numerous concepts for help

[ps_ricky_son]

hi all

Grateful if there can be concise explanation on the below concepts.

1. translation invariance P(c+R) = P(R) - c , why is it "mins" and what does the term P(c + R) mean?

2. In the chapter of " The Greeks" , it mentions a stop-loss strategy which is to buy the asset when it is above strike price and sell it when it is below strike price. Why is it so?

3. what is wrong-way risk?

4. what is Law of one price

5. what is law of large numbers

6. sometimes it mentions trading cheap/rich, is the calcualted price > observed price = trading cheap and we should buy the asset?

Thanks a lot.

 

[ami44]

hi ps_ricky_son,

let me try to answer two of your questions

1. translation invariance P(c+R) = P(R) - c , why is it "mins" and what does the term P(c + R) mean?

I assume you talk about coherent risk measures.
R is a random variable representing the loss of a Portfolio. ρ(.) is a risk measure assigning a number to each possible portfolio. That means the higher ρ(R) is the higher is the risk of the portfolio R. The important concept here is, that ρ(.) assigns a number to each random variable (not to their outcomes). As an example you can think of the expected shortfall as a risk measure ρ(.) if that helps.

The translation invariance ρ(R + c) = ρ(R) - c now is a property of ρ(.). It means if you add an amount of c to every possible outcome of the portfolio loss, than the risk ρ(.) will increase by the amount of c either. Here c ist just a number and if you're choosing a negative c the risk will decrease instead of increase.
Or in other words, if you have calculated the Expected Shortfall of a portfolio and now you look at another portfolio with the same loss distribution only that for every outcome we add a additional loss of amount c, then also the Expected Shortfall of that second portfolio will be higher by the amount of c than of the first portfolio.

I don't know what you are referring to with "mins" though.

5. what is law of large numbers

To learn about the mathematical concept i can recommend http://www.math.uah.edu/stat/sample/LLN.html or any other statistics tutorial, but maybe an example might be more helpful:
The expected value of a dice throw is (1+2+3+4+5+6)/6 = 3.5
The law of large numbers says, that if you throw a dice n times, than the arithmetric average of that tries,
(X1 + X2 + .... + Xn) / n
will converge against the expected value 3.5 of a single dice throw, if n gets large.

I hope I could help you a little.

 

[ShaktiRathore]

Hi
1)Translation invariance P(R+c)=P(R)-c where P() is risk function that is risk of cash + position in risky asset P(R+c) is =Risk in risky asset-cash. E.g. Var of say a portfolio with R=$10 in risky asset and c=1$ in cash ,If var of risky asset is $2 then max portfolio can lose is $2-$1=$1 that is 2$ loss can be compensated by 1$ cash position so that net overall loss is just $1. So Var of risky position with cash is reduced by inclusion of cash by amount of the cash,implying net total Var of just $1 instead of $2 of the overall position. To meet $2 loss we just need $1 from risky portfolio and rest is compensated by cash. Var(R)=$2 but when cash position is included Var is $1 so Var(R+c)=1=2-1=Var(R)-c.
2)In stop loss strategy as soon as asset price falls below a strike price the investor wants to prohibit further losses that is wants insurance against further losses as he wants to limit the losses but wants to buy during upside when price rises above strike price therefore in this strategy investor buys insurance from downside at same time capitalising from upside
3) https://forum.bionicturtle.com/threads/gregory-chapter-15-wrong-way-risk.8390/#post-34119
4)Law of one price says assets with identical risk and future cash flows should have the same one price. If market has single discounting function for risk free securities and risk free security A has identical cash flow with risk free security B then both A and B must sell at the same price.
5) law of large numbers as sample size gets larger the sample mean approaches the true population mean.
6)yes when calculated price>observed price that means asset has more intrinsic value as compared to observed price therefore observed price should equal to its true value to prevent arbitrage opportunity. Asset is trading cheap because it is trading below its true value and arbitrage would force price to rise to its true value.
Thanks

 

[ps_ricky_son]

many thanks for your help

BTW, my typo in question 1 -> 1. translation invariance P(c+R) = P(R) - c , why is it "minus" and what does the term P(c + R) mean?

 

[ami44]

many thanks for your help

BTW, my typo in question 1 -> 1. translation invariance P(c+R) = P(R) - c , why is it "minus" and what does the term P(c + R) mean?

Ops I messsed it up.

The problem is, if R is the loss or the profit of the portfolio. The two only differ in the sign.
If R is the loss than adding to that loss increases the risk, if R is profit, than adding to the profit, decreases risk.
For P(c+R) = P(R) - c to be true, R must be the profit and not the loss of the portfolio. I got it wrong in my first post.

 

[ps_ricky_son]

thx again~~