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This paper proposes a framework for the valuation and the management of complex life insurance contracts, whose design can be described by a portfolio of embedded options, which are activated according to one or more triggering events. These events are in general monitored discretely over the life of the policy, due to the contract terms. Similar designs can also be found in other contexts, such as counterparty credit risk for example. The framework is based on Fourier transform methods as they allow to derive convenient closed analytical formulas for a broad spectrum of underlying dynamics. Multidimensionality issues generated by the discrete monitoring of the triggering events are dealt with efficiently designed Monte Carlo integration strategies. We illustrate the tractability of the proposed approach by means of a detailed study of ratchet variable annuities, which can be considered a prototypical example of these complex structured products.

The Singapore Savings Bonds (SSB) is a unique investment program offered by the Singapore government whereby retail investors can earn risk-free tax-free step-up interest closely matched to Treasury bond rates for up to 10 years and can redeem on any business day prior to maturity without any early redemption penalty. This study analyses the unique design of the SSB and provides a valuation of the Bermudan option for early redemption that is embedded in the SSB. The Black–Derman–Toy model is used to build the interest rate tree, and an iterative method is employed to avoid arbitrary specification of the pre-determined short rate volatility function. This bespoke Bermudan option can have changing strike prices over time. It also has a novel characteristic whereby the value of exercise to a buyer need not equal to the cost of being exercised to a seller. Better understanding of embedded options within government savings bonds leads to innovative designs that may encourage effective citizens’ savings.

This paper discusses the valuation of credit default swaps, where default is announced when the reference asset price has gone below certain level from the last record maximum, also known as the high-water mark or drawdown. We assume that the protection buyer pays premium at a fixed rate when the asset price is above a pre-specified level and continuously pays whenever the price increases. This payment scheme is in favour of the buyer as she only pays the premium when the market is in good condition for the protection against financial downturn. Under this framework, we look at an embedded option which gives the issuer an opportunity to call back the contract to a new one with reduced premium payment rate and slightly lower default coverage subject to paying a certain cost. We assume that the buyer is risk neutral investor trying to maximize the expected monetary value of the option over a class of stopping time. We discuss optimal solution to the stopping problem when the source of uncertainty of the asset price is modelled by Levy process with only downward jumps. Using recent development in excursion theory of Levy process, the results are given explicitly in terms of scale function of the Levy process. Furthermore, the value function of the stopping problem is shown to satisfy continuous and smooth pasting conditions regardless of regularity of the sample paths of the Levy process. Optimality and uniqueness of the solution are established using martingale approach for drawdown process and convexity of the scale function under Esscher transform of measure. Some numerical examples are discussed to illustrate the main results.

This study addresses interest rate sensitivity of emerging market corporate debt. Previous research suggests that interest rate sensitivity of corporate bonds depends on residual maturity of issues, creditworthiness of issuers, embedded options and other idiosyncratic factors. However, the dependence of interest rate sensitivity on phases of the business cycle has not received an appropriate academic attention. This paper provides empirical evidence and theoretical interpretation of a dichotomy of interest rate sensitivity across the phases of the cycle, and sheds light on how credit spreads respond to interest rates. The historical span of the research covers the period of 2004–2016. The fndings imply that hedging interest rate risk ought to be a dynamic process and take into consideration where the economy is positioned in the current business cycle. This research provides important insights on the nature of interest rate sensitivity, capable of enhancing fnancial stability and improving efciency of fnancial system. info:eu-repo/semantics/publishedVersion

