That day
I was walking along the beach at Zandvoort; it was fairly cloudy and gray, but I could still see several seagulls in the sky. As I walked I thought about myself and asked a funny question of God. I asked, Well just where do I stand in the great scheme of things, how do I rate? Just then the sky opened up and a beam of sun shone squarely on a seagull above me and towards the sea, about 120 feet away. The gull was white but as the sun’s ray fell forcefully on his chest a massive, golden array of rippled muscles forming his chest, back, and left wing captured my soul. As his wings gracefully moved, the golden array continued to shine brightly in the newly forming blue sky; and with great freedom and command of his world, he flew off.
And I thought to myself,
so that’s it, I’m no better than a lowly gull.
Pickwick’s Umbrella
In London, half of the days have some rain. The weather forecaster is correct 2/3 of the
time, i.e., the probability that it rains, given
that she has predicted rain, and the
probability that it does not rain, given that she has predicted that it won’t rain, are both
equal to 2/3. When rain is forecast, Mr.
Pickwick takes his umbrella. When rain is not forecast, he takes it with probability 1/3.
Can you find
(a) the probability that Pickwick has no umbrella, given that it rains.
(b) the probability it doesn’t rain, given that he brings his umbrella.
This problem apears in
K. L. Chung, Elementary Probability Theory With Stochastic Processes, 3rd ed. (New York:
Springer-Verlag, 1979), p. 152
NOTE: You may not assume that the probability that she predicts rain is 1/2
or that the probability that she predicted rain given that it rains is 2/3; however, you need that, so prove it first.
HINT: Show that for any two events R & F such that a = P(R|F) = P(~R|~F)
P(F) = 1 - (a +P(R)) ,
1 - 2a
a ≠ ½ and under the further constraints that place the RHS between
0 and 1 inclusive.
When P(R) = ½ a drops out of the RHS and P(F) = ½ .
As we face the worst financial crisis since the Great Depression, we cannot help but ask what role mathematics played in this. Over the past 20 years the practice of finance has been revolutionized by the widespread adoption of mathematical models for pricing ever more exotic derivative securities. Mortgage-backed securities, which triggered the financial collapse, were priced using the Gaussian copula model. What would Albert Einstein have to say about all this? I suppose Einstein would remind us that to build a mathematical model, it is necessary to make assumptions. At best, these assumptions are idealizations of reality. The art of model building is to choose assumptions so that the model is unencumbered by less relevant considerations in order to illuminate more important phenomena. A good model provides insight and a guide to action. A good user of a model understands its limitations. We have seen all this play out in the financial markets.
Replication. The granddaddy of financial models is due to Black, Merton and Scholes. This model (catch up on the Black Scholes Merton - Option Pricing in Continuous -Time) begins with the assumption that a stock price evolves continuously in time and has a constant volatility, a parameter characterizing the risk of the stock. A European call on such a stock is the option to buy a share of the stock at a future “expiration” date in exchange for a payment set at the initial time but paid at expiration. Since the owner of the call has a potential gain at expiration, the call has a positive initial value or price Black and Scholes provided a formula for the price of the European call. Merton provided the replication argument we now use to understand it. Consider an investor who invests in the stock and borrows from a money-market account to finance this. It turns out that if the investor can trade continuously, then she can start with a certain initial capital and trade in such a way that the value of her portfolio at expiration agrees with the payoff of the call. The initial capital that permits this must be the initial price of the call, and it is given by the Black-Scholes formula. The Black-Scholes-Merton analysis contains the insight of pricing the call by replication rather than just computing the expected value of its payoff. The flip side of this is hedging. If one owns the call and uses the negative of the replicating strategy, then one has a hedged position. Any loss in the value of the call will be offset by a gain in the value of the portfolio held by this negative replication, and vice versa. An investment bank performing intermediation among parties must hold assets for a time, and during that time hedging protects the bank against loss. The replication argument involves a delicate interplay between the sensitivity of the call price to movements in the stock price and to the everdecreasing time to expiration, and this balancing act is where the stock volatility matters.
Risk-Neutral Pricing. In the early 1980s, M. Harrison, D. Kreps, and S. Pliska developed a risk-neutral pricing formula that greatly extended the applicability of pricing by replication. They pointed out that once a model is built, one can change the probability measure on the space on which the stock price is defined so that it has mean rate of return equal to the interest rate. This is akin to building a model based on tosses of a fair coin, and then pretending for computational purposes that the coin is biased. Under this socalled risk-neutral measure, both the stock and the money-market account held by the portfolio replicating a call have mean rate of return equal to the interest rate, and so the portfolio itself has this mean rate of return. Therefore, the initial value of the portfolio, which is the Black-Scholes price of the call, can be obtained by discounting the call payoff at the interest rate and taking the expected value under the risk-neutral measure. One can thus build a model with multiple primary assets, change to a risk-neutral measure,
... read on at their website www.analyticsmagazine.com
Recipe for Disaster: The Formula That Killed Wall Street
The formula that felled Wallstreet.
If you'd like more information about the copula,
Estimating VaR using Copula
Roger B. Nelsen, An Introduction to Copulas
mail@duport.com
Learning is tough because,
while a little knowledge that you're missing
can often bridge large gaps, it is not accessible to you - for if it was, you would reach out and grasp it. And no one else can provide it, because no one but you knows exactly what you know and don't know.
This is why we have few friends in our neighborhoods, just acquaintances and strangers.
Here is an easy to understand exposition of game theory and in particular, Nash Eguilibria
