Binary option price derivation

What are Binary Options? Binary Options Trading. Binary Option Example. Definition of Binary Options: Binary Options are like regular options in that they allow you to make a bet as to the future price of a stock. However, binary options are different in that if the "strike price" is met by the expiration date, the binary option has a fixed payoff of $100 per contract. It doesn't matter if the stock price is a penny over the "strike price" or if it is $100 over the strike price, they payoff from the binary option is the same--$100. They are called binary options for this very reason. Binary means "2" and binary options have only 2 possible payoffs--all or nothing ($100 or $0). In 2008 the AMEX (American Stock Exchange) and the CBOE started trading binary options on a few stocks and a few indices trading binary options is NOT available on very many stocks or indices just yet. The United States has been slow to accept binary option trading, but binary option trading has been quite popular in Europe for a few years, especially as they relate to FOREX. The best way to understand these relatively new type of securities is to look at the example below. Example of a "Binary Option" Suppose GOOG is at $590 a share and you believe GOOG will close at or above $600 this week. You could buy 5 GOOG Binary Options for a price of, say, $0.30. The multiplier on the binary options is also 100 so five of these options would cost 5 contracts x $0.30 * 100 multiplier=$150. If GOOG closes at $600 or higher by the expiration date then the binary option is worth $100 so five of these GOOG call options would be worth $500, for a profit of $350. It doesn't matter if GOOG closed at $600 or $650, the binary option is still worth $100.

If GOOG closes at $599.99 or lower, then the option expires worthless. Currently, all binary options are traded as European style, which means they can only be exercised or settled at expiration. In the U. S., the CBOE offers binary contracts on 2 indices, the SandP 500 Index (SPX) and the CBOE Volatility Index (VIX). The tickers for these binary contracts are BSZ and BVZ. If you want to trade them, there are not many popular brokers that have added them to their platform. The ETRADEs, TD Ameritrades, Schwabs, and Scottrades have not added them to their platform yet. If you follow some of the ads on the web, the brokers that trade them are not commonly known so there is great risk. Another Example of Binary Options: Unlike traditional calls and puts, binary options do not have set prices. The binary options trader decides the amount of money he wants to bet and invests that amount when he buys the binary option. If the price is $0.25 then he stands to make $0.75 if the underlying moves as much as the investor hopes.

The time of expiration for binary options is set at different time intervals throughout the day, such as expirations of 1 hour, 1 day, 1 month, etc. The short duration of these contracts makes them more attractive to speculators and risk takers. Here are the top 10 option concepts you should understand before making your first real trade: Options Resources and Links. Options trade on the Chicago Board of Options Exchange and the prices are reported by the Option Pricing Reporting Authority (OPRA): Binary option price derivation Contamos con las mas modernas instalaciones y equipamiento mЎs avanzado Leer mas. El Centro Integral de Ginecolog­a y Obstetricia del Dr. Millan Lasquetty cuenta con un equipo m©dico y param©dico que la colocan como un referente nacional. Leer MЎs. Ponemos a su disposiciіn dos localizaciones en Santander y Torrelavega, para su mayor comodidad y atenciіn personalizada. Leer MЎs. El doctor Rafael MillЎn contestarЎ a las preguntas que se vayan formulando. Esta tecnolog­a obtiene imЎgenes tridimensionales del feto.

їPor qu© negarles el derecho a ser madres?. Para ellas existe una soluciіn. Bienvenidos a Nuestro Centro Ginecolіgico. Calle Isabel II nє 13 1є A, en Santander. C Joaqu­n Hoyos, nє 12 -1є Izda, en Torrelavega. Santander Ђ“ 942282928 Torrelavega - 942897513 mіvil Ђ“ 600483848. Binary Option. What is a 'Binary Option' A binary option, or asset-or-nothing option, is 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. The success of a binary option is thus based on a yes or no proposition, hence “binary”. A binary option automatically exercises, meaning the option holder does not have the choice to buy or sell the underlying asset. BREAKING DOWN 'Binary Option' Difference Between Binary and Plain Vanilla Options. Binary options are significantly different from vanilla options. Plain vanilla options are a normal type of option that does not include any special features.

