Path-dependent Contingent Claims Research Articles

A Technical Note, Looback Options: a comparison between Monte Carlo Techniques

Looback options are path dependent contingent claims whose payoffs depend on the extrema of the underlying asset price over a certain time interval. In this note we compare the performance of two Monte Carlo techniques to price lookback options, a crude Monte Carlo estimator and Antithetic variate estimator. We find that the Antithetic estimator performs better under a variety of
 performance measures.

Quantile Hedging in a semi-static market with model uncertainty

With model uncertainty characterized by a convex, possibly non-dominated set of probability measures, the agent minimizes the cost of hedging a path dependent contingent claim with given expected success ratio, in a discrete-time, semi-static market of stocks and options. Based on duality results which link quantile hedging to a randomized composite hypothesis test, an arbitrage-free discretization of the market is proposed as an approximation. The discretized market has a dominating measure, which guarantees the existence of the optimal hedging strategy and helps numerical calculation of the quantile hedging price. As the discretization becomes finer, the approximate quantile hedging price converges and the hedging strategy is asymptotically optimal in the original market.

Bounds for path-dependent options

We develop new semiparametric bounds on the expected payoffs and prices of European call options and a wide range of path-dependent contingent claims. We first focus on the trinomial financial market model in which, as is well-known, an exact calculation of derivative prices based on no-arbitrage arguments is impossible. We show that the expected payoff of a European call option in the trinomial model with martingale-difference log-returns is bounded from above by the expected payoff of a call option written on an asset with i.i.d. symmetric two-valued log-returns. We further show that the expected payoff of a European call option in the multiperiod trinomial option pricing model is bounded by the expected payoff of a call option in the two-period model with a log-normal asset price. We also obtain bounds on the possible prices of call options in the (incomplete) trinomial model in terms of the parameters of the asset’s distribution. Similar bounds also hold for many other contingent claims in the trinomial option pricing model, including those with an arbitrary convex increasing payoff function as well as for path-dependent ones such as Asian options. We further obtain a wide range of new semiparametric moment bounds on the expected payoffs and prices of path-dependent Asian options with an arbitrary distribution of the underlying asset’s price. These results are based on recently obtained sharp moment inequalities for sums of multilinear forms and U-statistics and provide their first financial and economic applications in the literature. Similar bounds also hold for many other path-dependent contingent claims.

Static hedging under maturity mismatch

Can shorter maturity European options be statically hedged with longer maturity plain vanilla options? This problem appears, for example, when analysing options on forwards in relation to liquid options on the spot underlying. Under mild assumptions on the underlying security price process and the option’s payoff function, we show that approximate static hedges exist and we provide a recipe for constructing them. Examples illustrate the power of the hedge and its sensitivity to modelling assumptions. The results can be extended to formulating semi-static hedging strategies for discretely monitored path-dependent contingent claims.

Quantile Hedging in a Semi-Static Market with Model Uncertainty

With model uncertainty characterized by a convex, possibly non-dominated set of probability measures, the agent minimizes the cost of hedging a path dependent contingent claim with given expected success ratio, in a discrete-time, semi-static market of stocks and options. Based on duality results which link quantile hedging to a randomized composite hypothesis test, an arbitrage-free discretization of the market is proposed as an approximation. The discretized market has a dominating measure, which guarantees the existence of the optimal hedging strategy and helps numerical calculation of the quantile hedging price. As the discretization becomes finer, the approximate quantile hedging price converges and the hedging strategy is asymptotically optimal in the original market.

Static Hedging Under Maturity Mismatch

Can shorter maturity European options be statically hedged with longer maturity plain vanilla options? This problem appears for example when analyzing options on forwards in relation to liquid options on the spot underlying. Under mild assumptions on the underlying security price process and on the option's payoff function we show that approximate static hedges exist und we provide a recipe for constructing them. Examples illustrate the power of the hedge and its sensitivity to modelling assumptions. The results can be extended to formulating semi-static hedging strategies for discretely monitored path-dependent contingent claims.

