Double One-Touch Binary Options Explained

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Price double one-touch and double no-touch binary options using Black-Scholes option pricing model



Price = dbltouchbybls( RateSpec , StockSpec , Settle , Maturity , BarrierSpec , Barrier , Payoff ) calculates double one-touch and double no-touch binary options using Black-Scholes option pricing model.


Price a Double No-Touch Option

Compute the price of a double no-touch option using the following data:

Define the RateSpec using intenvset .

Define the StockSpec using stockspec .

Calculate the price of a double no-touch binary option.

Input Arguments

RateSpec — Interest-rate term structure

Interest-rate term structure (annualized and continuously compounded), specified by the RateSpec obtained from intenvset . For information on the interest-rate specification, see intenvset .

Data Types: struct

StockSpec — Stock specification for underlying asset

Stock specification for the underlying asset, specified by the StockSpec obtained from stockspec .

stockspec handles several types of underlying assets. For example, for physical commodities, the price is StockSpec.Asset , the volatility is StockSpec.Sigma , and the convenience yield is StockSpec.DividendAmounts .

Data Types: struct

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Settle — Settlement or trade date
serial date number | date character vector | datetime object

Settlement or trade date for the double touch option, specified as an NINST -by- 1 matrix using serial date numbers, date character vectors, or datetime objects.

Data Types: double | char | datetime

Maturity — Maturity date
serial date number | date character vector

Maturity date for the double touch option, specified as an NINST -by- 1 vector of serial date numbers or date character vectors.

Data Types: double | char | cell

BarrierSpec — Double barrier option type
cell array of character vectors with values of ‘DOT’ or ‘DNT’ | string array with values of “DOT” or “DNT”

Double barrier option type, specified as an NINST -by- 1 cell array of character vectors or string array with the following values:

‘DOT’ — Double one-touch. The double one-touch option defines two Barrier levels. A double one-touch option provides a Payoff if the underlying asset ever touches either the upper or lower Barrier levels.

‘DNT’ — Double no-touch. The double no-touch option defines two Barrier levels. A double no-touch option provides a Payoff if the underlying asset ever never touches either the upper or lower Barrier levels.

Data Types: char | cell | string

Barrier — Double barrier value

Double barrier value, specified as an NINST -by- 2 matrix of numeric values, where the first column is Upper Barrier(1)(UB) and the second column is Lower Barrier(2)(LB). Barrier(1) must be greater than Barrier(2).

Data Types: double

Payoff — Payoff value

Payoff value, specified as an NINST -by- 1 matrix of numeric values, where each element is a 1 -by- 2 vector in which the first column is Barrier(1)(UB) and the second column is Barrier(2)(LB). Barrier(1) must be greater than Barrier(2).

The payoff value is calculated for the point in time that the Barrier value is reached. The payoff is either cash or nothing. If you specify a double no-touch option using BarrierSpec , the payoff is at the Maturity of the option.

Data Types: double

Output Arguments

Price — Expected prices for double one-touch options

Expected prices for double one-touch options at time 0, returned as an NINST -by- 1 matrix.

More About

Double One-Touch and Double No-Touch Options

Double one-touch options and double no-touch options work the same way as one-touch options, except that there are two barriers.

A double one-touch or double no-touch option provides a payoff if the underlying spot either ever or never touches either the upper or lower Barrier levels. If neither barrier level is breached prior to expiration, the option expires worthless and the trader loses all the premium paid to the broker for setting up the trade. For example, if the current USD/EUR rate is 1.15, and the trader believes that this rate will change significantly over the next 15 days, the trader can use a double one-touch option with barriers at 1.10 and 1.20. The trader can profit if the rate moves beyond either of the two barriers.


[1] Haug, E. The Complete Guide to Option Pricing Formulas. McGraw-Hill Education, 2007.

[2] Wystup, U. FX Options and Structured Products. Wiley Finance, 2007.

5 types of binary options

We will discuss the main types of binary options – the classic put and call, one touch, Double one touch and others. What is the difference between the main types of binary options.

In the 79th issue of, we will continue to discover work with binary options. Recall that the last time we talked about the concept of this trading tool, about forecasting, strategies and types. This time we will expand the knowledge base and go a little deeper into the possibilities that binary options offer us.

