Nonmyopic optimal portfolios in viable markets

We provide a representation for the nonmyopic optimal portfolio of an agent consuming only at the terminal horizon when the single state variable follows a general diffusion process and the market consists of one risky asset and a risk-free asset. The key term of our representation is a new object that we call the “rate of macroeconomic fluctuation” whose properties are fundamental for the portfolio dynamics. We show that, under natural cyclicality conditions, (i) the agent’s hedging demand is positive (negative) when the product of his prudence and risk tolerance is below (above) two and (ii) the portfolio weights decrease in risk aversion. We apply our results to study a general continuous-time capital asset pricing model and show that under the same cyclicality conditions, the market price of risk is countercyclical and the price of the risky asset exhibits excess volatility.     

A closed form solution for vulnerable options with Heston’s stochastic volatility

Over-the-counter stock markets in the world have been growing rapidly and vulnerability to default risks of option holders traded in the over-the-counter markets became an important issue, in particular, since the global finance crisis and Eurozone crisis. This paper studies the pricing of European-type vulnerable options when the underlying asset follows the Heston dynamics. In this paper, we obtain a closed form analytic formula of the option price as a stochastic volatility extension of the classical Heston formula and find how the stochastic volatility effect on the Black–Scholes price as well as on the decreasing speed of the option price with credit risk depends on moneyness.     

Semiparametric bounds of mean and variance for exotic options

Finding semiparametric bounds for option prices is a widely studied pricing technique. We obtain closed-form semiparametric bounds of the mean and variance for the pay-off of two exotic (Collar and Gap) call options given mean and variance information on the underlying asset price. Mathematically, we extended domination technique by quadratic functions to bound mean and variances. This work was supported by National Science Foundation of the United States (Grant Nos. DMS-0720977 and DMS-0805929)     

Approximate valuation of average options

This paper concerns the valuation of average options of European type where an investor has the right to buy the average of an asset price process over some time interval, as the terminal price, at a prespecified exercise price. A discrete model is first constructed and a recurrence formula is derived for the exact price of the discrete average call option. For the continuous average call option price, we derive some approximations and theoretical upper and lower bounds. These approximations are shown to be very accurate for at-the-money and in-the-money cases compared to the simulation results. The theoretical bounds can be used to provide useful information in pricing average options.     

On the relative efficiency of nth order and DARA stochastic dominance rules

It is known that third order stochastic dominance implies DARA dominance while no implications exist between higher orders and DARA dominance. A recent contribution points out that, with regard to the problem of determining lower and upper bounds for the price of a financial option, the DARA rule turns out to improve the stochastic dominance criteria of any order. In this paper the relative efficiency of the ordinary stochastic dominance and DARA criteria for alternatives with discrete distributions are compared, in order to see if the better performance of DARA criterion is also suitable for other practical applications. Moreover, the operational use of the stochastic dominance techniques for financial choices is deepened.     

Sharp Upper and Lower Bounds for Basket Options

Given a basket option on two or more assets in a one‐period static hedging setting, the paper considers the problem of maximizing and minimizing the basket option price subject to the constraints of known option prices on the component stocks and consistency with forward prices and treat it as an optimization problem. Sharp upper bounds are derived for the general n‐asset case and sharp lower bounds for the two‐asset case, both in closed forms, of the price of the basket option. In the case n = 2 examples are given of discrete distributions attaining the bounds. Hedge ratios are also derived for optimal sub and super replicating portfolios consisting of the options on the individual underlying stocks and the stocks themselves.     

Robust option pricing

In this paper, we combine robust optimization and the idea of ?$?$-arbitrage to propose a tractable approach to price a wide variety of options. Rather than assuming a probabilistic model for the stock price dynamics, we assume that the conclusions of probability theory, such as the central limit theorem, hold deterministically on the underlying returns. This gives rise to an uncertainty set that the underlying asset returns satisfy. We then formulate the option pricing problem as a robust optimization problem that identifies the portfolio which minimizes the worst case replication error for a given uncertainty set defined on the underlying asset returns. The most significant benefits of our approach are (a) computational tractability illustrated by our ability to price multi-asset, American and Asian options using linear optimization; and thus the computational complexity of our approach scales polynomially with the number of assets and with time to expiry and (b) modeling flexibility illustrated by our ability to model different kinds of options, various levels of risk aversion among investors, transaction costs, shorting constraints and replication via option portfolios.     

