Risks and Vulnerabilities

Anti-Fragility Study Group

• The book antivulnerability was discussed

• How do you define antifragility?”

• Organize the knowledge on which the discussion is based

  • How to quantify the size of [risk
    • standard deviation
    • $\sqrt{\mathbb{E}[(X-\mu {)}^2]}$
    • $\sqrt{\sum (x-\mu {)}^{2}P(X=x)}$
    • 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 -$700million-$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,withastandarddeviationofabout$160,000.
    • reverse lottery
      • A gamble that gives you $300buttakes$700 million with a probability of 1/20 million.
      • This, of course, has the same degree of risk.
    • downside risk
  • $\sqrt{\mathbb{E}[(X-\mu {)}^2]1_{\{X\le \mu \}}}$
  • $\sqrt{\sum (X-\mu {)}^2{1}_{\{X\le \mu \}}}$
    • Incorrect: ${\mathbb{E}}_{\{X\le \mu \}}[(X-\mu {)}^2$]
      • 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.
    • Synonyms: upside risk.

• Concept of BETA

  • Some random variable X and a random variable Y
  • $Y=\alpha +\beta X+ϵ$
  • 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.
    • So-called “leverage”
  • Is that calculation really the right one? I’m beginning to wonder.
  • [Long Term Capital Management - Wikipedia https://ja.wikipedia.org/wiki/%E3%83%AD%E3%83%B3%E3%82%B0%E3%82%BF%E3%83%BC%E3%83%A0%E3%83%BB%E3%82% AD%E3%83%A3%E3%83%94%E3%82%BF%E3%83%AB%E3%83%BB%E3%83%9E%E3%83%8D%E3%82%B8%E3%83%A1%E3%83%B3%E3%83%88]
    • 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?
    • The higher the volatility, the lower the risk premium.
    • → 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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