The Black-Scholes Formula - Tim Worrall

Selling your Covered Call - Thoughts on How to Select Your Strike and Expiration

Congratulations! You are a bag holder of company XYZ which was thought to be the best penny stock ever. Instead of feeling sorry, you consider selling covered calls to help reduce your cost basis - and eventually get out of your bags with minimal loss or even a profit!
First - let's review the call option contract. The holder of the call option contract has the right but not the obligation to purchase 100 shares of XYZ at the strike price per share. This contract has an expiration date. We assume American style option contracts which means that the option can be exercised at any point prior to expiration. Thus, there are three parameters to the option contract - the strike price, the expiration date and the premium - which represents the price per share of the contract.
The holder of the call option contract is the person that buys the option. The writer of the contract is the seller. The buyer (or holder) pays the premium. The seller (or writer) collects the premium.
As an XYZ bag holder, the covered call may help. By writing a call contract against your XYZ shares, you can collect premium to reduce your investment cost in XYZ - reducing your average cost per share. For every 100 shares of XYZ, you can write 1 call contract. Notice that that by selling the contract, you do not control if the call is exercised - only the holder of the contract can exercise it.
There are several online descriptions about the covered call strategy. Here is an example that might be useful to review Covered Call Description
The general guidance is to select the call strike at the price in which you would be happy selling your shares. However, the context of most online resources on the covered call strategy assume that you either just purchased the shares at market value or your average cost is below the market price. In the case as a bag holder, your average cost is most likely over - if not significantly over - the current market price. This situation simply means that you have a little work to reduce your average before you are ready to have your bags called away. For example, you would not want to have your strike set at $2.50 when your average is above that value as this would guarantee a net loss. (However, if you are simply trying to rid your bags and your average is slightly above the strike, then you might consider it as the strike price).
One more abstract concept before getting to what you want to know. The following link shows the Profit/Loss Diagram for Covered Call Conceptually, the blue line shows the profit/loss value of your long stock position. The line crosses the x-axis at your average cost, i.e the break-even point for the long stock position. The green/red hockey stick is the profit (green) or loss (red) of the covered call position (100 long stock + 1 short call option). The profit has a maximum value at the strike price. This plateau is due to the fact that you only receive the agreed upon strike price per share when the call option is exercised. Below the strike, the profit decreases along the unit slope line until the value becomes negative. It is a misnomer to say that the covered call is at 'loss' since it is really the long stock that has decreased in value - but it is not loss (yet). Note that the break-even point marked in the plot is simply the reduced averaged cost from the collected premium selling the covered call.
As a bag holder, it will be a two-stage process: (1) reduce the average cost (2) get rid of bags.
Okay let's talk selecting strike and expiration. You must jointly select these two parameters. Far OTM strikes will collect less premium where the premium will increase as you move the strike closer to the share price. Shorter DTE will also collect less premium where the premium will increase as you increase the DTE.
It is easier to describe stage 2 "get rid of bags" first. Let us pretend that our hypothetical bag of 100 XYZ shares cost us $5.15/share. The current XYZ market price is $3/share - our hole is $2.15/share that we need to dig out. Finally, assume the following option chain (all hypothetical):
DTE Strike Premium Intrinsic Value Time Value
20 $2.5 $0.60 $0.50 $0.10
20 $5.0 $0.25 $0 $0.25
20 $7.5 $0.05 $0 $0.05
50 $2.5 $0.80 $0.50 $0.30
50 $5.0 $0.40 $0 $0.40
50 $7.5 $0.20 $0 $0.20
110 $2.5 $0.95 $0.50 $0.45
110 $5.0 $0.50 $0 $0.50
110 $7.5 $0.25 $0 $0.25
Purely made up the numbers, but the table illustrates the notional behavior of an option chain. The option value (premium) is the intrinsic value plus the time value. Only the $2.5 strike has intrinsic value since the share price is $3 (which is greater than $2.5). Notice that intrinsic value cannot be negative. The rest of the premium is the time value of the option which is essentially the monetary bet associated with the probability that the share price will exceed the strike at expiration.
According to the table, we could collect the most premium by selling the 110 DTE $2.5 call for $0.95. However, there is a couple problems with that option contract. We are sitting with bags at $5.15/share and receiving $0.95 will only reduce our average to $4.20/share. On expiration, if still above $2.5, then we are assigned, shares called away and we receive $2.50/share or a loss of $170 - not good.
Well, then how about the $5 strike at 110 DTE for $0.50? This reduces us to $4.65/share which is under the $5 strike so we would make a profit of $35! This is true - however 110 days is a long time to make $35. You might say that is fine you just want to get the bags gone don't care. Well maybe consider a shorter DTE - even the 20 DTE or 50 DTE would collect premium that reduces your average below $5. This would allow you to react to any stock movement that occurs in the near-term.