A plain vanilla option gives the holder the right to buy or sell an underlying asset at a specified price on the expiration date, which is also known as a plain vanilla European option. While a binary option has special features and conditions, as stated previously. Binary options are occasionally traded on platforms regulated by the Securities and Exchange Commission (SEC) and other regulatory agencies, but are most likely traded over the Internet on platforms existing outside of regulations. Because these platforms operate outside of regulations, investors are at greater risk of fraud. Conversely, vanilla options are typically regulated and traded on major exchanges. For example, a binary options trading platform may require the investor to deposit a sum of money to purchase the option. If the option expires out-of-the-money, meaning the investor chose the wrong proposition, the trading platform may take the entire sum of deposited money with no refund provided. Binary Option Real World Example. Assume the futures contracts on the Standard & Poor's 500 Index (S&P 500) is trading at 2,050.50. An investor is bullish and feels that the economic data being released at 8:30 am will push the futures contracts above 2,060 by the close of the current trading day. The binary call options on the S&P 500 Index futures contracts stipulate that the investor would receive $100 if the futures close above 2,060, but nothing if it closes below. The investor purchases one binary call option for $50. Therefore, if the futures close above 2,060, the investor would have a profit of $50, or $100 - $50. How to Understand Binary Options.

A binary option, sometimes called a digital option, is a type of option in which the trader takes a yes or no position on the price of a stock or other asset, such as ETFs or currencies, and the resulting payoff is all or nothing. Because of this characteristic, binary options can be easier to understand and trade than traditional options. Method One of Three: Understanding the Necessary Terms Edit. Trading Binary Options Edit. Method Three of Three: Understanding Costs and Where to Buy Edit. No, there is no insurance on trades. The closest you could come is to hedge your investments by putting money into a counterbalancing investment that would go up when your original investment goes down. There are a wide range of binary option online trading sites: 24options, EZTrader, and IQ Options, to name a few. Be sure to read the terms and conditions before you decide to trade with a site. It is not impossible, but neither is it very likely. Trading binary options involves little more than luck at hyper-speed.

So how lucky do you feel? You're as likely to lose money in binary options as you are to make it. No, you won't lose the money invested. If you win, you would get your return, which is the sum of any profit and the money invested. There is no fee in the usual sense, but brokers take your money, nonetheless. There are various ways brokers can manipulate trades so that they will reap rewards, and none of the ways benefit traders. Go to 7BinaryOptions. com and click on "Brokers" for reviews on many binary options brokers. See the wikiHow article, Trade Binary Options. Warnings Edit. Related wikiHows Edit. Understand Carbon Trading. Invest in the Stock Market.

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Binary Call Option Vega. Call option vega measures the change in the price of an option owing to a change in implied volatility and is the gradient of the slope of the binary call options price profile versus implied volatility. This page provides the derivation of the binary call option vega formula from first principles, illustrates the binary call option vega with respect to time to expiry and implied volatility, followed by the formula itself. Zero interest rates are assumed as usual. The vega has crucial importance when conducting binary options portfolio risk management or when simply taking a single speculative position. For the options market-maker who is conducting dynamic portfolio risk management the vega is in effect what the delta-neutral market-maker is trading, constantly buying and selling ‘vol’ and hedging away the deltas via trading the underlying. So for the market-maker, knowing ones vega is the same as a futures trader knowing how many futures contracts they are longshort. The trader using binary options to take directional views needs to understand the effect of vega since a purchase of binary calls might well be complemented with a rise in the underlying, but a change in implied volatility could negatively affect the value of the binary call option after the move. Binary Call Option Vega and Finite Vega. The vega V of any option is defined by: P = price of the option. σ = implied volatility. δP = a change in the value of P. δσ = a change in the value of σ. Figure 1 shows binary call option price profiles over different implied volatilities. Figure 2 shows how with seven static underlying prices, the binary call options change in value as the implied volatility rises from 1.0% to 45.0%, so in effect a profile from Figure 2 is a vertical cross section at that underlying price in Figure 1. What also might be recognised is that the legend is inverted from the same illustration in binary put option vega. This being because at 99.75 in the put option example the option is in-the-money, while with the call option version here, the option is out-of-the-money.