Despite the apparent uniformity of trading, the very structure of binary options, similar to a betting transaction, can significantly diversify approaches and strategies. Well, indeed, if we can “bet” that a currency pair will reach a mark in 60 seconds, then why not argue that it will not? In this regard, binary options trading in practice has five main types. We will consider them.

View 1. Binary option “Cash or nothing” / “All or nothing”

This view of Binary Options most popular among traders. At the time of purchase, he has a certain percentage of profit and price. This amount is the volume of the transaction, and it is to her that the trader risks it. Prior to the purchase of “Сash or nothing” binar, it is required to make a prediction in advance whether the selected asset will rise or fall, a Put / Put or Call / Call option is purchased (see the previous article), respectively.

If in the end, at the time of execution, the trader is right, then he receives a predetermined amount of profit, usually 70-85 %% plus the initial cost of the contract. If the forecast did not materialize, then he loses the invested amount in full. Sometimes companies make a refund when they lose in the amount of 10-15%, but this cannot be considered the rule, rather, it is just a bonus to cheer up.

View 2. Binary option “Asset or nothing” / “Asset or nothing”

Binary Option Asset or Nothing practically no different from the above, “All or nothing.” It can also be bought and sold by setting a certain level, above or below which there will be a price at the time of execution, also receive declared profits or lose the value of the contract. The difference is only in determining the profit, which is expressed in the value of the selected asset.

In fact, because profit calculation is automatic, many traders do not even see the difference between the binary options Cash or nothing and Asset or nothing.

View 3. Binary options “One touch” / “One touch”

We mentioned this type of binary options in the previous article, but it will not be harmful to repeat. One touch in execution and trading strategies differs significantly from the previous ones. Having all the same value and profitability, the contract also implies the presence of a certain level of asset price. Knowing him, the trader must predict whether the price reaches this mark in the allotted time period. At the same time, it doesn’t matter at what price the contract closes, the main thing is that asset touched fixed level.

View 4. Binary options “No touch” / “Inviolable”

The opposite in name and meaning for the “One touch” binary option is “No touch”. The only difference is the trader’s forecast of whether the price reaches the level specified in the contract. If he believes that he will not achieve, then you should buy the “No touch” option. If at the time of closing the asset did not touch the levelthen the trader gets the agreed profit. Otherwise, he loses the value of the contract.

View 5. Binary options “Double one touch” and “Double no touch” / “Double one touch” and “Double touch”

These are the most complex, and therefore the least popular types of binary options. In fact, this is a modification of the “One touch” and “No touch” options – the strategy and execution are identical. However, in order to achieve profit, two fixed levels are set, to reach or not to touch which the asset must until the expiration of the exercise time of the option.

Based on your own preferences, offers from binary options brokers., as well as market conditions, it is more profitable to trade different types of binary options. However, most traders prefer “Cash or nothing” as the most affordable and easy to understand.

Other binary options articles
FORTRADER magazine experts

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Pricing the Double-no-touch option

In this article by Balázs Márkus, coauthor of the book Mastering R for Quantitative Finance, you will learn about pricing and life of Double-no-touch (DNT) option.

(For more resources related to this topic, see here.)

A Double-no-touch (DNT) option is a binary option that pays a fixed amount of cash at expiry. Unfortunately, the fExoticOptions package does not contain a formula for this option at present. We will show two different ways to price DNTs that incorporate two different pricing approaches. In this section, we will call the function dnt1, and for the second approach, we will use dnt2 as the name for the function.

Hui (1996) showed how a one-touch double barrier binary option can be priced. In his terminology, “one-touch” means that a single trade is enough to trigger the knock-out event, and “double barrier” binary means that there are two barriers and this is a binary option. We call this DNT as it is commonly used on the FX markets. This is a good example for the fact that many popular exotic options are running under more than one name. In Haug (2007a), the Hui-formula is already translated into the generalized framework. S, r, b, s, and T have the same meaning. K means the payout (dollar amount) while L and U are the lower and upper barriers.

Implementing the Hui (1996) function to R starts with a big question mark: what should we do with an infinite sum? How high a number should we substitute as infinity? Interestingly, for practical purposes, small number like 5 or 10 could often play the role of infinity rather well. Hui (1996) states that convergence is fast most of the time. We are a bit skeptical about this since a will be used as an exponent. If b is negative and sigma is small enough, the (S/L)a part in the formula could turn out to be a problem.