Consider person A sells the 110 DTE $5 call and person B sells the 50 DTE $5 call. Suppose that the XYZ stock increases to $4.95/share in 50 days then goes to $8 in the next 30 days then drops to $3 after another 30 days. This timeline goes 110 days and person A had to watch the price go up and fall back to the same spot with XYZ stock at $3/share. Granted the premium collected reduced the average but stilling hold the bags. Person B on the other hand has the call expire worthless when XYZ is at $4.95/share. A decision can be made - sell immediately, sell another $5 call or sell a $7.5 call. Suppose the $7.5 call is sold with 30 DTE collecting some premium, then - jackpot - the shares are called away when XYZ is trading at $8/share! Of course, no one can predict the future, but the shorter DTE enables more decision points.
The takeaway for the second step in the 2-stage approach is that you need to select your profit target to help guide your strike selection. In this example, are you happy with the XYZ shares called away at $5/share or do you want $7.5/share? What is your opinion on the stock price trajectory? When do you foresee decision points? This will help determine the strike/expiration that matches your thoughts. Note: studies have shown that actively managing your position results in better performance than simply waiting for expiration, so you can adjust the position if your assessment on the movement is incorrect.
Let's circle back to the first step "reduce the average cost". What if your average cost of your 100 shares of XYZ is $8/share? Clearly, all of the strikes in our example option chain above is "bad" to a certain extent since we would stand to lose a lot of money if the option contract is exercised. However, by describing the second step, we know the objective for this first step is to reduce our average such that we can profit from the strikes. How do we achieve this objective?
It is somewhat the same process as previously described, but you need to do your homework a little more diligently. What is your forecast on the stock movement? Since $7.5 is the closest strike to your average, when do you expect XYZ to rise from $3/share to $7.5/share? Without PR, you might say never. With some PR then maybe 50/50 chance - if so, then what is the outlook for PR? What do you think the chances of going to $5/share where you could collect more premium?
Suppose that a few XYZ bag holders (all with a $8/share cost) discuss there outlook of the XYZ stock price in the next 120 days:
Person 10 days 20 days 30 days 40 days 50 days 100 days 120 days
A $3 $3 $3 $3 $3 $4 $4
B $4 $4 $5 $6 $7 $12 $14
C $7 $7 $7 $7 $7 $7 $7
Person A does not seem to think much price movement will occur. This person might sell the $5 call with either 20 DTE or 50 DTE. Then upon expiration, sell another $5 call for another 20-50 DTE. Person A could keep repeating this until the average is reduced enough to move onto step-2. Of course, this approach is risky if the Person A price forecast is incorrect and the stock price goes up - which might result in assignment too soon.
Person B appears to be the most bullish of the group. This person might sell the $5 call with 20 DTE then upon expiration sell the $7.5 call. After expiration, Person B might decide to leave the shares uncovered because her homework says XYZ is going to explode and she wants to capture those gains!
Person C believes that there will be a step increase in 10 days maybe due to major PR event. This person will not have the chance to reduce the average in time to sell quickly, so first he sells a $7.5 call with 20 DTE to chip at the average. At expiration, Person C would continue to sell $7.5 calls until the average at the point where he can move onto the "get rid of bags" step.
In all causes, each person must form an opinion on the XYZ price movement. Of course, the prediction will be wrong at some level (otherwise they wouldn't be bag holders!).
The takeaway for the first step in the 2-stage approach is that you need to do your homework to better forecast the price movement to identify the correct strikes to bring down your average. The quality of the homework and the risk that you are willing to take will dedicate the speed at which you can reduce your average.
Note that if you are unfortunate to have an extremely high average per share, then you might need to consider doing the good old buy-more-shares-to-average-down. This will be the fastest way to reduce your average. If you cannot invest more money, then the approach above will still work, but it will require much more patience. Remember there is no free lunch!
Advanced note: there is another method to reduce your (high) average per share - selling cash secured puts. It is the "put version" of a cover call. Suppose that you sell a XYZ $2.5 put contract for $0.50 with 60 DTE. You collect $50 from the premium of the contract. This money is immediately in your bank and reduces your investment cost. But what did you sell? If XYZ is trading below $2.50, then you will be assigned 100 shares of XYZ at $2.50/share or $250. You own more shares, but at a price which will reduce your average further. Being cash secured, your brokerage will reserve $250 from your account when you sell the contract. In essence, you reduce your buying power by $250 and conditionally purchase the shares - you do not have them until assignment. If XYZ is greater than the strike at expiration, then your broker gives back $250 cash / buying power and you keep the premium.

