Compound Option Pricing Research Articles

Introducing and testing the Carr model of default

The Merton (On the pricing of corporate debt: The risk structure of interest rates. J. Finance, 1974, 29, 449–470) model is often considered the simplest structural model of default. Its famous modification developed by KMV is also widely used and well understood by both practitioners and academics. However, as the underlying asset process is driven by a diffusion process, the Merton model suffers from the well-known issue of the vanishing of credit spreads for progressively shorter maturities. To overcome such shortfall, several modifications have been proposed in the literature; however, none of these, despite succeeding at better pricing spreads and explaining default dynamics, are able to retain the simplicity of the Merton model. In this paper, we propose a new structural model of default driven by an additive process whose pricing formulas are as simple as – if not even simpler than – those of the Merton model. We named such model the Carr model of default after Peter Carr, given his recent passing and the fact that the underlying asset distribution is the one introduced in Carr and Torricelli (Additive logistic processes in option pricing. Finance Stoch., 2021, 25, 689–724) and Carr and Maglione (Compound option pricing and the Roll-Geske-Whaley formula under the conjugate-power Dagum distribution. J. Deriv., 2022, 31, 1–32). We first provide pricing formulas for the firm's claims and credit spreads, and then discuss a general theory of change of measure for such processes in order to introduce a distance-to-default version of this newly-introduced model. Finally, we empirically test the Carr versus the Merton model and document the far better ability of the Carr model to reproduce CDS spreads as well as explain their cross-sectional and time-series variation.

Compound Option Pricing and the Roll-Geske-Whaley Formula under the Conjugate-Power Dagum Distribution

We explore the pricing of compound derivatives under the newly introduced conjugate-power Dagum distribution. Assuming a discrete-time multiplicative conjugate-power Dagum random walk, we first provide an alternative derivation of the price of a married put based on a change of measure, which is helpful for the pricing of compound options. Then, we apply these results to obtain the equivalent of the Roll-Geske-Whaley formula for the pricing of American options in presence of one known discrete dividend under this alternative distribution.

N-Fold compound option pricing with technical risk under fractional jump-diffusion model

The problem of generalizing the compound option pricing model to incorporate more empirical features becomes an urgent and necessary event. In this study, a new N-fold compound option pricing method is designed for the economic uncertainty and technical uncertainty. The economic uncertainty is modelled by a fractional jump-diffusion model, which incorporates the long-term dependence of financial markets, the kurtosis of returns and the unpredictable shocks of real world. The technical uncertainty is modelled as a simplified version Poisson-type jump process, which describes the catastrophic impact of the technical risk in multi-stage projects. The main contribution of this paper is that we firstly develop the N-fold compound option pricing model with the fractional Brownian motion and the technical risk variable. Further, the analytic solutions of pricing compound options are achieved and verified by the recursive formula of option price. Numerical examples are provided to support the theoretical results of this model. Moreover, from the sensitivity analysis, some results are presented to illustrate that the N-fold compound option price without considering the phase-specific characteristics of technical risk is improperly estimated. Thereby, it is essential for investors to take into account the technical risk when making decisions.

Default Risk and Cross Section of Returns

Prior research uses the basic one-period European call-option pricing model to compute default measures for individual firms and concludes that both the size and book-to-market effects are related to default risk. For example, small firms earn higher return than big firms only if they have higher default risk and value stocks earn higher returns than growth stocks if their default risk is high. In this paper we use a more advanced compound option pricing model for the computation of default risk and provide a more exhaustive test of stock returns using univariate and double-sorted portfolios. The results show that long/short hedge portfolios based on Geske measures of default risk produce significantly larger return differentials than Merton’s measure of default risk. The paper provides new evidence that mediates between the rational and behavioral explanations of value premium.

A geometric Levy model for [formula omitted]-fold compound option pricing in a fuzzy framework

In this paper, we study the problem of n-fold compound option valuation using the martingale method and the theory of fuzzy sets. We adopt a geometric Levy process for modeling the underlying asset price dynamics. The log-price is the sum of a Brownian motion with drift and a Poisson process describing jumps in the price. To obtain the pricing formula for n-fold compound option, we use the Esscher transformed martingale measure as the equivalent martingale measure. The major challenge in deriving the closed-form analytic expression for n-fold compound options is to calculate complex multivariate normal integrals; the complexity is due to the sophisticated structure of the options. In order to overcome this difficulty, we prove a mathematical expectation related to multivariate normal variables. To the best of our knowledge, this report describes the first time that such an approach has been adopted, and it is highly useful in the pricing of n-fold compound options and many other financial derivatives. Because the uncertainty in financial markets simultaneously involves randomness and fuzziness, we assume that the underlying asset price follows a fuzzy stochastic process. Under this assumption, the α-cut of the fuzzy price of n-fold compound call options is derived. Considering a decision maker’s subjective judgment, the permissible range of the expected prices of n-fold compound call options with uncertainty is given. Finally, numerical examples are presented to illustrate and analyze the theoretical results.

