Valuation Formula Research Articles

Asymptotic behavior of bond yields and volatilities for the extended 3/2 model under the real-world measure

Abstract The extended $3/2$ short rate model is a mean reverting model of the short rate which, for suitably chosen parameters, permits a sensible term structure of bond yields and closed-form valuation formulae of zero-coupon bonds and options. This article supplies proofs of the formulae for the expected present values of future cash flows under the real-world probability measure, known as actuarial valuation. Finally, we give formulae for asymptotic levels of bond yields and formulae for bond option prices for the extended $3/2$ model, under particular conditions on its parameters.

Intellectual Property Commercialization Policy to Expand Financing Options for Indonesian Creative Economy Businesses

This study aims to develop alternate strategies for completing the creative economy Intellectual Property collateral financing policy so that it may be implemented in Indonesia. The methods used descriptive qualitative research with SWOT analysis to analyze the implementation of Intellectual Property-based financing schemes' supporting and inhibiting factors. The preparation of alternative strategies based on the SWOT matrix produces nine strategies. The nine alternative strategies produced were evaluated and tested using four practical policy analysis criteria: Effectiveness, Efficiency, Legality, and Administrative Robustness. Three primary recommendations emerged from testing alternative approaches: first, a plan to fortify industry regulations related to financial services, which will serve as a model for financial institutions implementing credit provisions based on Intellectual Property collateral and expediting the realization of financing products backed by Intellectual Property Rights. Second, a plan to standard Intellectual Property valuation formula or economic value of Intellectual Property and establish an ecosystem for the Intellectual Property market to promote more consistent Intellectual Property valuations and third, managing credit risk through joint ventures with insurance companies, the debtor will pay the insurance costs that allow the Bank or Financial Institutions to transfer risks to third parties if the Intellectual Property value declines.

The capital-on-capital cost in solvency II risk margin

AbstractThis work contributes to the literature on time consistent valuation of insurance liabilities and to the ongoing discussion on revisions of risk margin (RM) calculation, by formally defining the concept of capital-on-capital cost. We describe the capital-on-capital as the amount required to cover unexpected variations in future regulatory capitals from the current time to liabilities maturity. That is, the capital-on-capital cost is the RM component dedicated to cover the risk of future RMs and not to cover variations of the best estimate of liabilities cash-flows. We mathematically formalize the capital-on-capital cost as the difference between two alternative time consistent liabilities valuation formula. The first is obtained through backward iteration of the one-period market-consistent valuation operator, by iterating the solvency capital requirement (SCR) risk measure. We propose a second alternative valuation formula for liabilities, based on a new time consistent dynamic formulation of the SCR risk measure, called additive-SCR (ASCR). The ASCR represents the expected total capital requirement from current time to liabilities maturity. We prove that the second valuation formula, based on ASCR, is time consistent, unless it is not based on iteration of the one-period SCR risk measure. Finally, we apply the proposed approach to a portfolio of long term equity linked life-insurance contracts. In particular, we estimate the capital-on-capital cost by calculating the difference between the two valuation formulas. Numerical results show that for liabilities with long maturities the capital-on-capital cost is a non negligible component of the RM.

Option Pricing in an Incomplete Market

The purpose of this paper is to illustrate the pricing of options in an incomplete market using the new consistent uplifted martingale measure methodology introduced by Grigorian and Jarrow [2024, Filtration Reduction and Incomplete Markets, Frontiers of Mathematical Finance, 3(1), 78–105; 2023, Filtration Reduction and Completeness in Brownian Motion Models. Working Paper, Cornell University; 2024, Filtration Reduction and Completeness in Jump-Diffusion Models. Working Paper, Cornell University]. We apply it to an incomplete market where a stock has stochastic volatility. Two valuation formulas are generated, depending upon whether the trader is more concerned about volatility or price risk in the construction of a partial replicating portfolio for the option’s payoff.

