Barrier Options Finance Essay Barrier options; uses and pricing. This occurs on the financial markets, which restructures risks towards those agents who are willing and able to assume them Markets for options and other derivatives are important as the agents who foresee future revenues or payments can ensure a profit above a certain level or insure themselves against a loss above a certain level. Due to their design, options allow for hedging against one-sided risk, options give the right, but not the obligation, to buy or sell a certain security in the future at a pre-specified price.
A barrier option is similar to a standard option, except that the payoff also depends on whether the asset price crosses a certain barrier level during the option’s life. Look back options and barrier options are two of the most popular types of path-dependent options (Davydov, D & Linetsky, V ). Look‐barrier options have many similarities with compound options and can be viewed as barrier options over [t, T 1] on a contract with T 1 payoff equal to the present value of a lookback option on the interval [T 1, T 2]. Pricing the look‐barrier options is a lot more difficult than pricing the extreme spread lookbacks of the previous section. Partial barrier-lookback options. In the current subsection, we combine the features of barrier-lookback options and of a class of partial barrier options introduced in. Consider the option .
We demonstrate advantages of our approach in terms of accuracy and convergence. AMS subject classifications: Introduction Lookback options are among the most popular path-dependent derivatives traded in exchanges worldwide. The payoffs of these options depend on the realized minimum or maximum asset price at expiration.
A standard European lookback call put gives the option holder the right to buy sell an asset at its lowest highest price during the life of the option. In addition to standard lookback options, paper [ 23 ] introduces fixed strike lookbacks. A fixed strike lookback call pays off the difference between the realized maximum price and some prespecified strike or zero, whichever is greater.
A lookback put with a fixed strike pays off the difference between the strike and the realized minimum price or zero, whichever is greater. Fixed strike lookback options can be priced also analytically in the Black-Scholes model [ 23 ].
In a discrete time setting the extremum of the asset price will be determined at discrete monitoring instants see e. The continuity corrections for discrete lookback options in the Black-Scholes setting are given in [ 16 ].
Other types of lookback options include exotic lookbacks, percentage lookback options in which the extreme values are multiplied by a constant [ 23 ], and partial lookback options [ 35 ] the monitoring interval for the extremum is a subinterval between the initial date and the expiry.
In recent years more and more attention has been given to stochastic models of financial markets which depart from the traditional Black-Scholes model. These models provide a better fit to empirical asset price distributions that typically have fatter tails than Gaussian ones, and can reproduce volatility smile phenomena in option prices.
The Wiener-Hopf factorization method is a standard tool for pricing path-dependent options. The probabilistic approach used in the paper allows, in particular, to recover the results for barrier options derived in [ 14 ] using the analytical form of the Wiener-Hopf factorization method.
In the case of jump diffusions with exponentially distributed Poisson jumps a double-exponential jump diffusion process DEJD and its generalization: The Laplace transform of the price having being calculated, one uses a suitable numerical Laplace inversion algorithm to recover the option price.
However, the problem of the inversion of the Laplace transform is non-trivial from the computational point of view. We refer the reader to [ 1 ] for a description of a general framework for related numerical methods.
The methods of these papers are computationally expensive when the monitoring is frequent e. As the number of monitoring times goes to infinity, discrete barrier lookback options converge to continuous barrier lookbacks.
However, the discrete options pricing methods described above converge to continuous prices rather slowly. The Richardson extrapolation is typically used to improve the rate of convergence of various numerical schemes. Unfortunately, even for the standard lookbacks, there are no theoretical results, which can be used to find a generalization of the Richardson extrapolation procedure appropriate for estimation of continuous values from discrete ones see the discussion in [ 30 ].
However, similar results for hybrid exotics are unavailabale at present. As well as in [ 44 ] we use Gaver-Stehfest algorithm see [ 1 ] and the Fast Wiener-Hopf factorization method developed in [ 42 ]. In contrast to pricing methods based on approximations by options with discrete monitoring, our pricing method converges very fast to prices of options with continuous monitoring.
The rest of the paper is organized as follows. Numerical examples in Section 4 demonstrate advantages of our method in terms of accuracy and speed. Section 5 gives details on the method implemented into Premia There are several forms of the Wiener-Hopf factorization.
The Wiener-Hopf factorization formula used in probability reads: Introducing the notation we can write 2. The name is due to the observation that, for a stream g Xt2. The operator form of the Wiener-Hopf factorization is written as follows see details in [ 59p.Essays on Lookback and Barrier Options: A Malliavin Calculus Approach [Hans-Peter Bermin] on metin2sell.com *FREE* shipping on qualifying offers.
This thesis consists of four theoretical essays on contingent claim analysis and its connection to Malliavin metin2sell.com: Hans-Peter Bermin. A lookback option allows the holder to exercise an option at the most beneficial price of the underlying asset, over the life of the option.
Lookback options do not trade on formal exchanges. A barrier option is similar to a standard option, except that the payoff also depends on whether the asset price crosses a certain barrier level during the option’s life. Look back options and barrier options are two of the most popular types of path-dependent options (Davydov, D & Linetsky, V ).
Essays on Lookback and Barrier Options - A Malliavin Calculus Approach Bermin, Hans-Peter LU () In Lund Economic Studies Mark; Abstract This thesis consists of four theoretical essays on contingent claim analysis and its connection to Malliavin calculus.
Look‐barrier options have many similarities with compound options and can be viewed as barrier options over [t, T 1] on a contract with T 1 payoff equal to the present value of a lookback option on the interval [T 1, T 2].
Pricing the look‐barrier options is a lot more difficult than pricing the extreme spread lookbacks of the previous section. lookback option, whose value is the average present value of the payo of the lookback option associated with each simulated path.
When the sampling frequency increases, the payo s of arithmetic average options and lookback options depend only on the daily or weekly closing prices. Next I will introduce how to employ the nite di erence method.