Participating life insurance contracts allow the policyholder to participate in the annual return of a reference portfolio. Additionally, they are often equipped with an annual (cliquet-style) return guarantee. The current low interest rate environment has again refreshed the discussion on risk management and fair valuation of such embedded options. While this problem is typically discussed from the viewpoint of a single contract or a homogeneous insurance portfolio, contracts are, in practice, managed within a heterogeneous insurance portfolio. Their valuation must then - unlike the case of asset portfolios - take account of portfolio effects: Their premiums are invested in the same reference portfolio; the contracts interact by a joint reserve, individual surrender options and joint default risk of the policy sponsor. Here, we discuss the impact of portfolio effects on the fair valuation of insurance contracts jointly managed in (homogeneous and) heterogeneous life insurance portfolios. First, in a rather general setting, including stochastic interest rates, we consider the case that otherwise homogeneous contracts interact due to the default risk of the policy sponsor. Second, and more importantly, we then also consider the case when policies are allowed to differ in further aspects like the guaranteed rate or time to maturity. We also provide an extensive numerical example for further analysis.

In recent years, with the rapid increase of the business volume of housing mortgage loans of commercial banks, the risk of prepayment is increasingly exposed. Prepayment will have a great impact on the duration and convexity of housing mortgage loans of commercial banks and then bring difficulties to the asset liability management of banks. Therefore, empirical research on the changes of duration and convexity of housing mortgage loans caused by prepayment when the market interest rate changes is of great significance for commercial banks to manage interest rate risk exposure. Based on the analysis of the option characteristics of prepayable housing mortgage loan, the CIR model with GARCH(1, 1) is selected to describe the interest rate change path, and the computer simulation method is used to calculate OAS and then calculate the effective duration and effective convexity of housing mortgage loan under different prepayment rates, so as to understand the interest rate risk of housing mortgage loan in the presence of embedded option.

Analytic tools used for decision support require high quality data inputs in order to arrive at precise estimates. Using an extensive database from US government procurement contracts, we document that an over-representation of accepted contract bids end in a series of zeros. These ending-zero contracts show systematic differences in contract outcomes. They require more extensive revision and are less likely to have embedded options exercised, indicating weaker contract performance and requiring additional resources for renegotiation. We find support for the presence of ending zeros being associated with uncertainty in the contract solicitation. Ending zero contracts are more frequent for certain services, particularly research and development, and less frequent when the contract refers to standardized goods or services. We explore whether the presence of ending-zero bids and their associated weak outcomes is driven by small contracts, potentially unsophisticated bidding firms, or particular government agencies. Our results suggest that the specificity of the contract solicitation itself and the information it provides may be a driving factor and that government allocation of resources could improve with additional detail provided in contract solicitations.

Hurricane bonds are parametric in nature as they have a dual-exercise structure: the first exercise is conditional on the hurricane’s physical landfall location and the second is conditional upon the embedded option ending in-the-money. We propose a coupled and physically-based hurricane bond pricing model via Monte Carlo simulation that resolves the dual exercise, which was not addressed in extant loss-based catastrophe bond pricing models. This coupled model is developed at the nexus of atmospheric science and finance by integrating hurricane risk modeling and option pricing. By applying this model to price a parametric hurricane bond, we demonstrate how a hurricane bond’s price is sensitive to its underlying hurricane’s physical parameters – genesis, heading, translation speed, velocity, and radius.

This paper examines the behavior of the interest rate risk management measures for bonds with embedded options and studies factors it depends on. The contingent option exercise implies that both the pricing and the risk management of bonds requires modelling future interest rates. We use the Ho and Lee (HL) and Black, Derman, and Toy (BDT) consistent interest rate models. In addition, specific interest rate measures that consider the contingent cash-flow structure of these coupon-bearing bonds must be computed. In our empirical analysis, we obtained evidence that effective duration and effective convexity depend primarily on the level of the forward interest rate and volatility. In addition, the higher the interest rate change and the lower the volatility, the greater the differences in pricing of these bonds when using the HL or BDT models.