Asset Pricing with Costly Short Sales

We study a dynamic general equilibrium model with costly-to-short stocks and heterogeneous beliefs. Costly short sales drive a wedge between the valuation of assets that promise identical cash flows but that are subject to different trading arrangements. In particular, we show that the price of an asset is given by the risk-adjusted present value of its future cash flows, which include both dividends and an endogenous lending yield that we characterize explicitly. This valuation formula implies that stocks with low and high shorting costs should offer similar risk-return trade-off once returns are appropriately adjusted for lending revenues and thus, sheds light on recent empirical findings about the explanatory power of shorting costs in the cross-section of returns. This paper was accepted by Lukas Schmid, finance. Supplemental Material: The data files are available at https://doi.org/10.1287/mnsc.2023.01887 .

The Pay‐for‐Success Contract: A Valuation Note

ABSTRACTPay‐for‐success contracts are social and financial innovations in social policy and capital markets, respectively. This paper argues that they exhibit option‐like payoffs and implements standard option‐pricing arguments in assessing the value of investing in pay‐for‐success contracts. Sensitivities vis‐à‐vis contract specifications are reflected in the valuation formula and help reach investment and social policy decisions. These sensitivities are demonstrated via a numerical application that uses parameters drawn from the Massachusetts Juvenile Justice Pay for Success Initiative, the largest pay‐for‐success initiative in the United States at the time of its launch.

Efficient valuation of guaranteed minimum accumulation benefits in regime switching jump diffusion models with lapse risk

We present a streamlined valuation method for the guaranteed minimum accumulation benefits incorporated within variable annuity contracts. At each contract renewal date, the insurer updates the policyholder's account value to the higher of the guaranteed value and the equity-linked investment value. In addition, we introduce lapse risks into the variable annuity contract, modeling the lapse decision under the assumption of stochastic intensity. Utilizing a combination of continuous-time Markov chain approximation and the Fourier cosine series expansion method, we derive closed-form valuation formulas under regime switching jump diffusion models. Numerical simulations showcase the precision and effectiveness of the proposed approach.

Option pricing with dynamic conditional skewness

AbstractIn this paper, we develop a discrete‐time affine option‐pricing model that explicitly incorporates the dynamics of conditional skewness. The new proposed model features different dynamics for conditional skewness and variance. To stress the difference in information, we use alternative realized measures constructed from high‐frequency historical returns to update skewness and variance dynamics. By Fourier inversion, we derive closed‐form option valuation formulas. Empirically, the flexibility that the model offers for conditional skewness as well as high‐frequency information from the underlying asset contribute to superior performance upon benchmark models using S&P 500 index options. Overall, the joint modeling of dynamic conditional skewness and realized measures leads to an out‐of‐sample gain of 12.25% in pricing accuracy. The improvements are more pronounced for deep in‐the‐money calls, options with shorter maturities, and during highly volatile periods.

THE JARROW AND TURNBULL SETTING REVISITED

We consider a financial market with zero-coupon bonds that are exposed to credit and liquidity risk. We revisit the famous Jarrow & Turnbull ( 1995 ) setting in order to account for these two intricately intertwined risk types. We utilize the foreign exchange analogy that interprets defaultable zero-coupon bonds as a conversion of nondefaultable foreign counterparts. The relevant exchange rate is only partially observable in the market filtration, which leads us naturally to an application of the concept of platonic financial markets as introduced by Cuchiero et al. ( 2020 ). We provide an example of tractable term structure models that are driven by a two-dimensional affine jump diffusion. Furthermore, we derive explicit valuation formulae for marketable products, e.g. for credit default swaps.

A static replication approach for callable interest rate derivatives: mathematical foundations and efficient estimation of SIMM–MVA

The computation of credit risk measures such as exposure and Credit Value Adjustments (CVA) requires the simulation of future portfolio prices. Recent metrics, such as dynamic Initial Margin (IM) and Margin Value Adjustments (MVA) additionally require the simulation of future conditional sensitivities. For portfolios with non-linear instruments that do not admit closed-form valuation formulas, this poses a significant computational challenge. This problem is addressed by proposing a static replication algorithm for interest rate options with early-exercise features under an affine term-structure model. Under the appropriate conditions, we can find an equivalent portfolio of vanilla options that replicate these products. Specifically, we decompose the product into a portfolio of European swaptions. The weights and strikes of the portfolio are obtained by regressing the target option value with interpretable, feed-forward neural networks. Once an equivalent portfolio of European swaptions is determined, we can leverage on closed-form expressions to obtain the conditional prices and sensitivities, which serve as an input to exposure and SIMM-driven MVA quantification. For a consistent forward sensitivity estimation, this involves the differentiation of the portfolio-weights. The accuracy and convergence of the method is demonstrated through several representative numerical examples, benchmarked against the established least-square Monte Carlo method.

