American Option Formula

Source: Global Derivatives

American options give the holder the right to exercise the option at or before the expiry date. This characteristic of American options render solutions to value them somewhat difficult.

The only case in which a closed-form solution to pricing an American option exists, is in the case where we are valuing an American call option with no dividends throughout its life. Exact pricing only exists for cases when a single known dividend exists via a pseudo-American formula.

Pricing:

1) For American Calls with No Dividends:

We can price American calls on a non-dividend paying stock because of an important attribute - it is never beneficial to exercise the option prior to expiry. We can briefly look at two primary reasons for this:

Firstly, holding the call option instead of exercising it and holding the stock is an insurance factor. An adverse stock movement (fall) would result in losses for the stock holder, but holding the call would enable the holder of the call to insure against any falls.

Secondly, there is the concept of time value of money. Paying the strike price earlier rather than later means that the holder of option loses out on the time value the money can achieve for the remainder of the option.

The attribute of non-exercise means that the American option can be priced via the standard Black-Scholes European call option formula and forcing dividends to 0.

2) American Options Approximation (Barone-Adesi, Whaley) 1987:

Barone-Adesi and Whaley (1987) gave a quadratic approximation to price American Options based on a quadratic approximation method proposed by MacMillan (1986), and the pricing of the option is essentially a European option adjusted for an early exercise premium.

if

and

if

The European option is valued using the Black-Scholes-Merton European formula.

Defining the variables as:

Where

and

The critical value of S* is defined as:

and can be solved using the Newton-Raphson method and specifying appropriate seed values.

For corresponding put values, we have a set of formulas to determine the value of an American put.

if

and

if

Where the variables are defined as:

Where

n and k being the same as for a call.

This approximation is suitable and fast for practical pricing of American options and gives a very close value when compared to closed form Black-76 (see Haug 93)

3) For American Calls with a single dividend (Roll, Geske & Whaley):

An American call option can be considered to be a series of call options which expire at the ex-dividend dates, and this case becomes a compound option or (an option on an option) with a closed-form solution as follows:

With the variables are defined as:

is the is bivariate cumulative normal distribution function and

S* is the critical stock price for which the following equation is satisfied:

The critical stock price can be solved iteratively via the Bisectional method.

4) American Option Approximation (Bjerksund, Stensland):

The Bjerksund-Stensland approximation assumes that the exercise is initiated to a corresponding 'flat' boundary, making use of a trigger price. This approximation is computational inexpensive and the method is fast, with evidence indicating that the approximation may be more accurate in pricing long dated options than the Barone-Adesi & Whaley model.

Where

The function phi is given as:

Where

The trigger price I is given as the following:

And to price an American put, we consider the Bjerksund-Stensland approximation for the call option and apply put-call parity in the form of:

The pricing found on this website is based on a plain vba function similar to that found in Haug (1998), but a MonteCarlo method or Hybrid Quasi-MonteCarlo method using Halton sequences. More details of QMC and MC can be found here soon.

5) American Put Option Approximation (Geske, Johnson) 1984

Geske & Johnson (1984) give an accurate approximation for an American put option by considering it as a series of Bermudan options, with the value of the American option given when the number of exercise dates for the Bermudan option tends to infinity or an infinite series of multivariate normal terms. The main breakthrough in the solution is a way to solve the free boundary problem for the case of American options.

They use Richardson's technique to improve the speed of convergence. This technique breaks down the problem so that the exercise of an American can happen at 3 points in time over the life of the option and therefore can be evaluated as 3 separated options:

a) At expiration (based off the Black-Scholes model)
b) Halfway through its life and at expiry, or;
c) One-third or two-thirds or at expiry during the life of the option

b) and c) require computing bivariate and trivariate normal probabilities. The weighted average of a) b) and c) is the value of an American put option under this method.

Where refer to a, b and c respectively.

Although effective in many cases, the setup is prone to the problem of non-uniform convergence - particularly for underlying assets which pay a dividend at point b) in the option life. As there are more possible exercise points in case c) relative to case b), one should expect that the value of c) is greater than that of b). However, if a dividend is paid at point b), leading to a high chance of exercise at point b), the value of b) will be greater than c). This ultimately results in non-uniform convergence.

Bunch & Johnson (1992) modify the above model to improve on the accuracy and computational efficiency of the original Geske & Johnson model by reducing the number of exercisable points to 2:

Where refers to a) again and is now the choice in exercise points (2 exercise points) which will maximise the value of the option.

6) American Option Approximation (Ju & Zhong) 1999

Based on the quadratic approximation of MacMillan (1986) and Barone-Adesi & Whaley (1987), Ju & Zhong (1999) looks to improve on the quadratic approximation - particularly for very short or very long dated option maturities.

The value of an American option under this model is given as:

Where if equal to 1 for a call option and -1 for a put option,

is equal to: ,
is the price of a European option under Black-Scholes,
is the price of a European option under Black-Scholes using S* as the underlying asset price.
S* is the critical stock price which satisfies the following:

Which can solved iteratively where:

And the differential equation:

is equal to:

Where:

Where N() is the cumulative normal distribution function and n() is the normal density function.

