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European Options

European options provide the fundamentals from which other options can be valued. We look at the foundations behind the Black-Scholes model, its pitfalls and how the framework can be applied to valution of exotic options.

 


Black-Scholes (No Dividends) || Black-Scholes-Merton (Continuous Dividends) || Binomial Trees || Trinomial Trees || Jump Diffusion || Adaptive Mesh || CEV || Finite Differences || Monte Carlo || Quasi-MCS || At-the-Money-Forward || Comparisons || Known Names / Variants || Additional Resources || Pricing Models

 

European Options are the foundations of the options universe. Presenting itself as the most basic type of option contract, these options all the holder or seller of the option the ability to exercise the option at (and only at) the expiry of the option.

1) European Options with No Dividends (Black-Scholes) 1973:

 

Continuing on from our introduction to the Black & Scholes model, we reiterate the application of the BS model to valuing European options on a non-dividend paying stock.

 

The respective call and put values are:

Where S is the stock price, X is the strike price, r is the risk free rate, (T-t) is the time to maturity and N(x) is the cumulative normal distribution of:

European call Option d2

Where d2 can be simplified to the following:

d2 Simplified

The final input to the model we need is the volatility of the underlying asset price. Choice of the volatility input is an entirely different discussion altogether, but for simple cases, the use of historical data in what is known as historical volatility can be used; however, a more practical and applied method is to use implied volatilities from the market.

 

2) European Options with a Constant Dividend Yield (Merton) 1973:


In reality, many assets pay some sort of 'dividend' over the life of the option, and Merton, also in 1973 extended the standard Black-Scholes model to incorporate a known annual dividend yield. His model follows the same foundations of the basic Black-Scholes model and the value of a call and put option is given as follows:

BSM Call

BSM Put

d1 and d2 are modified to:

BSM d1
BSM d2
Where d2 can again be simplified to:
d2 Simplified

3) European Options under a Binomial Lattice (Cox, Ross & Rubinstein) 1979:

The binomial method is a common method to price all types of options, both vanilla flavoured as well as the more exotic types due to it's flexibility. For pricing European options, it can be related to the Black-Scholes model for reasons which will be discussed shortly.

 

First, we consider the fundamental properties of a binomial lattice. It essentially models the movement of the stock price  and hence the option price over time by considering the movements of the asset at each time node. The asset is assumed to only be allowed to either take an "up" step or a "down" step, where these steps are given as:

Bin-Up

for an up movement, and:
Bin-Down

for a down movement.

 

As the asset can only take an up or a down move, and not both simultaneously, the down movement can be simplified to:

Bin-down2

We then need to assign a probability of the asset moving higher (i.e. increasing) as:

Bin-prob

and for the probability of a down movement, the combined probabilities must equal to 1, hence the probability of the down movement is simply (1-p).

An illustration of how the binomial tree works is shown below:
BinTree


If we start at the far left, "S" denotes the stock price today, which is, lets say 100. Assume that the up movement "u" is 1.0145. At the first node "Su", the stock price equals:

 

100 x 1.0145 = 101.45

 

If you consider the 4th node, assuming that stock price has moved up during all 4 periods, we can calculate the stock price at that node, given our knowledge of "Su4", the stock price at the 4th node would be:

 

Bin Illustration

 

Having found the value of stock price at the end of the nodes, we are able to calculate the option value by means of backwards induction, that is, working from the far right of the lattice, back to the origin. A simple computer algorithm is able to solve the option values by walking through the steps through an iteration count - even with a large number of nodes.

 

For European call and put options, the binomial model is given as:

Binomial Call

and

Binomial Put

j is given by the following condition:

Binomial J

Where Phi represents the next non-negative integer greater then the bracketed term. For example, if the bracketed term was 3.492, then j would be 4.

 

As we pointed out earlier, there is a relation between the binomial model and the Black-Scholes pricing model for the valuation of options. With a significant number of time nodes, the binomial method begins to converge, and the convergence of this value ultimately becomes the BS value. To illustrate this, we can take the following inputs, and we see that the binomial method converges to a value which is the Black-Scholes value.