When the underlying price is 100.00 the option is at-the-money and the changes in implied volatility has no effect on the price of the binary option as it is always 50. The 18.0% profile of Figure 1 is the highest of profiles when out-of-the-money (where S<100.00) but the lowest of the profiles when the binary call option is in-the-money (S>100.00). What this suggests is that as implied volatility rises the option increases in value when out-of-the-money (positive vega) and decreases in value when in-the-money (negative vega). Fig.1 – Binary Call Option Price profiles w. r.t. Implied Volatility. Figure 2 shows how the binary call options change value for a particular underlying price where implied volatility is shown on the horizontal axis. The gradient of an individual profile for a particular implied volatility will provide the vega for that binary call option. It is evident that below the Fair Value of 50, i. e. where the options are out-of-the-money, the value of the option increases as implied volatility rises along the lower axis, meaning positively sloping profiles and hence positive vegas. At the same time above the fair value price of 50 the options are falling in value as implied volatility rises, leading to negatively sloping profiles and negative vegas. As the implied volatility continues to rise to 45.0% all the profiles concertina around 50 and flatten out leading to very low vega at very high implied volatilities. Fig.2 – Binary Call Option Price profiles with Fixed Underlying Prices. The vega (as represented by the above formula Eq(1) measures the gradient of the slopes in Figure 2. Figure 3 is the S=99.75 price profile running from 4.0% implied volatility to 16.0% implied volatility, it is a section of the 99.75 profile of Fig.2. Chords have been added centred around 10.0% implied volatility so that, for example, the 6.0% chord stretches from 7.0% ‘vol’ to 13.0% ‘vol’. Since the price profile is increasing exponentially, the gradient of the chords decrease the longer the length of the chord. The gradient of the chord is defined by: Gradient = ( P2 – P1 ) ( σ2 – σ1 ) P2 = Binary Call value at σ2. P1 = Binary Call value at σ1. i. e. Gradient = (42.4366 ― 36.4953) (13 ‒ 7) = 0.9902.

as indicated in the δt = 6% row of the central column of Table 1. Fig.3 – Slope of the Vega at $99.75 plus approximating Vega ‘chords’ The gradients of the ’10.0% chord’ and ‘2.0% chord’ are calculated in the same manner and are also presented in the central column of Table 1. As the difference between implied volatilities narrows the gradient tends to the vega of 0.9056 at 10.0% implied volatility, i. e. where δσ = 0.0%. The vega is therefore the first differential of the binary call fair value with respect to implied volatility and can be stated mathematically as: as δσ → 0, V = dP dσ. which means that as δσ falls to zero the gradient approaches the tangent (vega) of the price profile of Figure 2 at 10.0% implied volatility. Binary Call Option Vega w. r.t. Implied Volatility. Figure 1 illustrates 4-day to expiry binary call profiles with Figure 4 providing the associated vegas for the same implied volatilities. Irrespective of the implied volatility the vega when at-the-money is always zero. When out-of-the-money the binary call option vega is always positive (as with out-of-the-money conventional call options) but when in-the-money the binary call option vega is negative (unlike in-the-money conventional call options). Fig.4 – Binary Call Option Vega w. r.t. Implied Volatility. As the implied volatility falls from 18.0% (where the absolute values of the vega are the lowest of the profiles) the peaks and troughs of the vegas increase absolutely while the peaks and troughs move closer to the strike. Binary Call Option Vega w. r.t. Time to Expiry. Figures 5 & 6 provide the binary call options price profiles over time to expiry with the associated binary call option vega. The maximum absolute vega in Figure 6 is fairly steady at around 2.43 irrespective of the time to expiry, although the time to expiry does determine how close to the strike the peak and trough in vega is. Fig.5 – Binary Call Option Price profiles w. r.t. Time to Expiry. Fig.6 – Binary Call Option Vega w. r.t. Time to Expiry.