Standard treatments of the impact of interest rate changes on bond prices include duration and convexity. These concepts derive from the first and second derivatives of a Taylor series expansion of an option-free bond price as a function of interest rates. Does the third derivative matter? This paper derives the formula for the third derivative- based term, which we call tilt, and explores its properties. Tilt tells how quickly the convexity changes. Similar to the properties of duration, a higher yield reduces tilt, a longer maturity increases tilt, and a higher coupon reduces tilt, ceteris paribus. Tilt adds little to the accuracy of the impact of small interest rate changes for default-free option-free bonds. Tilt becomes interesting for bonds with embedded options such as callable bonds and mortgage-backed securities. Such bonds can experience rapidly changing convexity and price under certain interest rate regimes.

Under the condition of continuous innovation of financial derivatives and marketization of interest rate, interest rates fluctuate more frequently and fiercely, and the measurement of interest rate risk also attracts more attention. Under the premise that the fluctuation of interest rate follows fuzzy stochastic process, based on the option characteristics of financial instruments with embedded option, this paper takes effective duration and effective convexity as tools to measure interest rate risk when embedded options exist, tries to choose CIR extended model as term structure model, and uses the Monte Carlo method for hybrid low deviation sequences (HPL-MC) to analyze the prepayment characteristics of MBS, a representative financial instrument with embedded options, when interest rates fluctuate; on this basis, the effectiveness of effective duration management of interest rate risk is demonstrated with asset liability management cases of commercial banks.

This paper considers pricing equity-linked notes (ELN) portfolio and related portfolio optimization. ELNs are the derivative instruments which can be viewed as bonds with floating coupons. The floating coupon is represented in terms of an embedded option that depends on the behavior of a certain underlying asset or a basket of them. We provide the new optimization problem by hyperbolic absolute risk aversion (HARA) utility function approach. We obtain the solution of this problem in terms of a dynamic programming equation.

In this paper, we consider three types of embedded options in pension benefit design. The first is the Florida second election (FSE) option, which has been offered to public employees in the state ...

Cybersecurity has become a key factor that determines the success or failure of companies that rely on information systems. Therefore, investment in cybersecurity is an important financial and operational decision. Typical information technology investments aim to create value, whereas cybersecurity investments aim to minimize loss incurred by cyber attacks. Admittedly, cybersecurity investment has become an increasingly complex one, since information systems are typically subject to frequent attacks, whose arrival and impact fluctuate stochastically. Furthermore, cybersecurity measures and improvements, such as patches, become available at random points in time making investment decisions even more challenging. We propose and develop an analytical real options framework that incorporates major components relevant to cybersecurity practice, and analyze how optimal cybersecurity investment decisions perform for a private firm. The novelty of this paper is that it provides analytical solutions that lend themselves to intuitive interpretations regarding the effect of timing and cybersecurity risk on investment behavior using real options theory. Such aspects are frequently not implemented within economic models that support policy initiatives. However, if these are not properly understood, security controls will not be properly set resulting in a dynamic inefficiency reflected in cycles of over or under investment, and, in turn, increased cybersecurity risk following corrective policy actions. Results indicate that greater uncertainty over the cost of cybersecurity attacks raises the value of an embedded option to invest in cybersecurity. This increases the incentive to suspend operations temporarily in order to install a cybersecurity patch that will make the firm more resilient to cybersecurity breaches. Similarly, greater likelihood associated with the availability of a cybersecurity patch increases the value of the option to invest in cybersecurity. However, the absence of an embedded investment option increases the incentive to delay the permanent abandonment of the company's operation due to the irreversible nature of the decision.

Purpose of the study : How to best structure an IP licensing agreement taking account of embedded optionalities and other terms negotiated between licensor and licensee via a case study involving a prototypical options-based business model (biotech industry). Methodology : Binomial lattice simulation. Findings : It shows how IP management practices would change depending on who pays for the development costs, controls the continuation/development or abandonment option and thereby appropriates most of the embedded option’s value. It presents alternative (iso-value) menu licensing term choices (different combinations of royalty vs. fixed upfront fee or milestone payments) that are fair and optimal in properly accounting for the optionality embedded in RD (b) consider licensor and licensee perspectives in negotiating the terms of the IP licensing agreement, providing guidelines on how to determine its optimal remuneration structure reflecting a fair sharing of project value and embedded optionality among the parties; (c) offer a tool for IP portfolio management that helps a licensor prioritize internal R&D projects accounting for managerial flexibility and optimal licensing design under uncertainty.