7) Binomial Trees (Cox, Ross & Rubinstein) 1979, (Rendlemann & Barrter) 1979

Binomial trees are widely used within finance to price American type options as it is easy to implement and handles American options relatively well. The binomial method constructs a tree lattice which represents the movements of the stock under geometric Brownian motion and prices the option relative to the stock price through means of backwards induction.

It effectively assigns a probability of an up movement and a down movement in the stock price based on the following:

The terms for the up and down movement are:

Where d can be simplified to:

The probability of the stock price increasing at the next time period (node) is given as.

And conversely, the probability of a down movement is given as 1-p.

We can now, via backwards induction, determine the price of an American call or put through the following:

Even though implementation of the binomial model for American options and beyond is fairly simple, the major disadvantage of using the model is that it often requires a large number of nodes to achieve a decent accuracy.

8) Accelerated Binomial Tree (Breen) 1991

Breen (1991) tries to improve the convergence of the CRR binomial model by basing a binomial tree off the Geske & Johnson (1984) American put approximation and the end result is a model somewhere in between the CRR binomial tree and the Geske-Johnson approximation.

The technique also makes use of what is known as Richardson's extrapolation, a method to used to improve the convergence of a sequence developed in 1910 (Richardson, 1910).

Subsequently, Chang, Chung & Stapleton (2001) also extend on Breen's model using a repeated Richardson extrapolation which the authors show to be similar in computational requirements to Breen's original model with a higher degree of accuracy. The repeated Richardson extrapolation looks to predict the interval of true option values.

However, similar to Gesk & Johnson's model, this accelerated binomial tree is also prone to the problems of non-uniform convergence and will be inaccurate in certain cases.

9) Trinomial Tree (Boyle)

The Trinomial tree is similar to the binomial method in that it employs a lattice-type method for pricing options. The exceptions are that the trinomial method arise at an accurate value faster than its Binomial counterpart due to the use of a 3-proned path compared to the 2-proned path seen with Binomial trees.

The probabilities of the price going up at the next time period is given as:

The respective American call and put can now be priced via backwards induction:

Call:

Put:

For further studies into the trinomial model, see Boyle (1986).

10) Jump Diffusion (Merton) 1976

The jump diffusion process which was suggested by Merton (1976) was aimed to price European options where exercise can only take place at maturity, and numerous authors have suggested ways to price American style options with Poisson jumps.

For options with a known finite number of jumps during its life, method of lines can be used to numerically solve the problem; see Meyer (1998), and finite differences is also often used (particularly explicit FD) to evaluation the differential equation.

By setting up the pricing problem as a linear complementarity problem, d"Halluin, Forsyth and Labahn (2003) makes use of an iterative process combined with a Fast Fourier Transform to evaluate the correlation integral while the early exercise feature of American options makes use of a penalty method. A penalty method is useful in solve constraint optimisation problems - in this case, the American constraint - which the authors show will converge readily, even if and when stochastic volatility is considered.

11) Monte Carlo Simulation

The use of Monte Carlo methods does not easily handle the pricing of American options due to their early exercise characteristic, and original research deemed pricing of American options to be not even possible using MCS. Simulation of option prices tend to employ a backwards induction technique, which will tend to overestimate the price of an option.

Various algorithms have been put forward to price American options using backwards induction, but many algorithms are computationally intensive in that it does not converge readily.

Longstaff & Schwartz (2001) combine the Least Squares method with Monte Carlo simulation to price American options, although computational time is high, accuracy is reasonable

A number of authors including Broadie & Glasserman (1997) and Fu, Laprise et al (2000) have suggested that the most flexible and easily implemented procedure is the simulated tree algorithm, but it does have its drawbacks, with the primary one being exponential growth in computation with the number of exercise opportunities. Rogers (2002) suggets a 'dual' way to price American options under a Monte Carlo framework by utilising a Lagrangian martingale optimisation method.

Attempts have also been done by combining the Least Squares method with Quasi-Monte Carlo simulation - see Longstaff & Schwartz (2001), however the rate of convergence for the simulation in higher-dimensions (e.g. with exotic American options) is an exponential function and will break down. However, improvements on this based on the work of Caflisch, Morokoff & Owen (1997) and Caflisch (1998) showing how a Brownian bridge can be used to reduce the convergence problem in higher dimensions.

12) Finite Differences Method (Brennan & Schwartz) 1977

The finite difference method detailed under the European Options section can be applied to the case of American options as well. By incorporating an early exercise 'test' within an algorithm, we can determine the value of an American option as given by the PDE and its initial and boundary conditions using explicit, implicit and Crank-Nicholson schemes.

Similar to tree models, the finite differences method(s) are commonly used in practice because of two reasons; they are relatively easy to implement computationally and generally converge to a solution (albeit with more timesteps).

