Monte Carlo Methods essay

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Monte Carlo methods are based on the Feynman–Kac. Originally introduced to the option pricing  literature in, they have found wide applicability ever since. An excellent survey may be found in, and we shall rely heavily on it here. This technique was introduced in finance by Boyle (1977). For a recent survey see Boyle et al. (1997). The value of an option is the risk-neutral expectation of its discounted pay-off. This expectation is estimated by computing the average of a large number of pay-offs. Tilley (1993) is the first who prices American options using this technique.

He proposes an algorithm in which, at each date, simulated paths are ordered by asset price and bundled into groups. Then, for each group, an optimal exercise decision is taken. As Broadie and Glasserman (1997a) indicate, there are no convergence results for this algorithm, all the simulated paths must be stored at one time and there is not a direct extension to deal with multiple state variables. Barraquand and Martineau (1995) propose to reduce the dimensionality of the valuation problem, grouping the simulated values into a set of “bins”. The transition probabilities between bins are determined by simulation and the option valuation is performed using each bin as a decision unit.

Broadie and Glasserman (1997a) present an algorithm that allows them to obtain point estimates and error bounds for American option prices. After showing that, under certain assumptions, there are not unbiased estimates of these prices, two (biased) estimates that converge asymptotically to the true price are generated. Combination of both estimates leads to a confidence interval for the American option price.

Comparison and Contrast Analysis:

The field of pricing derivative instruments using numerical methods is a large and confusing one, in part because the pricing problems vary widely, ranging from simple one factor European options through American options, barrier contracts, path-dependent options, interest rate derivatives, foreign exchange derivatives, contracts with discontinuous payoff functions, multidimensional problems, cases with different assumptions about the evolution of the underlying state variables, and all kinds of combinations of these items. The suitability of any particular numerical scheme depends largely on the context.

There are certain things which are quite similar in all these methods. All these methods can be used to price the derivatives. Usually banks and investment companies used different methods to value the derivatives. Generally, the Binomial model uses a "discrete-time" model of the varying price over time of the underlying financial instrument. Monte Carlo model and the Binomial Model can be comparing with each other because they both have similar characteristics both these methods value the derivatives according to the time frame. Due to its propensity to underlying the instrument over a specific period of time, both these methods can be easily applied. Monte Carlo methods are the only practical alternative for problems with more than three dimensions, but are relatively slow in other cases. Moreover, despite significant progress, early exercise remains problematic for simulation methods. Finite Different methods are fairly robust but cannot be used in high dimensions. They can also require sophisticated algorithms for solving large sparse linear systems of equations and are relatively difficult to code. Binomial and trinomial tree methods are just explicit type Finite Different methods that work very well on simple problems but can be quite hard to adapt to more complex situations because of the stability limitation tying the time step size and grid spacing together. In addition to the references, recent books such as are fine sources for additional information. Incorporating the idea behind the Moody’s KMV, one of the structural credit risk models, we develop a methodology for estimating the risk premium corresponding to the default risk originated from the structured product issuer. The framework of this method can be applied to all kinds of structured products, including interest rate related, foreign exchange rate related, and equity related ones. Similar to the nature of American binary put option, the credit risk is priced as an option sold by the investor (ELN holder) to the issuer (ELN writer). The option premium can be taken as the compensation for bearing issuer’s credit risk, which has been largely ignored. The default premium is estimated based on the contingent claim of the structured product and the capital structure of the issuer at the time of evaluation. Taking the “2-Year USD Equity Linked Note” issued by J.P. Morgan for illustration, the default probability within 2 years is about 1.70%, and the price of the structured notes is 0.9% lower than the default free counterpart. Furthermore, according to the outcome of comparative static analysis, the default probability rises as the asset to default boundary ratio becomes lower, and so does the asset to debt ratio. In addition, the asset volatility has a positive effect on the probability of default, and, hence, the default premium. One more contrast between these models is that Binomial and Monte Carlo Models can only value the American Option derivatives while the Finite Different Methods can value both American and European options in derivatives.  

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