 

Call Option: Stock Price: 100, Strike Price: 102, Volatility: 20%, Dividend Yield: 5%, Risk Free: 8%

Covergence

At the 200th time node for the binomial method, we get a call value of 8.0369. Under the Black-Scholes method, the value we get is 8.0310, an absolute difference of 0.0059. Taking the number of nodes to 10,000, we arise at a Binomial price of 8.0308, representing an absolute error of only 0.0002.

 

The binomial model described above is the original CRR binomial model. Variations and extensions of the model have appeared in literature since - including separation into additive and multiplicative recombining trees, additional factors and more.

 

4) Trinomial Method (Boyle) 1986


The trinomial method to price options was introduced in 1986 by Boyle and is an attempt to model stock price movements better than the binomial method. As one can guess by its name, the trinomial method is similar to the binomial lattice in that the stock price is modeled by a tree, but instead of 2 possible paths per node, the trinomial tree has 3; an up, down and stable path. The probabilities of each are given as:

Trinomial Down

Trinomial Down

Trinomial Mid

The probabilities can also be represented as:

Trinomial Prob Down - Hull

Trinomial Prob Up Hull

and hence:

Trinomial Prob Mid - Hull

We can show that the trinomial method converges much faster than the binomial method by an illustration:

Using the same inputs as before:

 

Call Option: Stock Price: 100, Strike Price: 102, Volatility: 20%, Dividend Yield: 5%, Risk Free: 8%

Trinomial

Transposing the two lines together, it is clear that the trinomial method converges much more rapidly than the binomial method.

Trinomial Binomial Convergence

An interesting property of the trinomial method is that it can be shown to be equivalent to the explicit finite difference model, which is discussed later on.

 

5) Jump Diffusion (Merton) 1976


In order to model fluctuations of the stock price process, Merton (1976) suggested a model in which the stock price following a geometric Brownian motion and a series of 'jumps' which assume are Poisson driven. You can picture each jump as a sudden movement in the stock price caused by any number of economic, industry or company factors. In addition to the standard pricing model, we define 2 additional variables in order to price under jumps:

 

1) The average number of jumps each year

2) The average jump size as a proportion of the stock price - or, alternatively the percentage of the stock volatility explained by the jumps.

The stochastic process of the stock price is given as:

Jump Diffusion Stock Process

Where Miu is the expected return of the stock, Lambda is the number of jumps per year and Gamma is the jump size as a proportion of the stock price. Furthermore, dp is the Poisson process which governs the jumps, dz is a Wiener process.

The respective call and put values respectively can be defined from the above equation to:

Jump Diffusion - Sum Call

and

Jump Diffusion - Put

Where cjump and pjump are the respective Black-Scholes values for a European call and put option. We must make an adjustment to the volatility used to calculate these BS values as follows:

Jump Diffusion - Sigma

Where Observed Vol is the observed volatility and Lambda & Gamma are the same as defined earlier.

 

6) Adaptive Mesh Model


The adaptive mesh model looks to improve on the standard lattice models in that additional lattices are transposed onto a standard lattice. For an adaptive mesh based on the trinomial model, the paths looks something like:

Trinomial Adaptive Mesh

Where the black grid is the standard trinomial lattice, and the orange grid towards the right side is an 'extended' version of the trinomial grid. This gives a faster convergence compared to standard lattice methods, and more accurate results.

 

Although the adaptive mesh is more common in pricing path-dependent options such as barrier options, it can be used to price European options.

 

7) Constant Elasticity of Variance Approach


The constant elasticity of variance approach (CEV) assumes that in the risk-neutral world, the process of the stock price is:

CEV Process

Compared to the geometric Brownian motion model risk-neutral process of:

Stock Process

Where Miu is the drift rate [which in the risk-neutral world is given as risk free rate (r) - dividend yield (D)], Sigma is the volatility of the stock price, dz is a Wiener Process and Alpha is a constant which takes on a value greater than 0.

 

Under the CEV approach, when Alpha is given as 1, then the CEV collapses to the standard log-normal stock price process.