First, we will try with normal parameters and see how quick the convergence is:

The following screenshot shows the result of the preceding code:

The Formula Error chart shows that after the seventh step, additional steps were not influencing the result. This means that for practical purposes, the infinite sum can be quickly estimated by calculating only the first seven steps. This looks like a very quick convergence indeed. However, this could be pure luck or coincidence.

What about decreasing the volatility down to 3 percent? We have to set N as 50 to see the convergence:

The preceding command gives the following output:

Not so impressive? 50 steps are still not that bad. What about decreasing the volatility even lower? At 1 percent, the formula with these parameters simply blows up. First, this looks catastrophic; however, the price of a DNT was already 98.75 percent of the payout when we used 3 percent volatility. Logic says that the DNT price should be a monotone-decreasing function of volatility, so we already know that the price of the DNT should be worth at least 98.75 percent if volatility is below 3 percent.

Another issue is that if we choose an extreme high U or extreme low L, calculation errors emerge. However, similar to the problem with volatility, common sense helps here too; the price of a DNT should increase if we make U higher or L lower.

There is still another trick. Since all the problem comes from the a parameter, we can try setting b as 0, which will make a equal to 0.5. If we also set r to 0, the price of a DNT converges into 100 percent as the volatility drops.

Anyway, whenever we substitute an infinite sum by a finite sum, it is always good to know when it will work and when it will not. We made a new code that takes into consideration that convergence is not always quick. The trick is that the function calculates the next step as long as the last step made any significant change. This is still not good for all the parameters as there is no cure for very low volatility, except that we accept the fact that if implied volatilities are below 1 percent, than this is an extreme market situation in which case DNT options should not be priced by this formula:

Now that we have a nice formula, it is possible to draw some DNT-related charts to get more familiar with this option. Later, we will use a particular AUDUSD DNT option with the following parameters: L equal to 0.9200, U equal to 0.9600, K (payout) equal to USD 1 million, T equal to 0.25 years, volatility equal to 6 percent,
r_AUD equal to 2.75 percent, r_USD equal to 0.25 percent, and b equal to -2.5 percent. We will calculate and plot all the possible values of this DNT from 0.9200
to 0.9600; each step will be one pip (0.0001), so we will use 2,000 steps.

The following code plots a graph of price of underlying:

The following output is the result of the preceding code:

It can be clearly seen that even a small change in volatility can have a huge impact on the price of a DNT. Looking at this chart is an intuitive way to find that vega must be negative. Interestingly enough even just taking a quick look at this chart can convince us that the absolute value of vega is decreasing if we are getting closer to the barriers.

Most end users think that the biggest risk is when the spot is getting close to the trigger. This is because end users really think about binary options in a binary way. As long as the DNT is alive, they focus on the positive outcome. However, for a dynamic hedger, the risk of a DNT is not that interesting when the value of the DNT is already small.

It is also very interesting that since the T-Bill price is independent of the volatility and since the DNT + DOT = T-Bill equation holds, an increasing volatility will decrease the price of the DNT by the exact same amount just like it will increase the price of the DOT. It is not surprising that the vega of the DOT should be the exact mirror of the DNT.

We can use the GetGreeks function to estimate vega, gamma, delta, and theta.
For gamma we can use the GetGreeks function in the following way:

The following chart is the result of the preceding code:

After having a look at the value chart, the delta of a DNT is also very close to intuitions; if we are coming close to the higher barrier, our delta gets negative, and if we are coming closer to the lower barrier, the delta gets positive as follows:

This is really a non-convex situation; if we would like to do a dynamic delta hedge, we will lose money for sure. If the spot price goes up, the delta of the DNT decreases, so we should buy some AUDUSD as a hedge. However, if the spot price goes down, we should sell some AUDUSD. Imagine a scenario where AUDUSD goes up 20 pips in the morning and then goes down 20 pips in the afternoon. For a dynamic hedger, this means buying some AUDUSD after the price moved up and selling this very same amount after the price comes down.

The changing of the delta can be described by the gamma as follows:

Negative gamma means that if the spot goes up, our delta is decreasing, but if the spot goes down, our delta is increasing. This doesn’t sound great. For this inconvenient non-convex situation, there is some compensation, that is, the value of theta is positive. If nothing happens, but one day passes, the DNT will automatically worth more.

