Option pricing excel model

Black-Scholes in Excel: The Big Picture

If you are not familiar with the Black-Scholes model, its assumptions, parameters, and (at least the logic of) the formulas, you may want to read those pages first (overview of all Black-Scholes resources is here).

Below I will show you how to apply the Black-Scholes formulas in Excel and how to put them all together in a simple option pricing spreadsheet. There are four steps:

1. Design cells where you will enter parameters.
2. Calculate d₁ and d₂.
3. Calculate call and put option prices.
4. Calculate option Greeks.

Black-Scholes Inputs

First you need to design six cells for the six Black-Scholes parameters. When pricing a particular option, you will have to enter all the parameters in these cells in the correct format. The parameters and formats are:

S = underlying price (USD per share)
K = strike price (USD per share)
σ = volatility (% p.a.)
r = continuously compounded risk-free interest rate (% p.a.)
q = continuously compounded dividend yield (% p.a.)
t = time to expiration (% of year)

Underlying price is the price at which the underlying security is trading on the market at the moment you are doing the option pricing. Enter it in dollars (or euros/yen/pound etc.) per share.

Strike price, also called exercise price, is the price at which you will buy (if call) or sell (if put) the underlying security if you choose to exercise the option. If you need more explanation, see: Strike vs. Market Price vs. Underlying Price. Enter it also in dollars per share (it must have same units as underlying price, also with the same contract or lot multipliers).

Volatility is the most difficult parameter to estimate (all the other parameters are more or less given). It is your job to decide how high volatility you expect and what number to enter – neither the Black-Scholes model, nor this page will tell you how high volatility to expect with your particular option (for more on that, see the volatility tutorials, particularly historical and implied volatility). Being able to estimate (= predict) volatility with more success than other people is the hard part and key factor determining success or failure in option trading. The important thing here is to enter it in the correct format, which is % p.a. (percent annualized).

Risk-free interest rate should be entered in % p.a., continuously compounded. The interest rate’s tenor (time to maturity) should match the time to expiration of the option you are pricing. You can interpolate the yield curve to get the interest rate for your exact time to expiration. Interest rate does not affect the resulting option price very much in the low interest environment that we’ve had in the recent years, but it can become very important when rates are higher (for more details on the effect of interest rates on option prices see the option rho tutorial).

Dividend yield should also be entered in % p.a., continuously compounded. If the underlying stock doesn’t pay any dividend, enter zero. If you are pricing an option on securities other than stocks, you may enter the second country interest rate (for FX options) or convenience yield (for commodities) here.

Time to expiration should be entered as % of year between the moment of pricing (now) and expiration of the option. For example, if the option expires in 24 calendar days, enter 24/365 = 6.58%. Alternatively, you can measure time in trading days rather than calendar days. If the option expires in 18 trading days and there are 252 trading days per year, you will enter time to expiration as 18/252 = 7.14%. You can also be more precise and measure time to expiration to hours or even minutes. In any case you must always express the time to expiration as % of year in order for the calculations to return correct results (it is very easy in Excel – just divide the number of days to expiration by the number of days per year).

I will illustrate the calculations on the example below. The parameters are in cells A44 (underlying price), B44 (strike price), C44 (volatility), D44 (interest rate), E44 (dividend yield), and G44 (time to expiration as % of year).

Note: It is row 44, because I am using the Black-Scholes Calculator for screenshots and it has charts in the rows above. You can of course start in row 1 or arrange your calculations in a column.

Black-Scholes d1 and d2

When you have the cells with parameters ready, the next step is to calculate d₁ and d₂, because these terms then enter all the calculations of call and put option prices and Greeks. The formulas for d₁ and d₂ are:

All the operations in these formulas are relatively simple mathematics. The only things that may be unfamiliar to some less savvy Excel users are the natural logarithm (LN Excel function) and square root (SQRT Excel function).

The hardest thing with the d₁ formula is making sure you put the brackets in the right places. This is why you may want to calculate individual parts of the formula in separate cells, as I do in the example below:

First I calculate the natural logarithm of the ratio of underlying price and strike price (this is why they must have the same units) in cell H44:

=LN(A44/B44)

Then I calculate the rest of the numerator of the d₁ formula in cell I44:

=(D44-E44+POWER(C44,2)/2)*G44

Then I calculate the denominator of the d₁ formula in cell J44. Another reason why you may want to calculate d₁ in separate parts is that this term will also enter the formula for d₂:

=C44*SQRT(G44)

Now I have all the three parts of the d₁ formula and I can combine them in cell K44 to get d₁:

=(H44+I44)/J44

Finally, I calculate d₂ in cell L44:

=K44-J44

Black-Scholes Option Price Excel Formulas

The Black-Scholes formulas for call option (C) and put option (P) prices are:

The two formulas are very similar. There are four terms in each formula. I will again calculate them in separate cells first and then combine them in the final call and put formulas.