Option pricing and managing equity linked insurance (ELI) require the proper modeling of stock return dynamics. Due to the long duration nature of equity-linked insurance products, a stock return model must be able to deal simultaneously with the preceding stylized facts and the impact of market structure changes. In response, this article proposes stock return dynamics that combine Levy processes in a regime-switching framework. We focus on a non-Gaussian, generalized hyperbolic distribution. We use the most popular linked equity of ELIs, the S&P 500 index, as an example. The empirical study verifies that the proposed regime-switching generalized hyperbolic (RSGH) model gives the best fit to data. In investigating the effects of stock return modeling on pricing and risk management for financial contracts, we derive the characteristic function, embedded option price, and risk measure of equity-linked insurance analytically. More importantly, we demonstrate that the regime-switching generalized hyperbolic ...

Bonds with embedded options are a subset of traditional fixed income instruments in which an option has the potential to influence the timing and amount of a security’s cash flows and the security’s valuation. The term embedded signifies that the option and the bond are inseparable. Unlike a warrant, which typically can be detached and traded independently of its underlying instrument, an embedded option cannot be split from the bond to create two distinct, investable assets—the bond and the option. The inseparability of the bond and option changes the risk-return profile for both issuers and investors alike, and therefore renders traditional bond metrics, such as yield-to-maturity, ineffective. This chapter explores the most common bonds with embedded options, which are callable, puttable, and convertible bonds, in addition to discussing some nontraditional embedded option bond structures including contingent convertibles, extendable bonds, combinations, and knock-in and knock-out options.

This chapter introduces the analysis and valuation of bonds with embedded options. For callable bonds, it discusses their unique reinvestment risk and negative convexity. For both callable bonds and puttable bonds, the chapter introduces two additional measures to gauge their risk: yield-to-call and yield-to-put, respectively. The chapter reviews the application of the spot rate curve in bond valuation and introduces the Z-spread to measure bond-specific risk more accurately. To model interest rate risk, the chapter builds a binomial interest rate model and calibrates it with on-the-run Treasury issues. The option-adjusted-spread (OAS) is introduced to measure the bond-specific risk excluding the option effect. The difference between Z-spread and OAS represents the option effect. Common measures of convertible bond risk and value are discussed including the possibility of valuating a convertible bond using option-pricing models and its drawbacks.

We introduce the Option Interpolation Model (OIM) for accurate approximation of embedded option values in insurance liabilities. Accurate approximation is required for ex-ante risk management applications. The OIM is based on interpolation with radial basis functions, which can interpolate scattered data, and does not suffer from the curse of dimensionality. To reduce computation time we present an inversion method to determine the interpolation function weights. The robustness, accuracy and efficiency of the OIM are investigated in several numerical experiments. We show that the OIM results in highly accurate approximations.

Crisis events have significantly changed the view that extreme events in financial markets have negligible probability. Especially in the life insurance market, the price of guaranteed participating life insurance contract will be affected by a change in asset volatility which leads to the fluctuations in embedded option value. Considering the correlation of different asset prices, MEGB2 (multivariate exponential generalized beta of the second kind) distribution is proposed to price guaranteed participating life insurance contract which can effectively describe the dependence structure of assets under some extreme risks. Assuming the returns of two different assets follow the MEGB2 distribution, a multifactor fair valuation pricing model of insurance contract is split into four components: the basic contract, the annual dividend option, the terminal dividend option, and the surrender option. This paper studies the effect of death rate, minimum guaranteed yield rate, annual dividend ratio, terminal dividend ratio, and surrender on the embedded option values and calculates the single premium of the insurance contract under different influence factors. The Least-Squares Monte Carlo simulation method is used to simulate the pricing model. This article makes a comparison in the sensitivity of the pricing parameters under the MEGB2 distribution and Multivariate Normal distribution asset returns. Finally, an optimal hedging strategy is designed to cover the possible risks of the underlying assets, which can effectively hedge the risks of portfolio.

