Stock options are called derivative securities because
Now there are three possible outcomes for the value of the stock , or. We assume that the option now expires at the end of the second period and we let , , be the value of the option in each of the three outcomes.
In order to price this option, first suppose that the stock went up during the first time period. Now consider how much stock and bond we should hold in order to replicate the payoff of the option at the expiry time. Using the same ideas, it is possible to extend this model to three, four, five, or any number of time periods.
In fact, using some mathematical notation, it is possible to write down a general formula for periods. The notation in this formula is explained here , but don't worry too much about the formula itself. The important point is that our technique of working backwards from the time of expiry works for any number of time periods. The brave reader might want to try and prove this result by induction. You can work out the option price in a multi-period model by working backwards from the last period.
The binomial model is a very simple model for understanding the ideas behind option pricing. However, so far the stock price can only take finitely many values and furthermore can only move at discrete time points.
Both of these features are somewhat undesirable, but there is a clever way around this problem. The basic idea is to divide the time to expiry of the option into equally-sized time periods and look at what happens to the model in the limit, as tends to infinity, in other words as the size of the time periods tends to zero.
This will move our model from discrete time to continuous time. The actual mathematics is a little too involved to be presented here though the keen reader may want to look at it in our appendix. In fact, it is not necessary to go into it: This is the famous Black-Scholes equation of financial mathematics. The only parameters it depends on are the strike price, , the time to expiry, , current stock price, , the interest rate, , and what is called the volatility. This parameter describes the variability of the stock price and has a precise mathematical definition.
The important message from our derivation is not so much the formula we end up with shown here on the right , but rather the way in which we got it. We saw in our discrete time model how we were able to exactly replicate the payoff of our option by holding the correct amount of stock and bond. This then told us that the price of the option at time zero must be the amount that it costs to replicate the option.
The underlying idea in continuous time is exactly the same: Of course, since we are working in continuous time, the amount held in the stock and bond will need to be adjusted continuously, rather than at discrete time steps. However, the idea of replication is exactly the same. The original Black-Scholes model assumed that stock price was a function of a random Brownian motion. The original paper written by Black and Scholes in used the idea of replication to work out the price of the European call option, though their approach was a little different from the one taken here.
They began directly with a continuous time model in which the stock price was a function of a Brownian motion: Mathematically, Brownian motion is a stochastic process which satisfies certain properties.
Black and Scholes' paper showed how the pricing of options can be transformed into a problem of solving partial differential equations with some given boundary conditions. Indeed, they were able to transform these partial differential equations and show that they were equivalent to solving the heat equation from physics. Unfortunately, the maths required to see the link with partial differential equation theory requires the machinery of stochastic calculus, which takes quite some effort to set up.
Although we have only shown how to price a European call option, we could use the same analysis to price any option whose payoff depends only on the terminal value of the stock. The Black-Scholes theory is indeed very general. However, there are also many other sorts of options, which don't fall under its remit. Pricing such exotic options creates many interesting problems in mathematics and keeps financial mathematicians in employment.
This post was really helpful! Thanks for posting this! Do you have any reading you recommend if you are a relative novice on this subject? Hi, I would like to know whether the price of a derivative can be zero? And when and why. We've had a suggestion from Chris Rogers for further reading for people new to the subject.
He says that John Hull's book "Options, Futures and other Derivative Securities" is a good accessible introduction to the area. Also, Chris answered Vaibhav's question: Yes, a derivative can have a value of zero. A down-and-out option will have zero value once the price of the underlying asset falls below a specified threshold.
Also an interest-rate swap could have zero value at inception, where one party swaps floating interest payments for payments at a fixed rate calculated to make the two payment streams exchanged of equal value. Skip to main content. Reckless trading of derivatives can cause huge losses.
What will the price of fuel be in a year's time? The stock price in a one-period binomial model. The stock price in a two-period binomial model. Comments Links don't work Permalink Submitted by vonjd on May 31, The links for the notation and for the appendix don't work. Thanks for spotting that, Permalink Submitted by Marianne on June 1, Thanks for spotting that, we've fixed the links! Permalink Submitted by Anonymous on June 7, The writer sells the put to collect the premium.
The put writer's total potential loss is limited to the put's strike price less the spot and premium already received. Puts can be used also to limit the writer's portfolio risk and may be part of an option spread.
That is, the buyer wants the value of the put option to increase by a decline in the price of the underlying asset below the strike price. The writer seller of a put is long on the underlying asset and short on the put option itself. That is, the seller wants the option to become worthless by an increase in the price of the underlying asset above the strike price.
Generally, a put option that is purchased is referred to as a long put and a put option that is sold is referred to as a short put.
A naked put , also called an uncovered put , is a put option whose writer the seller does not have a position in the underlying stock or other instrument. This strategy is best used by investors who want to accumulate a position in the underlying stock, but only if the price is low enough.
If the buyer fails to exercise the options, then the writer keeps the option premium as a "gift" for playing the game. If the underlying stock's market price is below the option's strike price when expiration arrives, the option owner buyer can exercise the put option, forcing the writer to buy the underlying stock at the strike price.
That allows the exerciser buyer to profit from the difference between the stock's market price and the option's strike price. But if the stock's market price is above the option's strike price at the end of expiration day, the option expires worthless, and the owner's loss is limited to the premium fee paid for it the writer's profit. The seller's potential loss on a naked put can be substantial.
If the stock falls all the way to zero bankruptcy , his loss is equal to the strike price at which he must buy the stock to cover the option minus the premium received. The potential upside is the premium received when selling the option: During the option's lifetime, if the stock moves lower, the option's premium may increase depending on how far the stock falls and how much time passes. If it does, it becomes more costly to close the position repurchase the put, sold earlier , resulting in a loss.
If the stock price completely collapses before the put position is closed, the put writer potentially can face catastrophic loss. In order to protect the put buyer from default, the put writer is required to post margin. The put buyer does not need to post margin because the buyer would not exercise the option if it had a negative payoff.
A buyer thinks the price of a stock will decrease. He pays a premium which he will never get back, unless it is sold before it expires.
The buyer has the right to sell the stock at the strike price. The writer receives a premium from the buyer. If the buyer exercises his option, the writer will buy the stock at the strike price. If the buyer does not exercise his option, the writer's profit is the premium. A put option is said to have intrinsic value when the underlying instrument has a spot price S below the option's strike price K. Upon exercise, a put option is valued at K-S if it is " in-the-money ", otherwise its value is zero.
Prior to exercise, an option has time value apart from its intrinsic value. The following factors reduce the time value of a put option: Option pricing is a central problem of financial mathematics.