 

From the above given process, we can price European options in a similar fashion to that of the standard Black-Scholes, but instead of using a cumulative normal distribution, a non-central chi-squared distribution is used, with variables defined as below. You can find algorithms for implementing the non-central chi-squared distribution in our code section.

 

The pricing of a European call under the CEV method can be shown in the following formulae.

 

Call Options:

Condition
Call
0 < < 1
> 1

 

Put Options:

Condition
Put
0 < < 1
> 1

 

The variables a, b and c are given as:

 

For more details on the non-central chi-squared distribution function, see the paper by Ding (1992).

 

8) Finite Differences


As with any class of option, the price of the derivative is governed by solving the underlying partial differential equation. The use of finite difference methods allows us to solve these PDEs by means of an iterative procedure.

 

We can start by looking at the Black-Scholes partial differential equation:

Where dV is the change in the value of an option, dt is a small change in time. Sigmais the volatility of the underlying, S is the underlying price and Sigma is the carry (r-D).

By specifying initial and boundary conditions, one can attain numerical solutions to all the derivatives of the Black-Scholes PDE using a finite difference grid. The grid is typically set up so that partitions in two dimensions - space and time (in our case, we would be looking at the asset price and the change in time):

Once the grid is set up, there are three methods to evaluate the PDE at each time step. The difference between each of the three methods is contingent on the choice of difference used for time (i.e. forward, backward or central differences - more details in our financial mathematics glossary here). Central differences is used for the space grid (S).

a) Explicit Finite Differences

Explicit FD uses forward differences at each time node t. By splitting the differential equation into the time element and space elements, we can apply forward differences to time as follows:

 

First of all, the PDE as a reminder:

if we substitute x = ln(S), the equation becomes:

Applying the finite differences method, the above equation can be broken down and approximated:

becomes

For the space grid, we can apply central differences for all order of derivatives:

becomes

and

becomes

and

becomes

Combining the terms gives:

Which is the same as:

where the probabilities of each of the nodes is:

This case is actually equivalent to the trinomial tree where probabilities can be assigned to the likelihood of an up move, a down move as well as no move. It can also be shown that the following approximation holds:

b) Implicit Finite Differences / Backward Difference


Similar to evaluating the PDE using an explicit finite difference method, the implicit method takes backward differences for the time derivative but still using central differences for the space derivatives:

Although similar in nature to the explicit finite differences method, the implicit FD method is typically more stable and convergent than the explicit FD method - however, it is often more computationally intensive.

The approximation to the PDE under an implicit FD method is given by the following:

Note that the main difference between the above equation and the one for the explicit FD method is in the selection of time step i.

The subsequent simplification of the approximation and the associated probabilities is similar to that of the explicit FD method.

c) Crank-Nicolson Scheme

An improvement over the implicit FD method is the Crank-Nicolson Scheme which uses central differences for both time and space dimensions. The result is that over smaller time steps dt, the method is more accurate, stable and convergent than both implicit and explicit methods - however, like the implicit FD method (which requires evaluating equations at each time step) it is more computationally intensive than the explicit FD method.

The approximation for the PDE is given as:

9) Monte Carlo Simulation

Monte Carlo Simulation (MCS) can be applied in various fields, particularly when one is try to model the behaviour of systems which have a degree of uncertainty in them. In our case, the purpose is to model the behaviour of the underlying asset price by generating many possible paths. To generate the paths, an assumption must be made as to the statistical process which governs the asset price - in the case of financial assets, this is often assumed to be a geometric Brownian motion.

 

Let us consider the model for the underlying asset price:

Where Miu is the drift rate, Sigma is the volatility of the stock price and dz is a Wiener Process. By using the model for the underlying asset price, we are able to define how the asset price changes with time based upon the parameters above.

 

In a practical case; these would be the steps to undertake a MCS for European options:

 

1. 'Simulate' the risk neutral random path of the asset by drawing on a standard normal random variable taken from a normal probability distribution. This variable will have a mean of 0 and a variance of 1 (as it has been drawn from a normal distribution).
2. Generate n number of random paths for the asset (this is typically anywhere from 1,000 to 1,000,000 times or more, the more paths which are generated, the more accurate the simulation becomes - however, this also implies a longer required time to run the simulation).
3 . Calculate the payoff of the option at the maturity date for each of these random paths. In this case, we will be looking at:

for all of the paths generated.
4. An average of all the option payoffs over all paths generated and then discount this value to get the present value:

This present value number is equal to the value of the option.