I present an analytical valuation framework for the management of fixed-income instruments traded in imperfectly competitive markets, like demand deposits and credit card loans in the banking book, inter alia, to stabilize the abnormal profit margin. Banking book instruments contain embedded options such as withdrawal rights, discretionary pricing, rate clustering and zero-based floors. Analytical solutions speed up computation time to calculate valuations, earnings and risk measures like closed-form expressions for margin spreads, hedge ratios and parameter sensitivities. Asymptotically, according to martingale central limit theorems and thanks to the long-term nature of the banking book, Gaussian approximations can be applied.

Although innovation and support schemes are among the main forces that drive investment in renewable energy (RE) technologies, both involve considerable uncertainty. We develop a real options framework to analyse the impact of technological, policy and electricity price uncertainty on the decision to invest sequentially in successively improved versions of a RE technology. Technological uncertainty is reflected in the random arrival of innovations, and policy uncertainty in the likely provision or retraction of a subsidy that takes the form of a fixed premium on top of the electricity price. We show that greater likelihood of subsidy retraction (provision) lowers (raises) the incentive to invest, and, by comparing a stepwise to a lumpy investment strategy, we show how an embedded option to adopt an improved technology version mitigates the impact of subsidy retraction on investment timing. Specifically, we show how stepwise investment facilitates earlier technology adoption compared to lumpy investment, and that, under stepwise investment, technological uncertainty accelerates technology adoption, thus further offsetting the incentive to delay investment in the light of subsidy retraction. This is an open access article distributed under the terms of the Creative Commons CC-BY license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A classical inventory problem is studied from the perspective of embedded options, reducing inventory-management to the design of optimal contracts for forward delivery of stock (commodity). Financial option techniques a la Black-Scholes are invoked to value the additional ‘option to expand stock’. A simplified approach which ignores distant time effects identifies an optimal ‘time to deliver’ and an optimal ‘amount to deliver’ for a production process run in continuous time modelled by a Cobb-Douglas revenue function. Commodity prices, quoted in initial value terms, are assumed to evolve as a geometric Brownian process with positive (inflationary) drift. Expected revenue maximization identifies an optimal ‘strike price’ for the expansion option to be exercised and reveals the underlying martingale in a truncated (censored) commodity price. The paper establishes comparative statics of the censor, using sensitivity analysis on the related censor functional equation; key here is that the censor, as a function of the drift and volatility of price, is the solution of a functional equation. Asymptotic approximation allows a tractable analysis of the optimal timing.

A framework for measuring and managing the risk of embedded options in non-traded credit is developed. For typical bank clients there is no market information related to their ability to pay (bond or CDS spreads) available. In this case a bank has to rely solely on statistical data to judge the credit quality of a borrower. To value a loan with embedded options like prepayment rights in this context, a model is proposed that combines a stochastic interest rate model with statistical information on default and recovery rates. Using this as the basis for evaluating the risk of embedded options in loans, it is shown how the concepts of credit risk management can be transferred to this framework after defining a suitable concept of risk. It turns out that this modeling approach combines the theories of derivatives pricing and credit risk modeling in the sense that derivatives pricing theory measures the costs for hedging optional components in loans while credit risk modeling measures the risk that these hedging costs turn out to be inadequate. This risk depends not only on the single loan's risk characteristics but also on the dependence structure and the granularity of the total loan portfolio.

