Abstracts

In mathematical finance a model of a financial market is said to be complete if any payoff can be obtained as the terminal value of a self-financing trading strategy. The completeness property is important because it allows the hedging of any non-traded derivative security. Yet, numerous models, such as stochastic volatility models or structural models in commodity markets are incomplete. We present conditions, in a general diffusion framework, which guarantee that in such cases the market of primitive assets enlarged with an appropriate number of traded derivative contracts is complete. Key to the proof is the analyticity in time and space of the coefficients of the model, allowing us to draw upon recent results on the analytic smoothing properties of linear parabolic partial differential equations.

In this paper, a new technique for the estimation of parameters for the stochastic volatility model is developed. Although several techniques including Markov chain Monte Carlo were applied and developed for this problem, most of their results were very sensitive for the initial value or random seed selection. Because of the problems mentioned above, it was difficult to apply those developed method directly to the real data. In the proposed method, unlikely in the original Markov chain Monte Carlo methods, the parameters are separated by some criteria and estimated iteratively like many other iterative methods such as expectation-maximization algorithm and this makes robust results. Applied to the artificial datasets, the proposed methods found the parameters and those results were robust compared to the parameters from the original Markov chain Monte Carlo. Additionally, it is successfully applied to the real market data including several indices and stock prices.

European interest rate markets use cash-settlement (CS) for swaptions, the main interest rate volatility instrument. In the market only at-the-money (ATM) straddles and out-of-the-money (OTM) payers/receivers are liquidly quoted and usable as calibration instruments. As a result all ITM payer/receiver swaption prices are model-based. CS-swaptions form the basis for all volatility-sensitive interest rate product valuations such as constant maturity swaps (CMS), Bermudan swaptions and callable libor exotics. Therefore, mispricings of CS-swaptions impacts the whole spectrum of interest rate volatility products.
As the difference between physical- and cash-settlement is often considered minor, a Black model based market-approximation (MA) formula is heavily used to value cash-settled swaptions across the industry. We show that at the forward swap rate (ATM) the value of cash-settled payer and receiver swaptions are economically significantly different, contrary to physical-settled swaptions. However, the MA formula implies equality for ATM payers/receivers. We will show that using the MA formula leads to economically significant mispricings of both ATM and ITM swaptions. Furthermore, we show problems in swaption portfolio risk management and significant mispricing of constant maturity swap (CMS) products using replication.
This paper provides an overview of the market formula followed by a detailed analysis of the key reasons for valuation/sensitivity differences. We propose a stochastic volatility model to accurately capture the convex/concave payoff profile. The model is capable of accounting for the non-linear payoff profile while being analytically tractable. Furthermore, it has enough degree of freedom to be calibrated accurately to the implied volatility smile and skew in the swaption market. We report several key findings. The implied volatilities for cash-settled payer and receiver swaptions are shown to be different due to the convex and concave payoff profile. ITM swaptions can be valued effectively using the semi-analytical model formulated in this work, allowing for efficient risk management of large swaption portfolio. In addition, it is observed that vega for cash-settled payer swaptions could become negative in high interest rates scenarios. Finally, we detail the implications of these mispricings for CMS products.