N(d1), N(d2), N(-d2), N(-d1)

Potentially unfamiliar parts of the formulas are the N(d₁), N(d₂), N(-d₂), and N(-d₁) terms.

N(x) denotes the standard normal cumulative distribution function:

For example, N(d₁) is the standard normal cumulative distribution function for the d₁ that we have calculated in the previous step.

In Excel you can easily calculate the standard normal cumulative distribution functions using the NORM.DIST function, which has four parameters:

NORM.DIST(x, mean, standard_dev, cumulative)

• x = link to the cell where you have calculated d₁ or d₂ (with minus sign for -d₁ and -d₂)
• mean = enter 0, because it is standard normal distribution
• standard_dev = enter 1, because it is standard normal distribution
• cumulative = enter TRUE, because it is cumulative

For example, I calculate N(d₁) in cell M44:

=NORM.DIST(K44,0,1,TRUE)

Note: There is also the NORM.S.DIST function in Excel, which is the same as NORM.DIST with fixed mean = 0 and standard_dev = 1 (therefore you enter only two parameters: x and cumulative). You can use either. NORM.S.DIST may not be available in some spreadsheet software.

The Terms with Exponential Functions

The exponents (e^-qt and e^-rt terms) are calculated using the EXP Excel function with -q*t or -r*t as parameter.

I calculate e^-rt in cell Q44:

=EXP(-D44*G44)

Then I use it to calculate K e^-rt in cell R44:

=B44*Q44

Analogically, I calculate e^-qt in cell S44:

=EXP(-E44*G44)

Then I use it to calculate S e^-qt in cell T44:

=A44*S44

Now I have all the individual terms and I can calculate the final call and put option price.

Call Option Price

I combine the four terms in the call formula to get call option price in cell U44:

=T44*M44-R44*O44

Put Option Price

I combine the four terms in the put formula to get put option price in cell U44:

=R44*P44-T44*N44

Black-Scholes Greeks in Excel

Here you can continue to the second part of this tutorial, which explains Excel calculation of the Greeks: delta, gamma, theta, vega, and rho:

Continue to Option Greeks Excel Formulas

Or you can see how all the Excel calculations work together in the Black-Scholes Calculator. Explanation of the calculator’s other features (parameter calculations and simulations of option prices and Greeks) are available in the calculator’s user guide.

Пояснение-дисклеймер от Excellentricks: автор сайта критично относится к криптовалютам и к используемым в статье ниже подходам (и явным ошибкам), статья приведена, как иллюстрация сопряжения элементов Excel и имитационного моделирования, поэтому некоторые формулы, написанные с применением  MathJax отображаются некорректно (полную версию можно прочитать по ссылке внизу статьи на хабре).

Введение

Определение опциона

• покупатель опциона хочет получить право купить 10 Ethereum (ETH) по цене $470 за 1 ETH через 30 дней.
• текущая цена Ethereum равна $450.

• в данном примере покупатель приобрел европейский call опцион в объеме 10 ETH со страйк-ценой $470 USD с экспирацией через 30 дней.
• “Европейский” — в данном контексте означает, что покупатель опциона может совершить сделку по указанной цене строго на определенную дату — на момент экспирации опциона. Существуют также “американские” опционы, где покупатель контракта может исполнить его в любое время до момента экспирации, но рассмотрение американских опционов выходит за рамки статьи.
• Покупатель опциона получает право купить актив (Ethereum) по указанной цене. Опцион на покупку актива определен как call-опцион, на продажу — put-опцион.
• Цена, по которой покупатель опциона имеет право приобрести актив — в нашем примере $470 — называется страйк-ценой.
• Сумма, которую покупатель опциона уплатит продавцу, называется премией.

Как продавец оценивает премию по опциону

Эталонный расчет — модель Блэка — Шоулза

Логнормальное распределение

• метод обратной функции позволяет получить СВ с произвольным, заданным функцией (таблицей), законом распределения, получая на входе СВ, равномерно распределенную в диапазоне от 0 до 1.

