Remember me

Register  |   Lost password?


Sign up here to let us know if you are interested in joining us for our Introduction to QuantLib Course later in the year.

 

Mathematical Finance's Blog

Mathematical Finance Header

OPTIMAL LIQUIDATION OF DERIVATIVE PORTFOLIOS

February 15, 2012 Comments (0)

We consider the problem facing a risk‐averse agent who seeks to liquidate or exercise a portfolio of (infinitely divisible) perpetual American‐style options on a single underlying asset. The optimal liquidation strategy is of threshold form and can be characterized explicitly as the solution of a calculus of variations problem. Apart from a possible initial exercise of a tranche of options, the optimal behavior involves liquidating the portfolio in infinitesimal amounts, but at times which are...

ON THE DYBVIG‐INGERSOLL‐ROSS THEOREM

February 15, 2012 Comments (0)

The Dybvig‐Ingersoll‐Ross (DIR) theorem states that, in arbitrage‐free term structure models, long‐term yields and forward rates can never fall. We present a refined version of the DIR theorem, where we identify the reciprocal of the maturity date as the maximal order that long‐term rates at earlier dates can dominate long‐term rates at later dates. The viability assumption imposed on the market model is weaker than those appearing previously in the literature.

ASYMPTOTICS OF IMPLIED VOLATILITY IN LOCAL VOLATILITY MODELS

February 15, 2012 Comments (0)

Using an expansion of the transition density function of a one‐dimensional time inhomogeneous diffusion, we obtain the first‐ and second‐order terms in the short time asymptotics of European call option prices. The method described can be generalized to any order. We then use these option prices approximations to calculate the first‐ and second‐order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate.

HAZARD PROCESSES AND MARTINGALE HAZARD PROCESSES

February 15, 2012 Comments (0)

In this paper, we build a bridge between different reduced‐form approaches to pricing defaultable claims. In particular, we show how the well‐known formulas by Duffie, Schroder, and Skiadas and by Elliott, Jeanblanc, and Yor are related. Moreover, in the spirit of Collin Dufresne, Hugonnier, and Goldstein, we propose a simple pricing formula under an equivalent change of measure.Two processes will play a central role: the hazard process and the martingale hazard process attached to a default...

ON THE EXISTENCE OF THE ENDOGENOUS MORTGAGE RATE PROCESS

February 15, 2012 Comments (0)

The mortgage rate is a major factor in the refinancing decision. The refinancing behavior influences cash flow and, therefore, mortgage price. The prices of mortgage instruments drives the mortgage rates. We consider a problem of the existence of a dynamic mortgage rate process which resolves this circular dependence. The existence is proved by constructing a solution using a newly proposed level set method.

VALUATION OF CONTINUOUSLY MONITORED DOUBLE BARRIER OPTIONS AND RELATED SECURITIES

February 15, 2012 Comments (0)

In this paper, we apply Carr's randomization approximation and the operator form of the Wiener‐Hopf method to double barrier options in continuous time. Each step in the resulting backward induction algorithm is solved using a simple iterative procedure that reduces the problem of pricing options with two barriers to pricing a sequence of certain perpetual contingent claims with first‐touch single barrier features. This procedure admits a clear financial interpretation that can be formulated in...

EQUILIBRIUM ASSET AND OPTION PRICING UNDER JUMP DIFFUSION

February 15, 2012 Comments (0)

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...

NONREPLICATION OF OPTIONS

February 15, 2012 Comments (0)

In this paper, we study the replication of options in security markets X with a finite number of states. Specifically, we prove that in security markets without binary vectors, for any portfolio, at most m − 3 options can be replicated where m is the number of states. This is an essential improvement of the result of Baptista where it is proved that the set of replicated options is of measure zero. Additionally, we extend the results of Aliprantis and Tourky on the nonreplication of options by...

RISK HORIZON AND REBALANCING HORIZON IN PORTFOLIO RISK MEASUREMENT

February 15, 2012 Comments (0)

This paper analyzes portfolio risk and volatility in the presence of constraints on portfolio rebalancing frequency. This investigation is motivated by the incremental risk charge (IRC) introduced by the Basel Committee on Banking Supervision. In contrast to the standard market risk measure based on a 10‐day value‐at‐risk calculated at 99% confidence, the IRC considers more extreme losses and is measured over a 1‐year horizon. More importantly, whereas 10‐day VaR is ordinarily calculated with a...

LIQUIDITY IN A BINOMIAL MARKET

February 15, 2012 Comments (0)

We study the binomial version of the illiquid market model introduced by Çetin, Jarrow, and Protter for continuous time and develop efficient numerical methods for its analysis. In particular, we characterize the liquidity premium that results from the model. In Çetin, Jarrow, and Protter, the arbitrage free price of a European option traded in this illiquid market is equal to the classical value. However, the corresponding hedge does not exist and the price is obtained only in L2‐approximating...