Valuing continuous-installment options

Abstract Installment options are path-dependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing European continuous-installment options written on dividend-paying assets in the standard Black–Scholes–Merton framework. The valuation of installment options can be formulated as a free boundary problem, due to the flexibility of continuing or stopping to pay installments. On the basis of a PDE for the initial premium, we derive an integral representation for the initial premium, being expressed as a difference of the corresponding European vanilla value and the expected present value of installment payments along the optimal stopping boundary. Applying the Laplace transform approach to this PDE, we obtain explicit Laplace transforms of the initial premium as well as its Greeks, which include the transformed stopping boundary in a closed form. Abelian theorems of Laplace transforms enable us to characterize asymptotic behaviors of the stopping boundary close and at infinite time to expiry. We show that numerical inversion of these Laplace transforms works well for computing both the option value and the optimal stopping boundary.

The Normal Class of Arbitrage-Free Spot-Rate Models - Implication and Implementation

In this paper we first show how to determine the T-forward adjusted risk-measure using the concept of fundamental solution to linear PDE’s. After that, relying on Fourier transformation we derive bond-and bond-option prices for the Extended Vasicek model from Hull and White (1990) and the Quadratic Interest Rate model. With respect to the Quadratic Interest Rate model we succeed in carrying the analysis much further than Jamshidian (1996).A special discrete time model - which in some cases is appropriate for the Quadratic Interest Rate model - is also derived.The last part of the paper analyse Monte Carlo techniques in connection with spot-rate models with a time-dependent drift. We also introduce a method - using the concept of forward induction - that constrain the Monte Carlo simulated spot-rate process for the matching of the initial yield-curve. For the pricing of path-dependent contingent claim, we deduce that, even though Monte Carlo is the natural method to use, it might not be the most efficient one - at least not when the spot-rate is Markovian.

American Continuous-Installment Options: Valuation and Premium Decomposition

Installment options are weakly path-dependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing American continuous-installment options written on dividend-paying assets. The setup is a standard Black–Scholes–Merton framework where the price of the underlying asset evolves according to a geometric Brownian motion. The valuation of installment options can be formulated as an optimal stopping problem, due to the flexibility of continuing or stopping to pay installments as well as the chance of early exercise. Analyzing cash flow generated by the optimal stop, we can characterize asymptotic behaviors of the stopping and early exercise boundaries close to expiry. Combining the PDE and Laplace transform approaches, we obtain the Laplace transform of the initial premium in an explicit form, which is decomposed into the value of the associated European vanilla option with the same payoff p...

Deposit guarantee evaluation and incentives analysis in a mutual guarantee system

This paper analyzes how the deposit guarantee value affects the risk incentives in a mutual guarantee system. We liken the guarantee’s value to that of a European-style contingent claims portfolio. The main feature emerging from our model is that a mutual guarantee system would give banks an adverse incentive to increase riskiness. To mitigate this incentive, we introduce a regulatory provision modelled using a path-dependent contingent claim. By comparing the mutual guarantee system with a non-mutual one, we show that the former is less expensive, but implies higher adverse incentives for the banks, especially for undercapitalized institutions.

Valuation and hedging of participating life-insurance policies under management discretion

The valuation and hedging of participating life insurance policies, also known as with-profits policies, is considered. Such policies can be seen as European path-dependent contingent claims whose underlying security is the investment portfolio of the insurance company that sold the policy. The fair valuation of these policies is studied under the assumption that the insurance company has the right to modify the investment strategy of the underlying portfolio at any time. Furthermore, it is assumed that the issuer of the policy does not setup a separate portfolio to hedge the risk associated with the policy. Instead, the issuer will use its discretion about the investment strategy of the underlying portfolio to hedge shortfall risks. In that sense, the insurer’s investment portfolio serves simultaneously as the underlying security and as the hedge portfolio. This means that the hedging problem can not be separated from the valuation problem. We investigate the relationship between risk-neutral valuation and hedging of these policies in complete and incomplete financial markets.