• Вероятность — то самое значение, от которого мы строим нашу функцию. Строго больше 0 и строго меньше 1. Сгенерируем 999 значений от 0.001 до 0.999 с шагом 0.001 в столбце A. Значения из столбца A и пойдут на вход функции НОРМ.ОБР.
• Среднее — математическое ожидание нашей СВ. Напомню, мы генерируем величину, пропорциональную динамике нашего ценового актива ABS/USD. Положительные значения соответствуют росту цены (pi>pi−1⟹ln(pi/pi−1)>0pi>pi−1⟹ln(pi/pi−1)>0), отрицательные — падению. Если мы зададим параметр “среднее” большим нуля, наш актив будет, скорее всего, расти (программист говорит: проведем миллион итераций и гарантированно увидим конечную цену, превышающую начальное значение). Наш выбор — среднее, равное 0. Что означает “нейтрально дрейфующую” цену ABS/USD.
• Стандартное отклонение. О нем тоже пойдет речь впоследствии. Величина, характеризующая волатильность нашего актива. Примем ее равной 0.5% или 0.005. Что примерно соответствует изменению цены в день на ± 0.5% в среднем.

• Формула ячейки B2: =НОРМ.ОБР(A2;0;0,005)
• Формула ячейки C2: =СЛУЧМЕЖДУ(0;999)
• Формула ячейки D2: =ДВССЫЛ(СЦЕПИТЬ(«B»;C2+2))

Δi=ln(pi/pi−1)∆i=ln(pi/pi−1)

Значит, цены ABS/USD, последующая и предшествующая, будут связаны функцией

pi=pi−1∗eΔpi=pi−1∗e∆

Расчет премии по формуле Б-Ш

• текущая цена (S) равна 1000,
• страйк (X) равен 1000. Страйк равен текущей цене, такой опцион на сленге именуется “ванильным”,
• экспирация через 30 дней (T),
• объем сделки — один контракт.

C=SN(d1)−Xe−rTN(d2)C=SN(d1)−Xe−rTN(d2)

d1=ln(SX)+(r+σ22)TσT−−√d1=ln(SX)+(r+σ22)TσT

d2=d1−σT−−√d2=d1−σT

• S и X нам уже известны — текущая (1000 USD) и страйк (1000 USD) цены актива, соответственно.
• T — время до экспирации, выраженное, как часть года. К примеру, наш контракт ABS торгуется 365 дней в году, подобно криптовалютным контрактам. Экспирация произойдет через 30 дней, следовательно, T = 30/365 ~ 0.082. Другой пример — опцион на EURUSD на Чикагской бирже, торгуется приблизительно 265 дней в году. Считаем количество торговых дней до конкретной даты — до дня экспирации. Скажем, мы посчитали 23 торговых дня. В таком случае параметр T будет равен 23/265 или примерно 0.087.
• r — безрисковая процентная ставка. Как мы уже отмечали, для актива ABS она равна 0.
• σ (сигма) — историческая волатильность актива. Здесь потребуется небольшое отступление.

Историческая волатильность

S0=1n−1∑i=1n(xi−x¯¯¯)2−−−−−−−−−−−−−−−√S0=1n−1∑i=1n(xi−x¯)2

• Столбец C содержит разность текущего и предыдущего значения цен ABS, поделенную на предыдущее значение и умноженную на 100%.
• Ячейка D2 содержит среднее значение из столбца C.
• Столбец E содержит квадраты разности ценового отклонения (столбец C) и среднего значения ценового отклонения (ячейка D2, вторая строка “зафиксирована” в формуле символом $).
• Ячейка G2 — дисперсия, сумма квадратов отклонений, деленная на количество значений за вычетом 1 (SIC!).
• Наконец, ячейка H2 содержит искомое СКО — корень из дисперсии (G2).

σY=σ365−−−√=0,4943%×19,105=9,443%σY=σ365=0,4943%×19,105=9,443%

Расчет премии в Excel

• Формула коэффициента d1 =(LN(B2/B3)+B4*(B6+B5*B5/2))/(B5*КОРЕНЬ(B4))
• d2 =H2-B5*КОРЕНЬ(B4)
• N(d1) =НОРМ.РАСП(H2;0;1; ИСТИНА)
• N(d2)=НОРМ.РАСП(H3;0;1; ИСТИНА)
• Наконец, премия CALL-опциона: =B2*СТЕПЕНЬ(2,71818;-B7*B4)*K2-B3*СТЕПЕНЬ(2,71818;-B6*B4)*K3

Расчет премии для “ненормального” ценового распределения

Моделирование “реальной” цены

1. сложить две (обратные) интегральные функции плотности нормального распределения,
2. подставить в получившуюся функцию аргумент — равномерно распределенную СВ.