 

Although MCS is more useful when the number of dimensions in the pricing problem are larger - in that it reduces the need to solve multi-dimensional partial differential equations, the use of MCS in pricing European options is not uncommon.

 

It must also be noted that the standard normal random number used in most MCS cases is actually a 'pseudo-random' number, i.e. it is not truly random. Although for most cases, the effect of using a pseudo random number does not affect the pricing of the option significantly, some users choose to utilise quasi 'random' numbers - which is where Quasi-Monte Carlo Simulation comes into play.

 

View MCS computational times for European options here.

 

10) Quasi-Monte Carlo Simulation (Low Discrepancy Sequences)


As the name suggests, Quasi-Monte Carlo Simulation is a variation of the MCS method to price options. While MCS uses what are known as pseudo-random numbers, QMCS employs what are known as low discrepancy sequences. If one can picture a box filled with points in a pseudo-random manner; because of the inherent qualities associated with points, although these points will fill the box seemingly randomly - groupings become evident.

With low discrepancy sequences, although true 'randomness' is not generated, these numbers are close to a random number and can be useful in Monte Carlo methods. As the name suggests, there are a number of low discrepancy sequences which can be used in QMCS;

 

Faure
Halton
Sobol
Niederreiter
van der Corput

 

The choice of sequence will be a topic of discussion in a future article, however - all sequences generate similar results.

 

View QMCS computational times for European options here.

 

11) At-The-Money Forward Approximation


The moneyness of an option refers to the relationship between the possibility of exercise and the underlying asset price and can be measured by the option greek, delta. For options which are trading at-the-money, the strike price is said to be the same as the current underlying asset price and the delta of the option will be given to be 0.50 for calls and -0.50 for puts.

 

As options have a maturity date at some point in the future, the moneyness is usually referenced to the underlying asset price at the expiry date (i.e. the forward price which matches the date of expiration). For example;

 

Lets assume we are looking to price a 6 month EURUSD call option with a strike of 1.342.

 

We find that the current underlying EURUSD is trading at 1.328, but the 6-month forward EURUSD rate is given at 1.342. Although the call strike is in-the-money based on the spot price, we say that it is at-the-money based on the forward price.

 

If an option is at-the-money forward, we can used Brenner & Subrahmanyam's approximation (1994) to approximate the value of a European option as follows:

In that their values are the same.

 

12) Comparisons


Here, we look at the pricing of several of the aforementioned methods and the magnitude of error compared to the BSM result.

 

Other Known Names / Variants


- Vanilla options
- Straight options

 

Additional/Useful List of resources


Papers:

 

Black, F. & Scholes, M. "The Pricing of Options & Corporate Liabilities", The Journal of Political Economy (May '73)
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)
Brenner, M., Subrahmanyam, M.G., "A Simple Approach to Option Valuation in the Black-Scholes Model" Financial Analysts Journal, March-April, pp. 25-28, 1994
Boyle, P., "Option Valuation Using a Three-Jump Process", International Options Journal 3, pages 7-12 (1986)
Cox, J., Ross, S., & Rubinstein M.,
“Option Pricing: A Simplified Approach." Journal of Financial Economics, 7. (Sept '79).
Cox, J., Ross, S., "The Valuation of Options for Alternative Stochastic Processes", Journal of Financial Economics, 3, pp 145-66 (Mar '76)
Ding, C.G., "Algorithm AS275: Computing the Non-Central Chi-Squared Distribution Function" Journal of the Royal Statistical Society, Series C, Applied Statistics, 41, pp 478-82 (1992)
Figlewski S., Gao, B., Ahn., D.H., "Pricing Discrete Barrier Options with an Adaptive Mesh Model", Working Paper, 1999
Haug, E., "Complete Guide to Option Pricing Formulas", 1998
Hull, J., "Options, Futures & Other Derivatives", 5th Edition 2002 - Chapter 12, 13, 20
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)

 

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