Defaultable perpetual American put option in a last passage time model

In this article we provide a valuation formula for a defaultable perpetual American put option in the Black-Schole’s market where the default time is modelled using the last passage time of the set [L,∞) by the stock price dynamics. In other words, we suppose that financial default happens if the stock price can no longer recover and rebound above the level L.

Term structure modeling with overnight rates beyond stochastic continuity

AbstractOvernight rates, such as the Secured Overnight Financing Rate (SOFR) in the United States, are central to the current reform of interest rate benchmarks. A striking feature of overnight rates is the presence of jumps and spikes occurring at predetermined dates due to monetary policy interventions and liquidity constraints. This corresponds to stochastic discontinuities (i.e., discontinuities occurring at ex ante known points in time) in their dynamics. In this work, we propose a term structure modeling framework based on overnight rates and characterize absence of arbitrage in a generalized Heath–Jarrow–Morton (HJM) setting. We extend the classical short‐rate approach to accommodate stochastic discontinuities, developing a tractable setup driven by affine semimartingales. In this context, we show that simple specifications allow to capture stylized facts of the jump behavior of overnight rates. In a Gaussian setting, we provide explicit valuation formulas for bonds and caplets. Furthermore, we investigate hedging in the sense of local risk‐minimization when the underlying term structures feature stochastic discontinuities.

Option pricing with overnight and intraday volatility

AbstractEfficiently exploiting the volatility information contained in price variations is important for pricing options and other derivatives. In this study, we develop a new and flexible option‐pricing model that explicitly specifies the joint dynamics of overnight and intraday returns. The application of multivariate Edgeworth–Sargan density enables us to derive analytical approximations for option valuation formulas. Empirically, the model improves significantly upon benchmark models using S&P 500 index options. In particular, its separate modeling of intraday and overnight return volatility leads to an out‐of‐sample gain of 7.24% in pricing accuracy compared with the modeling of the close‐to‐close return volatility as a whole. The improvements are more pronounced during highly volatile periods.

Multitouch Options

In this article, the multitouch option, also called the n-touch option (or the “baseball” option when n=3) is analyzed and valued in closed form. This is a kind of barrier option that has been traded for a long time on the markets, but that does not yet admit a known valuation formula. The multitouch option sets a gradual knock-out/knock-in mechanism based on the number of times the underlying asset has crossed a predefined barrier in various time intervals before expiry. The higher the number of predefined time intervals during which the barrier has been touched, the lower the value of a knock-out contract at expiry, and conversely for a knock-in one. Multitouch options can be viewed as an extension of step barrier options, preserving the ability of the latter to adjust the exposure to risk over time, while eliminating the notorious danger of “sudden death” that holders of step barrier options are faced with. They are thus less risky and more flexible than step barrier options, and all the more so when compared to standard barrier options. This article also provides closed-form valuation of multitouch options with nonstandard features such as an outside barrier or a barrier defined as a continuous function of time.

A Study on the Value of New Media Broadcast Rights for Major Sports Events

This paper adopts the deductive method to discuss in detail the application of the income approach in the valuation of event retransmission rights. By splitting the income hierarchy to estimate the expected income, applying the market comparison method to measure the share rate, reasonably calculating the discount rate with the help of a model and determining the duration of the income according to the characteristics of the event, constructing a new media retransmission rights asset valuation formula, creating a basic model for the valuation of sports event copyright assets, and applying the Winter Olympic Games The validity of the model was initially verified by applying the case of the Winter Olympic Games, and it was concluded that under the guidance of the valuation idea of the income approach, it was possible to assess the value of retransmission rights transactions.