I present an analytical valuation framework for the management of fixed-income instruments traded in imperfectly competitive markets, like demand deposits and credit card loans in the banking book, inter alia, to stabilize the abnormal profit margin. Banking book instruments contain embedded options such as withdrawal rights, discretionary pricing, rate clustering and zero-based floors. Analytical solutions speed up computation time to calculate valuations, earnings and risk measures like closed-form expressions for margin spreads, hedge ratios and parameter sensitivities. Asymptotically, according to martingale central limit theorems and thanks to the long-term nature of the banking book, Gaussian approximations can be applied.

The correlations between financial variables are always nonlinear and the financial instruments with several influence factors are therefore characteristized with complicated risk-return. Thus, this paper attempts to decompose the mezzanine financing into three parts to conveniently value them respectively. In order to dispose complicated relationship between variables during valuation, the support vector machine is introduced to give a nonparametric estimation of the equity embedded in the mezzanine financing. Therefore, a quantitative analysis framework for the valuation of the mezzanine financing is constructed to deal with the dual risk-return nature of both equity and debt of mezzanine financing. Experimental results display the proposed methodology can efficiently address sophisticated risk-return nature of mezzanine financing and deal with nonlinear dynamics embedded in sample data, which provides investors, issuers and regulators with constructive decision-making basis.

We price resettable convertible bonds (RCBs) whose embedded options contain conversion, call and put provisions and reset clause using complete path decomposition. The reset clause stipulates that the conversion price is adjusted downwards if the underlying stock price is lower than the conversion price by some pre-specified percentage. By assuming or identifying optimal strategies of RCBs’ issuers and investors, we completely decompose the value of RCBs into three parts: A sum of values of several path-dependent exotic options, the value of coupon payments, and the par value. Our method provides comprehensible insight of different provisions contained in RCBs.

Decision-making in construction projects often include options features. Such embedded options are difficult to value properly and many decision makers do not have experience in option analysis. The purpose of this paper is to demonstrate how real option analysis can be used to value capital expenditures on construction materials. We propose a real option framework to evaluate decision-making processes involving the purchase of construction materials. A case study was conducted by evaluating the purchase decision-making of solar cells, a good with high price volatility. Using real option analysis two strategies to improve the financial feasibility of installing a solar panel system were derived. The first strategy involves using a price cap that gives the project manager the right, but not obligation, to buy the modules for a predefined price during the next year. The second strategy is to defer the purchase of the solar cells until future price information becomes clearer. Both of the strategies in the case study were valued using the binominal model. This study will help to improve the financial feasibility of purchasing construction materials with high price volatility by including the value of managerial flexibility.

This paper deals with the pricing of convertible bond with call provision based on the traditional B-S formula. By applying the principle of no arbitrage, the partial differential equation for the bond is established with identified boundary conditions, which solution results in the closed form of the pricing formula.

In this financial engineering research, we study the behaviour of an option premium of a call/put option which is embedded in a typical fixed coupon bond with finite maturity. The contribution of the research is the conclusion about the dynamics of premium changes; represented by direction and sensitivity; with respect to the changes in credit rating and also risk-free interest rate development. The aim of the research is also to clearly demonstrate this theoretically complicated topic to the financial practitioners using a practical example. We are about to consider a 3-dimensional process where the dimensions are: time, rating development process and risk-free interest rate development. We use Standard & Poor’s rating transition matrix to create rating tree and Hull-White model for modelling of risk-free interest rate development. We add embedded call/put option to the bond structure and assume the call/put option to be exercised in case of interest rates decline/rise or rating worsening/improvement. For valuation, we use the risk-neutral concept. Using a numerical solution on the 3-dimensional tree (implemented in MATLAB), we avoid problems that appear while analytical solving of partial differential equations.

Electricity demand and renewables penetration are set to increase worldwide over the coming decades, thereby posing challenges related to increased peaks. Hence, significant network reinforcements may be needed over the coming years with a considerable challenge being the growing uncertainty that surrounds future connected load and generation capacity. This uncertainty inadvertently results in rising levels of stranding risk. Thus, new planning frameworks are needed that move away from the traditional deterministic approaches and utilize stochastic optimization techniques so as to encourage cost-efficient strategic investments through smart technologies by capturing their embedded option value. In this paper, the option value of Dynamic Line Rating and Storage technologies is computed and is shown to reflect the fact that such technologies can allow for generating considerable system savings through management of planning uncertainties.

