### Simulations Or Mathematics

I can't make this terribly long because I will need to get home soon, but I believe the relative merits of simulation-based and mathematics-based valuation are an important thing to think about.

To put it briefly, I think it's safe to say that mathematics, and closed form solutions/analytic tractability were the primary tool we used to value securities all of, say, 25 (30? 35? More?) years ago. Black Scholes is obviously an analytic equation derived from the heat equation with a few changes of variables having time run backwards. One can just as easily derive analytic equations for binary options. One can also value numerous other types of derivatives in a similar fashion. One simply needs to assume risk neutrality (that is, that the underlying has a risk neutral drift equal to the risk free rate and a volatility assumption) and you're good to go. But derivatives got a lot more complicated. How do you value a path dependent security like an asian option? How about a rainbow option? A range option? Or how about an option on a basis swap which pays a fraction of floating LIBOR and receives floating BMA, the municipal rate? Suddenly one is interleaving multiple stochastic processes in contorted ways, making it more and more difficult to value things with traditional mathematics.

Enter simulations. As derivatives were getting increasingly complicated, computers were getting increasingly powerful. Raw computing power isn't elegant, but it can surely get the job done. Rather than spend days or weeks searching for the proper way to value a CDS with a variable floating notional dependent on the level of interest rates, one can simply make an assumption on the laws governing its motion (as was done mathematically with the assumption of risk neutrality!), and simulate that stochastic process over and over and over again with a good random number generator. The simulated average terminal payoff is ones best guess for the value of that security. As the number of simulations, this should indeed converge to the theoretical value, assuming that ones assumptions on motion were correct. Furthermore with variance reduction methods, one can decrease the number of simulations necessary to converge to a solution that one is satisfied with; a solution with a standard error below some threshold amount (ie. .5% of the terminal value). Simulations do more than allow you to remain ignorant of mathematics while still getting approximately correct solutions, however. They also give you more flexibility regarding the laws of motion the underlying must follow. Stocks tend to jump up and down randomly at different points? Well, just add in jumps of random magnitude at random times (ie. jumps following a Poisson process with magnitudes that are Gaussian centered around some mean jump amount). Suddenly your model seems oh so much more accurate.

However the flexibility of simulations regarding laws of motion like those stated above are only half of the pie.

1) We cannot live for 1,000,000 years. We cannot possibly enter into 1,000,000 variable notional CDS contracts right now, so even if we assume the correct law of motion, hopefully we can see that the theoretically correct expected terminal value will in all likelihood diverge dramatically from realized terminal value. Over shorter time horizons, other factors become increasingly important. In fact it could be argued that they are so important that the simulated value is useless, to some degree. One must remember liquidity. As I've said before (see "A Refocusing On Liquidity Risk"), a security is worth the cost of hedging that security's risk. As a financial institution issuing a financial derivative, my job is not to take on positional risk. I want to avoid making directional bets on individual companies or asset classes. I am a business, and I want to make money regardless of what happens to the underlying stock. Therefore to me, the value of a derivative security is most definitely proportional to the basis risk I expect to incur while hedging that security. It is not (or should not) be a function of my expectation of what will happen to the underlying stock. Said differently, banks are financial intermediaries facilitating the transfer of risk from those who want to avoid risk to those who want to expose themselves to risk. To do so, I create financial derivatives which have very tailored risk profiles so that individuals or financial vehicles can expose themselves to specific forms of risk while remaining unexposed to others. Equities are an aggregated, somewhat clumsy way of exposing oneself to risk, because equities themselves represent huge multi-dimensional clusters of risk. So I create financial derivatives; I attempt to sell them to one party and then hedge off my risk by synthetically buying the same security somehow (or vice versa). Other people may drive the value of that security up or down, but when it comes down to it, the value of that security is then, once again, a function of the basis risk between my hedge and my short financial derivative.

(As a side note, deep value investors then may be able to successfully make money, year in and year out, by identifying fundamentally undervalued companies whose financial situation subsequently changes for the better. These investors are (hopefully) able to ride out the mark to market gains and losses which they may incur over shorter time intervals, and are thus willing to expose themselves more to mark to market risk. The reason is because they are able to carry that risk (hopefully) without blowing up, under a reasonable set of assumptions. Deep value investors typically play with equity because derivatives are in some sense more of a zero sum game, and their values are simply derivatives of that of the underlying, making an adjustment for liquidity. Not only do you get killed on the bid ask, but seriously, if you want to make a directional bet, wouldn't it make sense to take a position in the underlying?)

