Monte Carlo Valuation Research Articles

On the Valuation of Discrete Asian Options in High Volatility Environments

ABSTRACT In this paper, we are concerned with the Monte Carlo valuation of discretely sampled arithmetic and geometric average options in the Black-Scholes model and the stochastic volatility model of Heston in high volatility environments. To this end, we examine the limits and convergence rates of asset prices in these models when volatility parameters tend to infinity. We observe, on the one hand, that asset prices, as well as their arithmetic means converge to zero almost surely, while the respective expectations are constantly equal to the initial asset price. On the other hand, the expectation of geometric means of asset prices converges to zero. Moreover, we elaborate on the direct consequences for option prices based on such means and illustrate the implications of these findings for the design of efficient Monte-Carlo valuation algorithms. As a suitable control variate, we need among others the price of such discretely sampled geometric Asian options in the Heston model, for which we derive a closed-form solution.

Option pricing for rice by using Feynman path integral

The options are contracts that give a holder the right to buy or sell an underlying asset at the specific price before or on the expiration date. In return, the holder must pay the premium, the price of an option, to the seller in order to make contracts. As the financial problem occurs with many Thai farmers because of the fluctuation of agricultural product price. In this work, we try to price the American put option by using Feynman path integral in order to create the rice price insurance. The numerical procedures are separated into 2 parts. Firstly, the Atlantic path is determined by quasi-European method. Secondly, hybrid lattice Monte-Carlo method is used to simulate the underlying asset price path in order to evaluate the expectation profit of the option, or the premium. Moreover, the volatility rate of rice price is determined. As a result, the premium of American put option for rice is determined, and there is a deviation less than 8% comparing with Monte Carlo valuation with LSM algorithm in these examples. However, the result suggests that in this work’s assumption, the option pricing by using Feynman path integral is not effective to be used.

High Volatility Limits of Asset Prices with Applications to Option Pricing

In this paper, we are concerned with the Monte Carlo valuation of discretely sampled arithmetic and geometric average options in the Black-Scholes model and the stochastic volatility model of Heston in high volatility environments. To this end, we examine the limits and convergence rates of asset prices in these models when volatility parameters tend to infinity. We observe on the one hand that asset prices as well as their arithmetic means converge to zero almost surely, while the respective expectations are constantly equal to the initial asset price. On the other hand, the expectation of geometric means of asset prices converges to zero. Moreover, we elaborate on the direct consequences for option prices based on such means and illustrate the implications of these findings for the design of efficient Monte-Carlo valuation algorithms. As a suitable control variate, we need among others the price of such discretely sampled geometric Asian options in the Heston model, for which we derive a closed-form solution.

Business Valuation with ODDS: Odds as a Financial Ratio in Business Valuation Theory

Odds in sports betting is a common expression communicating the change and return of winning a bet. Odds as a ratio in business valuation theory is presented in this article. Using odds as a ratio in business valuation helps expressing the probability of a forecasted free cash flow. This might start further discussions and research on how to quantify the operational assumptions made and to bridge the gap between qualitative assumptions made in business valuation and quantitative concepts such as Monte Carlo valuation analysis, credit rating based on distance to default and real options. A literature review, within other areas of science addressing odds, could help in applying odds as a financial ratio in business valuation.

A Structural Model for the Fair Valuation of CDS Index Tranches

We present a Monte Carlo valuation of iTraxx IG index tranches, combining structural and reduced-form models. The aim is to find tranche fair values, taking into account full term structures and actual correlations between individual firms. For 125 firms in the index, 1, 3 and 5-year CDS are used to calibrate term structure parameters: initial asset/debt ratio, asset volatility, and Poisson jump-to-default intensity. Dependency is introduced using a 125x125 equity return correlation matrix, and used to value 0-3%, 3-6%, 6-9%, 9-12%, 12-22% and 22-100% tranches. Initial results overestimate the mezzanine versus equity tranches. In addition, heterogeneity in correlations and CDS has little impact on the equity-to-mezzanine ratio. We demonstrate two extensions towards more realistic pricing: i) a two-correlation approach that allows asset value correlations to differ from jump-time correlations, and ii) allowing random correlations in both asset values and jumps. To our knowledge, this is the first study to calibrate individual CDS term structures and use the parameters in combination with historical or random correlation estimates to price tranches. It provides a framework for combining empirical and market-implied data to value potentially illiquid products.

