Put Option Prices Research Articles

Currency option pricing in a credible exchange rate target zone

This article examines currency option pricing within a credible target zone arrangement where interventions at the boundaries push the exchange rate back into its fluctuation band. Valuation of such options is complicated by the requirement that the reflection mechanism should prevent the arbitrage opportunities that would arise if the exchange rate were to spend finite time on the boundaries. To prevent the latter, we superimpose instantaneously reflecting boundaries upon the familiar geometric Brownian motion (GBM) framework. We derive closed-form expressions for European call and put option prices and show that prices for the GBM model of Garman and Kohlhagen (1983) arise as the limit case for infinitely wide bands. We also illustrate that taking account of boundaries is of considerable economic value as erroneously using the unbounded-domain model of Garman and Kohlhagen (1983) easily overprices options by more than 100%.

Arbitrage bounds for prices of weighted variance swaps

We develop a theory of robust pricing and hedging of a weighted variance swap given market prices for a finite number of co-maturing put options. We assume the put option prices do not admit arbitrage and deduce no-arbitrage bounds on the weighted variance swap along with super- and sub-replicating strategies that enforce them. We find that market quotes for variance swaps are surprisingly close to the model-free lower bounds we determine. We solve the problem by transforming it into an analogous question for a European option with a convex payoff. The lower bound becomes a problem in semi-infinite linear programming which we solve in detail. The upper bound is explicit. We work in a model-independent and probability-free setup. In particular, we use and extend Follmer's pathwise stochastic calculus. Appropriate notions of arbitrage and admissibility are introduced. This allows us to establish the usual hedging relation between the variance swap and the “log contract” and similar connections for weighted variance swaps. Our results take the form of a FTAP: we show that the absence of (weak) arbitrage is equivalent to the existence of a classical model which reproduces the observed prices via risk-neutral expectations of discounted payoffs.

ARBITRAGE BOUNDS FOR PRICES OF WEIGHTED VARIANCE SWAPS

We develop a theory of robust pricing and hedging of a weighted variance swap given market prices for a finite number of co‐maturing put options. We assume the put option prices do not admit arbitrage and deduce no‐arbitrage bounds on the weighted variance swap along with super‐ and sub‐replicating strategies that enforce them. We find that market quotes for variance swaps are surprisingly close to the model‐free lower bounds we determine. We solve the problem by transforming it into an analogous question for a European option with a convex payoff. The lower bound becomes a problem in semi‐infinite linear programming which we solve in detail. The upper bound is explicit. We work in a model‐independent and probability‐free setup. In particular, we use and extend Föllmer's pathwise stochastic calculus. Appropriate notions of arbitrage and admissibility are introduced. This allows us to establish the usual hedging relation between the variance swap and the “log contract” and similar connections for weighted variance swaps. Our results take the form of a FTAP: we show that the absence of (weak) arbitrage is equivalent to the existence of a classical model which reproduces the observed prices via risk‐neutral expectations of discounted payoffs.

Solution of the Fractional Black-Scholes Option Pricing Model by Finite Difference Method

This work deals with the put option pricing problems based on the time-fractional Black-Scholes equation, where the fractional derivative is a so-called modified Riemann-Liouville fractional derivative. With the aid of symbolic calculation software, European and American put option pricing models that combine the time-fractional Black-Scholes equation with the conditions satisfied by the standard put options are numerically solved using the implicit scheme of the finite difference method.

A Feasibility Analysis of Black-Scholes-Merton Differential Equation Model for Stock Option Pricing by Using Historical Volatility : With Reference to Selected Stock Options Traded in NSE

In today’s financial world there is a great need to predict the value of the assets, using which strategic decisions can be made to make short term or long term capital gains. Due to the dynamic and uncertain nature of the financial markets, the prediction of the asset prices are really difficult. Many models have been developed to predict the option prices in the financial market. The certainity of these models to predict the option prices to the most accurate level or to the level of minimum deviation is questionable. This study is aimed at analyzing the feasibility of Black - Scholes – Merton differential equation model for stock option pricing in Indian stock exchanges. The result of this study can be used to predict the suitability of using Black - Scholes – Merton differential equation model to predict stock option prices in Indian market. Further the regression analysis has been used to see the impact of time to expiry over the option price and anova test has been used to check whether the mean difference between expected price as computed by Black - Scholes – Merton differential equation model and actual price have any significant difference. The result of analysis found that Black - Scholes – Merton model is more usefull in call option pricing than the put option pricing and also impact of timing is more relevenat for put option pricing than for call option pricing.

