Common Shocks and Relative Compensation

This paper studies qualitative properties of an optimal contract in a multi-agent setting in which agents are subject to a common shock. We derive a necessary and sufficient condition for the optimal reward of an agent producing an output level y to be a decreasing (increasing) function of the outputs of the other agents, under the assumption that the agents’ outputs are informative signals of the value of the common shock. The condition is that the likelihood ratio p(y, e, η)/p(y, e′, η), where e is a higher effort level than e′ and η is the value of the common shock, be a decreasing (increasing) function of η. We give examples of applications of the result and examine its consequences for CEO compensation.     

Local time and the pricing of time-dependent barrier options

A time-dependent double-barrier option is a derivative security that delivers the terminal value φ(S _T ) at expiry T if neither of the continuous time-dependent barriers b _±:[0,T]→ℝ₊ have been hit during the time interval [0,T]. Using a probabilistic approach, we obtain a decomposition of the barrier option price into the corresponding European option price minus the barrier premium for a wide class of payoff functions φ, barrier functions b _± and linear diffusions (S _t ) _t∈[0,T]. We show that the barrier premium can be expressed as a sum of integrals along the barriers b _± of the option’s deltas Δ _±:[0,T]→ℝ at the barriers and that the pair of functions (Δ ₊,Δ ₋) solves a system of Volterra integral equations of the first kind. We find a semi-analytic solution for this system in the case of constant double barriers and briefly discus a numerical algorithm for the time-dependent case.     

Smooth convergence in the binomial model

In this article, we consider a general class of binomial models with an additional parameter λ. We show that in the case of a European call option the binomial price converges to the Black–Scholes price at the rate 1/n and, more importantly, give a formula for the coefficient of 1/n in the expansion of the error. This enables us, by making special choices for λ, to prove that convergence is smooth in Tian’s flexible binomial model and also in a new center binomial model which we propose. Ken Palmer was supported by NSC grant 93-2118-M-002-002.     

Risk-neutral compatibility with option prices

A common problem is to choose a “risk-neutral” measure in an incomplete market in asset pricing models. We show in this paper that in some circumstances it is possible to choose a unique “equivalent local martingale measure” by completing the market with option prices. We do this by modeling the behavior of the stock price X, together with the behavior of the option prices for a relevant family of options which are (or can theoretically be) effectively traded. In doing so, we need to ensure a kind of “compatibility” between X and the prices of our options, and this poses some significant mathematical difficulties.     

Additive and multiplicative duals for American option pricing

We investigate and compare two dual formulations of the American option pricing problem based on two decompositions of supermartingales: the additive dual of Haugh and Kogan (Oper. Res. 52:258–270, 2004) and Rogers (Math. Finance 12:271–286, 2002) and the multiplicative dual of Jamshidian (Minimax optimality of Bermudan and American claims and their Monte- Carlo upper bound approximation. NIB Capital, The Hague, 2003). Both provide upper bounds on American option prices; we show how to improve these bounds iteratively and use this to show that any multiplicative dual can be improved by an additive dual and vice versa. This iterative improvement converges to the optimal value function. We also compare bias and variance under the two dual formulations as the time horizon grows; either method may have smaller bias, but the variance of the multiplicative method typically grows much faster than that of the additive method. We show that in the case of a discrete state space, the additive dual coincides with the dual of the optimal stopping problem in the sense of linear programming duality and the multiplicative method arises through a nonlinear duality.      

A probabilistic analysis of the trading the line strategy

We provide analytic models for which the appropriate statistics of the trading the line strategy, N _h , can be derived in closed form. In particular, we provide closed-form expressions concerning the average duration of the open position, E(N _h ), the variance of the open duration, Var(N _h ), the average of the stopped log price, E(S _{N h} ), the variance of the stopped log price, Var(S _{N h} ), the correlation, Corr(N _h , S _{N h} ), and the Laplace transform, E(e^{?s N _h} ). These results are obtained, in discrete time settings, for binomial and other price scenarios. Furthermore, when analytic results are not possible, such as the case of a normal distribution for log returns, we show by simulation that our general conclusions still hold. Using these statistics we point out some of the subtle features of the trailing stops strategy.     

