An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates

This paper proposes an asymptotic expansion scheme of currency options with a libor market model of interest rates and stochastic volatility models of spot exchange rates. In particular, we derive closed-form approximation formulas for the density functions of the underlying assets and for pricing currency options based on a third order asymptotic expansion scheme; we do not model a foreign exchange rate’s variance such as in Heston [(1993) The Review of Financial studies, 6, 327–343], but its volatility that follows a general time-inhomogeneous Markovian process. Further, the correlations among all the factors such as domestic and foreign interest rates, a spot foreign exchange rate and its volatility, are allowed. Finally, numerical examples are provided and the pricing formula are applied to the calibration of volatility surfaces in the JPY/USD option market.     

A quasi-analytical interpolation method for pricing American options under general multi-dimensional diffusion processes

We present a quasi-analytical method for pricing multi-dimensional American options based on interpolating two arbitrage bounds, along the lines of Johnson in J Financ Quant Anal 18(1):141–148 (1983). Our method allows for the close examination of the interpolation parameter on a rigorous theoretical footing instead of empirical regression. The method can be adapted to general diffusion processes as long as quick and accurate pricing methods exist for the corresponding European and perpetual American options. The American option price is shown to be approximately equal to an interpolation of two European option prices with the interpolation weight proportional to a perpetual American option. In the Black-Scholes model, our method achieves the same efficiency as the quadratic approximation of Barone-Adesi and Whaley in J Financ 42:301–320 (1987), with our method being generally more accurate for out-of-the-money and long-maturity options. When applied to Heston’s stochastic volatility model, our method is shown to be extremely efficient and fairly accurate.     

Optimal portfolio choice in the bond market

We consider the Merton problem of optimal portfolio choice when the traded instruments are the set of zero-coupon bonds. Working within a Markovian Heath–Jarrow–Morton model of the interest rate term structure driven by an infinite-dimensional Wiener process, we give sufficient conditions for the existence and uniqueness of an optimal trading strategy. When there is uniqueness, we provide a characterization of the optimal portfolio as a sum of mutual funds. Furthermore, we show that a Gauss–Markov random field model proposed by Kennedy [Math. Financ. 4, 247–258(1994)] can be treated in this framework, and explicitly calculate the optimal portfolio. We show that the optimal portfolio in this case can be identified with the discontinuities of a certain function of the market parameters.     

Duration, factor sensitivities, and interest rate Greeks

In this paper, we introduce for interest rate sensitive assets the natural analogs of delta and gamma for equity options by considering the derivatives of asset prices with respect to the directions along which the forward rate curve may evolve. Macaulay duration and convexity, as well as stochastic duration considered in Cox et al. (J Business 52:51–61, 1979) and Munk (Rev Derivat Res 3:157–181, 1999), are easily obtained as special cases of these in which the derivatives are computed along parallel shifts and the direction of the forward rate volatilities, respectively. Moreover, we demonstrate using the example of the Ritchken and Sankarasubramanian (Math Financ 5:55–72, 1995) model that the hedging strategy based on these sensitivity measures provides a superior performance in comparison to the traditional duration based hedging approaches.      

Empirically Effective Bond Pricing Model and Analysis on Term Structures of Implied Interest Rates in Financial Crisis

In his book (1993) Kariya proposed a government bond (GB) pricing model that simultaneously values individual fixed-coupon (non-defaultable) bonds of different coupon rates and maturities via a discount function approach, and Kariya and Tsuda (Financ Eng Japanese Mark 1:1–20, 1994) verified its empirical effectiveness of the model as a pricing model for Japanese Government bonds (JGBs) though the empirical setting was limited to a simple case. In this paper we first clarify the theoretical relation between our stochastic discount function approach and the spot rate or forward rate approach in mathematical finance. Then we make a comprehensive empirical study on the capacity of the model in view of its pricing capability for individual GBs with different attributes and in view of its capacity of describing the movements of term structures of interest rates that JGBs imply as yield curves. Based on various tests of validity in a GLS (Generalized Least Squares) framework we propose a specific formulation with a polynomial of order 6 for the mean discount function that depends on maturity and coupon as attributes and a specific covariance structure. It is shown that even in the middle of the Financial Crisis, the cross-sectional model we propose is shown to be very effective for simultaneously pricing all the existing JGBs and deriving and describing zero yields.     