The class of distortion risk measures is the one we should use if we stick to the law invariance and comonotonic additivity in addition to the four axioms of coherence, and is broad enough to fully express agents’ subjective assessment of risk (conservatism). If one is to apply this risk measure to practical financial risk management problems, it is necessary to pick one distortion function depending on his/her attitude towards risk.
In our previous work, we have shown that estimating distortion risk measure is possible based on general weakly dependent times series data. The next step in financial risk management is backtesting. The purpose of backtesting is twofold: to monitor the performance of the model and estimation methods for risk measurement, and to compare relative performance of the models and methods. It is a tool for the validation process which is indispensible for adequate risk management and is even required by the Basel 2 Capital Accord.
Statistically, it is just a form of cross validation; the ex ante risk measure forecast from the model is compared with the ex post realized portfolio loss. In the case of Value-at-Risk (VaR), a popular procedure for backtesting VaR depends on the number of VaR violations; we say that a VaR violation occurred when the loss exceeds VaR. Extending a backtesting procedure for the renowned expected shortfall (ES), we suggest, based on a simple observation, a backtesting procedure for distortion risk measures, and check its effectiveness in a simulation study using ES and also proportional odds distortion risk measure.
Many people claim that it is easier to backtest VaR than ES and other risk measures for the following reasons: (i) the existing tests for ES are based on parametric assumptions for the null distribution; (ii) asymptotic approximation is needed for the null distribution of the test statistics; (iii) testing an expectation is harder than testing a single quantile. Summing up these arguments, what they call ‘backtestability’of a risk measure means that it can be nonparametrically backtested with small samples.
It has recently been claimed that expected shortfall (and distortion risk measures) cannot be backtested because it fails to satisfy the so-called elicitability condition. Roughly speaking, a statistical functional is called elicitable if it is a unique minimizer of some expected loss function. While elicitability is useful when we wants to compare and rank several estimation procedures, there seems to be no clear connection with backtestability. We will try to illustrate this with simple examples including the expectiles.

We consider the stochastic control model with finite time horizon for a mixed duopoly R&D (Research and Development) race between the profit-maximizing private firm and welfare maximizing public firm. In our two-firm stochastic control R&D race model, the stochastic control variable is taken to be the private firm’s rate of R&D expenditure and the hazard rate of success of innovation has dependence on the R&D effort and knowledge stock. Given the fixed R&D effort of the public firm, the optimal control is determined so as to maximize the private firm’s value function subject to market uncertainty arising from the stochastic profit flow of the new innovative product. Our R&D race model also incorporates the impact of input and output spillovers. We apply the Bellman optimality condition to construct the Hamilton-Jacobi-Bellman equation of the stochastic control model. Finite difference schemes together with policy iteration procedure are constructed for the numerical solution of the value function and optimal control of R&D expenditure of the private firm. We conduct various sensitivity tests with varying model parameters to analyze the effects of input spillover, output spillover and knowledge stock on the optimal control policy and the value function of the profit-maximizing private firm. The R&D effort of the private firm is found to increase when the profit flow rate increases. Moreover, the optimal R&D effort level may decrease with increasing private firm’s knowledge stock and output spillover. The effects of input spillover on the optimal control policy and value function are seen to be relatively small. We examine the robustness of various observed phenomena of the two-firm R&D race with varying values of the fixed R&D effort of the public firm. With regard to public policy issue, we examine the level of the fixed public firm’s R&D effort so that social welfare is maximized.

It is a well-established empirical fact that there are jumps in real world assets prices. These complicate calibration due to the additional parameters and they also make hedges based on Taylor expansions (delta, Gamma etc. ) perform suboptimal, see Brodén and Tankov [2011].
It was found in Lindström et al. [2008] that the calibration can be written as a non-linear filter problem, which is faster and more robust than (penalized) weighted least squares. The filter idea was extended to simultaneous calibration and quadratic hedging (computed under the risk neutral measure) in Lindström and Guo [2013], a result that was achieved by augmenting the filter problem with the underlying asset as an additional state variable.
There are two main contributions in this paper. The primary contribution is the computational complexity, as we are unable to see that calibration can be done with fewer function evaluations (price calculations) than what is done here. The second contribution is that parameter uncertainty; cf. Lindström [2010] is taken into account when calculating the hedges. This is important as the price for having a flexible model often is many parameters, and hence large parameter uncertainty.
We find that the resulting algorithm, implemented using an unscented Kalman filter and Fourier based option valuation, is computationally very competitive, with hedges that often are similar to the ordinary quadratic hedges when these can be computed. A nice feature is that quadratic hedges using several hedge instruments are obtained with very few (and inexpensive) additional computations, since the required covariances are already calculated in the filter. Another feature of the proposed algorithm is that only prices are required to compute hedges for exotic options, simplifying quadratic hedging of these options substantially.