The Intraday and Overnight Behavior of SPY Options and Adjusted Delta Hedging

This study examines the intraday and overnight performance of S&P 500 exchange traded fund (SPY) put options from May 2005 through April 2010. The basic findings indicate that deviations of put option price changes from projections based on Black–Scholes–Merton (BSM) deltas are often substantial and can be explained mostly by the negative relation between implied volatility and SPY returns. The study then examines the impact of adjusting deltas with out of sample estimates of the relationship between implied volatility and SPY returns on the profitability of intraday and overnight delta hedged short SPY put option positions. The findings indicate that these adjustments lower P&L volatility and improve measures of downside risk relative to both BSM deltas and deltas adjusted with the volatility skew. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:443‐468, 2013

A STOCK MODEL WITH JUMPS FOR UNCERTAIN MARKETS

Uncertain differential equation with jumps is a type of differential equation driven by two classes of uncertain processes, namely canonical process and renewal process. Based on uncertain differential equation with jumps, this paper proposes a stock model with jumps for uncertain financial markets. Furthermore, the European call and put option pricing formulas for the stock model are formulated and some mathematical properties of them are studied. Finally, some generalized uncertain stock models with jumps are discussed.

Stock Return Predictability of Cross-Market Deviations in Option Prices and Credit Default Swap Spreads

Cross-market deviations in equity put option prices and credit default swap spreads are temporal and revert to their usual level shortly after they occur, on average within about one week. The process of reversion involves predictable and economically significant changes also in the equity values of the reference firm, thus empirically supporting the theoretical linkage of the three markets. Cross-market deviations occur for firms with high information uncertainty and are not created equally. Firms with unusually more expensive credit than equity insurance are likely to experience informed trading in their credit insurance contract. However, we attribute instances of unusually expensive equity insurance to belief heterogeneity.

The Fear of Rare Economic Disasters Reflected in Option Prices

The paper studies the asset pricing implications of economic disaster risk in the financial markets. We consider rare events as jumps in aggregate consumption with time-varying intensity. We use a general equilibrium framework with recursive preferences and allow the elasticity of intertemporal substitution (EIS) to deviate from one. The prices are obtained imposing an affine structure on the state variables. In order to assess the ex-ante risk perceived by investors we compute put option prices. The empirical estimation is based on S&P500 options to imply the time series of jump intensity and the utility parameters. We find that volatility smiles flatten out with maturity, which captures the long-term influence of disasters. We explore the effect of an EIS different from one and show that while an EIS > 1 fits better option data, an EIS

The Use of Statistical Tests to Calibrate the Black-Scholes Asset Dynamics Model Applied to Pricing Options with Uncertain Volatility

A new method for calibrating the Black-Scholes asset price dynamics model is proposed. The data used to test the calibration problem included observations of asset prices over a finite set of (known) equispaced discrete time values. Statistical tests were used to estimate the statistical significance of the two parameters of the Black-Scholes model: the volatility and the drift. The effects of these estimates on the option pricing problem were investigated. In particular, the pricing of an option with uncertain volatility in the Black-Scholes framework was revisited, and a statistical significance was associated with the price intervals determined using the Black-Scholes-Barenblatt equations. Numerical experiments involving synthetic and real data were presented. The real data considered were the daily closing values of the S&P500 index and the associated European call and put option prices in the year 2005. The method proposed here for calibrating the Black-Scholes dynamics model could be extended to other science and engineering models that may be expressed in terms of stochastic dynamical systems.

Error Estimates for Approximations of American Put Option Price

Abstract Finite difference approximations to multi-asset American put option price are considered. The assets are modelled as a multi-dimensional diffusion process with variable drift and volatility. Approximation error of order one quarter with respect to the time discretisation parameter and one half with respect to the space discretisation parameter is proved by reformulating the corresponding optimal stopping problem as a solution of a degenerate Hamilton-Jacobi-Bellman equation. Furthermore, the error arising from restricting the discrete problem to a finite grid by reducing the original problem to a bounded domain is estimated.

Weak convergence for approximation of American option prices

Based on a sequence of discretized American option price processes under the multinomial model proposed by Maller, Solomon and Szimayer (2006), the sequence converges to the counterpart under the original Levy process in distribution for almost all time. We prove a weak convergence in this case for American put options for all time. By adapting Skorokhod representation theorem, a new sequence of approximating processes with the same laws with the multinomial tree model defined by Maller, Solomon and Szimayer (2006) is obtained. The new sequence of approximating processes satisfies Aldous’ criterion for tightness. And, the sequence of filtrations generated by the new approximation converges to the filtration generated by the representative of Levy process weakly. By using results of Coquet and Toldo (2007), we give a complete proof of the weak convergence for the approximation of American put option prices for all time.