The price of risk and ambiguity in an intertemporal general equilibrium model of asset prices

We consider a version of the intertemporal general equilibrium model of Cox et?al. (Econometrica 53:363–384, 1985) with a single production process and two correlated state variables. It is assumed that only one of them, Y ₂, has shocks correlated with those of the economy’s output rate and, simultaneously, that the representative agent is ambiguous about its stochastic process. This implies that changes in Y ₂ should be hedged and its uncertainty priced, with this price containing risk and ambiguity components. Ambiguity impacts asset pricing through two channels: the price of uncertainty associated with the ambiguous state variable, Y ₂, and the interest rate. With ambiguity, the equilibrium price of uncertainty associated with Y ₂ and the equilibrium interest rate can increase or decrease, depending on: (i) the correlations between the shocks in Y ₂ and those in the output rate and in the other state variable; (ii) the diffusion functions of the stochastic processes for Y ₂ and for the output rate; and (iii) the gradient of the value function with respect to Y ₂. As applications of our generic setting, we deduct the model of Longstaff and Schwartz (J Financ 47:1259–1282, 1992) for interest-rate-sensitive contingent claim pricing and the variance-risk price specification in the option pricing model of Heston (Rev Financ Stud 6:327–343, 1993). Additionally, it is obtained a variance-uncertainty price specification that can be used to obtain a closed-form solution for option pricing with ambiguity about stochastic variance.     

Extreme Events and the Copula Pricing of Commercial Mortgage-Backed Securities

Commercial mortgage-backed securities (CMBS), as a portfolio-based financial product, have gained great popularity in financial markets. This paper extends Childs, Ott and Riddiough’s (J Financ Quant Anal, 31(4), 581–603, 1996) model by proposing a copula-based methodology for pricing CMBS bonds. Default on underlying commercial mortgages within a pool is a crucial risk associated with CMBS transactions. Two important issues associated with such default—extreme events and default dependencies among the mortgages—have been identified to play crucial roles in determining credit risk in the pooled commercial mortgage portfolios. This article pays particular attention to these two issues in pricing CMBS bonds. Our results show the usefulness and potential of copula-based models in pricing CMBS bonds, and the ability of such models to correctly price CMBS tranches of different seniorities. It is also important to sufficiently consider complex default dependence structure and the likelihood of extreme events occurring in pricing various CMBS bonds.     

A super-replication theorem in Kabanov’s model of transaction costs

We prove a general version of the super-replication theorem, which applies to Kabanov’s model of foreign exchange markets under proportional transaction costs. The market is described by a matrix-valued càdlàg bid-ask process evolving in continuous time. We propose a new definition of admissible portfolio processes as predictable (not necessarily right- or left- continuous) processes of finite variation related to the bid-ask process by economically meaningful relations. Under the assumption of existence of a strictly consistent price system (SCPS), we prove a closedness property for the set of attainable vector-valued contingent claims. We then obtain the super-replication theorem as a consequence of that property, thus generalizing to possibly discontinuous bid-ask processes analogous results obtained by Kabanov (Financ. Stoch. 3, 237–248, 1999), Kabanov and Last (Math. Financ. 12, 63–70, 2002) and Kabanov and Stricker (Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann, pp 125–136, 2002). Rásonyi’s counter-example (Lecture Notes in Mathematics 1832, 394–398, 2003) served as an important motivation for our approach.     

The pricing of Bermudan-style options on correlated assets

In this paper, we present a methodology for approximating a correlated multivariate-lognormal process with a recombining or “simple” multivariate-binomial process. The method represents an extension and implementation of previous work by Nelson and Ramaswamy (1990) and Ho, Stapleton and Subrahmanyam (1995) on diffusion approximation. The general method is illustrated by pricing a Bermudan-style put option on the minimum of three asset prices, and by pricing Bermudan-style options on bonds, where the value of the bond at a point in time depends upon the interest rate in two currencies and the foreign exchange rate. This type of structure, known as the “Power Reverse Dual” is a popular product in the case of Japanese Yen-US Dollar currencies. This revised version was published online in June 2006 with corrections to the Cover Date.     

A risk-sensitive stochastic control approach to an optimal investment problem with partial information

We consider an infinite time horizon optimal investment problem where an investor tries to maximize the probability of beating a given index. From a mathematical viewpoint, this is a large deviation probability control problem. As shown by Pham (in Syst. Control Lett. 49: 295–309, 2003; Financ. Stoch. 7: 169–195, 2003), its dual problem can be regarded as an ergodic risk-sensitive stochastic control problem. We discuss the partial information counterpart of Pham (in Syst. Control Lett. 49: 295–309, 2003; Financ. Stoch. 7: 169–195, 2003). The optimal strategy and the value function for the dual problem are constructed by using the solution of an algebraic Riccati equation. This equation is the limit equation of a time inhomogeneous Riccati equation derived from a finite time horizon problem with partial information. As a result, we obtain explicit representations of the value function and the optimal strategy for the problem. Furthermore we compare the optimal strategies and the value functions in both full and partial information cases.

