Home Business & Finance Frequently Asked Questions in Quantitative Finance

# What is Meant by the 'Value' of a Contract?

Value usually means the theoretical cost of building up a new contract from simpler products, such as replicating an option by dynamically buying and selling stock.

Example

Wheels cost \$10 each. A soapbox is \$20. How much is a go-cart? The value is \$60.

To many people the value of a contract is what they see on a screen or comes out of their pricing software. Matters are actually somewhat more subtle than this. Let's work with the above go-cart example.

To the quant the value of the go-cart is simply \$60, the cost of the soapbox and four wheels, ignoring nails and suchlike, and certainly ignoring the cost of manpower involved in building it.

Are you going to sell the go-cart for \$60? I don't think so. You'd like to make a profit, so you sell it for \$80. That is the price of the go-cart.

Why did someone buy it from you for \$80? Clearly the \$80 must be seen by them as being a reasonable amount to pay. Perhaps they are going to enter a go-carting competition with a first prize of \$200. Without the go-cart they can't enter, and they can't win the \$200. The possibility of winning the prize money means that the go-cart is worth more to them than the \$80. Maybe they would have gone as high as \$100.

This simple example illustrates the subtlety of the whole valuation/pricing process. In many ways options are like go-carts and valuable insight can be gained by thinking on this more basic level.

The quant rarely thinks like the above. To him value and price are the same, the two words often used interchangeably. And the concept of worth does not crop up.

When a quant has to value an exotic contract he looks to the exchange-traded vanillas to give him some insight into what volatility to use. This is calibration. A vanilla trades at \$10, say. That is the price. The quant then backs out from a Black-Scholes valuation formula the market's implied volatility. By so doing he is assuming that price and value are identical.

Related to this topic is the question of whether a mathematical model explains or describes a phenomenon. The equations of fluid mechanics, for example, do both. They are based on conservation of mass and momentum, two very sound physical principles. Contrast this with the models for derivatives.

Prices are dictated in practice by supply and demand. Contracts that are in demand, such as out-of-the-money puts for downside protection, are relatively expensive. This is the explanation for prices. Yet the mathematical models we use for pricing have no mention of supply or demand. They are based on random walks for the underlying with an unobservable volatility parameter, and the assumption of no arbitrage. The models try to describe how the prices ought to behave given a volatility. But as we know from data, if we plug in our own forecast of future volatility into the option-pricing formulae we will get values that disagree with the market prices. Either our forecast is wrong and the market knows better, or the model is incorrect, or the market is incorrect. Common-sense says all three are to blame. Whenever you calibrate your model by backing out volatility from supply-demand driven prices using a valuation formula you are mixing apples and oranges.

To some extent what the quant is trying to do is the same as the go-cart builder. The big difference is that the go-cart builder does not need a dynamic model for the prices of wheels and soapboxes, his is a static calculation. One go-cart equals one soapbox plus four wheels. It is rarely so simple for the quant. His calculations are inevitably dynamic, his hedge changes as the stock price and time change. It would be like a go-cart for which you had to keep buying extra wheels during the race, not knowing what the price of wheels would be before you bought them. This is where the mathematical models come in, and errors, confusion, and opportunities appear.

And worth? That is a more subjective concept. Quantifying it might require a utility approach. As Oscar Wilde said ''A cynic is a man who knows the price of everything but the value of nothing.''

References and Further Reading

Wilde, O 2003 The Complete Works of Oscar Wilde. Harper Perennial

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