for anything from car repairs to overdue bills to medical emergencies and more.
Comments Off 
NOVEMBER 4TH, 2009
By ADMIN
The most logical starting place to look for an estimate of future volatility is past volatility. When the underlying is a publicly traded asset, we usually can collect some data over a recent past period and estimate the standard deviation of the continuously compounded return.
We convert these prices to returns, convert the returns to continuously compounded returns, find the variance of the series of continuously compounded returns, and then convert the variance to the standard deviation. In this example, the data are monthly returns, so we must annualize the variance by multiplying it by 12. Then we take the square root to obtain the historical estimate of the annual standard deviation or volatility.
The historical estimate of the volatility is based only on what happened in the past. To get the best estimate, we must use a lot of prices, but that means going back farther in time. The farther back we go, the less current the data become, and the less reliable our estimate of the volatility. We now look at a way of obtaining a more current estimate of the volatility, but one that raises questions as well as answers them.
Comments Off
OCTOBER 27TH, 2009
By ADMIN
With only a few very advanced variations, the Black-Scholes-Merton model does not price American options. Users of the model must keep this in mind, or they may badly misprice these options. For pricing American options, the best approach is the binomial model with a large number of time periods.
Comments Off
OCTOBER 20TH, 2009
By ADMIN
We have made this assumption all along in pricing all types of derivatives.Taxes and transaction costs greatly complicate our models and keep us from seeing the essential financial principles involved in the models. It is possible to relax this assumption, but we shall not do so here.
There are no cash flows on the underlying
We have discussed this assumption at great length in pricing futures and forwards and earlier in this series of posts in studying the fundamentals of option pricing. The basic form of the Black-Scholes-Merton model makes this assumption, but it can easily be relaxed.
Comments Off
OCTOBER 13TH, 2009
By ADMIN
The volatility of the underlying asset, specified in the form of the standard deviation of the log return, is assumed to be known at all times and does not change over the life of the option. This assumption is the most critical, and we take it up again in a later section. In reality, the volatility is definitely not known and must be estimated or obtained from some other source. In addition, volatility is generally not constant. Obviously, the stock market is more volatile at some times than at others. Nonetheless, the assumption is critical for this model. Considerable research has been conducted with the assumption relaxed, but this topic is an advanced one and does not concern us here.
Comments Off
OCTOBER 6TH, 2009
By ADMIN
This assumption is probably the most difficult to understand, but in simple terms, the underlying price follows a lognormal probability distribution as it evolves through time. lognormal probability distribution is one in which the log return is normally distributed For example, if a stock moves from 100 to 110, the return is 10 percent but the log retun is ln(l.lO) = 0.0953 or 9.53 percent. Log returns are often called continuously corn pounded returns. If the log or continuously compounded return follows the familiar nor ma1 or bell-shaped distribution, the return is said to be lognormally distributed. Tb distribution of the return itself is skewed, reaching further out to the right and truncated 01 the left side, reflecting the limitation that an asset cannot be worth less than zero.
The lognormal distribution is a convenient and widely used assumption. It is almost surely not an exact measure in reality, but it suffices for our purposes.
Comments Off
SEPTEMBER 29TH, 2009
By ADMIN
Suppose we are pricing a one-year option. If we use only one binomial period, it will give only two prices for the underlying, and we are unlikely to get a very good result. If we use two binomial periods, we will have three prices for the underlying at expiration. This result would probably be better but still not very good. But as we increase the number of periods, the result should become more accurate. In fact, in the limiting case, we are likely to get a very good result. By increasing the number of periods, we are moving from discrete time to continuous time. Consider the following example of a one-period binomial model for a nine-month
option. The asset is priced at 52.75. It can go up by 35.41 percent or down by 26.15 per- cent, so u = 1.3541 and d = 1 – 0.2615 = 0.7385. The risk-free rate is 4.88 percent. A call option has an exercise price of 50 and expires in nine months. Using a one-period binomial model would obtain an option price of 10.0259. The manner in which we fit the binomial tree is not arbitrary, however, because we have to alter the values of u, d, and the risk-free rate so that the underlying price move is reasonable for the life of the option. How we alter u and d is related to the volatility, a topic we cover in the next section. In fact, we need not concern ourselves with exactly how to alter any of these values. We need only to observe that our binomial option price appears to be converging to a value of around 8.62.
In the same way a sequence of rapidly taken still photographs converges to what appears to be a continuous sequence of a subject’s movements, the binomial model converges to a continuous-time model, the subject of which is in our next section.
Comments Off
SEPTEMBER 22ND, 2009
By ADMIN
In the examples above, the applications were appropriate for options on a stock, currency, or commodity. Now we take a brief look at options on bonds and interest rates. A model for pricing these options must start with a model for the one-period interest rate and the
prices of zero-coupon bonds. Note that this binomial tree is the first one we have seen with more than two time periods. At each point in the tree, we see a group of numbers. The first number is the one-period interest rate. The second set of numbers, which are in parentheses, represents the prices of $1 face value zero-coupon bonds of var- ious maturities. At time 0,0.9048 is the price of a one-period zero-coupon bond, 0.8106 is the price of a two-period zero-coupon bond, 0.7254 is the price of a three-period zero- coupon bond, and 0.6479 is the price of a four-period zero-coupon bond. The one-period bond price can be determined from the one-period rate-that is, 0.9048 = 111.1051, subject to some rounding off. The other prices cannot be determined solely from the one- period rate; we would have to see a tree of the two-, three-, and four-period rates. As we move forward in time, we lose one bond as the one-period bond matures. Thus, at time 1, when the one-period rate is 13.04 percent, the two-period bond from the previous
period, whose price was 0.8106, is now a one-period bond whose price is 1/1.1304 = 0.8846. Although we present these prices and rates here without derivation, they were determined using a model that prevents arbitrage opportunities in buying and selling bonds. We do not cover the actual derivation of the model here.
Comments Off
SEPTEMBER 15TH, 2009
By ADMIN
Until now, we have looked only at some basic principles of option pricing. Other than put-call parity, all we examined were rules and conditions, often suggesting limitations, on option prices. With put-call parity, we found that we could price a put or a call based on the prices of the combinations of instruments that make up the synthetic version of the instrument. If we wanted to determine a call price, we had to have a put; if we wanted to determine a put price, we had to have a call. What we need to be able to do is price a put or a call without the other instrument. Now, we introduce a simple means of pricing an option. It may appear that we oversimplify the situation, but we shall remove the simplifying assumptions gradually, and eventually reach a more realistic scenario.
The approach we take here is called the binomial model. The word “binomial” refers to the fact that there are only two outcomes. In other words, we let the underlying price move to only one of two possible new prices. As noted, this framework oversimplifies things, but the model can eventually be extended to encompass all possible prices. In addition, we refer to the structure of this model as discrete time, which means that time moves in distinct increments. This is much like looking at a calendar and observing only the months, weeks, or days. Even at its smallest interval, we know that time moves forward at a rate faster than one day at a time. It moves in hours, minutes, seconds, and even fractions of seconds, and fractions of fractions of seconds. When we talk about time moving in the tiniest increments, we are talking about continuous time. We will see that the discrete time model can be extended to become a continuous time model.