Here, we use theta as minus 1 times the partial derivative, since if (T-t) is the time left, we check how the value changes as t increases by one day:

The more negative the gamma, the more positive our theta. This is how time compensates for the potential losses generated by the negative gamma.

Risk-neutral pricing also implicates that negative gamma should be compensated by a positive theta. This is the main message of the Black-Scholes framework for vanilla options, but this is also true for exotics. See Taleb (1997) and Wilmott (2006).

We already introduced the Black-Scholes surface before; now, we can go into more detail. This surface is also a nice interpretation of how theta and delta work. It shows the price of an option for different spot prices and times to maturity, so the slope of this surface is the theta for one direction and delta for the other. The code for this is as follows:

The preceding code gives the following output:

We can see what was already suspected; DNT likes when time is passing and the spot is moving to the middle of the (L,U) interval.

Another way to price the Double-no-touch option

Static replication is always the most elegant way of pricing. The no-arbitrage argument will let us say that if, at some time in the future, two portfolios have the same value for sure, then their price should be equal any time before this. We will show how double-knock-out (DKO) options could be used to build a DNT. We will need to use a trick; the strike price could be the same as one of the barriers. For a DKO call, the strike price should be lower than the upper barrier because if the strike price is not lower than the upper barrier, the DKO call would be knocked out before it could become in-the-money, so in this case, the option would be worthless as nobody can ever exercise it in-the-money. However, we can choose the strike price to be equal to the lower barrier. For a put, the strike price should be higher than the lower barrier, so why not make it equal to the upper barrier. This way, the DKO call and DKO put option will have a very convenient feature; if they are still alive, they will both expiry in-the-money.

Now, we are almost done. We just have to add the DKO prices, and we will get a DNT that has a payout of (U-L) dollars. Since DNT prices are linear in the payout, we only have to multiply the result by K*(U-L):

Now, we have two functions for which we can compare the results:

For a DNT with a USD 1 million contingent payout and an initial market value of over 48,000 dollars, it is very nice to see that the difference in the prices is only 14 cents. Technically, however, having a second pricing function is not a big help since low volatility is also an issue for dnt2.

We will use dnt1 for the rest of the article.

The life of a Double-no-touch option – a simulation

How has the DNT price been evolving during the second quarter of 2020?
We have the open-high-low-close type time series with five minute frequency for AUDUSD, so we know all the extreme prices:

The following is the output for the preceding code:

The price of a DNT is shown in red on the right axis (divided by 1000), and the actual AUDUSD price is shown in grey on the left axis. The green lines are the barriers of 0.9200 and 0.9600. The chart shows that in 2020 Q2, the AUDUSD currency pair was traded inside the (0.9200; 0.9600) interval; thus, the payout of the DNT would have been USD 1 million. This DNT looks like a very good investment; however, reality is just one trajectory out of an a priori almost infinite set. It could have happened differently. For example, on May 02, 2020, there were still 59 days left until expiry, and AUDUSD was traded at 0.9203, just three pips away from the lower barrier. At this point, the price of this DNT was only USD 5,302 dollars which is shown in the following code:

Compare this USD 5,302 to the initial USD 48,564 option price!

In the following simulation, we will show some different trajectories. All of them start from the same 0.9266 AUDUSD spot price as it was on the dawn of April 01, and we will see how many of them stayed inside the (0.9200; 0.9600) interval. To make it simple, we will simulate geometric Brown motions by using the same 6 percent volatility as we used to price the DNT:

The following is the output for the preceding code:

Here, the only surviving trajectory is the red one; in all other cases, the DNT hits either the higher or the lower barrier. The line set.seed(214) grants that this simulation will look the same anytime we run this. One out of five is still not that bad; it would suggest that for an end user or gambler who does no dynamic hedging, this option has an approximate value of 20 percent of the payout (especially since the interest rates are low, the time value of money is not important).

However, five trajectories are still too few to jump to such conclusions. We should check the DNT survivorship ratio for a much higher number of trajectories.

The ratio of the surviving trajectories could be a good estimator of the a priori real-world survivorship probability of this DNT; thus, the end user value of it. Before increasing N rapidly, we should keep in mind how much time this simulation took. For my computer, it took 50.75 seconds for N = 5, and 153.11 seconds for N = 15.