The Authors propose a personal pension product, consisting of a non-traditional profit sharing life insurance contract where the insured is allowed to share the profit of the pension’s invested funds all along the contract duration, that is from the issue time till the insured’s death. In its concrete realization, the idea comes true as a sequence of premiums characterized by a level cap, followed by the sequence of benefits characterized by a level floor. The two embedded options are inserted in the basic structure of a pension annuity. Due to the negligibility of the pooling effect in such kind of portfolios, the impact of the accidental demographic risk source is investigated.

Solvency assessing is a compelling issue for the insurance industry, also in light of the current international risk-based regulations. Internal models have to take into account risk/profit indicators, in order to provide flexible tools aimed at valuing solvency. We focus on a variable annuity with an embedded option involving a participation level which depends on the period financial result. We realize a performance evaluation by means of a suitable indicator, which properly captures both financial and demographic risk drivers. In fact, in the case of life annuity business, assessing solvency has to be framed within a wide time horizon, where specific financial and demographic risks are realized. In this order of ideas, solvency indicators have to capture the amount of capital to cope with the impact of those risk sources over the considered period. The analysis is carried out in accordance with a management perspective, apt to measure the business performance, which requires a correct risk control; in particular we present a study of the dynamics of the the profit realized per unit of the total financial value of the contract. On the other hand, the consumer profitability is also measured by means of an utility-equivalent fixed life annuity. According to the insureds’ point of view, we measure their perception of the contract profitability within the expected utility approach.

Renewable Identification Number (RIN) is a floor-and-trade mechanism to enforce renewable energy standards in the US transportation fuels market. Motivated by several real-world cases of gasoline producers' bankruptcy or financial stress caused by volatile RINs prices, we offer a dynamic stochastic optimization framework to determine the price behavior of RIN certificates and their connections to firms' operational decisions. One key novel aspect of our work is that we explicitly identify and model the embedded option value of RINs under various scenarios. We derive some closed-form solutions for the price dynamics of RINs, and also propose accurate numerical methods for when the closed-form solution is not available. Among other results, we show that the price of RINs has a U-shape relationship with the volatility of ethanol and gasoline prices and a negative relationship with the correlation between the two price processes. We compare the empirical performance of the model with real historical data, and find that our model allows a significant improvement over existing pricing models. We also discuss the interaction of operational features and constraints of blenders and the market price of RINs. We show how the analytic framework can be used to better inform operational decisions of firms active in markets with a floor-and-trade permit mechanism.

This paper studies the Continuous Workout Mortgage (CWM), a two in one product: a fixed rate home loan coupled with negative equity insurance, to advocate its viability in mitigating financial fragility. In order to tackle the many issues that CWMs embrace, we perform a range of tasks. We optimally price CWMs and take a systemic market-based approach, stipulating that mortgage values and payments should be linked to housing prices and adjusted downward to prevent negative equity. We illustrate that amortizing CWMs can be the efficient home financing choice for many households. We price CWMs as American option style, defaulting debt in conjunction with prepayment within a continuous time, analytic framework. We introduce random prepayments via the intensity approach of Jarrow and Turnbull (1995) . We also model the optimal embedded option to default whose exercise is motivated by decreasing random house prices. We adapt the Barone-Adesi and Whaley (1987) (BAW) approach to work within amortizing mortgage context. We derive new closed-form and new analytical approximation methodologies which apply both for pricing CWMs, as well as for pricing the standard US 30-year Fixed Rate Mortgage (FRM).