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Issuers’ incentives and tests of Baron’s model of IPO underpricing

This paper reexamines the validity of Baron’s (J Financ 37:955–976, 1982) model of IPO underpricing, in which IPO underpricing is caused by asymmetric information between issuers and investment bankers. Muscarella and Vetsuypens (J Financ Econ 24:125–135, 1989) find that lead-manager IPOs are significantly more underpriced than non-self-marketed IPOs and conclude that their empirical results do not support Baron’s model. We compare self-marketed underwriters’ IPOs with non-self-marketed underwriters’ IPOs and with IPOs they lead. Our empirical results show that it is premature to reject Baron’s model of IPO underpricing when we take issuer incentives into account.     

The Pricing Formula for Commodity-Linked Bonds with Stochastic Convenience Yields and Default Risk

At the maturity, the owner of a commodity-linked bond has the right to receive the face value of the bond and the excess amount of spot market value of the reference commodity bundle over the prespecified exercise price. This payoff structure is an important characteristic of the commodity-linked bonds. In this paper, we derive closed pricing formulae for the commodity-linked bonds. We assume that the reference commodity price and the value of the firm (bonds' issuer) follow geometric Brownian motions and that the net marginal convenience yield and interest rate follow Ornstein–Uhlenbech processes. In the appendix, we derive pricing formulae for bonds which are the same as the above commodity-linked bonds, except that the reference commodity price in the definition of the payoff at the maturity is replaced by the value of a special asset which depends on the convenience yield. This revised version was published online in August 2006 with corrections to the Cover Date.     

A forward started jump-diffusion model and pricing of cliquet style exotics

In this paper we present an alternative model for pricing exotic options and structured products with forward-starting components. As presented in the recent study by Eberlein and Madan (Quantitative Finance 9(1):27–42, 2009), the pricing of such exotic products (which consist primarily of different variations of locally/globally, capped/floored, arithmetic/geometric etc. cliquets) depends critically on the modeling of the forward–return distributions. Therefore, in our approach, we directly take up the modeling of forward variances corresponding to the tenor structure of the product to be priced. We propose a two factor forward variance market model with jumps in returns and volatility. It allows the model user to directly control the behavior of future smiles and hence properly price forward smile risk of cliquet-style exotic products. The key idea, in order to achieve consistency between the dynamics of forward variance swaps and the underlying stock, is to adopt a forward starting model for the stock dynamics over each reset period of the tenor structure. We also present in detail the calibration steps for our proposed model.     

The supermartingale property of the optimal wealth process for general semimartingales

We consider an incomplete stochastic financial market where the price processes are described by a vector valued semimartingale that is possibly non locally bounded. We face the classical problem of utility maximization from terminal wealth, under the assumption that the utility function is finite-valued and smooth on the entire real line and satisfies reasonable asymptotic elasticity. In this general setting, it was shown in Biagini and Frittelli (Financ. Stoch. 9, 493–517, 2005) that the optimal claim admits an integral representation as soon as the minimax σ-martingale measure is equivalent to the reference probability measure. We show that the optimal wealth process is in fact a supermartingale with respect to every σ-martingale measure with finite generalized entropy, thus extending the analogous result proved by Schachermayer (Financ. Stoch. 4, 433–457, 2003) for the locally bounded case.      

Using Nonnormal Distributions to Analyze the Relationship Between Stock Returns in Japan and the US

The Pearson distribution system is researched and applied to financial engineering (Nagahara, Financ Eng Jpn Mark 2(2):139–154 in 1995, Financ Eng Jpn Mark 3(2):121–149 in 1996, Stat Prob Lett 43:251–264 in 1999, J Time Ser Anal 24(6):721–738 in 2003, A method of fitting multivariate nonnormal distributions to financial data. Discussion paper of Institute of Social Sciences, F-2006-2, Meiji University in 2006, Asia Pac Financial Markets 15(3–4):175–184 in 2008a). And a method of fitting multivariate nonnormal distributions by using random numbers from the Pearson distribution system was developed (Nagahara, Comput Stat Data Anal 47(1):1–29 in 2004). This method uses the grid search of the parameters for the maximum likelihood. In this paper, we adopt Grid-Computing and its middleware for the parameter sweep in order to reduce the computational time and the workload of this method. In the area of the financial risk management, it is very important to analyze the relationship between stock returns in Japan and the US. We analyze the data based on the same date and the following date because Japanese stock market opens before the US stock market opens in a day. We compare these returns by means of the multivariate nonnormal distributions by using this method. And we test the international transmission of stock markets movement. Furthermore, we obtain the optimal job schedule for our computer system using the middleware in order to reduce the computational time.     