Irrespective of time to expiry the binary call option vega travels through zero for the now familiar reason that at-the-money binaries are priced at 50, or very close to it. Points of note are: 1) Whereas conventional call option vegas are always positive as an increase in implied volatility always increases the value of the option, the effect of an increase in implied volatility with binary call options can be positive or negative dependent on whether they are in - or out-of-the-money. 2) Whereas with conventional call options vega is always at its absolute highest when at-the-money, the binary call option vega when at-the-money is always zero. 3) Out-of-the-money binary call options have positive or zero vega, in-the-money binary call options have zero or negative vega. This formula is based on binary call option prices that range between 0 and 1. Should a vega be required for binary call option prices that range between 0 and 100 then the vega should be multiplied by 100. Vega is an indispensible metric for the binary options market-maker but can also be used proficiently by the speculator, especially the speculator who is trading one-touch calls and puts and double no-touch strategies. Assessing the change of vega due to a move in the underlying can be critically important so that when buying and selling options it is sometimes just not good enough to forecast the direction of the underlying, it is also important to forecast what implied volatility will do should your directional forecast prove correct. Binary Call Option Theta. The Binary Call Option Theta measures the change in the price of a binary call option over time and is the gradient of the slope of the binary options price profile versus time decay. This section on binary call option theta, as with the binary put option theta section, is in two parts: i. the first section covers the derivation of the formula (which can be found immediately above the Summary) from first principles, plus the binary call options theta with respect to time to expiry and implied volatility, ii. while the second section analyses the theta as reflected by the formula as a useful analytical tool, discusses its drawbacks and provides an alternative ‘practical’ theta, followed by the formula. Binary Call Option Theta and Finite Theta. The theta ϴ of any option is defined by: P = price of the option. t = time in years to expiry. δP = a change in the value of P. δt = a change in the value of t. N. B. The equation for the binary call options theta can be found at the bottom of the page.

Figure 1 shows binary call option price profiles at different times to expiry. Figure 2 shows how with seven static underlying prices, the binary call options change in value as the days to expiry fall from 25 to 0, so in effect a profile from Figure 2 is a vertical cross section at that underlying price in Figure 1. When the underlying price is 100.00 the option is at-the-money and the passing of time has no effect on the price of the binary option as it is always 50. When the underlying price is above 100.00 the price profiles all slope upwards reflecting a positive theta, whereas the out-of-the-money profiles, i. e. where S < 100.00, the price profiles all slope down meaning a negative theta. Fig.1 – Binary Call Option Price profiles w. r.t. Time to Expiry. Fig.2 – Binary Call Option Price profiles w. r.t. Time to Expiry. The theta (as represented by the above formula) measures the gradient of the slopes in Figure 2. When there is over 20 days to expiry price decay (whether negative or positive) is very low as time passes the theta increases in absolute value with that increase dependent on how close to the strike the underlying is. Figure 3 is the S=99.75 price profile over the last 11 days of its life. Chords have been added centred around five days to expiry so that, for example, the five-day chord stretches from 7.5 days to 2.5 days to expiry. Since the price profile is decreasing exponentially, the gradient of the chords decrease the longer the length of the chord. The gradient of the chord is defined by: Gradient = ‒ ( P2 – P1 ) ( t2 – t1 ) P2 = Binary Call value at t2. P1 = Binary Call value at t1. i. e. Gradient = ― (37.3446 ― 16.9094) (9 ‒ 1) = ― 2.5544. Fig.3 – Slope of the Theta at $99.75 plus approximating Theta ‘chords’ as indicated in the bottom row of the central column of Table 1. The gradients of the ‘5 day chord’ and ‘2 day chord’ are calculated in the same manner and are also presented in the central column of Table 1. As the time difference narrows (as reflected by δt = 5 and δt = 2) the gradient tends to the theta of ―1.5446 at 5 days to expiry, i. e. where δt = 0. The theta is therefore the first differential of the binary call fair value with respect to time to expiry and can be stated mathematically as: as δt → 0, ϴ = dP dt. which means that as δt falls to zero the gradient approaches the tangent (theta) of the price profile of Figure 2 at 5 days. Binary Call Option Theta w. r.t. Time to Expiry. Figure 1 illustrates 5.0% implied volatility binary call profiles with Figure 4 providing the associated thetas for the same days to expiry. Irrespective of the days to expiry the theta when at-the-money is always zero.