13) Other Models

Given the difficulty in finding closed form pricing methods for American options, a large amount of research has put into developing approximations to the pricing problem. Carr & Faguet (1996) utilise the method of lines to discretise the time derivative in the partial differential equation. Gaporale and Cerrato (2005) look at a polynomial approximation using Chebyshev nodes and which is similar the Gaussian quadrature method used suggested by Sullivan (2000) . Figlewski & Gao (1999) uses an adaptive mesh model to greatly improve on a standard binomial or trinomial tree by adjusting the trees near the strike price region. Regression based models have been suggested by Johnson (1983) as well as Broadie & Detemple (1996), although the use of these typically requires a large initial dataset of options to generate regression coefficients and are therefore not that practical.

Other Known Names / Variants

- Bermudan options
- Mid-Atlantic options
- Early Exercise options
- Hawaiian options

Papers:

Amin, K., "Jump Diffusion Option Valuation in Discrete Time", Journal of Finance, 48, pp. 1833-1863 (1993)
Barone-Adesi, G. & Whaley, R. "Efficient Analytic Approximation of American Option Values", Journal of Finance (June '87)
Bjerksund, P. & Stensland, G. "Closed-Form Approximation of American Options", Scandinavian Journal of Management, vol.9, pp.87-99 (1993)
Black, F. & Scholes, M. "The Pricing of Options & Corporate Liabilities", The Journal of Political Economy (May '73)
Breen, R., "The Accelerated Binomial Option Pricing Model", The Journal of Financial and Quantitative Analysis, Vol. 26, No. 2, pp. 153-164 (1991)
Brennan, M. & Schwartz, E., "Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims". Journal of Finance and Quantitative Analysis, 13:462--474 (1978)
Broadie, M., Detemple, J., "American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods, Review of Financial Studies, Vol. 9, No. 4, pp. 1211-1250 (1996)
Boyle, P., "Option Valuation Using a Three-Jump Process", International Options Journal 3, pages 7-12 (1986)
Bunch, D.S. & Johnson, "A Simple Numerically Efficient Valuation Method for American Puts Using a Modified Geske-Johnson Approach", Journal of Finance, 47, pp. 809-816 (Jun 1992)
Caflisch, R., "Monte Carlo and Quasi-Monte Carlo Methods", Acta Numerica, pp. 1-49 (1998)
Caflisch, R., Morokoff, W., & Owen, A.B., "Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension", Journal of Computational Finance, 1, pp. 27-46 (1997)
Caporale, G. & Cerrato, M., "Valuing American Put Options Using Chebyshev Polynomial Approximation) (2005)
Carr, P., Hirsa, A., "Why Be Backward. Forward Equations for American Options" (2002)
Carr, P. & Faguet, D., "Fast Accurate Valuation of American Options", working paper, Cornell University (1996)
Chang, C.C., Chung, S.L, & Stapleton, R., "Richardson Extrapolation Techniques for Pricing American-style Options", Working Paper (2001)
Cox, J., Ross, S., & Rubinstein M., “Option Pricing: A Simplified Approach." Journal of Financial Economics, 7. (Sept '79).
d'Halluin, Y., Forsyth, P.A., Labahn, G., "A Penalty Method For American Options with Jump Diffusion Processes" (2003)
Figlewski S., Gao, B., Ahn., D.H., "Pricing Discrete Barrier Options with an Adaptive Mesh Model", Working Paper, 1999
Geske, R. & Johnson, H.E. "The American Put Option Valued Analytically" Journal of Finance, 39, 1511-1524. (1984)
Hull, J., "Options, Futures & Other Derivatives", 5th Edition 2002 - Chapter 12
Johnson, H., "An Analytical Approximation for the American Put Price", Journal of Financial & Quantitative Analysis, 18, pp. 141-148 (1983)
Ju, N., & Zhong, R., "An Approximate Formula for Pricing American Options" Journal of Derivatives, 7, 2, 31-40, 1999
Lewis, A., "American Options under Jump-diffusions: An Introduction" Wilmott Mag (Mar' 03)
Longstaff, F.A. & Schwartz, E.S., "Valuing American Options by Simulation: A Simple Least-Squares Approach", "Review of Financial Studies, 14, 1, pp. 113-147 (2001)
MacMillan, W., "Analytic Approximation for the American Put Option" in Advances in Futures and Options Research, 1, 119-139. (1986)
Merton, R.,(a) "Option Pricing when Underlying Stock Returns Are Discontinuous", Journal of Financial Economics 3, pp. 125-144 (Jun '73)
Merton, R.,(b) "Theory of Rational Option Pricing", Bell Journal of Economics & Management (June '73)
Meyer, G.H., "The Numerical Valuation of Options with Underlying Jumps", Acta Math univ Comemianae, pp. 69-82 (1998)
Rendleman, R. & Bartter, B., "Two-State Option Pricing", Journal of Finance 34, 1093–1110 (1979)
Richardson, L. F., "The approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam". Philosophical Transactions of the Royal Society of London, Series A 210: 307–357 (1910)
Sullivan, A.M., "Valuing American Put Options Using Gaussian Quadrature", Review of Financial Studies, 1, pp. 75-94 (2000)