A Second-Order Difference Scheme for the Penalized Black–Scholes Equation Governing American Put Option Pricing

In this paper we present a stable finite difference scheme on a piecewise uniform mesh along with a power penalty method for solving the American put option problem. By adding a power penalty term the linear complementarity problem arising from pricing American put options is transformed into a nonlinear parabolic partial differential equation. Then a finite difference scheme is proposed to solve the penalized nonlinear PDE, which combines a central difference scheme on a piecewise uniform mesh with respect to the spatial variable with an implicit time stepping technique. It is proved that the scheme is stable for arbitrary volatility and arbitrary interest rate without any extra conditions and is second-order convergent with respect to the spatial variable. Furthermore, our method can efficiently treats the singularities of the non-smooth payoff function. Numerical results support the theoretical results.

Pricing perpetual American options under a stochastic-volatility model with fast mean reversion

In this paper, we present a “correction” to Merton’s (1973) well-known classical case of pricing perpetual American puts by considering the same pricing problem under a general fast mean-reverting SV (stochastic-volatility) model. By using the perturbation method, two analytic formulae are derived for the option price and the optimal exercise price, respectively. Based on the newly obtained formulae, we conduct a quantitative analysis of the impact of the SV term on the price of a perpetual American put option as well as its early exercise strategies. It shows that the presence of a fast mean-reverting SV tends to universally increase the put option price and to defer the optimal time to exercise the option contract, had the underlying been assumed to be falling. It is also noted that such an effect could be quite significant when the option is near the money.

Crash risk of the euro in the sovereign debt crisis of 2009–2010

The economic-political instability of a country, which is tied to its credit risk, often leads to sharp depreciation and heightened volatility in its currency. This paper shows that not only the creditworthiness of the euro-area countries with weaker fiscal positions but also that of the member countries with more sound fiscal positions are important determinants of the deep out-of-the-money euro put option prices, which embedded information on the euro crash risk during the sovereign debt crisis of 2009–2010. We also find evidence of information flow from the sovereign credit default swap market to the currency option market during the crisis.

On the perpetual American put options for level dependent volatility models with jumps

We prove that the perpetual American put option price of level dependent volatility model with com- pound Poisson jumps is convex and is the classical solution of its associated quasi-variational inequality, that it is C 2 except at the stopping boundary and that it is C 1 everywhere (i.e. the smooth pasting condition always holds).

A new predictor–corrector scheme for valuing American puts

In this paper, we present a new numerical scheme, based on the finite difference method, to solve American put option pricing problems. Upon applying a Landau transform or the so-called front-fixing technique [19] to the Black–Scholes partial differential equation, a predictor–corrector finite difference scheme is proposed to numerically solve the nonlinear differential system. Through the comparison with Zhu’s analytical solution [35], we shall demonstrate that the numerical results obtained from the new scheme converge well to the exact optimal exercise boundary and option values. The results of our numerical examples suggest that this approach can be used as an accurate and efficient method even for pricing other types of financial derivative with American-style exercise.

Recovering a Piecewise Constant Volatility from Perpetual Put Option Prices

In this paper we present a method to recover a time-homogeneous piecewise constant volatility from a finite set of perpetual put option prices. The whole calculation process of the volatility is decomposed into easy computations in many fixed disjoint intervals. In each interval, the volatility is obtained by solving a system of nonlinear equations.

Recovering a Piecewise Constant Volatility from Perpetual Put Option Prices

In this paper we present a method to recover a time-homogeneous piecewise constant volatility from a finite set of perpetual put option prices. The whole calculation process of the volatility is decomposed into easy computations in many fixed disjoint intervals. In each interval, the volatility is obtained by solving a system of nonlinear equations.

Option Theory Does Not Refute Time Diversification

It is often recommended that asset allocation, in particular the proportion of stocks in a portfolio, should be influenced by the time for which the portfolio is expected to be held. Short investment horizons are seen as unsuitable for stocks. Long-term portfolios are regarded as suitable for high proportions of stocks. This is partly because long investment periods are seen as moderating the relative risk of stocks without distracting from the relatively high expected returns of stocks. Time diversification is one factor that ameliorates the long-run risk of stocks. Over the long term there will be periods of relatively good returns and periods of relatively poor returns; good periods and bad periods have a tendency to offset each other over long time spans. In consequence, risk increases less than proportionately with time, whereas returns have a compounding effect over time. Some academics have suggested that stock market risk increases as the investment horizon lengthens, thus refuting the concept of time diversification. That argument has been based on an apparent increase in the cost of hedging stock price risk, as indicated by theoretical put option prices, which appear to increase when the investment horizon extends. This article argues that those views are based on a restrictive use of the Black-Scholes option pricing model. When the model is used less restrictively, put option prices are found to decrease as the investment horizon lengthens when investment growth exceeds a particular rate. Risk, as measured by the cost of hedging, may decline as the holding period lengthens. Depending on expected investment growth, option theory supports time diversification rather than refuting it. <b>TOPICS:</b>Risk management, portfolio construction, performance measurement