Risks and Vulnerabilities
Anti-Fragility Study Group
How do you define antifragility?"
Organize the knowledge on which the discussion is based
How to quantify the size of [risk
$ \sqrt{\mathbb{E}[(X-\mu)^2]}
$ \sqrt{\sum (x - \mu)^2 P(X=x)}
No change Zero
10,000 is either 0 or 20,000.
Standard deviation 10,000 yen
lottery
Example: 1/20 million chance of 700 million yen
Ignore other minor awards because the math is too tedious.
Expected value 35 yen
1/2 billion chance of -$700 million -$35 plus, -$265 for the rest of the probability.
sqrt(2.45e9 + 70225)
Standard deviation is about 50,000 yen
It's like betting 50,000 becomes 0 or 100,000.
In general, they say the expected value is about $150, with a standard deviation of about $160,000.
reverse lottery
A gamble that gives you $300 but takes $700 million with a probability of 1/20 million.
This, of course, has the same degree of risk.
$ \sqrt { \mathbb{E}[(X-\mu)^{2}]1_{\{X\leq\mu\}} }
$ \sqrt{\sum (X - \mu) ^2 1_{\{X\leq\mu\}}}
Incorrect: $ \mathbb{E}_{\{X\leq\mu\}}[(X-\mu)^{2}]
I was wrong.
In general, we discuss not only $ \mu , but also the downside risk when the point of interest k is defined
The downside risk of lottery tickets is...
The downside risk of betting 50,000 becomes 0 or 100,000...
The downside risk of a reverse lottery is...
The lower you go, the bigger it gets.
Concept of BETA
Some random variable X and a random variable Y
$ Y = \alpha + \beta X + \epsilon
at this time
$ \beta = \frac{Cov(X, Y)}{Var(Y)}
Assume base asset X is a stock.
If you borrow money and buy 10 times the amount of stock, BETA will increase by about 10 times.
Is that calculation really the right one? I'm beginning to wonder.
High investment performance was achieved, but this was due to leverage
It is now being discussed whether it might be better to think about downside with respect to this Variance as well.
→The emergence of the concept of downside beta
The concept of volatility
Standard deviation with fixed time horizon
I thought about the standard deviation "after the winning numbers are announced" in the discussion of lottery risk.
Often the context assumes a change in the option price.
Consider "the distribution of prices after a certain number of days" since prices fluctuate continuously on a daily basis.
Often write $ \sigma .
vega concept
$ \frac{\partial V}{\partial \sigma}
Option price differentiated by volatility
For example, when considering an option transaction that says, "I will give you 10,000 yen if the price in one month's time is within 10% of the current price, but I will take 10,000 yen if it is not," the higher the volatility of the underlying asset, the lower the price of this option.
→ negative vega
stock option
Option to exchange "the right to earn X shares in the future" for a current cash salary of 1,000,000 yen.
If the stock becomes a piece of paper - 1,000,000 yen
If the price of X share is 2 million, +1 million
The higher the volatility, the higher the expected value (because it does not go below 0).
Stock options increase in value.
→ positive vega
Cybozu's Shareholding Association
Option to earn 2X shares of stock at the present time in exchange for X amount of cash at the present time
If the stock price does not change, +X yen
If the stock price goes up, well, I'll be happy.
Losses if the stock price drops below half its original price.
Ignoring the probability of going from the current price to zero, the expected value remains the same.
What are the option prices?
→ negative vega
What is Fragility?
It is a left-side vega
In other words, the same composition as "risk -> downside risk" and "beta -> downside beta" is used for vega.
Reverse lottery example
Risks that are not downside could not distinguish between "mostly gains but very rarely large losses" and "mostly losses but very rarely large gains".
Similarly, can't a non-downside vega handle the very rare "very high volatility conditions" (base asset spikes and crashes) well?
left-side means that the downside is to the left when the probability distribution is drawn with the target value on the X-axis.
What is ROBUST?
Less fragility?
We're talking about distribution shape stability.
What is antifragility?
Antifragility is not the simple opposite of fragility, as we saw in Table 1. Measuring antifragility, on the one hand, consists of the flipside of fragility on the right-hand side, but on the other hand requires a control on the robustness of the probability distribution on the left-hand side. From that aspect, unlike fragility, antifragility cannot be summarized in one single figure but necessitates at least two of them.
Fragility (downside vega) folded to the right and Robustness combined concept
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