Long/Short Vega and Option Positions

Why do you get long vega when you buy an option and short vega when you sell an option?

I would have thought that for both buying and selling options the vega would change according to whether the option was ITM or OTM. This would be because as an option participant an increase/decrease in volatility would change the chance of expiring at a financially beneficial position. However this not appear to explain the first statement (which I believe is correct).

Answer

The risk exposures/sensitivities of long and short positions always have different signs. This has to hold since derivatives are zero sum games.

Vega is always positive for a long position in a European plain vanilla option (or any convex payoff in general). This is true even when the option is already in-the-money. As volatility increases, the probability of very positive and very negative returns increases. As the holder of the option, you are protected against moves in one direction but participate in the other.

You can construct a very simple binomial example to illustrate this. Consider a one period setting. You are long a call with strike 90. The current stock price is 100 and there are no rates or dividends.

1. First consider a "low volatility" scenario, where the stock either goes up to 105 or down to 95. The payoff is either 15 or 5 and the initial price is 10 with a time value of 0.


2. Now consider a "high volatility" scenario, where the stock either goes up to 120 or down to 80. They payoff is either 30 or 0 and the initial price is 15 with a time value of 5.

You see that due to the convexity of the payoff, a higher volatility is advantageous even when the option is already in-the-money since losses are limited. European vanilla options derive their time value (ignoring rates, dividends, ...) from the possibility of crossing the strike, no matter whether they are already in-the-money or not.

mathematics - Is there a formula for future value of a growing annuity with yearly payment growth and monthly payments?

My example is saving for college:

• assume a start of 0 balance


• deposits of 200 made monthly, every year they increase by (g) 2% to account for salary increases, first deposit made at the end of the first month


• Interest Rate (r) is constant at 8% (effective rate)


• Goes for (n=15) years

What is the future value?

Even though I can convert the yearly rate into a compounded monthly rate to match the yearly rate, I can't use the "future value of a growing annuity" formula, that assumes timing of growth and payment are the same.

It is acceptable to make it a two or three steps (like use equation 1 to solve for a new value for payment to plug that into equation 2), I am just trying to avoid making calculations for each and every year as I'm doing now.

n(1) = 2486
n(2) = 5222.23

n(15)= 75693

Update
I found my own answer as well below that combines well known formulas to get to the same answer (and I presume, with substitution, would be equivalent to the accepted answer)

Answer

You can calculate it with the formula below, which is produced from a double sum.

P. S. The initial examples are for an annuity due (savings type annuity).

Future value = (r*(-1 + r^y)*(-b^(1 + a) + r^((1 + a)*y))*z)/((-1 + r)*(-b + r^y)) 

where

r = 1 + monthly rate = 1.08^(1/12) = 1.00643

y = months per year = 12
a = years - 1 = 14
b = deposit increase rate + 1 = 1.02
z = initial deposit amount = 200

(r*(-1 + r^y)*(-b^(1 + a) + r^((1 + a)*y))*z)/((-1 + r)*(-b + r^y)) = 76180.4

Mathematica was used to produce the formula from the double sum:

The double sum is produced from the workings below.

Edit

To illustrate the robustness of the formula here is another example with different period parameters: a twice-yearly deposit of 200 for three years, again incrementing annually by 2%, with 8% interest rate.

Running the calculation in four forms produces the same result. This proves the formula's robustness.

r = 1 + six-monthly rate = 1.08^(1/2) = 1.03923
y = periods per year = 2
a = years - 1 = 2
b = deposit increase rate + 1 = 1.02
z = initial deposit amount = 200

(r*(-1 + r^y)*(-b^(1 + a) + r^((1 + a)*y))*z)/((-1 + r)*(-b + r^y)) = 1402.25

2nd Edit

Recalculation for ordinary annuity (loan type), rather than annuity due (savings). - ref. Calculating The Present And Future Value Of Annuities

((-1 + r^y)*(-b^(1 + a) + r^((1 + a)*y))*z)/((-1 + r)*(-b + r^y)) = 1349.32

Create engraving in GIMP/Inkscape

I want to turn a photo in a engraving style picture, like for example this one using GIMP and, if possible, then make an SVG out of it with Inkscape.

Can you recommend any filters/tools/plugins/tutorials for this?

terminology - How broad is the definition of affordance?