The shift from defined benefit (DB) to defined contribution (DC) is pervasive among pension funds, due to demographic changes and macroeconomic pressures. In DB all risks are borne by the provider, while in plain vanilla DC all risks are borne by the beneficiary. However, for DC to provide income security some kind of guarantee is required. A minimum guarantee clause can be modeled as a put option written on some underlying reference portfolio and we develop a discrete model that selects the reference portfolio to minimize the cost of a guarantee. While the relation DB–DC is typically viewed as a binary one, the model shows how to price a wide range of guarantees creating a continuum between DB and DC. Integrating guarantee pricing with asset allocation decision is useful to both pension fund managers and regulators. The former are given a yardstick to assess if a given asset portfolio is fit-for-purpose; the latter can assess differences of specific reference funds with respect to the optimal one, signaling possible cases of moral hazard. We develop the model and report numerical results to illustrate its uses.

This paper is concerned with the valuation of equity-linked annuities with mortality risk under a double regime-switching model, which provides a way to endogenously determine the regime-switching risk. The model parameters and the reference investment fund price level are modulated by a continuous-time, finite-time, observable Markov chain. In particular, the risk-free interest rate, the appreciation rate, the volatility and the martingale describing the jump component of the reference investment fund are related to the modulating Markov chain. Two approaches, namely, the regime-switching Esscher transform and the minimal martingale measure, are used to select pricing kernels for the fair valuation. Analytical pricing formulas for the embedded options underlying these products are derived using the inverse Fourier transform. The fast Fourier transform approach is then used to numerically evaluate the embedded options. Numerical examples are provided to illustrate our approach.

The value of a conventional convertible bond is the value of a straight bond plus the value of the option to exchange it for a specified number of shares of common stock. First, I develop a closed-form contingent-claims convertible bond valuation model that quantifies the value of the exchange option when the short-term riskless rate, the firm's credit spread, and its share price are stochastic. I model the firm's decision to force early conversion as a stopping time problem in which the firm forces conversion as soon as the conversion value reaches the forced conversion barrier. I empirically validate the model by comparing model and market prices for a sample of 148 corporate convertible bonds issued between 2006 and 2010. The average median and mean pricing errors are − 0.18% and 0.21%, respectively, which are within the average bid–ask spread for convertible bonds during the sample period. I use the model to quantify the disruptive impact that the prohibition on short selling during the recent financial crisis had on convertible bond prices.

We propose a model for valuing equity linked annuity (ELA) products under a generalized gamma model with a Markov-switching compensator. We suppose that the market interest rate and all the parameters of the underlying reference portfolio switch over time according to the state of an economy, which is modelled by a continuous-time Markov chain. The model considered here can provide market practitioners with flexibility in modelling the dynamics of the reference portfolio. We price the ELA by pricing its embedded options, for which we employ the regime-switching version of Esscher transform to determine the pricing kernel. A system of coupled partial-differential-integral equations satisfied by the embedded option prices is derived. Simulation results of the model have been presented and discussed

Non-maturity deposits like savings accounts or demand deposits contain significant option risks caused by the bank’s discretionary pricing and the customers’ withdrawal right. Option risks follow from inherent non-linear factor exposures. I propose an ordinal response model for deposit rate jumps to identify non-linear factor exposures and a discrete-time term structure model to value the resulting option risks and to derive hedge measures “outside the model”. My delta profile resembles a constant maturity swap, but vega and gamma are more pronounced, which demonstrates that the widespread practice of static hedging with zero bonds is inadequate.

Home Equity Conversion Mortgages ("HECMs'') implicitly bundle put options on borrowers' homes with non-defaultable credit lines. Put proceeds are bequeathable and insure longevity and home prices. Credit use is elective, so the put's expected net present value bounds HECM's value to borrowers below. Older homeowners' weak demand is commonly attributed to "high costs,'' and the government sets prices intending to avoid subsidy. Simulation results make low demand more puzzling and illustrate risks to federal insurance. HECM has offered many borrowers favorably priced puts, even using backward-looking expectations near the recent price cycle peak and ignoring maintenance and termination options.