1,720,967 research outputs found
Efficient valuation of barrier options under equity and interest rate risks
No full textIn this paper we study European and American equity derivatives with barrier features within a generic market model characterized by correlated equity and interest rate risk factors. First of all, we provide general algorithms to price discretely monitored European and American knock-in and knock-out options. Secondly, we adapt these techniques to a fairly general market model characterized by local volatility and a correlated mean-reverting process for the interest rate and the detail how to improve its efficiency. In particular, we discuss how to improve the precision of lattice-based pricing techniques in case of barrier options and we assess the computational efficiency of the proposed algorithms with respect to standard Monte Carlo-based approaches. Finally, we test our algorithms for two particular sets of barrier contracts retrieving also the optimal exercise policies of their American counterparts in the form of critical surfaces
Seasonality and spikes in the natural gas market
Full text linkIn this paper we propose and examine an arbitrage-free model for the natural gas spot price and its convenience yield. Performing an empirical analysis of the European natural gas spot and futures markets, we observe that log spot prices are non-stationary, exhibit mild seasonality, and display almost continuous behaviour. In contrast, the implied convenience yield is stationary, shows strong seasonality, and experiences frequent spikes. Motivated by this evidence, we model the spot convenience yield as a combination of a deterministic seasonal component and a mean-reverting stochastic process with jumps. By assuming an appropriate distribution for the jump component, we derive a closed-form expression for futures prices. Our model demonstrates an excellent fit to European data, both before and after the COVID-19 pandemic and the Russia–Ukraine war
Linking futures and options pricing in the natural gas market
Full text linkA robust model for natural gas prices should simultaneously capture the observed prices of both futures and options. While incorporating a seasonal factor in the convenience yield of the spot price effectively replicates forward curves, it proves insufficient for ac- curately modelling the options price surface. The latter is more sensitive to the volatility structure of the spot price process, which has a limited impact on futures pricing. In this paper, we analyse European natural gas spot, futures, and options prices throughout 2024 and propose a no-arbitrage model that integrates both a seasonal stochastic convenience yield and a local volatility factor. This framework enables a simultaneous and accurate fit of both forward curves and options prices
Essays on American Options
Full text linkThis thesis deals with the pricing of American equity options exposed to correlated interest rate
and equity risks.
The first article, American options on high dividend securities: a numerical investigation by F.
Rotondi, investigates the Monte Carlo-based algorithm proposed by Longstaff and Schwartz (2001)
to price American options. I show how this algorithm might deliver biased results when valuing
American options that start out of the money, especially if the dividend yield of the underlying is
high. I propose two workarounds to correct for this bias and I numerically show their strength.
The second article, American options and stochastic interest rates by A. Battauz and F. Rotondi
introduces a novel lattice-based approach to evaluate American option within the Vasicek model,
namely a market model with mean-reverting stochastic interest rates. Interestingly, interest rates
are not assumed to be necessarily positive and non standard optimal exercise policy of American
call and put options arise when interest rates are just mildly negative. The third article, Barrier
options under correlated equity and interest rate risks by F. Rotondi deals with derivatives with
barrier features within a market model with both equity and interest rate risk. Exploiting latticebased
algorithm, I price European and American knock-in and knock-out contracts with both a
discrete and a continuous monitoring. Then, I calibrate the model to current European data and
I document how models that assume either a constant interest rate, or strictly positive stochastic
interest rates or uncorrelated interest rates deliver sizeable pricing errors
American Options on High Dividend Securities: A Numerical Investigation
No full textI document a sizeable bias that might arise when valuing out of the money American options via the Least Square Method proposed by Longstaff and Schwartz (2001). The key point of this algorithm is the regression-based estimate of the continuation value of an American option. If this regression is ill-posed, the procedure might deliver biased results. The price of the American option might even fall below the price of its European counterpart. For call options, this is likely to occur when the dividend yield of the underlying is high. This distortion is documented within the standard Black−Scholes−Merton model as well as within its most common extensions (the jump-diffusion, the stochastic volatility and the stochastic interest rates models). Finally, I propose two easy and effective workarounds that fix this distortion