Estimating Tranche Spreads by Loss Process Simulation

A credit derivative is a path dependent contingent claim on the aggregate loss in a portfolio of credit sensitive securities. We estimate the value of a credit derivative by Monte Carlo simulation of the affine point process that models the loss. We consider two algorithms that exploit the direct specification of the loss process in terms of an intensity. One algorithm is based on the simulation of intensity paths. Here discretization introduces bias into the results. The other algorithm facilitates exact simulation of default times and generates an unbiased estimator of the derivative price. We implement the algorithms to value index and tranche swaps, and we calibrate the loss process to quotes on the CDX North America High Yield index.

Return smoothing mechanisms in life and pension insurance: Path-dependent contingent claims

Traditional with-profits pension saving schemes have been criticized for their opacity, plagued by embedded options and guarantees, and have recently created enormous problems for the solvency of the life insurance and pension industry. This has fueled creativity in the industry’s product development departments, and this paper analyzes a representative member of a family of new pension schemes that have been introduced in the new millennium to alleviate these problems. The complete transparency of the new scheme’s smoothing mechanism means that it can be analyzed using contingent claims pricing theory. We explore the properties of this pension scheme in detail and find that in terms of market value, smoothing is an illusion, but also that the return smoothing mechanism implies a dynamic asset allocation strategy which corresponds with traditional pension saving advice and the recommendations of state-of-the-art dynamic portfolio choice models.

Return Smoothing Mechanisms in Life and Pension Insurance: Path-Dependent Contingent Claims

Traditional with-profits pension saving schemes have been criticized for their opacity, plagued by embedded options and guarantees, and have recently created enormous problems for the solvency of the life insurance and pension industry. This has fueled creativity in the industry's product development departments, and this paper analyzes a representative member of a family of new pension schemes that have been introduced in the new millennium to alleviate these problems. The complete transparency of the new scheme's smoothing mechanism means that it can be analyzed using contingent claims pricing theory. We explore the properties of this pension scheme in detail and find that in terms of market value, smoothing is an illusion, but also that the return smoothing mechanism implies a dynamic asset allocation strategy which corresponds with traditional pension saving advice and the recommendations of state-of-the-art dynamic portfolio choice models.

A CONTINUOUS-TIME REEXAMINATION OF DOLLAR-COST AVERAGING

The widespread practice of dollar-cost averaging (DCA) amongst the investing public, has puzzled most financial economists, ever since Constantinides [2] demonstrated the dynamic inefficiency of this strategy under very general conditions. This enduring phenomena has forced researchers, such as Statman [12], to suggest behavioral explanations for DCA's popularity, predicated on the prospect theory of Kahneman and Tversky [4]. In this paper we reexamine the payoff structure of DCA via continuous-time financial mathematics and then ask the question: Is it possible to reconcile the theory and practice of dollar-cost averaging? To answer this question, we take a slightly different approach to the issue by using the tools of stochastic calculus and Brownian bridges. We demonstrate that engaging in a dollar-cost averaging strategy is akin to purchasing a zero strike arithmetic Asian option on the underlying security. In other words, people who engage in dollar-cost averaging are implicitly purchasing a path-dependent contingent claim. We then prove that the expected return from this exotic option — i.e. the DCA strategy — conditional on knowing the final value of the security will uniformly exceed the return from the underlying security for all sufficiently large volatilities. This leads us to argue that investors may be dollar-cost averaging because they have "target prices" for the underlying asset price. The strategy of dollar-cost averaging would then exceed the returns from lump-sum investing, based on their subjective conditional expectation. In fact, the more volatile the underlying security, the greater is the benefit to dollar-cost averaging — conditional on knowing the final value — which is consistent with common practice.

