced, for example, in Barrios, Ferreira and Nemeth (2015).