Heterogeneous time varying transaction costs and asset pricing in international equity markets

In this paper, we examine the time variation in transaction costs relative to excess returns, in a panel consisting of 10 international equity indices over the time period 1984–2005. This is undertaken by extending the consumption CAPM (CCAPM) model proposed by Campbell and Shiller (Rev. Financ. Stud. 1:195–228, 1988) to incorporate time varying proportional transaction costs. We rigorously address both the cross-country heterogeneity in the estimated model and endogeneity. We find strong evidence that suggests transaction costs should be included as an additional explanatory variable in the CCAPM. This leads to the conclusion that transaction costs should be included in asset pricing models as their stochastic process impacts directly on private consumption expenditure.

Are the Fama–French factors proxying news related to GDP growth? The Australian evidence

Inspired by Vassalou (J Financ Econ 68:47–73, 2003), we investigate the contention that the Fama and French (J Financ Econ 33:3–56, 1993) model’s ability to explain the cross sectional variation in equity returns is because the Fama–French factors are proxying for risk associated with future GDP growth in the Australian equities market. To assess the validity of Vassalou’s findings, we augment the CAPM and the Fama–French model with a GDP growth factor and run system regressions of the GDP-enhanced models using the GMM approach. Our results suggest that news about future GDP growth is not priced in equity returns and that any ability that SMB and HML exhibit in explaining equity returns is not because they contain information about future GDP growth.

A Note on Gaussian Estimation of the CKLS and CIR Models with Feedback Effects for Japan

In this note we extend the Gaussian estimation of two factor CKLS and CIR models recently considered in Nowman, K. B. (2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34) to include feedback effects in the conditional mean as was originally formulated in general continuous time models by Bergstrom, A. R. (1966, Non-recursive models as discrete approximations to systems of stochastic differential equations, Econometrica 34, 173–182) with constant volatility. We use the exact discrete model of Bergstrom, A. R. (1966, Non-recursive models as discrete approximations to systems of stochastic differential equations, Econometrica 34, 173–182) to estimate the parameters which was first used by Brennan, M. J. and Schwartz, E. S. (1979, A continuous time approach to the pricing of bonds, J. Bank. Financ. 3, 133–155) to estimate their two factor interest model but incorporating the assumption of Nowman, K. B. (1997, Gaussian estimation of single-factor continuous time models of the term structure of interest rates, J. Financ. 52, 1695–1706; 2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34). An application to monthly Japanese Euro currency rates indicates some evidence of feedback from the 1-year rate to the 1-month rate in both the CKLS and CIR models. We also find a low level-volatility effect supporting Nowman, K. B. (2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34).     

Pricing Property Index Linked Swaps with Counterparty Default Risk

This paper extends Bjork and Clapham (Journal of Housing Economics 11:418–432, 2002) model for pricing real estate index total return swaps. Our extension considers counterparty default risk within a first passage contingent claims model. We price total return swaps on property indices with different levels of default risk. We develop this model under same assumptions as Bjork and Clapham (Journal of Housing Economics 11:418–432, 2002) and find that total return swap price is no longer zero. Total return swap payer must charge a spread over the market interest rate that compensates him for the exposure to this additional risk. Based on commercial property indices in the UK, we observe that computed spreads are much lower than a sample of quotes obtained from one of the traders in the market.     

Does knowing the volatility states affect the market risk premium?

Mayfield (J Financ Econ 73:465–496, 2004) has devised a method for estimating the market risk premium, based on a variant of Merton’s ICAPM wherein volatility is specified as a two-state Markov process. In this study, we assess Mayfield’s key assumption that investors know the current volatility state with certainty, via empirical testing of the assumption of exogenous Markov-switching in Mayfield’s model. We detect strong evidence of endogenous switching. This indicates that investors infer the current volatility state, as opposed to simply observing it. We also find that the risk premium estimates are affected by the switching type.