Numerical Methods

Numerical methods are needed for derivatives pricing in cases where analytic solutions are either unavailable or not easily computable. Examples of the former case include American-style options and most discretely observed path-dependent options. Examples of the latter type include the analytic formula for valuing continuously observed Asian options in, which is very hard to calculate for a wide range of parameter values often encountered in practice, and the closed form solution for the price of a discretely observed partial barrier option in, which requires high dimensional numerical integration. The subject of numerical methods in the area of derivatives valuation and hedgingis very broad. A wide range of different types of contracts are available, and in many cases there are several candidate models for the stochastic evolution of the underlying state variables. Many subtle numerical issues can arise in various contexts. A complete description of these would be very lengthy, so here we will only give a sample of the issues involved. The main prospective of this report is to compare and contrast different methods of derivatives available. There are three methods on which this entire report will be based on, which predominantly are, Binomial Methods, Monte Carlo Simulations and Numerical PDE Methods. First we will describe what the derivative is all about and then all the methods and then shift our gears towards the main prospective of this study.  Let’s start our analysis.

What is Derivative?

A derivative is a monetary instrument (or more easily, a concord between two people/two parties) that has a regard determined by the hope cost of something besides. Derivatives can be thought of as bets on the degree of something. Suppose you bet with your helper on the cost of a mountain of corn. If the outlay in one year is less than $3 your comrade pays you $1. If the cost is more than $3 you pay your friend $1. Thus, the underlying in the pact is the esteem of corn and the quantity of the bargain to you depends on that underlying. So derivatives is the collective name worn for a broad style of economic instruments that stem their regard from other pecuniary instruments (known as the underlying), dealings or conditions. Essentially, a derivative is a shrink between two parties where the value of the deal is related to the assess of another fiscal instrument or by a specified affair or situation.
Derivatives are usually broadly categorized by the:
• relationship between the underlying and the derivative (e.g. farther, choice, swap)
• nature of underlying (e.g. equity derivatives, strange switch derivatives, benefit rate derivatives, commodity derivatives or character derivatives)
• promote in which they trade (e.g., chat traded or over-the-offset)
• pay-off profile (Some derivatives have non-linear induce diagrams due to embedded optionality)
Another chance distinction is between:
• vanilla derivatives (clean and more shared) and
• exotic derivatives (more complicated and specialized)
There is no definitive statute for distinguishing one from the other, so the distinction is regularly a trouble of custom.

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The Binomial Method.

Perhaps the easiest numerical method to understand is the binomial method that was originally developed in. We begin with the simplest model of uncertainty; there is one period, and the underlying stock price can take on one of two possible values at the end of this period. Let the current stock price be S, and the two possible values after one period be Su and Sd, where u and d are constants with u > d. Denote one plus the risk free interest rate for the period as R. Absence of arbitrage requires that u > R > d (if u ≤ R, then the stock would be dominated by a risk free bond; if R < d, then the stock would dominate the risk free bond). Figure below provides an illustration of this model. We know the value of the option at expiry as a function of S. suppose we are valuing a European call option with strike price K. Then the payoff is Vu = max(Su − K, 0), if S = Su at T and Vd = max(Sd − K, 0), if S = Sd at that time. Consider an investment strategyinvolving the underlying stock and a risk free bond that is designed to replicate the option’s payoff. In particular, suppose we buy h shares of stock and invest C in the risk free asset. We pick h and C to satisfy The cost of entering into this strategy at the start of the period is hS + C. No-arbitrage considerations then tell us that the price of the option today must equal the cost of this alternative investment strategy that produces the same future payoffs as the option.

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The binomial method can easily be programmed to contain hundreds (if not thousands) of sub periods. The binomial method was introduced by Cox et al. (1979) and Rendlemanand Bartter (1979). The method is based on the random walk approximation to the Brownian motion and provides a simple and intuitive numerical solution. In this method, the partition {t0 = 0, t1, t2,, tN−1, tN = T } of the time interval [0, T ] is considered. At each point of this partition, it is assumed that the price of the underlying asset follows a multiplicative binomial process: it either jumps up by a proportion u or goes down by a proportion d. Both proportions u and d determine the mean and the volatility of the underlying asset. According to this evolution of the asset price, the call option price goes either to Cu = max{uS − K, 0} or to Cd = max{dS − K, 0}. As in the Black-Scholes model, a riskless portfolio is built and the price of a call option with one period to maturity is given by

C = e−r Δt (p Cu + (1 − p) Cd), p=er Δt − d u − d , Δt = T/N

Thus, the call option price can be interpreted as the (discounted) expectation of the future pay-offs under the risk-neutral probabilities.

Since the binomial model is an approximation to the continuous-time model of the asset price, we choose the values of the jump parameters (u and d) and the (risk-neutral) probability p to match the risk-neutral mean and variance of the process given in equation

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Finite Different Methods:

A more general approach is to value the derivatives using a numerical PDE approach. There are several different possibilities here including finite differences, finite elements, and finite volume methods. In a one-dimensional setting, the differences between these various techniques are slight; in many cases all of these approaches give rise to the same set of discrete equations and the same option values. We will consider the finite difference approach here as it is the easiest to describe. The basic idea is to replace the derivatives in with discrete differences calculated on a finite grid of values for S. This is given by Si , i = 0, . . ., p, where S0 = 0 and Sp = Smax, the highest value for the stock price on the grid. Boundary conditions will depend on the contract, and can be arrived at using obvious financial reasoning in most cases. For instance, for a European call option at Smax, we can use V (Sp) = S − Ke−rτ , reflecting the virtually certain exercise of the instrument at expiry. Similarly, a put option is almost surely not going to be exercised if S is very large, so we would use V (Sp) = 0 in this situation. At S0 = 0, a European put option would be worth the present value of the exercise price, while a call would be worth zero. We can specify boundary conditions using these values, but it is not required. This is because at S = 0, the PDE (3) reduces to the easily solved ordinary differential equation Vτ = −rV . In most finance references, an equally spaced grid is assumed but this is not necessary for anything other than ease of exposition. In the following, we will consider the more general case of a no uniform grid since it offers considerable advantages in some contexts. For the moment, let us consider the derivatives with respect to S. Let Vi denote the option value at grid node i. Using a first-order Taylor’s series expansion, we can approximate VS at node i by either the forward difference

VS ≈ Vi+1 − Vi Si+1 − Si , (8)

or the backward difference VS ≈ Vi − Vi−1 Si − Si−1.

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