Pricing growth-rate risk

We characterize the compensation demanded by investors in equilibrium for incremental exposure to growth-rate risk. Given an underlying Markov diffusion that governs the state variables in the economy, the economic model implies a stochastic discount factor process S. We also consider a reference growth process G that may represent the growth in the payoff of a single asset or of the macroeconomy. Both S and G are modeled conveniently as multiplicative functionals of a multidimensional Brownian motion. We consider the pricing implications of parametrized family of growth processes G ^ε , with G ⁰=G, as ε is made small. This parametrization defines a direction of growth-rate risk exposure that is priced using the stochastic discount factor S. By changing the investment horizon, we trace a term structure of risk prices that shows how the valuation of risky cash flows depends on the investment horizon. Using methods of Hansen and Scheinkman (Econometrica 77:177–234, 2009), we characterize the limiting behavior of the risk prices as the investment horizon is made arbitrarily long.     

A Comparison of Option Prices Under Different Pricing Measures in a Stochastic Volatility Model with Correlation

This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result which says that prices for European options with convex payoffs are decreasing in the market price of volatility risk.As an example, and as our main motivation, we investigate option pricing under the class of q-optimal pricing measures. The q-optimal pricing measure is related to the marginal utility indifference price of an agent with constant relative risk aversion. Using the ordering result, we prove comparison theorems between option prices under the minimal martingale, minimal entropy and variance-optimal pricing measures. If the Sharpe ratio is deterministic, the comparison collapses to the well known result that option prices computed under these three pricing measures are the same.As a concrete example, we specialize to a variant of the Hull-White or Heston model for which the Sharpe ratio is increasing in volatility. For this example we are able to deduce option prices are decreasing in the parameter q. Numerical solution of the pricing pde corroborates the theory and shows the magnitude of the differences in option price due to varying q.JEL Classification: D52, G13     

The critical price for the American put in an exponential Lévy model

This paper considers the behavior of the critical price for the American put in the exponential Lévy model when the underlying stock pays dividends at a continuous rate. We prove the continuity of the free boundary and give a characterization of the critical price at maturity, generalizing a recent result of S.Z. Levendorskiǐ (Int. J. Theor. Appl. Finance 7:303–336, 2004).      

Short note on inf-convolution preserving the Fatou property

We model agents’ preferences by cash-invariant concave functionals defined on L ^∞, and formulate the optimal risk allocation problem as their infimal-convolution. We study the case of agents whose choice functionals are law-invariant with respect to different probability measures and show how, in this case, the value function preserves a desirable dual representation (equivalent to the Fatou property). Financial support from the European Science Foundation (ESF) “Advanced Mathematical Methods for Finance” (AMaMeF) under the exchange grant 1192 is gratefully acknowledged.     

On the Validity of the Wiener Process Assumption in Option Pricing Models: Contradictory Evidence from Taiwan

This study tests the validity of the critical assumption underlying the option pricing model that the log form of the stock price movements follows the Wiener process, i.e., stock price movements follow a geometric Brownian motion. Using data compiled from the Taiwan Stock Exchange (TSE), this study's major empirical findings are as follows: first, the null hypothesis that the log of the stock prices is normally distributed is rejected; second, the null hypothesis that the stock price in log form has mean [ln P _s + (µ- ²)_t] and variance t is rejected; third, the null hypothesis that successive non-overlapping increments of the log of the stock price are independent from each other is also rejected. These empirical findings undermine the validity of the Wiener process assumption which is fundamental to many option pricing models.     

Mean square error for the Leland–Lott hedging strategy: convex pay-offs

Leland’s approach to the hedging of derivatives under proportional transaction costs is based on an approximate replication of the European-type contingent claim V _T using the classical Black–Scholes formula with a suitably enlarged volatility. The formal mathematical framework is a scheme of series, i.e., a sequence of models with transaction cost coefficients k _n =k ₀ n ^−α , where α∈[0,1/2] and n is the number of portfolio revision dates. The enlarged volatility [^(s)]_n\widehat{\sigma}_{n} in general depends on n except for the case which was investigated in detail by Lott, to whom belongs the first rigorous result on convergence of the approximating portfolio value Vⁿ_TV^{n}_{T} to the pay-off V _T . In this paper, we consider only the Lott case α=1/2. We prove first, for an arbitrary pay-off V _T =G(S _T ) where G is a convex piecewise smooth function, that the mean square approximation error converges to zero with rate n ^−1/2 in L ² and find the first order term of the asymptotics. We are working in a setting with non-uniform revision intervals and establish the asymptotic expansion when the revision dates are t_iⁿ=g(i/n)t_{i}^{n}=g(i/n), where the strictly increasing scale function g:[0,1]→[0,1] and its inverse f are continuous with their first and second derivatives on the whole interval, or g(t)=1−(1−t) ^β , β≥1. We show that the sequence n^1/2(V_Tⁿ-V_T)n^{1/2}(V_{T}^{n}-V_{T}) converges in law to a random variable which is the terminal value of a component of a two-dimensional Markov diffusion process and calculate the limit. Our central result is a functional limit theorem for the discrepancy process.     