• Столбцы C (R1), D (R2) содержат обратную функцию нормального распределения с параметрами 0.005 и 0.0015 соответственно.
• Столбец E (R ) — взвешенную сумму двух этих величин — 0,8 x R1 + 0,65 x R2.
• Столбец F (RND) — случайное число в диапазоне от 2 до 1000.
• Наконец, столбец G(∆) — случайным образом выбранную из столбца E (R ) ячейку. То распределение, которого мы добивались.

xi=xi−1∗eΔixi=xi−1∗e∆i

На этом этапе возможностей MS Excel мне уже недостает (их не достаёт автору статьи, а не автору https://Excellentricks.ru), пора переходить к программированию.

The Binomial Option Pricing Model Excel is available as a template with MarketXLS. The Binomial Option Pricing Model is a popular model for stock options evaluation, and to calculate the options premium.

The Binomial Options Pricing Model provides investors with a tool to help evaluate stock options. The model uses multiple periods to value the option.  For each period, the model simulates the options premium at two possibilities of price movement (up or down). The periods create a binomial tree — In the tree, each tree shows the two possible outcomes or the movement of the price.

The model creates a binomial distribution of possible stock prices for the option. It creates possible paths that the stock price could go until the expiration date and the resulting impact on the options premium. Unlike the Black Scholes model of valuation of the option premium, the Binomial model gives you a view of an option contract at different prices at different periods until the expiration date.

Black-Scholes Vs Binomial Model

Black-Scholes model assumes that the option contract you are pricing is a European style option contract. A European style option contract is the one that can only be exercised at the date of the Expiry. The Americal style options contracts are the ones that can be exercised on any day until the expiry. Unlike, the Black Scholes model the Binomial option pricing model excel calculates the price of the option at various periods until the expiry. Since most of the exchange-traded options are American style options, the Black Scholes model seems to have a limitation.

If you were to assume that each period (days/weeks/months) until the expiry is the expiry date itself, you could also use the Black Scholes model to calculate a similar pay off table showing the value of the option for each period until expiry.

See the example below, where I use the Black Scholes model to generate a payoff for an option contract until the expiry date by assuming each day until the expiry is the expiry date. You can refer to our Options Profit Calculator template here.

How do you calculate the Option Premiums using the Binomial Model?

The Binomial Option Pricing Model Excel takes the following as the Inputs. For example, I have taken a Call Option of American Airlines expiring on August 7th, 2020 and today is 29th of July 2020.  So, there are 10 days left until the expiry. The variable T as shown below in the days to expiry and n is the number of steps that we need in our Binomial tree. The current price of this option is 0.54 per contract. And the stock price is at 11.77. The following table shows other values and assumptions.

S = 11.77 #underlying pricek = 12 #Strike pricer = .04 #Riskfree ratev = .81 #VolatilityT = 10./365 #Time to maturityn = 10 #StepsUn= 1 #1 Unit is 100 stocksPC = 0 #Call option

The first step in the calculation is to create a binomial tree. This tree will have a specified amount of time that ends at the expiration date. Each point on the tree is a node. And each node is the price the stock can go at. The following image shows the binomial tree for the stock price movement(in table 1). So, for each period the table below shows the possible price movement on the underlying stock.

This chart below is the table for the price of the stock and the one below it is the table for the price of the option contract at corresponding prices (in table 2). And finally we have a table that shows the expected payoffs (in dollars) at these prices (in table 3) until the expiry when we buy 1 contract of this call option.

TABLE 1:

TABLE 2:

TABLE 3:

The binomial tree diagram represents the option payoff and probability at different nodes. Nodes outline the paths the price of the underlying asset may take over time. The following binomial tree represents the general one-period call option.

The option value using the one-period binomial option pricing model can be worked out using the following formula:

The put option uses the same formula as the call option:

Where:

C+ is the payoff of an up move;

C- is the payoff of the down move;

π is the probability of an up move;

1-π is the probability of the down move;

r is the discount rate.

Where π is the probability of an up move which is determined using the following equation:

Where:

t is the period multiplier (time to maturity);

r is the discount rate;

d is the down factor;

u is the up factor.

The binomial option pricing model excel is useful for options traders to help estimate the theoretical values of options. Price movements of the underlying stocks provide insight into the values of options premium. The model offers a calculation of what the price of an option contract could be worth today.