Compound option pricing under stochastic volatility

The paper proposes a flexible and computationally efficient lattice-based approximation for evaluating European and American compound options under stochastic volatility models. In comparison with the existing evaluation procedures, the method is more flexible because it may accommodate several stochastic volatility specifications of the asset price process, and more efficient because it is computationally faster in computing accurate compound option prices. The method is obtained as an extension of Costabile et al. (2012) discretisation, which consists in approximating the stochastic volatility process by a recombining binomial lattice, and considers the asset value as an auxiliary variable whose dynamics is captured by generating subsets of representative realisations to cover the range of possible asset prices at each time slice. The backward induction scheme based on a linear interpolation technique is adapted to compute both the underlying daughter option and the compound option prices. Numerical experiments confirm the method efficiency and accuracy.

Compound option pricing under stochastic volatility

The paper proposes a flexible and computationally efficient lattice-based approximation for evaluating European and American compound options under stochastic volatility models. In comparison with the existing evaluation procedures, the method is more flexible because it may accommodate several stochastic volatility specifications of the asset price process, and more efficient because it is computationally faster in computing accurate compound option prices. The method is obtained as an extension of Costabile et al. (2012) discretisation, which consists in approximating the stochastic volatility process by a recombining binomial lattice, and considers the asset value as an auxiliary variable whose dynamics is captured by generating subsets of representative realisations to cover the range of possible asset prices at each time slice. The backward induction scheme based on a linear interpolation technique is adapted to compute both the underlying daughter option and the compound option prices. Numerical experiments confirm the method efficiency and accuracy.

Venture capital, staged financing and optimal funding policies under uncertainty

Our paper presents a dynamic model of entrepreneurial venture financing under uncertainty based on option exercise games between an entrepreneur and a venture capitalist (VC). In particular, we analyze the impact of multi-staged financing and both economic and technological uncertainty on optimal contracting in the context of VC-financing. Our novel approach combines compound option pricing with sequential non-cooperative contracting, allowing us to determine whether renegotiation will improve the probability of coming to an agreement and proceed with the venture. It is shown that both sources of uncertainty positively impact the VC-investor's optimal equity share. Specifically, higher uncertainty leads to a larger stake in the venture, and renegotiation may result in a dramatic shift of control rights in the venture, preventing the venture from failure. Moreover, given ventures with low volatility, situations might occur where the VC-investor loses his first-mover advantage. Based on a comparative-static analysis, new testable hypotheses for further empirical studies are derived from the model.

The evaluation of compound options based on RBF approximation methods

Recently, real options have gained more importance in computational finance studies. It has already been shown that the compound option pricing can be formulated as a two-pass boundary PDE arising from Black–Scholes model. Radial basis function (RBF) as a meshfree approximation method is widely used for numerical study of the time dependent PDEs. In this paper, the aim is to introduce the robust numerical approach based on RBF-QR to compute the price of European compound options such as the popular put on put options. We also extend the proposed approach to American compound option pricing. The numerical experiments will show the efficiency of the performance for European and American compound option with single asset and multi-asset cases.

Where Would the EUR/CHF Exchange Rate be Without the SNB's Minimum Exchange Rate Policy?

AbstractSince its announcement made on September 6, 2011, the Swiss National Bank (SNB) has been pursuing the goal of a minimum EUR/CHF exchange rate of 1.20, promising to intervene on currency markets to prevent the exchange rate from falling below this level. We use a compound option pricing approach to estimate the latent exchange rate that would prevail in the absence of the SNB's interventions, together with the market's confidence in the SNB's commitment to this policy. © 2014 Wiley Periodicals, Inc. Jrl Fut Mark 35:1103–1116, 2015

A comparative study on time-efficient methods to price compound options in the Heston model

The primary purpose of this paper is to provide an in-depth analysis of a number of structurally different methods to numerically evaluate European compound option prices under Heston’s stochastic volatility dynamics. Therefore, we first outline several approaches that can be used to price these type of options in the Heston model: a modified sparse grid method, a fractional fast Fourier transform technique, a (semi-)analytical valuation formula using Green’s function of logarithmic spot and volatility and a Monte Carlo simulation. Then we compare the methods on a theoretical basis and report on their numerical properties with respect to computational times and accuracy. One key element of our analysis is that the analyzed methods are extended to incorporate piecewise time-dependent model parameters, which allows for a more realistic compound option pricing. The results in the numerical analysis section are important for practitioners in the financial industry to identify under which model prerequisites (for instance, Heston model where Feller condition is fulfilled or not, Heston model with piecewise time-dependent parameters or with stochastic interest rates) it is preferable to use and which of the available numerical methods.

Compound Option Pricing under Fuzzy Environment

Considering the uncertainty of a financial market includes two aspects: risk and vagueness; in this paper, fuzzy sets theory is applied to model the imprecise input parameters (interest rate and volatility). We present the fuzzy price of compound option by fuzzing the interest and volatility in Geske’s compound option pricing formula. For eachα, theα-level set of fuzzy prices is obtained according to the fuzzy arithmetics and the definition of fuzzy-valued function. We apply a defuzzification method based on crisp possibilistic mean values of the fuzzy interest rate and fuzzy volatility to obtain the crisp possibilistic mean value of compound option price. Finally, we present a numerical analysis to illustrate the compound option pricing under fuzzy environment.

