Derivative trading and option pricing theory
The value of the option at time zero must be , for if it traded at any other value, then there would be two entities the replicating portfolio and the option in our model that have the same payoff, yet cost different amounts. If this happened, everyone would buy one and sell the other in order to make a riskless profit. Hence, is the price of the option. Rather surprisingly, the probability of a good period plays no part in this formula. A similar expression does indeed appear, but with replaced by , however is determined only by the interest rate , and the possible changes in the stock price and.
The quantities and are accordingly termed equivalent probabilities. Remarkably, these equivalent probabilities suffice to determine the option price: Our model only lasts one time period, and we would like to extend it to a multi-period case. So let us suppose now that there are two periods, both of equal length.
In each period the stock can either go up or down, as shown in the diagram on the right. 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. This is called the Time value. Time value is the amount the option trader is paying for a contract above its intrinsic value, with the belief that prior to expiration the contract value will increase because of a favourable change in the price of the underlying asset.
The longer the length of time until the expiry of the contract, the greater the time value. There are many factors which affect option premium. These factors affect the premium of the option with varying intensity. Some of these factors are listed here:. Apart from above, other factors like bond yield or interest rate also affect the premium. This is because the money invested by the seller can earn this risk free income in any case and hence while selling option; he has to earn more than this because of higher risk he is taking.
Because the values of option contracts depend on a number of different variables in addition to the value of the underlying asset, they are complex to value. There are many pricing models in use, although all essentially incorporate the concepts of rational pricing , moneyness , option time value and put-call parity.
Post the financial crisis of , the "fair-value" is computed as before, but using the Overnight Index Swap OIS curve for discounting. The OIS is chosen here as it reflects the rate for overnight unsecured lending between banks, and is thus considered a good indicator of the interbank credit markets.
Relatedly, this risk neutral value is then adjusted for the impact of counterparty credit risk via a credit valuation adjustment , or CVA, as well as various other X-Value Adjustments which may also be appended. From Wikipedia, the free encyclopedia. This article does not cite any sources.
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