# Binary option pricing

As we saw in the precedent pages, Binary options are simply a bet between two players called Traders.

• When a Trader bets on the underlying to go up then he has to buy a Binary Call
• When a Trader bets on the underlying to go down then he has to buy a Binary Put

At expiry of the option the buyer of Call or Put will get either

1. \$0
2. The full amount of the bet (\$100 for example)

There are only two possibility of results this is why we called those option Binary.

Binary options have however a special characteristic that differs from a normal bet. In a simple bet there is a Winner and and Loser but no one wins is the score is null. For Binary options is a little bit different because the buyer pay a premium to the seller of the option that he can keep if the score is null or if the buyer lose the bet. In fact the seller of the option will never win more than the paid premium by the buyer while the buyer can win the full amount of the bet.

To evaluate what the premium is there are différent mathematical methods but we will concentrate on one intuitive explanation.

Let’s imagine the following:

During a bet, a buyer A buy a binary call at \$50 that a stock will go up. If A wins he get \$100. The seller B of the option will get the \$50 premium but hold the risk to give \$100 to A if the stock rises. See below recapitulation.

For the Buyer A:

• 50% chance to win
• 50% chance to lose

For the seller B there two possibilities to win and one chance:

• One chance to lose \$100 if the stock goes up
• Two chances to win \$50 if the stock stay flat or decrease

We know the value of the option at the beginning i.e. \$50 and at the end (\$0 or \$100) but we need to know how the option price move during the time of the trader t=0 and the expiration time T. To well understand how a binary option is priced it is necessary to first understand that its value is moving with respect to several parameters that are the value of its underlying, time to expiry, the underlying volatility and finally the free risk interest rate.

1. The value of a Binary option varies with respect to his underlying. Let’s imagine that the price of a stock is \$1,000 and that a Trader bet \$50 that a stock move above \$1,000 by buying a Binary Call. The below table shows the price of the option versus the price of its underlying during a trading day.
 Underlying price Underlying price change in % Option price Option price change in % 1000 49.42 999 -0.10% 48 -2.76% 998 -0.10% 47 -2.84% 1000 0.20% 49 5.84% 1002 0.20% 52 5.51% 1005 0.30% 56 7.77% 1004 -0.10% 55 -2.39% 1000 -0.40% 49 -9.90% 999 -0.10% 48 -2.76%

Note that the Option price change in % est well above its underlying change.

Below a recapitulation chart.

It is logical that the value of an option increases because the probability to win the bet (Underlying > \$1,000) increases when the underlying (blue line) increases. Until here this is nothing like rocket science.

In trading we often look at how much an option price move with respect to its underlying move up or down in order to know how much we win or lose. For example if the underlying is worth \$1,000 and rises to \$1,005 then the Trader wishes to know how much he needs to add to its binary call option price to know how much money he won or lost (depending if he bought or sold this option). This measure is called the Delta.

For example if a binary call with a strike of \$1,000 (the strike is the underlying price at wish he starts to win) was bought for \$50 and has a 50% delta. This means that if the underlying price move from \$1,000 to \$1,005 i.e. up \$5 then its option price moves by \$5 times %50 = \$2.5. See below recapitulation.

1. The underlying moved from \$1000 to \$1005, this is up \$5
2. The call option price will move up as well by \$5 * 50% (delta) = \$2.5
3. The call option will be 50 + 2.5 = \$52.5 (The buyer will win \$2.5 and the seller will lose \$2.5)

The knowledge of the delta is very important because it allows to know at what speed the option price will move with respect to its underlying.

Let’s take another example with a 10% delta.

1. The underlying moved from \$1000 to \$1005, this is up \$5
2. The call option price will move up as well by \$5 * 10% (delta) = \$0.5
3. The call option will be 50 + 0.5 = \$50.5 (The buyer will win \$0.5 and the seller will lose \$0.5)

The below chart shows the delta of an option with a \$50 strike at three given times of its life.

As we can see the delta changes with respect to the underlying bu also with respect to time. The orange line represents the delta five days before expiry,  the closer the option expiry time the more the option delta increases (i.e. the option prices move rapidly). Longer term option will move slower as the result of the bet is still very uncertain 25 and 40 days before the final result.

We can notice as well than the delta stays very low when far from the strike. When the delta is far from the strike its means that the bet is already won or lost by far then it makes sense that the value of the option doesn’t move a lot.

In trading terms we say that the call option is in the money when the underlying is above to the strike, out the money when the underlying is below the strike and at the money when the underlying equal the strike.

For the put, the option is in the money when the underlying is below to the strike, out the money when the underlying is above the strike and at the money when the underlying equal the strike.

Below we added all the same chart of delta than above (with a strike of 10 instead of 50) with all the period of time to give a 3 dimension chart.

2. The value of an option varies with respect to time. We can intuitively understand this as the longer the time the expiry the less predictable is the future. Let’s imagine that the price of a stock is \$1,000 and the time to expiry 5 days. If a Trader buy a 1,000 strike call on this stock at \$50, he basically bet that the stock will move above \$1,000 in 5 days. Now if the stock goes to \$999 the same day then the option will move from \$50 to \$49.8 due to the delta move.