Analytic Solution of Black-Scholes-Merton European Power Put Option Model on Dividend Yield with Modified-Log-Power Payoff Function

This paper proposes a framework based on the celebrated transform of Mellin type (MT) for the analytic solution of the Black-Scholes-Merton European Power Put Option Model (BSMEPPOM) on Dividend Yield (DY) with Modified-Log-Power Payoff Function (MLPPF) under the geometric Brownian motion. The MT has the capability of tackling complex functions by means of its fundamental properties and it is closely related to other well-known transforms such as Laplace and Fourier types. The main goal of this paper is to use MT to obtain a valuation formula for the European Power Put Option (EPPO) which pays a DY with MLPPF. By means of MT and its inversion formula, the price of EPPO on DY was expressed in terms of integral equation. Moreover, the valuation formula of EPPO was obtained with the help of the convolution property of MT and final time condition. The MT was tested on an illustrative example in order to measure its performance, effectiveness and suitability. The MLPPF was compared with other existing payoff functions. Hence, the effect of DY on the pricing of EPPO with MLPPF was also investigated.

Short Communication: Caplet Pricing in Affine Models for Alternative Risk-Free Rates

Alternative risk-free rates (RFRs) play a central role in the reform of interest rate benchmarks. We study a short-rate model for RFRs driven by a general affine process. In this context, under minimal assumptions, we derive explicit valuation formulas for forward-looking and backward-looking caplets/floorlets as well as term-basis caplets.

ALTERNATIVE APPROACH FOR SOLVING A EUROPEAN POWER PUT OPTION MODEL WITH MODIFIED-LOG-POWER PAYOFF FUNCTION

The main goal of this paper is to propose an alternative approach based on the celebrated transform of Mellin type (MT) for solving a European Power Put Option Model (EPPOM) in the sense of Modified-Log-Power Payoff Function (MLPPF) under the geometric Brownian motion. The MT has the capability of tackling complex functions by means of its fundamental properties and it is closely related to other well-known transforms such as Laplace and Fourier types. Using the MT inversion formula, convolution property and final time condition, the fundamental valuation formula of EPPOM was obtained. Moreover, the proposed MLPPF was also compared with other existing payoff functions. Also, the prices of EPPO generated by means of MT were compared with that of the celebrated Black-Scholes model. Hence, it is assumed that the underlying asset price pays no dividend yield.

Randomization and the valuation of guaranteed minimum death benefits

In this article, we focus on death-linked contingent claims (GMDBs) paying a random financial return at a random time of death in the general case where financial returns follow a regime switching model with two-sided phase-type jumps. We approximate the distribution of the remaining lifetime by either a series of Erlang distributions or a Laguerre series expansion, whose capability to fit the tail of the observed mortality data turns out to be much better than the commonly used series of exponential distributions.More precisely, we develop a Laurent series expansion of the discounted Laplace transform of the subordinated process at an Erlang distributed time, which leads to explicit formulae for European-type GMDB as well as related risk measures such as the Value-at-Risk (VaR) and the Conditional-Tail-Expectation (CTE). We further concentrate upon path-dependent GMDBs with lookback features like dynamic fund protection or dynamic withdrawal benefits, by relying on a Sylvester equation approach. The advantage of our approaches is that our results are of semi-closed form, avoiding numerical Fourier inversion or Monte-Carlo simulation, leading to fast evaluation. This is necessary in risk-management, in particularly for nested simulation in the framework of Solvency II. Several numerical experiments are included.Our results have implications beyond life-insurance and GMDBs, namely in all situations where randomization or Erlangization replaces known quantities, like, for example, model parameters, by random variables. In Finance, it is for example well-known that a random maturity time leads to much more convenient valuation formulas that well approximate its non-random counterpart.

An Option Valuation Formula for Stochastic Volatility Driven by GARCH Processes

An Option Valuation Formula for Stochastic Volatility Driven by GARCH Processes