Valuation of variable long-term care Annuities with Guaranteed Lifetime Withdrawal Benefits: A variance reduction approach

This paper proposes a new product, the Variable Life Care Annuity with Guaranteed Lifetime Withdrawal Benefits (LCA-GLWB), and designs an efficient valuation algorithm. This innovative product provides a comprehensive retirement solution for both longevity risk and long-term care protection. It includes the benefits of guaranteed income streams with downside risk protection and long-term care expenses for retirees. However, the valuation of this type of product is both complex and time-consuming. In this paper, we propose a Monte Carlo valuation algorithm that uses the variance reduction technique. The numerical results indicate that the proposed valuation algorithm is very efficient under a broad range of asset return models. The proposed algorithm provides a general approach for the rapid valuation of similar products and can help provide life insurance companies offering innovative products with an appropriate valuation tool.

An improved least squares Monte Carlo valuation method based on heteroscedasticity

Longstaff–Schwartz’s least squares Monte Carlo method is one of the most applied numerical methods for pricing American-style derivatives. We examine the algorithms regression step, demonstrating that the OLS regression is not the best linear unbiased estimator because of heteroscedasticity. We prove the existence of heteroscedasticity for single-asset and multi-asset payoffs numerically and theoretically, and propose weighted-least squares MC valuation method to correct for it. An extensive numerical study shows that the proposed method produces significantly smaller pricing bias than the Longstaff–Schwartz method under several well-known price dynamics. An empirical pricing exercise using market data confirms the advantages of the improved method.

Practical Valuation of Options on Durable Goods

Manufacturers and retailers in markets with a secondary market continually struggle to differentiate their products and increase the perceived value of their offerings to consumers. Prior literature presented the marketing rationale for manufacturers and dealers in the car industry to offer new car buyers options to buy a new car at a predetermined price or a discount, or to sell the car they buy now at a guaranteed fixed price in the future, on the next purchase event. Previous publications often avoided a systematic valuation of such options. An exception is Afik, Lowengart, and Yosef (2012), (ALY), presenting rigorous mathematical expressions for mathematically tractable price processes. When the valuation of more complex options is required, developing mathematical tools such as those of ALY becomes a Herculean effort and often mathematically impossible. In this paper, on the other hand, we focus our attention on the development and application of Monte Carlo valuation schemes for such options, presenting a very flexible and easy methodology to apply numerical solution to this mathematical challenge.

Theory and Validation of Replicating Portfolios in Insurance Risk Management

The Solvency II framework challenges insurers to evaluate and manage their embedded balance sheet risks appropriately. However, insurances hold balance sheet items, for which closed-form solutions and market prices are not available. Pure Monte Carlo valuation requires nested simulations, which are too time-intensive. Therefore, methods that project these balance sheet items into functional representations, which simplify and enhance risk analysis, have been suggested. Among these, replicating portfolios are widely applied in practice, though their validity and properties have not been fully examined yet. This paper corrects this shortcoming and proposes a mathematical framework within which the asymptotic properties of replicating portfolios are analyzed. It is shown that the replicating portfolio problem is mathematically well-defined and asymptotically converges to the true solution. Hence, this paper provides a general mathematical validation for replicating portfolios applied in insurance. A typical path-dependent insurance policy is discussed within the framework and numerical results are presented.

Technology investment decision-making under uncertainty

Innovations involving information technology (IT) provide potentially valuable investment opportunities for industry and government organizations. Significant uncertainties are associated with decision-making for IT investment though, a problem that senior executives have been concerned about for a long time. The uncertainties include consumer, market and regulatory responses, IT-driven changes in operational and transactional performance, technology standards and competition, and future market conditions. All these things have an impact on organizations’ willingness to adopt. As a result, traditional capital budgeting, investment experience, and intuition have not been very effective in IT investment decision-making. We propose a new option-based stochastic valuation modeling approach for IT investment under uncertainty that incorporates a mean reversion process to capture cost and benefit flow variations over time. We apply the proposed approach in two industry settings: to a large-scale IT investment in the consolidation of data marts at a major airline, and to a mobile payment system infrastructure investment on the part of a start-up. The applications supported the evaluation of the proposed methods, and offered some illustrations about the kinds of managerial insights that can be obtained. We also report on several extensions that demonstrate how the creation of useful management findings from the modeling approach can be supplemented with project value sensitivity analysis and the use of simulation-based least-squares Monte Carlo valuation. The findings are useful to assess the power and value of the approach.