The following is the output for N = 15:

Now, 3 out of 15 survived, so the estimated survivorship ratio is still 3/15, which is equal to 20 percent. Looks like this is a very nice product; the price is around 5 percent of the payout, while 20 percent is the estimated survivorship ratio. Just out of curiosity, run the simulation for N equal to 200. This should take about 30 minutes.

The following is the output for N = 200:

The results are shocking; now, only 12 out of 200 survive, and the ratio is only 6 percent! So to get a better picture, we should run the simulation for a larger N.

The movie Whatever Works by Woody Allen (starring Larry David) is 92 minutes long; in simulation time, that is N = 541. For this N = 541, there are only 38 surviving trajectories, resulting in a survivorship ratio of 7 percent.

What is the real expected survivorship ratio? Is it 20 percent, 6 percent, or 7 percent? We simply don’t know at this point. Mathematicians warn us that the law of large numbers requires large numbers, where large is much more than 541, so it would be advisable to run this simulation for as large an N as time allows. Of course, getting a better computer also helps to do more N during the same time. Anyway, from this point of view, Hui’s (1996) relatively fast converging DNT pricing formula gets some respect.


We started this article by introducing exotic options. In a brief theoretical summary, we explained how exotics are linked together. There are many types of exotics. We showed one possible way of classification that is consistent with the fExoticOptions package. We showed how the Black-Scholes surface (a 3D chart that contains the price of a derivative dependent on time and the underlying price) can be constructed for any pricing function.

Resources for Article:

Further resources on this subject:

Binary Options Trading Explained

Interested in binary options trading? Want to learn more about it? Want to know how to get started? Want to know about the risks and the strategies? Want to know about binary options trading platforms. Then, this article is exactly right for you!

One of the most popular investment arenas in recent years has been trading the world’s currencies, due primarily to its flexibility, ease of access, and trading software that assimilates mountains of data to guide your every move in the market. Casualty rates for beginners, however, have been high and for good reason. Trading forex is very high risk. A great deal of preparation and practice trading are necessary if one wants to win in this genre. Most newcomers grow impatient, resort to “gut” gambling, and soon lose.

Learning and applying prudent risk and money management principles can be difficult, but the forex market has responded to these issues by offering “binary options”, a new way to play the game with currencies, as well as with stocks, commodities, and indexes. Trading binary options requires an entirely different approach, where much of the “headache” has been removed so that an investor can focus on the moment and directly on the price behavior for his chosen investment vehicle. Your downside risk exposure is “fixed” up front, as well as the amount of your position and your potential payoff.

What are Binary Options?

Binary options are now gaining in popularity more quickly than nearly any other area due to their simplicity. They may go by many names – barrier options, digital options, two-way-options, all-or-nothing options, and fixed-return options, to name a few. A basic definition from follows:

“A type of option in which the payoff is structured to be either a fixed amount of compensation if the option expires in the money, or nothing at all if the option expires out of the money.”

These options allow the investor an opportunity for instant gains of from 70% to 85%, depending on the investment type offered and the marketing bias of the broker. Investors need only guess the correct direction of the market within a defined time period to cash in, or retain anywhere from zero to 15% of his capital at risk. The simplest form is a pure “high/low” or “Call/Put” bet, but “one-touch”, “no-touch”, and “double-touch” options allow for typical trending and ranging strategies, where technical competence may provide the trader with a competitive edge if he can use his charts and indicators prudently to support his decision making.

For a simple “high/low” example, the guesswork of making a trade has been taken care of for you. You are offered a special screen view of the pricing behavior for your chosen asset for the recent past and asked to predict where it will be at the end of a specified time limit, the “expiration point”. The potential “payoff” is stated on the screen, say 85% for example, and you decide the amount of your position. If you wagered $100 and the price finished in line with your prediction, you win $85 plus your $100 investment. If not, then you may lose $100 or, in some cases, you may receive as much as $15 back.

The other types mentioned above allow for some variation on this basic theme, but you can never lose more than you specify. There is no need for complicated risk management strategies or worries about leverage and its financial implications. There are no margin calls or fees, either. The rules are simple and straightforward, the reason why this type of investing is gaining widespread popularity.

