# Binary Call Option Black Scholes  Mathematical model

The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments.

From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-styleoptions and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. It is widely used, although often with some adjustments, by options market participants.:751

Based on works previously developed by market researchers and practitioners, such as Louis Bachelier, Sheen Kassouf and Ed Thorp among others, Fischer Black and Myron Scholes demonstrated in the late 1960s that a dynamic revision of a portfolio removes the expected return of the security, thus inventing the risk neutral argument. In 1970, after they attempted to apply the formula to the markets and incurred financial losses due to lack of risk management in their trades, they decided to focus in their domain area, the academic environment. After three years of efforts, the formula — named in honor of them for making it public — was finally published in 1973 in an article entitled "The Pricing of Options and Corporate Liabilities", in the Journal of Political Economy.Robert C.

Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model, and coined the term "Black–Scholes options pricing model". Merton and Scholes received the 1997 Nobel Memorial Prize in Economic Sciences for their work, the committee citing their discovery of the risk neutral dynamic revision as a breakthrough that separates the option from the risk of the underlying security. Although ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish Academy.

The key idea behind the model is to hedge the option by buying and selling the underlying asset in just the right way and, as a consequence, to eliminate risk.

## Black–Scholes model

This type of hedging is called "continuously revised delta hedging" and is the basis of more complicated hedging strategies such as those engaged in by investment banks and hedge funds.

The model's assumptions have been relaxed and generalized in many directions, leading to a plethora of models that are currently used in derivative pricing and risk management.

It is the insights of the model, as exemplified in the Black–Scholes formula, that are frequently used by market participants, as distinguished from the actual prices.

These insights include no-arbitrage bounds and risk-neutral pricing (thanks to continuous revision).

## Black Scholes Calculator

Further, the Black–Scholes equation, a partial differential equation that governs the price of the option, enables pricing using numerical methods when an explicit formula is not possible.

The Black–Scholes formula has only one parameter that cannot be directly observed in the market: the average future volatility of the underlying asset, though it can be found from the price of other options. Since the option value (whether put or call) is increasing in this parameter, it can be inverted to produce a "volatility surface" that is then used to calibrate other models, e.g. for OTC derivatives.

## The Black–Scholes world

The Black–Scholes model assumes that the market consists of at least one risky asset, usually called the stock, and one riskless asset, usually called the money market, cash, or bond.

Now we make assumptions on the assets (which explain their names):

• (riskless rate) The rate of return on the riskless asset is constant and thus called the risk-free interest rate.
• (random walk) The instantaneous log return of stock price is an infinitesimal random walk with drift; more precisely, the stock price follows a geometric Brownian motion, and we will assume its drift and volatility are constant (if they are time-varying, we can deduce a suitably modified Black–Scholes formula quite simply, as long as the volatility is not random).
• The stock does not pay a dividend.[Notes 1]

Assumptions on the market:

• There is no arbitrage opportunity (i.e., there is no way to make a riskless profit).
• It is possible to borrow and lend any amount, even fractional, of cash at the riskless rate.
• It is possible to buy and sell any amount, even fractional, of the stock (this includes short selling).
• The above transactions do not incur any fees or costs (i.e., frictionless market).

With these assumptions holding, suppose there is a derivative security also trading in this market. We specify that this security will have a certain payoff at a specified date in the future, depending on the value(s) taken by the stock up to that date. It is a surprising fact that the derivative's price is completely determined at the current time, even though we do not know what path the stock price will take in the future.

## Option Type: Call Option

For the special case of a European call or put option, Black and Scholes showed that "it is possible to create a hedged position, consisting of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock". Their dynamic hedging strategy led to a partial differential equation which governed the price of the option. Its solution is given by the Black–Scholes formula.

Several of these assumptions of the original model have been removed in subsequent extensions of the model. Modern versions account for dynamic interest rates (Merton, 1976),[citation needed]transaction costs and taxes (Ingersoll, 1976),[citation needed] and dividend payout.