On the game interpretation of a shadow price process in utility maximization problems under transaction costs

To any utility maximization problem under transaction costs one can assign a frictionless model with a price process S ^?, lying in the bid/ask price interval $[\underline{S}, \overline{S}]$ . Such a process S ^? is called a shadow price if it provides the same optimal utility value as in the original model with bid-ask spread. We call S ^? a generalized shadow price if the above property is true for the relaxed utility function in the frictionless model. This relaxation is defined as the lower semicontinuous envelope of the original utility, considered as a function on the set $[\underline{S}, \overline{S}]$ , equipped with some natural weak topology. We prove the existence of a generalized shadow price under rather weak assumptions and mark its relation to a saddle point of the trader/market zero-sum game, determined by the relaxed utility function. The relation of the notion of a shadow price to its generalization is illustrated by several examples. Also, we briefly discuss the interpretation of shadow prices via Lagrange duality.     

An endogenous volatility approach to pricing and hedging call options with transaction costs

Standard delta hedging fails to exactly replicate a European call option in the presence of transaction costs. We study a pricing and hedging model similar to the delta hedging strategy with an endogenous volatility parameter for the calculation of delta over time. The endogenous volatility depends on both the transaction costs and the option strike prices. The optimal hedging volatility is calculated using the criterion of minimizing the weighted upside and downside replication errors. The endogenous volatility model with equal weights on the up and down replication errors yields an option premium close to the Leland [J. Finance, 1985 Leland, HE. 1985. Option pricing and replication with transaction costs. J. Finance, 40: 1283–1301. [Crossref], [Web of Science ®] , [Google Scholar], 40, 1283–1301] heuristic approach. The model with weights being the probabilities of the option's moneyness provides option prices closest to the actual prices. Option prices from the model are identical to the Black–Scholes option prices when transaction costs are zero. Data on S&P 500 index cash options from January to June 2008 illustrate the model.     

Symmetry and duality in Lévy markets

The aim of this paper is to introduce the notion of symmetry in a Lévy market. This notion appears as a particular case of a general known relation between prices of put and call options, of both the European and the American type, which is also reviewed in the paper, and that we call put–call duality. Symmetric Lévy markets have the distinctive feature of producing symmetric smile curves, in the log of strike/futures prices.

Put–call duality is obtained as a consequence of a change of the risk neutral probability measure through Girsanov's theorem, when considering the discounted and reinvested stock price as the numeraire. Symmetry is defined when a certain law before and after the change of measure through Girsanov's theorem coincides. A parameter characterizing the departure from symmetry is introduced, and a necessary and sufficient condition for symmetry to hold is obtained, in terms of the jump measure of the Lévy process, answering a question raised by Carr and Chesney (American put call symmetry, preprint, 1996 Carr, P and Chesney, M. 1996. American put call symmetry. preprint [Google Scholar]). Some empirical evidence is shown, supporting that, in general, markets are not symmetric.     

Efficient estimation of drift parameters in stochastic volatility models

We study the parametric problem of estimating the drift coefficient in a stochastic volatility model , where Y is a log price process and V the volatility process. Assuming that one can recover the volatility, precisely enough, from the observation of the price process, we construct an efficient estimator for the drift parameter of the diffusion V. As an application we present the efficient estimation based on the discrete sampling with δ _n →0 and n δ _n →∞. We show that our setup is general enough to cover the case of ‘microstructure noise’ for the price process as well.      

Comparison of Option Prices in Semimartingale Models

In this paper we generalize the recent comparison results of El Karoui et al. (Math Finance 8:93–126, 1998), Bellamy and Jeanblanc (Finance Stoch 4:209–222, 2000) and Gushchin and Mordecki (Proc Steklov Inst Math 237:73–113, 2002) to d-dimensional exponential semimartingales. Our main result gives sufficient conditions for the comparison of European options with respect to martingale pricing measures. The comparison is with respect to convex and also with respect to directionally convex functions. Sufficient conditions for these orderings are formulated in terms of the predictable characteristics of the stochastic logarithm of the stock price processes. As examples we discuss the comparison of exponential semimartingales to multivariate diffusion processes, to stochastic volatility models, to Lévy processes, and to diffusions with jumps. We obtain extensions of several recent results on nontrivial price intervals. A crucial property in this approach is the propagation of convexity property. We develop a new approach to establish this property for several further examples of univariate and multivariate processes.