Static super-replicating strategies for a class of exotic options

In this paper, we investigate static super-replicating strategies for European-type call options written on a weighted sum of asset prices. This class of exotic options includes Asian options and basket options among others. We assume that there exists a market where the plain vanilla options on the different assets are traded and hence their prices can be observed in the market. Both the infinite market case (where prices of the plain vanilla options are available for all strikes) and the finite market case (where only a finite number of plain vanilla option prices are observed) are considered. We prove that the finite market case converges to the infinite market case when the number of observed plain vanilla option prices tends to infinity.We show how to construct a portfolio consisting of the plain vanilla options on the different assets, whose pay-off super-replicates the pay-off of the exotic option. As a consequence, the price of the super-replicating portfolio is an upper bound for the price of the exotic option. The super-hedging strategy is model-free in the sense that it is expressed in terms of the observed option prices on the individual assets, which can be e.g. dividend paying stocks with no explicit dividend process known. This paper is a generalization of the work of Simon et al. [Simon, S., Goovaerts, M., Dhaene, J., 2000. An easy computable upper bound for the price of an arithmetic Asian option. Insurance Math. Econom. 26 (2–3), 175–184] who considered this problem for Asian options in the infinite market case. Laurence and Wang [Laurence, P., Wang, T.H., 2004. What’s a basket worth? Risk Mag. 17, 73–77] and Hobson et al. [Hobson, D., Laurence, P., Wang, T.H., 2005. Static-arbitrage upper bounds for the prices of basket options. Quant. Fin. 5 (4), 329–342] considered this problem for basket options, in the infinite as well as in the finite market case.As opposed to Hobson et al. [Hobson, D., Laurence, P., Wang, T.H., 2005. Static-arbitrage upper bounds for the prices of basket options. Quant. Fin. 5 (4), 329–342] who use Lagrange optimization techniques, the proofs in this paper are based on the theory of integral stochastic orders and on the theory of comonotonic risks.     

Portfolio selection and duality under mean variance preferences

This paper uses duality to analyze an investor’s behavior in a n-asset portfolio selection problem when the investor has mean variance preferences. The indirect utility and wealth requirement functions are used to derive Roy’s identity, Shephard’s lemma and the Slutsky equation. In our simple Slutsky equation the income effect is characterized by decreasing absolute risk aversion (DARA) and the substitution effect is always positive [negative] with respect to an asset’s holding if the asset’s mean return [risk] increases. Substitution effect and income effect work in the same direction presupposed mean variance preferences display DARA.     

Estimation of risk-neutral density surfaces

Option price data is often used to infer risk-neutral densities for future prices of an underlying asset. Given the prices of a set of options on the same underlying asset with different strikes and maturities, we propose a nonparametric approach for estimating risk-neutral densities associated with several maturities. Our method uses bicubic splines in order to achieve the desired smoothness for the estimation and an optimization model to choose the spline functions that best fit the price data. Semidefinite programming is employed to guarantee the nonnegativity of the densities. We illustrate the process using synthetic option price data generated using log-normal and absolute diffusion processes as well as actual price data for options on the S&P 500 index. We also used the risk-neutral densities that we computed to price exotic options and observed that this approach generates prices that closely approximate the market prices of these options.     

An option-theoretic model of a reputation network

This paper constructs a new theory of social networks based on reputation. The model assumes that reputation is an asset and that individuals connect by buying options on the reputation of others. In networking, individuals construct portfolios of call options to leverage the reputations of others and put options to hedge the connections with others. A network then consists of portfolios of reputation options. The option model confers advantages not present in existing models. First, the payoff to connecting is the payoff on a portfolio of reputation options. Second, the network forms as individuals take option positions; the network evolves as individuals adjust those positions. Third, networking strategies become option strategies. The model allows for insights into network structure, the price of connecting and the value of connecting.     

