Jumping into Black-Scholes I: Motivation

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Have you vaguely heard of Black-Scholes and want a concise explanation? As in, this name reminds you of pricing options and not, for example, a shoal of color deficient fish? Do you know your basics of Probability? Then you have come to the right place. I'll demonstrate the common idea behind pricing an option using one of the simplest of option: The european options. First, let me motivate you.

Let's say you are a farmer and want to sell an asset (possibly a cow) in T number of years from now on a fixed date (known as the maturity date). Your target cow market is offshore, where it's all settled  in a currency foreign to the one you use to buy bread. You know the price changes from day to day, and depending on the price T years from now, it might be worth to keep the cows for milking instead.
Furthermore, often you see the paper and read about the volatility of foreign exchange and the financial sector as a whole. The stress starts to pile on. You drop your morning read of "Seeds and Hot Tractors" for "Financial Times". Questions start to emerge in your countryside head: How can I plan my future budget when I don't know how much my cow sale will pull in?

Fortunately, Black-Scholes comes to the rescue. You find out that you can fix this future sell price (known as the strike K) today, to a particular buyer. Better yet, you can buy the right to sell at the price K, and only exercise it if it's packing moo (worth it, in farmer tongue). How much does this right cost? This right is known as a Plain Vanilla option, and the mystery of how much this option should cost boggled many a mind for many a decade. Black and Scholes come along, and, to the horror of many an economist, use some continuous mathematics with probability to give a simple, transparent model to price this thing. For the farmer, it costs a small percentage of the price K.

Finally you can rest at ease. You know exactly how many bucks you could get for your moos T years from now. You drop Financial Times and it's preposterously pompous highly adjectivised text, and pick up something far more earthy, like "Big Udders". Your attention turns to further specializing yourself in your field (of knowledge and land) and planning the purchase of expensive machinery to up your productivity. Who said Finance was futile and not fertile?

After germinating purpose to the pricing of an option, let's bring in precision. Next, we carefully deduce the Black-Scholes price for this option. Though this model is no longer in use, instead one typically would use the Market price, i.e., aggregated opinion of everybody else, it still serves a purpose. It teaches us how to formally deduce prices of such financial stuff. It is also employed in pricing other complicated financial contracts (look-up Phoenix rainbow basket options), when more technically sophisticated models become intractable, and one must fall back on Black-Scholes framework.