So I suppose this is something of a paradigm shift. One must make a fundamental choice between establishing a position and taking on position risk, and exposing yourself to basis risk. They are quite different, and it seems that in some ways simulations may be left a bit lacking. Mathematics may be lacking as well, of course :)

-Danny

Categories: theoretical_quant portfolio_management theoretical_intersections

To put it briefly, I think it's safe to say that mathematics, and closed form solutions/analytic tractability were the primary tool we used to value securities all of, say, 25 (30? 35? More?) years ago. Black Scholes is obviously an analytic equation derived from the heat equation with a few changes of variables having time run backwards. One can just as easily derive analytic equations for binary options. One can also value numerous other types of derivatives in a similar fashion. One simply needs to assume risk neutrality (that is, that the underlying has a risk neutral drift equal to the risk free rate and a volatility assumption) and you're good to go. But derivatives got a lot more complicated. How do you value a path dependent security like an asian option? How about a rainbow option? A range option? Or how about an option on a basis swap which pays a fraction of floating LIBOR and receives floating BMA, the municipal rate? Suddenly one is interleaving multiple stochastic processes in contorted ways, making it more and more difficult to value things with traditional mathematics.

Enter simulations. As derivatives were getting increasingly complicated, computers were getting increasingly powerful. Raw computing power isn't elegant, but it can surely get the job done. Rather than spend days or weeks searching for the proper way to value a CDS with a variable floating notional dependent on the level of interest rates, one can simply make an assumption on the laws governing its motion (as was done mathematically with the assumption of risk neutrality!), and simulate that stochastic process over and over and over again with a good random number generator. The simulated average terminal payoff is ones best guess for the value of that security. As the number of simulations, this should indeed converge to the theoretical value, assuming that ones assumptions on motion were correct. Furthermore with variance reduction methods, one can decrease the number of simulations necessary to converge to a solution that one is satisfied with; a solution with a standard error below some threshold amount (ie. .5% of the terminal value). Simulations do more than allow you to remain ignorant of mathematics while still getting approximately correct solutions, however. They also give you more flexibility regarding the laws of motion the underlying must follow. Stocks tend to jump up and down randomly at different points? Well, just add in jumps of random magnitude at random times (ie. jumps following a Poisson process with magnitudes that are Gaussian centered around some mean jump amount). Suddenly your model seems oh so much more accurate.

However the flexibility of simulations regarding laws of motion like those stated above are only half of the pie.

1) We cannot live for 1,000,000 years. We cannot possibly enter into 1,000,000 variable notional CDS contracts right now, so even if we assume the correct law of motion, hopefully we can see that the theoretically correct expected terminal value will in all likelihood diverge dramatically from realized terminal value. Over shorter time horizons, other factors become increasingly important. In fact it could be argued that they are so important that the simulated value is useless, to some degree. One must remember liquidity. As I've said before (see "A Refocusing On Liquidity Risk"), a security is worth the cost of hedging that security's risk. As a financial institution issuing a financial derivative, my job is not to take on positional risk. I want to avoid making directional bets on individual companies or asset classes. I am a business, and I want to make money regardless of what happens to the underlying stock. Therefore to me, the value of a derivative security is most definitely proportional to the basis risk I expect to incur while hedging that security. It is not (or should not) be a function of my expectation of what will happen to the underlying stock. Said differently, banks are financial intermediaries facilitating the transfer of risk from those who want to avoid risk to those who want to expose themselves to risk. To do so, I create financial derivatives which have very tailored risk profiles so that individuals or financial vehicles can expose themselves to specific forms of risk while remaining unexposed to others. Equities are an aggregated, somewhat clumsy way of exposing oneself to risk, because equities themselves represent huge multi-dimensional clusters of risk. So I create financial derivatives; I attempt to sell them to one party and then hedge off my risk by synthetically buying the same security somehow (or vice versa). Other people may drive the value of that security up or down, but when it comes down to it, the value of that security is then, once again, a function of the basis risk between my hedge and my short financial derivative.

(As a side note, deep value investors then may be able to successfully make money, year in and year out, by identifying fundamentally undervalued companies whose financial situation subsequently changes for the better. These investors are (hopefully) able to ride out the mark to market gains and losses which they may incur over shorter time intervals, and are thus willing to expose themselves more to mark to market risk. The reason is because they are able to carry that risk (hopefully) without blowing up, under a reasonable set of assumptions. Deep value investors typically play with equity because derivatives are in some sense more of a zero sum game, and their values are simply derivatives of that of the underlying, making an adjustment for liquidity. Not only do you get killed on the bid ask, but seriously, if you want to make a directional bet, wouldn't it make sense to take a position in the underlying?)

So I suppose this is something of a paradigm shift. One must make a fundamental choice between establishing a position and taking on position risk, and exposing yourself to basis risk. They are quite different, and it seems that in some ways simulations may be left a bit lacking. Mathematics may be lacking as well, of course :)

-Danny

Categories: theoretical_quant portfolio_management theoretical_intersections

## 1 Comments:

Cool, finally a blog written by somebody really smart!

By Gebeleizis, at 11:31 AM

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