Early assignment - one concern is the chance of early assignment. The American style option contract allows the holder the opportunity to exercise the contract at any time prior to expiration. Early assignment almost never occurs. There are special cases that typically deal with dividends but most penny stocks are not in the position to hand out dividends. Aside from that, the holder would be throwing away option time value by early exercise. It possibly can handle - probably won't - it actually would be a benefit when selling covered calls as you would receive your profit more quickly!


This post has probably gone too long! I will stop and let's discuss this matter. I will add follow-on material with some of the following topics which factors into this discussion:
Open to other suggestions. I'm sure there are some typos and unclear statements - I will edit as needed!
\I'm not a financial advisor. Simply helping to 'coach' people through the process. You are responsible for your decisions. Do not execute a trade that you do not understand. Ask questions if needed!**
submitted by x05595113 to pennystockoptions [link] [comments]

ELI5 IVx on TastyWorks?

Hey all,
Still getting used to working on TT and had a question regarding the IVx number listed next to each set of contracts. I've read the TastyWorks definitely of IVx but was hoping somebody could ELI5 it for me.
Using AMZN as an example:
On Tastytrade it shows an IVx for 2/1 Options Contracts of: 51.3%(+/-115.54)
On Schwab (my other broker) they break down Imp Vol. on a strike by strike basis for the contracts expiring 2/1 and each strike has an Imp Vol of <50.
Can anyone explain the difference and how I can use it as I become more familiar with how options pricing drives IV?
submitted by Xayde26 to wallstreetbets [link] [comments]

How to really, truly calculate expected move?

Hi everybody. Hopefully this post doesn't sound too rant-y but I'm pretty frustrated by the amount of info out there that I'm not able to pick up on. There just seems to be a million ways to do calculated expected move. Here's what I've gathered so far.
There seems to be two general methods:
First Method: IV-Based
where P = price, IV = annualized implied volatility, DTE = days to expiration [0]
This means that there is a 68% probability that the stock in question will be between -1 and +1 sigma at the date of expiration, a 95% probability between -2 and +2, and a 99% probability between -3 and +3.
Sometimes 250-252 is used instead of 365, which seems to be the case when DTE refers to market days until expiration. Is that correct?
There are a number of ways to calculate IV. I would appreciate it if somebody could elaborate on which might be best and the differences between them:
  1. ThinkOrSwim uses the Bjerksund-Stensland Model [1] - I assume this is the "annualized" implied volatility aforementioned, because it is an IV value assigned to the stock as a whole ... what does that mean? I thought IV values were only calculated for a specific option contract??
    1. As an aside, ToS in particular confuses me because none of the IVs seem to correlate - Exhibit A
  2. I thought I might look into how VIX was priced off of SPY [2], as an analog, and use it as a basis for finding IV for any other stock as a whole. I don't know where they got their formula from
  3. Backsolve for IV using Black-Scholes [3]. This would only gives one value for IV, which I think only applies to that specific option contract and not to the stock as a whole??
  4. Some websites say to use the IV given that is closest to the desired time period [4] - of course I have no idea how the IV is calculated in the first place (Bjerksund-Stensland again? Black-Scholes?) What's the difference between using the IV of a weekly or a yearly option?
  5. Brenner and Subrahmanyam [5] - understood that this seems to be just an approximation. Should I be looking at formulas from 1988, however?
A very big question of mine is why there is an implied volatility for the stock as a whole and an implied volatility for every other options contract. I can kind of understand it both ways - why should a later-expiry contract have the same IV as an earlier-expiry contract? On the other hand, why should they be different? Why isn't there just one IV for the stock as a whole?

