RISK INDIFFERENCE PRICING IN JUMP DIFFUSION MARKETS

We study the risk indifference pricing principle in incomplete markets: The (seller's)  risk indifference price        is the initial payment that makes the  risk  involved for the seller of a contract equal to the risk involved if the contract is not sold, with no initial payment. We use stochastic control theory and PDE methods to find a formula for       and similarly for      . In particular, we prove that  where    p _low   and    p _up   are the lower and upper hedging prices, respectively.     

EQUILIBRIUM ASSET AND OPTION PRICING UNDER JUMP DIFFUSION

This paper develops an equilibrium asset and option pricing model in a production economy under jump diffusion. The model provides analytical formulas for an equity premium and a more general pricing kernel that links the physical and risk‐neutral densities. The model explains the two empirical phenomena of the negative variance risk premium and implied volatility smirk if market crashes are expected. Model estimation with the S&P 500 index from 1985 to 2005 shows that jump size is indeed negative and the risk aversion coefficient has a reasonable value when taking the jump into account.     

A NEW METHOD OF PRICING LOOKBACK OPTIONS

A new method for pricing lookback options (a.k.a. hindsight options) is presented, which simplifies the derivation of analytical formulas for this class of exotics in the Black-Scholes framework. Underlying the method is the observation that a lookback option can be considered as an integrated form of a related barrier option. The integrations with respect to the barrier price are evaluated at the expiry date to derive the payoff of an equivalent portfolio of European-type binary options. The arbitrage-free price of the lookback option can then be evaluated by static replication as the present value of this portfolio. We illustrate the method by deriving expressions for generic, standard floating-, fixed-, and reverse-strike lookbacks, and then show how the method can be used to price the more complex partial-price and partial-time lookback options. The method is in principle applicable to frameworks with alternative asset-price dynamics to the Black-Scholes world.     

PRICING A CLASS OF EXOTIC OPTIONS VIA MOMENTS AND SDP RELAXATIONS

We present a new methodology for the numerical pricing of a class of exotic derivatives such as Asian or barrier options when the underlying asset price dynamics are modeled by a geometric Brownian motion or a number of mean-reverting processes of interest. This methodology identifies derivative prices with infinite-dimensional linear programming problems involving the moments of appropriate measures, and then develops suitable finite-dimensional relaxations that take the form of semidefinite programs (SDP) indexed by the number of moments involved. By maximizing or minimizing appropriate criteria, monotone sequences of both upper and lower bounds are obtained. Numerical investigation shows that very good results are obtained with only a small number of moments. Theoretical convergence results are also established.     

BESSEL PROCESSES, ASIAN OPTIONS, AND PERPETUITIES

Using Bessel processes, one can solve several open problems involving the integral of an exponential of Brownian motion. This point will be illustrated with three examples. The first one is a formula for the Laplace transform of an Asian option which is "out of the money." The second example concerns volatility misspecification in portfolio insurance strategies, when the stochastic volatility is represented by the Hull and White model. The third one is the valuation of perpetuities or annuities under stochastic interest rates within the Cox-Ingersoll-Ross framework. Moreover, without using time changes or Bessel processes, but only simple probabilistic methods, we obtain further results about Asian options: the computation of the moments of all orders of an arithmetic average of geometric Brownian motion; the property that, in contrast with most of what has been written so far, the Asian option may be more expensive than the standard option (e.g., options on currencies or oil spreads); and a simple, closed-form expression of the Asian option price when the option is "in the money," thereby illuminating the impact on the Asian option price of the revealed underlying asset price as time goes by. This formula has an interesting resemblance with the Black-Scholes formula, even though the comparison cannot be carried too far.     

TUG‐OF‐WAR,MARKET MANIPULATION,AND OPTION PRICING

We develop an option pricing model based on a tug‐of‐war game. This two‐player zero‐sum stochastic differential game is formulated in the context of a multidimensional financial market. The issuer and the holder try to manipulate asset price processes in order to minimize and maximize the expected discounted reward. We prove that the game has a value and that the value function is the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the nonlinear and completely degenerate infinity Laplace operator.     

PROPERTIES OF OPTION PRICES IN MODELS WITH JUMPS

We study convexity and monotonicity properties of option prices in a model with jumps using the fact that these prices satisfy certain parabolic integro–differential equations. Conditions are provided under which preservation of convexity holds, i.e., under which the value, calculated under a chosen martingale measure, of an option with a convex contract function is convex as a function of the underlying stock price. The preservation of convexity is then used to derive monotonicity properties of the option value with respect to the different parameters of the model, such as the volatility, the jump size, and the jump intensity.     

PRICING CALLABLE BONDS BY MEANS OF GREEN'S FUNCTION1

This paper derives a closed-form solutin for the price of the European and semi-Amirican callable bond for two popular one-factor models of the term structure of interest rates which have been proposed by Vasicek as well as Cox, Ingersoll, and Ross. the price is derived by means of repeated use of Green's function, which, in turn, is derived from a series solution of the partial differential equation to value a discount bond. the boundary conditions which lead to the well-known formulae for the price of a discount bond are also identified. the algorithm to implement the explicit solution relies on numerical quadrature involving Green's function. It offers both higher accuracy and higher speed of computation than finite difference methods, which suffer from numerical instabilites due to discontinuous boundary values. For suitably small time steps, the proposed algorithm can also be applied to American callable bonds or to any American-type option with Green's function being explicitly known.