Robust portfolio optimization with derivative insurance guarantees

Robust portfolio optimization aims to maximize the worst-case portfolio return given that the asset returns are allowed to vary within a prescribed uncertainty set. If the uncertainty set is not too large, the resulting portfolio performs well under normal market conditions. However, its performance may substantially degrade in the presence of market crashes, that is, if the asset returns materialize far outside of the uncertainty set. We propose a novel robust optimization model for designing portfolios that include European-style options. This model trades off weak and strong guarantees on the worst-case portfolio return. The weak guarantee applies as long as the asset returns are realized within the prescribed uncertainty set, while the strong guarantee applies for all possible asset returns. The resulting model constitutes a convex second-order cone program, which is amenable to efficient numerical solution procedures. We evaluate the model using simulated and empirical backtests and analyze the impact of the insurance guarantees on the portfolio performance.     

On pricing and reserving with-profits life insurance contracts

As a first approximation, asset and liability management issues faced by life insurance companies originate from the sale of with-profits contracts. These contracts are bond-type products with several rate guarantees and other interestsensitive embedded options. Benefits paid out to policyholders mostly depend on the investment performance of a given asset portfolio in which premiums are invested. Thus, guarantees and options granted to policyholders may become effective when the investment performance of the asset portfolio is poor. Issuing a with-profits contract is therefore not equivalent to issuing plain-vanilla debt. The purpose of this paper is to value with-profits liabilities in a consistent option-pricing framework and to develop efficient asset or liability strategies to manage profitability and variability of shareholder value.     

面对成比例交易成本的情况下外汇期权的定价

The purpose of present work is to examine the financial problem of finding the universal reservation prices of a European call option written on exchange rate when there is proportional transaction costs of trading foreign currency in the market. An approach is suggested to compute the reservation bid-ask price of foreign currency call option based on maximizing the investor's expected utility. Option prices are determined from the investor's basic portfolio selection problem, without the need to solve a more complex optimization problem involving the insertion of the option payoffs into the terminal value function. Option prices are computed numerically in a Markov chain approximation for the case of exponential utility.Numerical results show that the option price bounds are almost independent of the alternative risk aversion parameter, but the bounds of NT region becomes narrower and the range of values of the initial holding for which the fair price lies within the bid-ask spread is shifted to a lower value when the risk aversion parameter increases.     

FOREIGN CURRENCY OPTION PRICING WITH PROPORTIONAL TRANSACTION COSTS

The purpose of present work is to examine the financial problem of finding the universal reservation prices of a European call option written on exchange rate when there is proportional transaction costs of trading foreign currency in the market. An approach is suggested to compute the reservation bid-ask price of foreign currency call option based on maximizing the investor＇s expected utility. Option prices are determined from the investor＇s basic portfolio selection problem, without the need to solve a more complex optimization problem involving the insertion of the option payoffs into the terminal value function. Option prices are computed numerically in a Markov chain approximation for the case of exponential utility. Numerical results show that the option price bounds are almost independent of the alternative risk aversion parameter, but the bounds of NT region becomes narrower and the range of values of the initial holding for which the fair price lies within the bid-ask spread is shifted to a lower value when the risk aversion parameter increases.     