While discussing the need for affordances to be consistent, comments between me and JeroenEijkhof indicate we don't have a similar interpretation of the term 'affordance'.

I claimed that something being 'toggleable' (selectable) is a perceived affordance, while JeroenEijkhof draws the line at a perceived affordance of being clickable, selected/unselected only being a result of that.

The sources I found don't really help in knowing where to draw the line.

According to wikipedia:

An affordance is a quality of an object, or an environment, which allows an individual to perform an action. For example, a knob affords twisting, and perhaps pushing, while a cord affords pulling.

A further clarification by Don Norman, who introduced the term affordance to design, doesn't give a final answer either.

How broad is the definition of affordance? Could 'selectable' or 'toggleable' be considered a perceived affordance?

Answer

My understanding is that originally affordances meant the interactions that are physically possible owing to the form of an object. When we say a doorknob two inches in diameter affords single-hand grasping we simply mean it possible to grasp it. If it were two feet in diameter, it would not afford one-hand grasping. From our experiences, we humans learn to recognize affordance by appearance, so this can be used to guide interactions.

The original concept of affordances falls apart when you get to software user interfaces because there are no physical interactions and anything is in principle possible. A user can click anywhere. It’s just a question of if and how something will respond. In the essay you linked to, Norman attempted to clarify and expand the concept of affordances to include “perceived affordances” in addition to the real affordances that are only possible with physical interactions.

Perceive affordances, according to Norman, are “cultural constraints” and “learned conventions.” Underlined colored text “affords” clicking because users have learned that’s what indicates a link. In other words, having perceived affordances is simply being consistent with what everyone else does (and therefore what users have learned). It merely means following tradition and well-known standards. Can something afford toggling? If it looks like how toggling is traditionally distinguished from non-toggling (e.g., a checkbox), then yes. Likewise for selectable.

The concept of perceived affordance lacks rigor. How conventional does a convention have to be to qualify as an affordance? The usefulness of the concept is questionable. Why use the term “affordance” when we already have the term “consistency”? I sometimes find “affordance” useful to refer semi-formally to a metaphor, convention, or standard so ingrained in user habits as they respond to it automatically, almost compulsively. But arguments over precisely what is or isn’t an affordance are irresolvable.

The vagueness of the affordance concept can also be counterproductive. I get the impression the concept has led some UX designers to believe there are certain elemental perceptual properties that inherently communicate interactions to the user, such as gradient shading affords clicking to select, or a list of objects (on a mobile device) affords swiping to scroll. However, if we recognize affordances as synonymous with external consistency, we realize there are no elemental perceptual properties. Instead there is the holistic image and its similarity on multiple dimensions with previous experience. Gradient shading only works reliably on a smallish long rectangularish shape with centered text label, for example.

I would not object to “affordance” being deleted from our professional vocabulary like “user friendly” has been.

equities - Hedging stocks with VIX futures

It seems that VIX futures could be a great hedge for a long-only stock portfolio since they rise when stocks fall. But how many VIX futures should I buy to hedge my portfolio, and which futures expiration should I use?

Answer

VIX measures volatility. It doesn't always go up if stocks go down.

typography - What is this style of text-as-image called?

I'm seeing a trend (especially on pinterest) of these posters where the image and subject is the text. THe text itself isn't really that important either, it's mainly the typefaces used and composition.

Does this style have a name to it? Is it just called text-as-image?

Answer

I'd generally call this kind of graphics typographic artwork or, from the times when these advertising graphics and commercial signage were drawn by hand, lettering artwork. This is especially true for the second example (blue). The first one maybe not so much.

See also: letter art.

There is also a different genre of graphics called 'typographic art'.

correlation - Time-series similarity measures

Suppose I have two time series $X$ and $Y$ of stock prices. How do I measure the "similarity" of $X$ and $Y$?

(I'm being deliberately vague as I don't have a particular application, and I'm curious about different approaches in general. But I guess you can imagine that there's some stock x that I don't want to trade directly, for whatever reason, so I want to find a similar stock y to trade in its place.)

One method is to take a Pearson or Spearman correlation. To avoid problems of spurious correlation (since the price series likely contain trends), I should take these correlations on the differenced or returns series (which should be more stationary).

What are other similarity methods and their pros/cons?