When out-of-the-money the binary call theta is always negative (as with out-of-the-money conventional call options) but when in-the-money the binary call options theta is positive (unlike in-the-money conventional call options). With sufficient days to expiry (25 days in Figure 4) the binary call option theta is almost flat at close to zero. As time passes the absolute maximum value of the theta increases with the peak and trough progressively closing on the strike. This can be explained by the case where there is just 0.5 days to expiry where at an underlying price of 99.90 the binary call option is worth 29.4059 which is the amount that the option will decrease by over the next half-day if the underlying remains at 99.90. Fig.4 – Binary Call Option ‘Theoretical’ Theta w. r.t. Time to Expiry. Although at 99.90 and 1-day to expiry the binary call option is worth 35.0638 (5.6579 more than at the half-day to expiry) the binary call theta is lower as the theta is an annual measurement, not necessarily a practical one. Binary Call Option Theta w. r.t. Implied Volatility. Figures 5 & 6 provide the binary call options price profiles over a range of implied volatilities with the associated binary call theta. As is usual the implied volatility has a similar effect on the price profiles but there are some subtle differences between the binary call theta profiles of Figs. 4 & 6. The maximum absolute theta in Figure 6 is fairly steady at around 2.43 irrespective of the implied volatility, although the implied volatility does determine how close to the strike the peak and trough in theta is. Fig.5 – Binary Call Option Price profiles w. r.t. Implied Volatility. Fig.6 – Binary Call Option ‘Theoretical’ Theta w. r.t. Implied Volatility. Irrespective of implied volatility the binary call theta travels through zero for the now familiar reason that at-the-money binaries are priced at 50, or very close to it. ‘Theoretical’ Theta and ‘Practical’ Theta. From Figure 3 above it is (hopefully) visually apparent that an equal measure of time backwards provides an increase in call option value which is less than the decrease in option value for an equivalent jump forwards in time, e. g. at time 5 days to expiry the binary call option fair value is 33.3357, so using the example with δt=2, the 6-day and 4-day options are worth respectively 34.6912 and 31.5315. So from the 6th day to the 5th day the option loses: Price decay from Day 6 to Day 5 = (34.6912―33.3357) = 1.3555.

while from the 5th day to the 4th day the option loses: Price decay from Day 5 to Day 4 = (33.3357―31.5315) = 1.8042. Table 2 presents the option value at days to expiry from 7 to 0 with the daily difference plus the ‘theoretical’ theta it is apparent that the actual decay from one day to the next is greater than the theoretical theta. The ‘theoretical’ binary call theta in this instance is derived from the formula of Eq(1) above divided by 365 (Eq(1) provides an annual rate) and multiplied by 100 (Eq(1) assumes a binary option price range between 0 and 1, not 0 and 100). This begs the question as to the efficacy of using the formula of Eq(1) when might it not be simpler to compute the theta as calculated from the ‘Day’s Decay’ row of Table 2. Not particularly mathematically elegant, but there are a number of equally inelegant adjustments made by market practitioners to ‘elegant’ mathematical models in order to make them work, with volatility ‘skew’ being one of the more obvious. To be even deeper, the CAPM financial model is dependent on a ‘risk-free’ rate of interest…………is there such a thing as a ‘risk-free’ rate of interest?: what if the IMF was downgraded by Moody’s over the PIGS?! Figures 7a-f offer graphical illustrations of the difference between ‘theoretical’ theta and ‘practical’ theta, a term I’ve coined to simply describe the actual change in price from one day to the next. Figure 7a shows that as the binary call option price decay (either positive or negative) is negligible then the theoretical theta almost overlaps the practical theta, especially when implied volatility is low. Fig.7a – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 25-Days to Expiry w. r.t. Implied Volatility. With 10 and 4 days to expiry the theoretical theta gradually becomes more inaccurate as a measure of actual option price change with the actual time decay being absolutely greater at the peaks and troughs of the theta binary call options theta profiles but becoming lesser as the underlying moves away from the strike. This ‘smoothing’ is what might be expected when comparing the actual price changes of the ‘practical’ theta and the notional price changes portrayed by the ‘theoretical’ theta which itself is an annualised rate and in effect has a built in averaging mechanism. The left hand scales of Figures 7a-c are gradually increasing in value as the theta increases over time. Fig.7b – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 10-Days to Expiry w. r.t. Implied Volatility.