Canonical Least-Squares Monte Carlo Valuation of American Options: Convergence and Empirical Pricing Analysis

The paper by Liu (2010) introduces a method termed the canonical least-squares Monte Carlo (CLM) which combines a martingale-constrained entropy model and a least-squares Monte Carlo algorithm to price American options. In this paper, we first provide the convergence results of CLM and numerically examine the convergence properties. Then, the comparative analysis is empirically conducted using a large sample of the S&P 100 Index (OEX) puts and IBM puts. The results on the convergence show that choosing the shifted Legendre polynomials with four regressors is more appropriate considering the pricing accuracy and the computational cost. With this choice, CLM method is empirically demonstrated to be superior to the benchmark methods of binominal tree and finite difference with historical volatilities.

LOOKBACK OPTION PRICES UNDER A SPECTRALLY NEGATIVE TEMPERED-STABLE MODEL

We perform a Laplace transform inversion in the time parameter on the two Wiener-Hopf factors for a spectrally negative tempered stable Lévy process. This yields the issuing price of continuously monitored lookback options. We also propose a simulation technique for the purpose of Monte-Carlo valuation and discuss the convergence rate to continuous prices when the number of discretization steps (i.e. monitoring dates) goes to infinity.

Valuing Investments in Multi-Energy Conversion, Storage, and Demand-Side Management Systems Under Uncertainty

In this paper, a financial valuation method for energy hubs with conversion, storage, and demand-side management (DSM) capabilities is proposed. An energy hub is an integrated system of units, e.g., a combined heat and power plant and a heat storage, which allows the conversion and storage of multiple energy carriers. In this paper, an extended energy hub model is presented which additionally takes into account the possibility of performing DSM with the load(s) connected to the hub output. Taking into account the energy hub's flexibility to change its output power(s), its economic value is determined with a method based on Monte Carlo simulation. This method calculates an optimal dispatch of the hub for a large amount of possible price paths of the input and output energy carriers. By means of the proposed energy hub Monte Carlo valuation model, integrated systems of multi-energy conversion and storage devices can be valued together with load management schemes. In doing so, the energy hub's ability to flexibly adapt its output to uncertain and volatile market prices is explicitly considered.

Unbiased Monte Carlo valuation of lookback, swing and barrier options with continuous monitoring under variance gamma models

We introduce fast and unbiased methods for Monte Carlo valuation of lookback, swing, and barrier options under variance gamma models. For the valuation of lookback and swing options, a procedure to draw samples of the final value, infimum, and supremum of variance gamma processes with an additional drift term up to arbitrary precision is developed; for barrier options, a separate method is designed to improve the performance by exploiting the particular payout structure. All algorithms substantially rely on adaptive difference-of-gammas bridge sampling, a newly introduced enhancement of the truncated difference-of-gammas bridge sampling method originally presented by Avramidis et al (2003) and successfully applied to lookback and barrier option valuation by Avramidis (2004) and Avramidis and L'Ecuyer (2006). By applying our algorithms to examples treated by Avramidis (2004) and Avramidis and L'Ecuyer (2006) we observe considerable reductions in computational effort and memory requirements. Furthermore, our algorithms feature a priori bounds on the bias, whereas Avramidis and L'Ecuyer (2006) are restricted to a posteriori bounds.

