Financial Option Example

A common problem in financial engineering is to determine the value of a financial option, that is, the right to collect an uncertain future payoff. In this example, we value a path-dependent option known as the Asian (or average value) call option.

Problem Description

A stock's price S(t_i) is observed at a finite number m of increasing times t_i. The average price of this stock is:

avgPrice = [S(t₁) + S(t₂) + … + S(t_m)]/m.

At time t_m (the option's maturity), the option holder (buyer) receives a final payoff that equals the amount by which the observed average price exceeds a given exercise price E. In Hava, this is expressed as:

payoff = max(avgPrice - E, 0).

As an example, consider a six-month option with end-of-month observations and an exercise price E = 9.50. If the observed stock prices are 10.00, 10.80, 11.25, 10.28, 9.59, 8.92, then the final payoff is max(10.14-9.50, 0) = 0.64.

Stochastic Model

We use a binomial lattice to describe the evolution of stock prices: Next period's stock price either goes up by a factor U with probability q or goes down by a factor 1/U with probability 1-q.

The values for U and q are derived from basic parameters in the conventional way:

S0 = 10;             // initial price
sigma = 0.5;         // annual volatility
dt = 1/52;           // period length in years
K = 26;              // number of periods in lattice
r = 0.02;            // annual interest rate

The option's market value today is simply the discounted expected value of the final payoff, where the discount factor is determined from these same parameters:

beta = exp(-r*K*dt); // discount for option duration

Monte Carlo Simulation

Since the evaluation of the option's value requires a computation of an expectation (or average) of a function of a price path, Monte Carlo simulation is a convenient approach. The steps in a Monte Carlo simulation are:

Generate a collection of sample price paths.
For each path, compute its average price and final payoff.
Take the average of the final payoffs as an estimate of the option's value.

Generating Sample Paths

The random event at each period is either "Stock price goes up" or "Stock price goes down". We represent each of these outcomes as a Hava structure that contains both the probability that the outcome will occur, and the price multiplier with which it is associated. Then we place the two outcomes in a list:

struct Event(prob, multiplier); 
EventList = (Event(q, U), Event(1-q, 1/U));

The variable EventList serves as a probability mass function. A single random event may be expressed as random(EventList). A single random multiplier is therefore:

function rm = random(EventList).multiplier;

Note that rm must be a function, not a variable, since it will be computed more than once. (A variable retains its value the first time it is computed, and retrieves the earlier value each time it is subsequently requested.)

To generate a single sample price path, we proceed in two steps.

Generate a list rlm of (random) multipliers, one for each period:
function rlm = collect(k=1 to K) {rm};

Construct the (random) price path rpp from the list of multipliers rlm:

function rpp = buildPP(rlm, 1, collect(S0)); 
function buildPP(lm, k, pp) = 
  if (k > K) {pp} 
  else {buildPP(lm, k+1, join(pp, lm[k]*pp[k]))};

Click here to view the Hava program so far. This sample code contains a additional line (at the bottom) to generate three sample paths.

Sampling

The value of an Asian option for a single random price path rpp is calculated as:

F_Asian(rpp) = max((1/(K+1))*sum(i=1 to K+1) {rpp[i]} - E, 0);

The average payoff over a sample of N such paths is given by:

stats = sum(i=1 to N) {F_Asian(rpp)}; 
sampleMeanPayoff = stats/N;

Finally, the discounted (present) value is obtained by multiplying the expected value by the discount rate for the duration of the option:

optionValue = beta*sampleMeanPayoff;

Click here to view the complete Hava program.

Extensions

It is possible to gather sample statistics associated with multiple functions of the sample path. For example, a holder of this option might well want to know the probability that an Asian option payoff exceeds a given "tail" value. It is also straightforward to add more events to the multiplier probability mass function, or to add moments to the statistics. Finally, the methods described here can be applied to other options problems, such as Asian put option, Max or Min price option, or European call/put options.