Fig.7c – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 4-Days to Expiry w. r.t. Implied Volatility. When there is one day to expiry (Figure 7d) the undervaluation of time decay as generated by the ‘theoretical’ theta is at its most pronounced because at this point the ‘practical’ theta is in fact the binary call option premium when out-of-the-money and 100 less the binary call option premium when in-the-money. Fig.7d – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 1-Day to Expiry w. r.t. Implied Volatility. Finally Figures 7e & 7f illustrate the absolute ‘theoretical’ theta rising aggressively while the absolute ‘practical’ theta is now falling, the latter due to the lower premium of the option. Fig.7e – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 0.4-Days to Expiry w. r.t. Implied Volatility. Fig.7f – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 0.1-Days to Expiry w. r.t. Implied Volatility. The scales of Figures 7e & 7f are worth noting, in particular Fig 7f where the ‘theoretical’ theta now rises above 100, which is an interesting concept since the maximum range of the binary call option is limited to 100! Points of note are: 1) Whereas conventional call option thetas are always negative as time value is always positive, time value with binary call options can be positive or negative dependent on whether they are in - or out-of-the-money. 2) Whereas with conventional call options theta is always at its absolute highest when at-the-money, the binary call options theta when at-the-money is always zero. 3) Out-of-the-money binary call options have negative or zero theta, in-the-money binary call options have a zero or positive theta. 4) Using Eq(1) to calculate theta can generate theta in excess of 100. (i) The theta generated by the above equation is an annualised number, so should a daily theta be required as an approximation then the theta needs to be divided by 365.

(ii) This formula is based on binary call option prices that range between 0 and 1. Should a theta be required for binary call option prices that range between 0 and 100 then the theta should be multiplied by 100. If theta is solely represented by the results of Eq(1) then it is a useful tool for establishing daily time decay if divided by 365 plus there is sufficient time to expiry. But as time to expiry falls this ‘theoretical’ theta becomes increasingly inaccurate as a tool for forecasting the binary option price change over time. The delta can be hedged away by trading the underlying until time itself becomes a tradable entity (a future?) hedging theta can only be achieved by trading other options. As with deltas, as expiry approaches the theta can reach ludicrously high numbers so one should always observe the tenet: “Beware Greeks bearing silly analysis numbers…” (as ever). Binary option price derivation So you thought binaries were all about Overs and Unders, with the odd Range and Touch thrown in? Think again! Recommended Binary Providers. An exclusive club of brokers that this site feels confident in recommending. Binary Options Insights. Delve into the nether regions of binary options and discover more valuable nuggets of information. ETX Capital Review ETX Capital is a UK company based in Broadgate in the financial centre of London.

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The NZDUSD is trading lower in sympathy. The dollar is on the &hellip Are EZTrader Trading Insolvently? Are EZTrader Trading Insolvently? EZTD’s Accounts for Year-Ends 2014 and 2015 were both qualified. The first six months of the current year saw them generate a pre-tax loss of $8.3m. Are they bust? Are EZTrader trading insolvently as at 31st March 2016 EZTD’s cash & cash equivalents stood at $2.275m yet for the three months ending &hellip TechFinancials Share Price Up 20% on H1 Report. TechFinancials Share Price Up 20% on H1 Report On July 29th TechFinancials’ share price slumped to 8.5p but now on the release of their H1 2016 report the shares are now trading at 15.5p middle, up a storming 82%! The main performer driving the TechFinancials share price up has been the B2C division where DragonFinancials has started &hellip Binary Call Option Theta. The Binary Call Option Theta measures the change in the price of a binary call option over time and is the gradient of the slope of the binary options price profile versus time decay. This section on binary call option theta, as with the binary put option theta section, is in two parts: i. the first section covers the derivation of the formula (which can be found immediately above the Summary) from first principles, plus the binary call options theta with respect to time to expiry and implied volatility, ii. while the second section analyses the theta as reflected by the formula as a useful analytical tool, discusses its drawbacks and provides an alternative ‘practical’ theta, followed by the formula. Binary Call Option Theta and Finite Theta. The theta ϴ of any option is defined by: P = price of the option. t = time in years to expiry. δP = a change in the value of P. δt = a change in the value of t. N. B. The equation for the binary call options theta can be found at the bottom of the page. Figure 1 shows binary call option price profiles at different times to expiry.