If the stock stays at \$999 the following day then the option price will lose some value again because it has 1 day less chance to go above the strike of \$1,000. The real option price will be \$46.

If the stock is stil at \$999 the fifth day until the very last second of its expiry then the option will be worth nothing as the probability for the stock to rise above \$1,000 is close to zero.

The below table recapitulates in details the option price move for the above losing case, I also added the wining case for education purpose:

 Day to expiry Option price for the losing scenario (i.e. stock price < call strike) Same option price for the winning scenario (i.e. stock price > call strike) 5 49.8 49.8 4.5 46.2 53.4 4 46.0 53.6 3.5 45.7 53.9 3 45.4 54.2 2.5 45.0 54.6 2 44.5 55.2 1.5 43.7 56.1 1 42.3 57.5 0 0.0 100.0

Above values charted below.

Note that the binary option wins or loses a bit part of its value the very last days before its expiry. If we zoom on the last day to get the time to expiry in hours instead of days we would see the below.

The price of a binary option is function of the time to expiry and as we saw above its value moves a lot when the option is closed to expiry and the underlying close to the strike. Like the delta there is a measure that allows us to know how much the option vale will increase or decrease by day, this measure  is called the Theta. For example his an option has a theta of -\$1 its means that the option will lose -\$1 overnight when the market is closed. The below 3D chart show the theta of a binary call option during all his life. This chart recapitulates all the above explanations.

3.  The value of a binary call varies with respect to the volatility to its underlying. We call the volatility the speed at which the underlying move up or down. The below chart shows the price of an underlying with three different volatility. Which one do you think is the more volatile?

If you replied the orange C line then you were right. Looking at the chart the blue line is definitely the flatter one and then the less volatile followed by the red one and by the orange line that goes much lower and higher than all other lines.

If someone has to bet on one of these three stocks to go up by buying a binary call then clearly he will choose the line that move up the most (the C orange line). Why? because the stock C is so volatile that the buyer of the call has more chance than the stock will go much higher than the other two stocks given him more chance to resell the option with a larger profit than with stock A or B. (However note that if you are forced to keep the option to expiry then the most volatile stocks is not always the optimal choice. On the above chart the stock collapses below its starting point just before the end of the trading session meaning that the binary call will lose all his value if its strike is \$100 or above. One way to solve this problem is to buy a put to protect your downside when the market is above your strike. See the trading strategies pages.)

Obviously the seller of a binary option will have the opposite reasoning, he will try to sell an option on the less volatile stocks as he makes money even if the stock doesn’t move. The logic wants that one option with the same strike on A, B and C will be worth more on the most volatile market, why? Simply because the seller has more chance to lose and will ask for more money to bet are the odds are against him.

When pricing binary option the underlying volatility is a very important parameter especially for the long term trading, why? Because if an option expires in a few minutes and the underlying price is \$100 and the call strike \$110 then there is very little chance than the underlying price will move 10% in a few minutes while if the expiry is one year then a move of 10% is more probable. An option price incorporate a larger volatility value when the time to expiry is longer. Like the Delta and Theta an option also has a measure of how much the option price move due to an increase of decrease of the underlying volatility, this measure is called the Vega.

The below 3D chart shows the Vega of a binary call with respect to the underlying price and time to expiry. The highest the Vega the more sensitive the option price is to the underlying volatility.

4. The value of an option varies with respect to the risk free interest rate.  It is very important to understand that trading is all about comparing investment vehicle. Why bothering investing in an underlying that performed less than the risk free interest rate that your banker is giving you? Like all other investment option price need to incorporate in its pricing the value of the interest rate. To get the intuition let’s imagine that you invest \$1,000,000 at 10% risk free at the bank and \$1,000,000 on an option that expiry in one year. After a month your bank account increased by approximatively \$8,300 and your option didn’t move because the underlying is not moving. The opportunity cost of this investment is then \$8,300 because it is the money you could have made risk free at the bank. The option pricing model takes into account this amount of money therefore your option lost \$8,300 in one month. This money is going to the seller of the option. (remember that one of the reason to the sell an option is because you think that the underlying will not move a lot or will go the opposite direction than the buyer thinks the underlying goes)

Binary option price is also function to the risk free rate, the measure that allows to know how much the option price will move with respect to the risk free rate is called Rho.

The below chart show a \$50 strike with 25 days to expiry call option Rho.

The below chart shows a \$10 strike Rho during a 100 days time to expiry period.

5. As we saw above the price of an option is moving with respect to

1. Its underlying price
2. Its time to expiry
3. The volatility of the underlying
4. The risk free rate
5. The strike of the option

To be able to price a binary option you need those five parameters (For stocks you should also use the dividend rate and substrat it to the risk free rate). In the 70’s three mathematician Black, Merton and Scholes developed an analytical formula, the formula is the following.

For a binary call:

For a binary put:

with:

S = Underlying price

K = Strike

r = Risk free rate

σ = Standard deviation of the underlying return (annualised volatility)

(T -t)= Time to expiry

et

Example of pricing:

S = 50

K = 50

r = 1%

σ = 25%

T – t = 25 days

T-t needs to be annualised 25/365 = 0.0685 year.

Binary call price = 0.4908

The below 3D chart e graphique 3D shows the price of a binary call with respect to the underlying and time to expiry