## Notation , the price of the underlying asset at time t.; , the price of the option as a function of the underlying asset S, at time t; , the price of a European call option and the price of a European put option; , the strike price of the option, also known as the exercise price; , the annualized risk-free interest rate, continuously compounded Also known as the force of interest; , the drift rate of , annualized; , the standard deviation of the stock's returns; this is the square root of the quadratic variation of the stock's log price process; , a time in years; we generally use: now , expiry ; , the value of the portfolio.

We will use to denote the standard normalcumulative distribution function,  will denote the standard normal probability density function, ## Black–Scholes equation

Main article: Black–Scholes equation

As above, the Black–Scholes equation is a partial differential equation, which describes the price of the option over time.

The equation is: The key financial insight behind the equation is that one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk".[citation needed] This hedge, in turn, implies that there is only one right price for the option, as returned by the Black–Scholes formula (see the next section).

## Black–Scholes formula

The Black–Scholes formula calculates the price of Europeanput and call options. This price is consistent with the Black–Scholes equation as above; this follows since the formula can be obtained by solving the equation for the corresponding terminal and boundary conditions.

## How to interpret N(d1) and N(d2) in Black Scholes Merton (FRM T4-12)

The value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is: The price of a corresponding put option based on put–call parity is: For both, as above:

### Alternative formulation

Introducing some auxiliary variables allows the formula to be simplified and reformulated in a form that is often more convenient (this is a special case of the Black '76 formula): The auxiliary variables are:

with d+ = d1 and d = d2 to clarify notation.

Given put–call parity, which is expressed in these terms as: the price of a put option is: ### Interpretation

The Black–Scholes formula can be interpreted fairly handily, with the main subtlety the interpretation of the (and a fortiori ) terms, particularly and why there are two different terms.

The formula can be interpreted by first decomposing a call option into the difference of two binary options: an asset-or-nothing call minus a cash-or-nothing call (long an asset-or-nothing call, short a cash-or-nothing call).

A call option exchanges cash for an asset at expiry, while an asset-or-nothing call just yields the asset (with no cash in exchange) and a cash-or-nothing call just yields cash (with no asset in exchange).

The Black–Scholes formula is a difference of two terms, and these two terms equal the values of the binary call options. These binary options are much less frequently traded than vanilla call options, but are easier to analyze.

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Thus the formula: breaks up as: where is the present value of an asset-or-nothing call and is the present value of a cash-or-nothing call.

The D factor is for discounting, because the expiration date is in future, and removing it changes present value to future value (value at expiry).

Thus is the future value of an asset-or-nothing call and is the future value of a cash-or-nothing call.

## The Black-Scholes Option Pricing Formula

In risk-neutral terms, these are the expected value of the asset and the expected value of the cash in the risk-neutral measure.

The naive, and not quite correct, interpretation of these terms is that is the probability of the option expiring in the money , times the value of the underlying at expiry F, while is the probability of the option expiring in the money times the value of the cash at expiry K. This is obviously incorrect, as either both binaries expire in the money or both expire out of the money (either cash is exchanged for asset or it is not), but the probabilities and are not equal.

In fact, can be interpreted as measures of moneyness (in standard deviations) and as probabilities of expiring ITM (percent moneyness), in the respective numéraire, as discussed below.

Simply put, the interpretation of the cash option, , is correct, as the value of the cash is independent of movements of the underlying, and thus can be interpreted as a simple product of "probability times value", while the is more complicated, as the probability of expiring in the money and the value of the asset at expiry are not independent. More precisely, the value of the asset at expiry is variable in terms of cash, but is constant in terms of the asset itself (a fixed quantity of the asset), and thus these quantities are independent if one changes numéraire to the asset rather than cash.

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If one uses spot S instead of forward F, in instead of the term there is  Simulated geometric Brownian motions with parameters from market data A European call valued using the Black–Scholes pricing equation for varying asset price and time-to-expiry .

In this particular example, the strike price is set to 1.