Figure 2 shows how with seven static underlying prices, the binary call options change in value as the days to expiry fall from 25 to 0, so in effect a profile from Figure 2 is a vertical cross section at that underlying price in Figure 1. When the underlying price is 100.00 the option is at-the-money and the passing of time has no effect on the price of the binary option as it is always 50. When the underlying price is above 100.00 the price profiles all slope upwards reflecting a positive theta, whereas the out-of-the-money profiles, i. e. where S < 100.00, the price profiles all slope down meaning a negative theta. Fig.1 – Binary Call Option Price profiles w. r.t. Time to Expiry. Fig.2 – Binary Call Option Price profiles w. r.t. Time to Expiry. The theta (as represented by the above formula) measures the gradient of the slopes in Figure 2. When there is over 20 days to expiry price decay (whether negative or positive) is very low as time passes the theta increases in absolute value with that increase dependent on how close to the strike the underlying is. Figure 3 is the S=99.75 price profile over the last 11 days of its life. Chords have been added centred around five days to expiry so that, for example, the five-day chord stretches from 7.5 days to 2.5 days to expiry. Since the price profile is decreasing exponentially, the gradient of the chords decrease the longer the length of the chord. The gradient of the chord is defined by: Gradient = ‒ ( P2 – P1 ) ( t2 – t1 ) P2 = Binary Call value at t2. P1 = Binary Call value at t1. i. e. Gradient = ― (37.3446 ― 16.9094) (9 ‒ 1) = ― 2.5544. Fig.3 – Slope of the Theta at $99.75 plus approximating Theta ‘chords’ as indicated in the bottom row of the central column of Table 1. The gradients of the ‘5 day chord’ and ‘2 day chord’ are calculated in the same manner and are also presented in the central column of Table 1. As the time difference narrows (as reflected by δt = 5 and δt = 2) the gradient tends to the theta of ―1.5446 at 5 days to expiry, i. e. where δt = 0. The theta is therefore the first differential of the binary call fair value with respect to time to expiry and can be stated mathematically as: as δt → 0, ϴ = dP dt. which means that as δt falls to zero the gradient approaches the tangent (theta) of the price profile of Figure 2 at 5 days. Binary Call Option Theta w. r.t. Time to Expiry. Figure 1 illustrates 5.0% implied volatility binary call profiles with Figure 4 providing the associated thetas for the same days to expiry. Irrespective of the days to expiry the theta when at-the-money is always zero.

When out-of-the-money the binary call theta is always negative (as with out-of-the-money conventional call options) but when in-the-money the binary call options theta is positive (unlike in-the-money conventional call options). With sufficient days to expiry (25 days in Figure 4) the binary call option theta is almost flat at close to zero. As time passes the absolute maximum value of the theta increases with the peak and trough progressively closing on the strike. This can be explained by the case where there is just 0.5 days to expiry where at an underlying price of 99.90 the binary call option is worth 29.4059 which is the amount that the option will decrease by over the next half-day if the underlying remains at 99.90. Fig.4 – Binary Call Option ‘Theoretical’ Theta w. r.t. Time to Expiry. Although at 99.90 and 1-day to expiry the binary call option is worth 35.0638 (5.6579 more than at the half-day to expiry) the binary call theta is lower as the theta is an annual measurement, not necessarily a practical one. Binary Call Option Theta w. r.t. Implied Volatility. Figures 5 & 6 provide the binary call options price profiles over a range of implied volatilities with the associated binary call theta. As is usual the implied volatility has a similar effect on the price profiles but there are some subtle differences between the binary call theta profiles of Figs. 4 & 6. The maximum absolute theta in Figure 6 is fairly steady at around 2.43 irrespective of the implied volatility, although the implied volatility does determine how close to the strike the peak and trough in theta is. Fig.5 – Binary Call Option Price profiles w. r.t. Implied Volatility. Fig.6 – Binary Call Option ‘Theoretical’ Theta w. r.t. Implied Volatility. Irrespective of implied volatility the binary call theta travels through zero for the now familiar reason that at-the-money binaries are priced at 50, or very close to it. ‘Theoretical’ Theta and ‘Practical’ Theta. From Figure 3 above it is (hopefully) visually apparent that an equal measure of time backwards provides an increase in call option value which is less than the decrease in option value for an equivalent jump forwards in time, e. g. at time 5 days to expiry the binary call option fair value is 33.3357, so using the example with δt=2, the 6-day and 4-day options are worth respectively 34.6912 and 31.5315. So from the 6th day to the 5th day the option loses: Price decay from Day 6 to Day 5 = (34.6912―33.3357) = 1.3555. while from the 5th day to the 4th day the option loses: Price decay from Day 5 to Day 4 = (33.3357―31.5315) = 1.8042.