American options and stochastic interest rates
Full text linkWe study finite-maturity American equity options in a stochastic mean-reverting diffusive interest rate framework. We allow for a non-zero correlation between the innovations driving the equity price and the interest rate. Importantly, we also allow for the interest rate to assume negative values, which is the case for some investment grade government bonds in Europe in recent years. In this setting we focus on Amer- ican equity call and put options and characterize analytically their two-dimensional free boundary, i.e. the underlying equity and the interest rate values that trigger the optimal exercise of the option before maturity. We show that non-standard double continuation regions may appear, extending the findings documented in the litera- ture in a constant interest rate framework. Moreover, we contribute by developing a bivariate discretization of the equity price and interest rate processes that converges in distribution as the time step shrinks. This discretization, described by a recombin- ing quadrinomial tree, allows us to compute American equity options’ prices and to analyze their free boundaries with respect to time and current interest rate. Finally, we document the existence of non-standard optimal exercise policies for American call options on a non-dividend-paying equity
Optimal liquidation policies of redeemable shares
No full textIn this paper we explore the optimal issuance and liquidation of redeemable shares. Redeemable shares are those that the issuer can repurchase, or redeem, at a predetermined price, known as the call price, as soon as a given barrier event is triggered. We first determine the optimal call price for the issuer by stating and solving a stylized earning per share maximization problem from the point of view of a company. Once the call price is determined, we focus on the valuation of both perpetual and finite-maturity redeemable shares and we examine the problem of their optimal liquidation from the point of view of a shareholder holding them. Along with the few closed-form results that can be obtained in a lognormal continuous-time framework, we propose an intuitive and flexible method to retrieve the optimal liquidation policy in the form of a liquidation boundary, thanks to a parsimonious Markovianization of the evaluation problem in a binomial framework. Numerical tests using alternative market models and different dividend formulations confirm the robustness of our results
Valuation of general GMWB annuities in a low interest rate environment
Full text linkVariable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) entitle the policy holder to periodic withdrawals together with a terminal payoff linked to the performance of an equity fund. In this paper, we consider the valuation of a general class of GMWB annuities, allowing for step-up, bonus and surrender features, taking also into account mortality risk and death benefits. When dynamic withdrawals are allowed, the valuation of GMWB annuities leads to a stochastic optimal control problem, which we address here by dynamic programming techniques. Adopting a Hull-White interest rate model, correlated with the equity fund, we propose an efficient tree-based algorithm. We perform a thorough analysis of the determinants of the market value of GMWB annuities and of the optimal withdrawal strategies. In particular, we study the impact of a low/negative interest rate environment. Our findings indicate that low/negative rates profoundly affect the optimal withdrawal behaviour and, in combination with step-up and bonus features, increase significantly the fair values of GMWB annuities, which can only be compensated by large management fees
On horizon-consistent mean-variance portfolio allocation
No full textWe analyze the problem of constructing multiple buy-and-hold mean-variance portfolios over increasing investment horizons in continuous-time arbitrage-free stochastic interest rate markets. The orthogonal approach to the one-period mean-variance optimization of Hansen and Richard (Econometrica 55(3):587-613, 1987) requires the replication of a risky payoff for each investment horizon. When many maturities are considered, a large number of payoffs must be replicated, with an impact on transaction costs. In this paper, we orthogonally decompose the whole processes defined by asset returns to obtain a mean-variance frontier generated by the same two securities across a multiplicity of horizons. Our risk-adjusted mean-variance frontier rests on the martingale property of the returns discounted by the log-optimal portfolio and features a horizon consistency property. The outcome is that the replication of a single risky payoff is required to implement such frontier at any investment horizon. As a result, when transaction costs are taken into account, our risk-adjusted mean-variance frontier may outperform the traditional mean-variance optimal strategies in terms of Sharpe ratio. Realistic numerical examples show the improvements of our approach in medium- or long-term cashflow management, when a sequence of target returns at increasing investment horizons is considered
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