American Option


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American Option Explained and Valuation


FinPricing provides valuation models for the following equity vanilla options:

  • European option
  • American option
  • Asian option
  • Basket option
  • Binary Option
  • VIX option
  • Spread option
  • Quanto option
  • FX American option
  • Check FinPricing valuation models

All the equity models in FinPricing take volatility skew/smile and dividend into account.


1. American Equity Option Introduction

An American option give an investor the right but not the obligation to buy a call or sell a put at a set strike price at any time prior to the contract’s expiry date. Since investors have the freedom to exercise their American options at any point during the life of the contract, they are more valuable than European equity options, which can only be exercised at maturity. An investor holding an American-style option and seeking optimal value prefer to sell it on, rather than exercise it before maturity under certain circumstances.

Investors and traders can use equity options to take a long or short position in a stock without actually buying or shorting the stock. This is advantageous because taking a position with options allows the investor/trader more leverage in that the amount of capital needed is much less than a similar outright long or short position on margin. Investors/traders can therefore profit more from a price movement in the underlying stock.

American options provide investors a way to hedge risk or speculate. Also option trading can limit an investor’s risk and leverage investing potential. Stock option investors have a number of strategies they can utilize, depending on risk tolerance and expected return.Buying call options allows you to benefit from an upward price movement. The right to buy stock at a fixed price becomes more valuable as the price of the underlying stock increases. Put options may provide a more attractive method than shorting stock for profiting on stock price declines. If you have an established profitable long stock position, you can buy puts to protect this position against short-term stock price declines. An option seller earns the premium if the underlying stock price would not change much.


2. American Equity Option Valuation

In general, American options do not have a closed-form solution. There are several methods to approximate the price of an American option: Roll-Geske-Whaley, Barone-Adesi and Whaley, Bjerksund and Stensland. Those approximations are quite inaccurate in dealing with discrete dividends.

To accurately value an American option, one needs to use a numerical approach. The most popular numerical methods are tree, lattice, partial differential equation (PDE) and Monte Carlo. FinPricing is using the Black-Scholes PDE plus finite difference method to price an American equity option. The finite difference model is one of the most widely used methods of approximation to solve the PDE equation for American options.

The three finite difference approximations most widely used for pricing American options are the Explicit, Fully Implicit and Crank-Nicolson models. Among these three, the last is unconditionally stable with respect to domain discretization and is the most accurate. Compared with the binomial tree method or the Whaley model, Crank-Nicolson has obvious advantages of computation speed and accuracy.

The Black-Scholes PDE describes the evolution of any derivative whose underlying asset satisfies the Black-Scholes assumptions, and can be used to price American options. The main difference between European option and American options is that the latter can be executed any time prior to the expiry date. The European option under Black-Scholes assumptions has an analytical solution for the fair price. The American option, in general, does not have an analytical solution so the PDE has to be solved using other techniques such as finite difference numerical methods.

The Black-Scholes PDE for American option is:


The Crank-Nicolson finite difference model converts a differential equation into a set of difference equations and then solves it iteratively. It is computationally more efficient to use finite difference with ln(S) than to use S as the underlying variable. Let’s define Z = ln(S), then the above PDE becomes


Practical Notes


To solve the equation using the finite difference method, we need to define the grid as follows: Divide the time span into N equally spaced intervals of length Δt=T/N where T is the option expiry date. Define Z into a total of M + 1 equally spaced values with . The points Δt and ΔZ define a grid consisting of a total of (M+1)(N+1) points.

On this grid, the point (i,j) corresponds to time iΔt and stock price jΔZ. Let the value of the option at (i,j) be denoted by f_(i,j). Values for N and M are chosen based on the life of the option and the desired level of accuracy.

The Crank-Nicolson finite difference model replaces the space and time derivatives with finite differences centered at an imaginary time step at (i+1/2). Therefore, we obtain the following formula:


Since American options can be exercised at any time, we must add an extra condition at each point on the grid to verify if it is optimal to do so. Therefore, there is an extra verification for the put option


and one for the call option


One of the uncertainties in the finite difference method is the definition of the grid, especially in the case of discrete dividends. When the dividend date does not fall on one of the discretized time steps, the price should be interpolated or an extra grid point inserted. Also, if the underlying asset step and time step for the grid construction are chosen ambiguously, then the price of the option might significantly differ from the fair value.

For the case when the underlying stock pays discrete dividends, it might be optimal to exercise the option right before the dividend date, only if


Therefore, at time t_d the new asset price and option price have to be calculated on the grid discounted with D. FinPricing system dynamically adds an extra step to the grid, if necessary, to ensure that discrete dividends dates are mapped on the grid.

Some Greeks, in particular delta (Δ), gamma (Γ) and theta (Θ), can be calculated directly from the f_(i,j) values on the grid. To calculate Vega (V), it is necessary to slightly change the volatility and recalculate the option price using the same grid. FinPricing platform calculates option prices as well as Greeks for the call options and put options for the underlying stocks with no dividends, continuous yield dividends and discrete dividends.


3. Related Topics