Table 2 presents the option value at days to expiry from 7 to 0 with the daily difference plus the ‘theoretical’ theta it is apparent that the actual decay from one day to the next is greater than the theoretical theta. The ‘theoretical’ binary call theta in this instance is derived from the formula of Eq(1) above divided by 365 (Eq(1) provides an annual rate) and multiplied by 100 (Eq(1) assumes a binary option price range between 0 and 1, not 0 and 100). This begs the question as to the efficacy of using the formula of Eq(1) when might it not be simpler to compute the theta as calculated from the ‘Day’s Decay’ row of Table 2. Not particularly mathematically elegant, but there are a number of equally inelegant adjustments made by market practitioners to ‘elegant’ mathematical models in order to make them work, with volatility ‘skew’ being one of the more obvious. To be even deeper, the CAPM financial model is dependent on a ‘risk-free’ rate of interest…………is there such a thing as a ‘risk-free’ rate of interest?: what if the IMF was downgraded by Moody’s over the PIGS?! Figures 7a-f offer graphical illustrations of the difference between ‘theoretical’ theta and ‘practical’ theta, a term I’ve coined to simply describe the actual change in price from one day to the next. Figure 7a shows that as the binary call option price decay (either positive or negative) is negligible then the theoretical theta almost overlaps the practical theta, especially when implied volatility is low. Fig.7a – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 25-Days to Expiry w. r.t. Implied Volatility. With 10 and 4 days to expiry the theoretical theta gradually becomes more inaccurate as a measure of actual option price change with the actual time decay being absolutely greater at the peaks and troughs of the theta binary call options theta profiles but becoming lesser as the underlying moves away from the strike. This ‘smoothing’ is what might be expected when comparing the actual price changes of the ‘practical’ theta and the notional price changes portrayed by the ‘theoretical’ theta which itself is an annualised rate and in effect has a built in averaging mechanism. The left hand scales of Figures 7a-c are gradually increasing in value as the theta increases over time. Fig.7b – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 10-Days to Expiry w. r.t. Implied Volatility. Fig.7c – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 4-Days to Expiry w. r.t. Implied Volatility.

When there is one day to expiry (Figure 7d) the undervaluation of time decay as generated by the ‘theoretical’ theta is at its most pronounced because at this point the ‘practical’ theta is in fact the binary call option premium when out-of-the-money and 100 less the binary call option premium when in-the-money. Fig.7d – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 1-Day to Expiry w. r.t. Implied Volatility. Finally Figures 7e & 7f illustrate the absolute ‘theoretical’ theta rising aggressively while the absolute ‘practical’ theta is now falling, the latter due to the lower premium of the option. Fig.7e – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 0.4-Days to Expiry w. r.t. Implied Volatility. Fig.7f – Binary Call Option Theta, ‘Theoretical’ & ‘Practical’, 0.1-Days to Expiry w. r.t. Implied Volatility. The scales of Figures 7e & 7f are worth noting, in particular Fig 7f where the ‘theoretical’ theta now rises above 100, which is an interesting concept since the maximum range of the binary call option is limited to 100! Points of note are: 1) Whereas conventional call option thetas are always negative as time value is always positive, time value with binary call options can be positive or negative dependent on whether they are in - or out-of-the-money. 2) Whereas with conventional call options theta is always at its absolute highest when at-the-money, the binary call options theta when at-the-money is always zero. 3) Out-of-the-money binary call options have negative or zero theta, in-the-money binary call options have a zero or positive theta. 4) Using Eq(1) to calculate theta can generate theta in excess of 100. (i) The theta generated by the above equation is an annualised number, so should a daily theta be required as an approximation then the theta needs to be divided by 365. (ii) This formula is based on binary call option prices that range between 0 and 1. Should a theta be required for binary call option prices that range between 0 and 100 then the theta should be multiplied by 100. If theta is solely represented by the results of Eq(1) then it is a useful tool for establishing daily time decay if divided by 365 plus there is sufficient time to expiry.

But as time to expiry falls this ‘theoretical’ theta becomes increasingly inaccurate as a tool for forecasting the binary option price change over time. The delta can be hedged away by trading the underlying until time itself becomes a tradable entity (a future?) hedging theta can only be achieved by trading other options. As with deltas, as expiry approaches the theta can reach ludicrously high numbers so one should always observe the tenet: “Beware Greeks bearing silly analysis numbers…” (as ever).