Comment on ‘Correcting for Simulation Bias in Monte Carlo Methods to Value Exotic Options in Models Driven by Lévy Processes’ by C. Ribeiro and N. Webber

Ribeiro and Webber (2006) propose a method to correct for simulation bias in the Monte Carlo valuation of options with pay-offs depending on the extreme value(s) of the underlying which is driven by a special Lévy process, namely a normal inverse Gaussian (NIG) or a variance gamma (VG) process. The proposed method was already successfully used by Beaglehole et al. (1997) and El Babsiri and Noel (1998) when the underlying follows a Brownian motion. Unfortunately, Ribeiro and Webber, in their attempt to exploit well-known subordinator representations of NIG and VG processes, overlook the fact that these subordinator representations lead to discontinuous subordinators. Therefore their correction method ‘overcorrects’ the simulation bias by magnitudes, resulting in a much bigger simulation bias with reversed sign. We point out where the assumption of a continuous subordinator is implicitly used in the paper of Ribeiro and Webber (2006). Furthermore, by applying the unbiased Monte Carlo valuation approach for Barrier options under VG models of Becker (2009) to the barrier and lookback options considered in Ribeiro and Webber (2006), we show that the newly introduced simulation bias exceeds the corrected simulation bias by far.

Analytical approximations for prices of swap rate dependent embedded options in insurance products

Life insurance products have profit sharing features in combination with guarantees. These so-called embedded options are often dependent on or approximated by forward swap rates. In practice, these kinds of options are mostly valued by Monte Carlo simulations. However, for risk management calculations and reporting processes, lots of valuations are needed. Therefore, a more efficient method to value these options would be helpful. In this paper analytical approximations are derived for these kinds of options, based on an underlying multi-factor Gaussian interest rate model. The analytical approximation for options with direct payment is almost exact while the approximation for compounding options is also satisfactory. In addition, the proposed analytical approximation can be used as a control variate in Monte Carlo valuation of options for which no analytical approximation is available, such as similar options with management actions. Furthermore, it’s also possible to construct analytical approximations when returns on additional assets (such as equities) are part of the profit sharing rate.

Monte Carlo valuation of natural gas investments

In this evaluation of energy assets related to natural gas, our particular focus is on a base load natural gas combined cycle power plant and a liquefied natural gas facility in a realistic setting. We also value several American-type investment options following the least squares Monte Carlo approach. We calibrate mean-reverting stochastic processes for gas and electricity prices by using data from NYMEX NG futures contracts and the Spanish wholesale electricity market, respectively. Additional sources of uncertainty concern the initial investment outlay, or the option's time to maturity, or the cost of CO 2 emission permits.

Valuation and incentive effects of hurdle rate executive stock options

Traditional executive stock options are often criticized for inherently weak links between pay and performance. Hurdle rate executive stock options represent a viable improvement. However, valuing these options presents extraordinary analytic difficulties. With a constant dividend yield the strike price becomes a path-dependent function of the stock price and exact analytic valuation is intractable. To solve this problem, we apply the Monte Carlo valuation approach developed by Longstaff and Schwartz (Rev Financ Stud 4:113–147, 2001) to estimate the value of path-dependent American options. We also extend the methodology to incorporate the theoretical framework by Ingersoll (J Bus 79:453–487, 2006) to permit subjective valuation influenced by an executive’s risk aversion.

Barrier Options on Spot LIBOR Rates under Multi–Factor Gaussian HJM Models

Plain vanilla caps and floors already present problems for valuation and risk management because the distributions of forward rates generated by the underlying interest rate process must be evaluated for multiple future dates at the same time. Introducing exotic features like barriers only makes things harder. But using a variety of techniques, including change of numeraire, stochastic time-change, and an approximation to the Radon-Nikodym derivative, Nunes is able to obtain approximate pricing models in closed-form for single-barrier caps and floors in a multivariate Gaussian Heath-Jarrow-Morton framework. Comparisons against a Monte Carlo solution and an alternative quasi-analytic technique demonstrate that this technique can produce a huge improvement in performance, in some cases achieving better accuracy in under a second than Monte Carlo valuation reaches in more than 24 hours.

American options and the LSM algorithm: quasi-random sequences and Brownian bridges

The least-squares Monte Carlo (LSM) algorithm of Longstaff and Schwartz (2001) is a method for Monte Carlo valuation of the price of American options. We use quasi-random sequences to generate asset prices which follow geometric Brownian motions, and obtain a significant increase in the rate of convergence of American min-puts and American-Bermuda-Asian calls. Using the Brownian-bridge formula, we present a method for reducing the memory requirements from O(N x M x d) to O(Nd log(M)) for quasi-random sequences, where N is the number of paths, M is the number of timesteps and d is the number of assets, for example, in a min-put.