Bounds for the price of a European-style Asian option in a binary tree model

Inspired by the ideas of Rogers and Shi [J. Appl. Prob. 32 (1995) 1077], Chalasani et al. [J. Comput. Finance 1(4) (1998) 11] derived accurate lower and upper bounds for the price of a European-style Asian option with continuous averaging over the full lifetime of the option, using a discrete-time binary tree model. In this paper, we consider arithmetic Asian options with discrete sampling and we generalize their method to the case of forward starting Asian options. In this case with daily time steps, the method of Chalasani et al. is still very accurate but the computation can take a very long time on a PC when the number of steps in the binomial tree is high. We derive analytical lower and upper bounds based on the approach of Kaas et al. [Insurance: Math. Econ. 27 (2000) 151] for bounds for stop-loss premiums of sums of dependent random variables, and by conditioning on the value of underlying asset at the exercise date. The comonotonic upper bound corresponds to an optimal superhedging strategy. By putting in less information than Chalasani et al. the bounds lose some accuracy but are still very good and they are easily computable and moreover the computation on a PC is fast. We illustrate our results by different numerical experiments and compare with bounds for the Black and Scholes model [J. Pol. Econ. 7 (1973) 637] found in another paper [Bounds for the price of discretely sampled arithmetic Asian options, Working paper, Ghent University, 2002]. We notice that the intervals of Chalasani et al. do not always lie within the Black and Scholes intervals. We have proved that our bounds converge to the corresponding bounds in the Black and Scholes model. Our numerical illustrations also show that the hedging error is small if the Asian option is in the money. If the option is out of the money, the price of the superhedging strategy is not as adequate, but still lower than the straightforward hedge of buying one European option with the same exercise price.     

Gain�Closs based convex risk limits in discrete-time trading

We present an approach for pricing and hedging in incomplete markets, which encompasses other recently introduced approaches for the same purpose. In a discrete time, finite space probability framework conducive to numerical computation we introduce a gain–loss ratio based restriction controlled by a loss aversion parameter, and characterize portfolio values which can be traded in discrete time to acceptability. The new risk measure specializes to a well-known risk measure (the Carr–Geman–Madan risk measure) for a specific choice of the risk aversion parameter, and to a robust version of the gain–loss measure (the Bernardo–Ledoit proposal) for a specific choice of thresholds. The result implies potentially tighter price bounds for contingent claims than the no-arbitrage price bounds. We illustrate the price bounds through numerical examples from option pricing.     

双跳-扩散过程下的脆弱期权定价

本文考虑含有交易对手违约风险的衍生产品的定价,以公司价值信用风险模型为基础,在标的资产价格和公司价值均服从跳-扩散过程的情况下,运用结构化的方法对脆弱期权定价进行建模,建立了双跳-扩散过程下的脆弱期权定价模型,分别在公司负债固定和随机的情况下推导出了脆弱期权的定价公式.     

Options on constant proportion portfolio insurance with guaranteed minimum equity exposure

In the present paper we study a new exotic option offering participation in a dynamic asset allocation strategy, which is an extension of the well‐known Constant Proportion Portfolio Insurance (CPPI) strategy. Our novel approach consists in assuming that the percentage of wealth invested in stocks cannot go under a fixed level, called guaranteed minimum equity exposure (GMEE). In particular, our proposal ensures to overcome the so‐called cash‐in risk, typically related to a standard CPPI technique, simultaneously guaranteeing the equity market participation. We look deeper into the valuation of call and put options linked to this new CPPI‐GMEE strategy. A particular attention is devoted to the analysis of key parameters' value as to gain a better understanding of the sensitivities of the option prices, when changing, for example, the embedded guarantee level. To show the effectiveness of our proposal we provide a detailed computational analysis within the Heston‐Vasicek framework, numerically comparing the evaluation of the price of European plain vanilla options when the underlying is either a purely risky asset, a standard CPPI portfolio and a CPPI with GMEE.     

Martingale optimal transport and robust hedging in continuous time

The duality between the robust (or equivalently, model independent) hedging of path dependent European options and a martingale optimal transport problem is proved. The financial market is modeled through a risky asset whose price is only assumed to be a continuous function of time. The hedging problem is to construct a minimal super-hedging portfolio that consists of dynamically trading the underlying risky asset and a static position of vanilla options which can be exercised at the given, fixed maturity. The dual is a Monge–Kantorovich type martingale transport problem of maximizing the expected value of the option over all martingale measures that have a given marginal at maturity. In addition to duality, a family of simple, piecewise constant super-replication portfolios that asymptotically achieve the minimal super-replication cost is constructed.