Second Method: Straddle-Based
My understanding is that this is more used for binary events like earnings, but in general I've found two methods:
I have no idea where [5] comes from and I can sort of understand 6 but not really.

In the end, I'm just trying to be as accurate as possible. Is there a best, preferred method to calculating the expected move of a stock in a given timeframe? Is there a best, preferred method to calculating IV (I'm inclined to go with ToS's model simply because they're large and trusted). Is there some Python library out there that already does this? For a retail trader like me, does it even matter??
Any help is appreciated. Thanks!
submitted by hatitat to options [link] [comments]

Looking for some feedback on a University Project

Hi everyone,
I've been working on a project for my Bachelor Thesis in Finance with Python for quite a while now and I'd love to have some feedback from you.
The project focuses on Option Pricing using the Black Scholes Model for Plain Vanilla and Binary Options. It allows the user to perform a series of tasks like computing and plotting greeks, option payoffs and the implied volatility surface and skew.
The script requires Chrome and quite a few modules to properly work, and the code is macOS native, meaning that it may not work on other operating systems (sorry for that).
My go-to IDE is PyCharm, but I guess any other IDE will work fine.
Here you can find the link to the GitHub repository where the project is located.
I will leave also some links to resources about all the theory behind the computations I do in the project for the ones that are not familiar with this topic.
Options Theory)
Black-Scholes Model
Binary Options
Greeks)
Options Strategies
Implied Volatility
Volatility Smile (Skew)

If you have any question let me know.
Thank you!
submitted by rcrmlt to learnpython [link] [comments]

Research papers I'm reading this month

Hi all, was doing searching for some research papers like I do every few months, and decided I'd throw them up here if anyone is interested in them.
Most of these link directly to pdfs (view, not instant-download).
bolded = you should read them
If anyone else reads these, I'm sure lots of the guys here would appreciate a quick review, summary points, or just your thoughts on any of them.
  1. Forecasting Volatility in Financial Markets: a Review (60 pages)
  2. Option Strategies: Good Deals and Margin Calls (40 pages)
  3. Option trading strategies based on semiparametric implied volatility surface prediction (30 pages)
  4. Short Term Variations and Long-term Dynamics in Commodity Prices (20 pages)
  5. Success and failure of technical trading strategies in the cocoa futures market (40 pages)
  6. The Information Content of the S&P 500 Index and VIX Options on the Dynamics of the S&P 500 Index (45 pages)
  7. The Performance of Model Based Option Trading Strategies (25 pages)
  8. evidence on the efficiency of index option markets (15 pages)
  9. OPTIONS EVALUATION - BLACK-SCHOLES MODEL VS. BINOMIAL OPTIONS PRICING MODEL (10 pages)
  10. ECB: risk, uncertainty, and monetary policy (40 pages)
  11. TIMING STRATEGY PERFORMANCE IN THE CRUDE OIL FUTURES MARKET (30 pages)
  12. An Anatomy of Futures Returns: Risk Premiums and Trading Strategies (40 pages)
  13. Roll strategy efficiency in commodity futures markets (40 pages)
  14. Spread trading strategies in the crude oil futures markets (35 pages)
  15. Commodity Strategies Based on Momentum, Term Structure and Idiosyncratic Volatility (20 pages)
  16. AN EXAMINATION OF MOMENTUM STRATEGIES IN COMMODITY FUTURES MARKETS (30 pages)
  17. understanding crude oil prices (45 pages)
  18. BONUS BOOK: The Bond and Money Markets: Strategy, Trading, Analysis (1150 pages): a comprehensive textbook on bonds, interest-rate derivatives, money markets, credit derivatives, yield curve analysis, structured products, CDOs
submitted by ObviousTwist to thewallstreet [link] [comments]

