Ready To Risk It All Part II, Continuous Processes

The dog days of summer are fast approaching. Even from my air-conditioned office cubicle, I can feel the temperature outside rising – and my productivity levels decreasing with each additional degree. But I’m looking forward to the heat for more reasons than just excuses to borrow my friend’s pool. As the dog days of summer approach, I literally see more dogs out and about when I go outside. As a dog-deprived graduate student, this is the time of year that I live for. In addition to creeping on the dogs of D.C., I am constantly obsessing over my friend’s dogs. Whenever I see my friends’ puppies, I’m always amazed at the dramatic amount of change that has occurred since I saw them last. It’s a truly incredible phenomenon how quickly puppies seem to grow when you only see them in discrete time intervals! Living with a pup that you interact with on the daily, however, you barely notice the change.

 

I could gush on and on about dogs for pages. But you’re probably wondering by now – what does this have to do with finance?

In a continuous process, values change from moment to moment and the change occurs constantly, even though the change may seem so small you don’t notice it. In other words, values cannot make instantaneous jumps. Puppies constant grow little by little every second, not in drastic jumps between the time intervals that you see them. Stock prices and securities also display continuous process behavior.

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Path of continuous time motion versus S&P 500 Index

Last time, we talked about Myron Scholes, Fisher Black, and their dynamic hedging strategy to price options. There was one practical problem with their formula, however. It assumed that markets were always in equilibrium, that supply is always equal to demand. It is a frequent criticism of mathematical economics that emphasis on the compactness and presentation of models loses the richness that makes these models “real.” In the real world, Black and Scholes calculations for dynamic hedging might be thrown off by fast-moving changes in the stock market. They needed a way to instantly rebalance a portfolio of stocks and options to constantly offset fluctuations, rather than an unmoving equilibrium state.

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Temptation is always lurking

Using the notion of continuous time, the value of the option could be constantly recalculated. Robert Merton used this insight to reframe and reformulate Black and Scholes’ model. The continuous-time model has proved to be a versatile and productive tool in the development of finance. Although mathematically more complex than its discrete-time counterpart, the advantage of the continuous-time framework is that it produces more precise solutions. At any given moment, we can balance our portfolio against changes. Being a dog owner comes with the advantage of constant exposure to the growth of your beloved pet, and protects against the shock of suddenly seeing a pup all grown up.

 

Special thanks for my friend’s dogs featured above for being so ADORABLE.

 

 

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