Compound Option Pricing Model Applied in Overseas Mineral Investment

According to characteristics of selecting an overseas mineral project，such as a long period, high-risk and investment decisions in different phased, several stages is divided in the process of entire mineral investment decision-making. Using the method of the multi-stage real option pricing method, the compound option pricing model is to construct the investment in mineral based on the compound call option pricing formula of Geske model. Taking a company for example, we introduce the probability of success in different stages for sensitivity analysis. Selecting the lower successful probability at the exploration stage and higher at the mining stage to illustrate the availability and stability of the model, helping investors to make the right choice.

Investigating Time-Efficient Methods to Price Compound Options in the Heston Model

The primary purpose of this paper is to provide an in-depth analysis of a number of structurally different methods to numerically evaluate European compound option prices under Heston’s stochastic volatility dynamics. Therefore, we first outline several approaches that can be used to price these type of options in the Heston model: a modified sparse grid method, a fractional fast Fourier transform technique, a (semi-)analytical valuation formula using the Green’s function of logarithmic spot and volatility and a Monte Carlo simulation. Then we compare the methods on a theoretical basis and report on their numerical properties with respect to computational times and accuracy. One key element of our analysis is that the analyzed methods are extended to incorporate piecewise time-dependent model parameters, which allows for a more realistic compound option pricing.

Research on European Option and Compound Option Pricing

Research on European Option and Compound Option Pricing

VALUATION OF COMPOUND OPTION WHEN THE UNDERLYING ASSET IS NON-TRADABLE

After Geske (1979), compound options — options on options — have been employed in many fields in which real options are applied. The formula for a compound option is convenient to use in real project investment, but it has one drawback — the assets that underlie the compound options are usually non-tradable. This article addresses this issue and proposes two new compound option pricing formulae to overcome this drawback.

RohnerChem to scale EnCat

Our paper presents a dynamic model of entrepreneurial venture financing under uncertainty based on option exercise games between an entrepreneur and a venture capitalist (VC). In particular, we analyze the impact of multi-staged financing and both economic and technological uncertainty on optimal contracting in the context of VC-financing. Our novel approach combines compound option pricing with sequential non-cooperative contracting, allowing us to determine whether renegotiation will improve the probability of coming to an agreement and proceed with the venture. It is shown that both sources of uncertainty positively impact the VC-investor's optimal equity share. Specifically, higher uncertainty leads to a larger stake in the venture, and renegotiation may result in a dramatic shift of control rights in the venture, preventing the venture from failure. Moreover, given ventures with low volatility, situations might occur where the VC-investor loses his first-mover advantage. Based on a comparative-static analysis, new testable hypotheses for further empirical studies are derived from the model.

Explaining the Level of Credit Spreads: Option-Implied Jump Risk Premia in a Firm Value Model

Prices of equity index put options contain information on the price of systematic downward jump risk. We use a structural jump-diffusion firm value model to assess the level of credit spreads that is generated by option-implied jump risk premia. In our compound option pricing model, an equity index option is an option on a portfolio of call options on the underlying firm values. We calibrate the model parameters to historical information on default risk, the equity premium and equity return distribution, and S&P 500 index option prices. Our results show that a model without jumps fails to fit the equity return distribution and option prices, and generates a low out-of-sample prediction for credit spreads. Adding jumps and jump risk premia improves the fit of the model in terms of equity and option characteristics considerably and brings predicted credit spread levels much closer to observed levels.

Valuation of the minimum revenue guarantee and the option to abandon in BOT infrastructure projects

The real option approach is used to value the minimum revenue guarantee (MRG) and the option to abandon in Build‐Operate‐Transfer infrastructure projects. The option to abandon is formulated under an investment option held by the concessionaire at contract signing and to expire before construction commencement. MRG is formulated as a series of European style put options in a single option pricing model. When combined with the option to abandon in the pre‐construction phase, MRG is reconstructed as a series of European style call options to develop a compound option pricing formula. The Taiwan High‐Speed Rail Project is chosen as a numerical case to apply the formulas. The results show both MRG and the option to abandon can create values. When MRG and the option to abandon are combined, they will counteract each other and their values will thus be reduced. Increasing the MRG level will decrease the value of the option to abandon, and, at a certain MRG level, the option to abandon will be rendered worthless.

Explaining the Level of Credit Spreads: Option-Implied Jump Risk Premia in a Firm Value Model

Prices of equity index put options contain information on the price of systematic downward jump risk. We use a structural jump-diffusion firm value model to assess the level of credit spreads that is generated by option-implied jump risk premia. In our compound option pricing model, an equity index option is an option on a portfolio of call options on the underlying firm values. We calibrate the model parameters to historical information on default risk, the equity premium and equity return distribution, and S&P 500 index option prices. Our results show that a model without jumps fails to fit the equity return distribution and option prices, and generates a low out-of-sample prediction for credit spreads. Adding jumps and jump risk premia improves the fit of the model in terms of equity and option characteristics considerably and brings predicted credit spread levels much closer to observed levels.