How Do You Execute a Trade?

Binary options require a customized approach, quite unlike the typical Metatrader4 platform or any other general trading support software. Not all brokers offer these instruments because they must first develop a proprietary trading system that has been customized just for this primary task. Thankfully, most binary option brokers have followed a similar theme. Your trading “dashboard” will typically resemble the diagram presented below:

The five steps have been added for clarification purposes to illustrate how easy it is to execute a trade. In this example, the position is for $25, and the potential payoff is $43, the sum of $25 plus $18, or 72%. If you are wrong, then $2.5, or 10%, will be returned to your account. The two arrows on the left give you some sense of what others have predicted, and the pricing behavior chart gives you a basis for making your own prediction of what will transpire by the expiration time chosen in “Step 2”.

Is It a Good Time to Consider Binary Options Trading?

The reason for the apparent popularity of this genre is due to its inherent simplicity. Risk and reward variables are fixed at the outset. There is no need to set protective stop-loss orders or worry about margin calls. Your downside risk is known, based on the amount you choose to wager, and your potential return is also defined when the order is executed. For investment beginners, many of the complexities of risk and money management principles are removed from the investment decision upon execution.

Is now the time to jump in with both feet? As always, the answer to this question depends on your personal tolerance level for risk and your appraisal of the state of this industry. With each passing month, the number of new broker offerings hitting the market continues to soar. Competition is a good thing since it will improve payout criteria and your odds for winning, but you must educate yourself first and perform the necessary due diligence before choosing your specialized broker. There are many websites that can assist you with this task, and be sure to take a “test run” first by practicing with “free” broker demo systems before risking your personal capital. On this page we give you our list of trusted brokers where you can start binary trading with a demo account.

What Are the Key Factors for Success When Trading Binary Options?

Like any other investment medium, the key factors for success are three in number –

  1. Knowledge
  2. Experience
  3. Emotional control

Newcomers typically fail in the trading arena primary due to the last factor, emotional control. It can be easy to establish a position in the market, but then waver when it comes time to close it, whether it is a winner or loser. The goal is to maximize your “winners” and minimize your “losers”, but, unfortunately, beginners tend to get it the other way around.

Basic binary options remove the threats of emotional intervention, so to speak. The expiration time fixes the endpoint. There is no decision to make. For traders that desire more flexibility, brokers often offer “Rollover” or “Double-Down” features that allow the trader to extend time periods or increase his position if it appears to be a winner, but these decisions require an action on your part. You have time to think about the actions you might take, without changing a thing. You are in control of your position. Your risks only grow if you decide to allow them to do so.

The first two factors can be easily addressed. There are many tutorials, trading guides, and information available on the Internet today to acquire the knowledge necessary to understand and win with binary options. Most brokers take a great deal of pride in the instructional materials that they provide. With competition running so high, every broker wants to provide the best trading experience around, supplying all manner of tools to assist you in the process. Market data, commentaries, and fundamental event calendars are standard offerings in today’s market.

The “middle” key to winning is experience. Seasoned veterans generally swear by their practice regimens. Trading binary options is not the latest form of Internet “gambling” or an amusing video game. You must develop a disciplined approach to the market, utilizing the same analytical skills required in any trading market. Never risk any funds in this market that you cannot afford to lose. Your position sizes should never exceed 2% to 3% of your account value. You will have losing trades. Accept them, and move on. The goal is consistency with “net” gains where winners exceed losers over time.

What Should I Do Now?

If this medium has piqued your interest, then it is time to do some homework. Read up on the topic. Read our article about binary options trading strategy and signals. Study the various offerings of various firms and be sure to perform your own due diligence before selecting one for initial testing. Trading platforms are often proprietary, but easy to understand with online access from the Internet. Brokers tend to be offshore, but there are a few with offices in the United States.

Ever since this OTC mode of investing acquired SEC approval in 2008, brokers and investors have literally leapt into the space, leading to increasing popularity that has only continued without abatement into the current year. A few leaders have emerged, and many firms have added unique “twists” to differentiate themselves from their competitors, but caution is the watchword to keep in mind at all times. Stay focused on your personal objectives. Invest the time practicing with “demo” systems, and, when you feel ready, go slowly at first. No reason to rush, and enjoy the process, too.

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