Window double barrier options

This paper examines a path-dependent contingent claim called the window double barrier option, including standard but also more exotic features such as combinations of single and double barriers. Price properties and hedging issues are discussed, as well as financial applications.Explicit formulae are provided, along with simple techniques for theirimplementation. Numerical results show that they compare very favourablywith alternative pricing approaches in terms of accuracy and efficiency.

MEAN-REVERTING STOCHASTIC VOLATILITY

We present derivative pricing and estimation tools for a class of stochastic volatility models that exploit the observed "bursty" or persistent nature of stock price volatility. An empirical analysis of high-frequency S&P 500 index data confirms that volatility reverts slowly to its mean in comparison to the tick-by-tick fluctuations of the index value, but it is fast mean-reverting when looked at over the time scale of a derivative contract (many months). This motivates an asymptotic analysis of the partial differential equation satisfied by derivative prices, utilizing the distinction between these time scales. The analysis yields pricing and implied volatility formulas, and the latter is used to "fit the smile" from European index option prices. The theory identifies the important group parameters that are needed for the derivative pricing and hedging problem for European-style securities, namely the average volatility and the slope and intercept of the implied volatility line, plotted as a function of the log-moneyness-to-maturity-ratio. The results considerably simplify the estimation procedure, and the data produces estimates of the three important parameters which are found to be stable within periods where the underlying volatility is close to being stationary. These segments of stationarity are identified using a wavelet-based tool. The remaining parameters, including the growth rate of the underlying, the correlation between asset price and volatility shocks, the rate of mean-reversion of the volatility and the market price of volatility risk can be roughly estimated, but are not needed for the asymptotic pricing formulas for European derivatives. The extension to American and path-dependent contingent claims is the subject of future work.

Financial Modeling in a Fast Mean-Reverting Stochastic Volatility Environment

We present a derivative pricing and estimation methodology for a class of stochastic volatility models that exploits the observed 'bursty' or persistent nature of stock price volatility. Empirical analysis of high-frequency S&P 500 index data confirms that volatility reverts slowly to its mean in comparison to the tick-by- tick fluctuations of the index value, but it is fast mean- reverting when looked at over the time scale of a derivative contract (many months). This motivates an asymptotic analysis of the partial differential equation satisfied by derivative prices, utilizing the distinction between these time scales. The analysis yields pricing and implied volatility formulas, and the latter provides a simple procedure to 'fit the skew' from European index option prices. The theory identifies the important group parameters that are needed for the derivative pricing and hedging problem for European-style securities, namely the average volatility and the slope and intercept of the implied volatility line, plotted as a function of the log- moneyness-to-maturity-ratio. The results considerably simplify the estimation procedure. The remaining parameters, including the growth rate of the underlying, the correlation between asset price and volatility shocks, the rate of mean-reversion of the volatility and the market price of volatility risk are not needed for the asymptotic pricing formulas for European derivatives, and we derive the formula for a knock-out barrier option as an example. The extension to American and path-dependent contingent claims is the subject of future work.

Valuation of Path-Dependent Contingent Claims with Multiple Exercise Decisions over Time: The Case of Take-Or-Pay

Hull and White (1993) introduced a method to extend lattice-based contingent claim val? uation techniques to problems with path-dependent payoffs. This paper builds upon their method in the process of valuing two types of commodity purchase contracts that ex? hibit a form of timing flexibility not previously analyzed in the literature. The techniques demonstrated take advantage of characteristics of claims that simplify incorporation of path dependencies. Possible uses for the methods demonstrated include the valuation of: port? folios of mortgages, capital projects where intensity of use affects productivity ofthe asset, and capital projects with storage characteristics.

Path Dependent Options: The Case of Lookback Options

ABSTRACTLookback options are path dependent contingent claims whose payoffs depend on the extrema of a given security's price over a certain period of time. Using probabilistic tools, we derive explicit formulas for various European lookback options, and provide some results about their American counterparts.