Can she squirt? The tale of Taylor Swift and her vibrator (I meant to post this yesterday)

This is how messed up my mind is (and why the title works):
Vibrator -> "Good Vibrations" -> The Beach Boys -> "Smile" Album -> Volatility Smile
Why Taylor Swift? She is innocent and naive just like most retail.
Now that we have that out of the way let's talk options....
So one of things that came out of the 87' crash was what we call the volatility smile which is simply the skew of puts and calls relative to their moniness. So what the fuck does that mean? Prior to the crash a modified black scholes model was used to price options; the problem is that black scholes assumes that volatility is constant - it isn't. When the black swan event occurred vol shot up and everyone lost their asses as both puts and calls were not skewed and thus always under priced.
The smile name is derived because the volatility surface resembles a smile or smirk where OTM puts and calls are skewed higher than ATM puts and calls. Here is an example:
http://www.optionsideacentral.com/wp-content/uploads/2013/10/AAPL-Skew.gif
You can also see the vol skewness on think or swim or ib under the "implied vol" section and see how as you get closer and closer to the ATM vol contracts and then goes up again.
the "smirk" (or the look I give a chick after I am caught looking at here clevage)
You will see that most smiles are askew to the downside (google volatility smile mother fucker can't do everything for you) - why is this? Isn't it wrong? Max loss to the downside is zero and max loss to the upside is infinity so why is it like this?
Volatility on the downside skew is greater than the upside because:
  1. They are insurance of a black swan event
  2. If the skew is noticeably higher then they are getting pounded by buyers
  3. Puts are more expensive to trade
So what brought this lesson on? This question:
What's happening with options prices on SPY? looking at sep 26 calls $202 strike. options price was trading higher today when SPY was 200.60 than it is now, when it's 201.21. will things normalize after this volatility? I'm red on my calls when i feel like i should be well in the money
Skew is the reason - with a binary event volatility is raised significantly more on the wings (further OTM options) then the ATM (regardless the whole vol surface moves) so when the market doesn't move the skew flattens taking out the higher vol component even though the stock delta may be rising.
saavy???
submitted by midgetginger to options [link] [comments]

TRADING OPTIONS With Using Best Binary Options Strategy 2017 online stock trading BINARY OPTIONS TRADING - Profit with Binary Options in 2019 Exotic options: binary (aka, digital) option (FRM T3-44) Pricing Options with Mathematical Models  CaltechX on edX  Course About Video

On Black-Scholes Equation, Black-Scholes Formula and Binary Option Price Chi Gao 12/15/2013 Abstract: I. Black-Scholes Equation is derived using two methods: (1) risk-neutral measure; (2) - hedge. II. The Black-Scholes Formula (the price of European call option is calculated) is calculated Definition of the Option Pricing Model: The Option Pricing Model is a formula that is used to determine a fair price for a call or put option based on factors such as underlying stock volatility, days to expiration, and others. The calculation is generally accepted and used on Wall Street and by option traders and has stood the test of time since its publication in 1973. examining digital or binary options which are easy and intuitive to price. We shall show how the Black-Scholes formula can be derived and derive and justify the Black-Scholes-Merton partial di erential equation. Keywords: Black-Scholes formula, Black-Scholers-Merton partial di eren-tial equation, replication, self- nancing portfolio, martingale This is an updated version of my "Black-Scholes Model and Greeks for European Options" indicator, that i previously published. I decided to make this updated version open-source, so people can tweak and improve it. The Black-Scholes model is a mathematical model used for pricing options. From this model you can derive the theoretical fair value of an options contract. History. The Black Scholes pricing model is named after the American economists Fischer Black and Myron Scholes. In 1970 Black, a mathematical physicist, and Scholes, a professor of finance at Stanford University, wrote a paper titled “The Pricing of Options and Corporate Liabilities.”

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TRADING OPTIONS With Using Best Binary Options Strategy 2017

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