Black–Scholes model

The Black- Scholes model (pronounced ˌ blæk ʃoʊlz ) is a financial mathematical model for the valuation of financial options, which was ( after two rejection by prestigious journals) published by Fischer Black and Myron Samuel Scholes in 1973 and is considered a landmark in the financial industry. Robert C. Merton was also involved in the preparation, however, published a separate article. To be fair, the model should therefore also bear his name, but never sat down. However, Merton was honored along with Scholes for the development of this model with the price of the Swedish realm bank for economics in 1997, Black had died in 1995. However, Black also put different emphasis than Scholes and Merton.

Assumptions

The original model meets some idealized assumptions:

• There is a perfect and complete capital market. This means, among other things, transaction costs freedom, no restrictions on short selling and arbitrage free.
• The returns of the underlying asset, ie the relative changes its course are identical normally distributed, stochastically independent random variables. This means in particular that the volatility of returns is constant. In the basic model shares are considered, which do not pay dividends.
• There exists a constant interest rate at which at any time any can be borrowed and invested money.

In model extensions, dividends, stochastic interest rates and stochastic volatility are considered.

Model framework

The model is based on the assumption that the natural logarithm of the underlying asset ( eg a stock price ) of an option is followed by a so-called Wiener process, so that stock prices behave similarly to a geometric Brownian motion:

This is the expected return of the stock price. About the argument of the risk- neutral valuation or arbitrage, however, is not even included in the evaluation of the options. Next denote the volatility, the time and a standard Wiener process. Clearly, as an infinitesimal increase of be considered, that is, as a normally distributed random variable with mean 0 and variance. The stock prices are thus assumed to be log normally distributed.

In addition, the model assumes that the market deposits and borrowings with continuous compounding allows for constant interest rate and no additional transaction costs. That is, there exists a risk-free investment opportunity whose value is derived according to the ( ordinary ) differential equation

Developed, so applies.

Objective is to assess ( Price determination) of a European call and put options on the underlying with an exercise price, a total running time and a remaining maturity of at the time.

The options provide at the end of the term ( in ) capital flows:

Respectively

Black and Scholes show in their article that under the assumption of a constant interest rate and volatility of the option by developing a suitable portfolio consisting of the underlying asset and an investment or a loan with the interest rate may be duplicated dynamic. The value of the option must then comply at all times with the value of Duplikationsportfolios.

The fair price of the option can be derived through different arguments. It can be represented as the discounted expected value of the payments in, where the expectation value is to be formed with respect to the risk-neutral measure.

Another way to derive an explicit formula for the option price is the application of the lemma of Itō and the adoption of the absence of arbitrage. They lead to the Black- Scholes differential equation:

In this case denotes the value of an option. This differential equation is valid under the given assumptions for European equity options. Taking the cash flows to maturity of the option as an initial value for the differential equation, the evaluation function is the solution of the initial value problem.

Option prices in the Black- Scholes model

The Black- Scholes differential equation can be transformed by appropriate substitutions in the form of a heat equation. There arise as explicit solutions, the values ​​of call and put:

In which

The distribution function of the standard normal distribution respectively.

The value of an option is therefore determined by five parameters:

• : Current share price
• : Congruent with the remaining term of the interest rate option
• : The future volatility of the underlying. This is in the contract, the only unknown quantity, and thus ultimately the subject of pricing between the Parties.
• : Remaining term of the option with total duration at the time
• : Base price, determined as part of the contract

The Greeks Black-Scholes

As Greeks (English Greeks ), the partial derivatives of the option price on the respective model parameters are known. The advantage of the explicit formula for the option prices - as opposed to a numerical solution - lies in the fact that these derivatives can be easily calculated.

Delta

The delta indicates by how much the price of the option changes when the price of the underlying changes by one unit and all other factors remain the same. For example, a " deep in the money " lying option to buy (call) a delta of 1, a " deep in the money " lying put option ( put) of -1.

In the Black-Scholes model, the delta is calculated as:

For the European call, or

For the put.

Gamma

The Gamma is the second derivative of the option price according to the price of the underlying. It is for call and put in the Black Scholes model and the same, although

The gamma is not so negative, that is, the option price is always changing in the same direction ( rise / fall ) as the volatility. Is the money " option" at the (" in the money " ), the gamma can grow with decreasing residual maturity of all barriers. The letter stands for the density function of the normal distribution, cf distribution function.

Vega (lambda, kappa )

The Vega, also called lambda or kappa, denotes the derivative of the option price with respect to volatility and thus indicates how much an option responds to changes in ( in the Black- Scholes model constant ) volatility. The Vega is the same for a European call and put, namely

Vega is not a Greek letter. Sigma is awarded as a sign already for the standard deviation. Volatility is used as an estimate for future standard deviation.

Theta

The theta denotes the derivative with respect to the elapsed time t, there is the sensitivity of the option to changes in time. Since ceteris paribus approaches with time the value of an option to the payoff the maturity date, the theta of a European call option is never positive, the option loses value over time. It is also called a value of the option. In the Black- Scholes model, it is

Or

Rho

With the sensitivity of the Rho option for small changes in the interest rate is called.

Omega

The option elasticity is a percentage sensitivity:

Application

The Greeks are important for risk management. They make it easier here is to analyze the influence of individual risk factors. This is especially true at the level of a portfolio of financial instruments where the influence of individual risk factors - namely the model parameters - should be estimated on the total portfolio. An example would be a portfolio of options and positions in the associated underlying, eg options on the Euro Bund Future and Euro -Bund futures positions as such. About the Delta, the ( linear ) effect of a change in the futures price to the overall portfolio are presented.

Therefore, the Greeks can also be used for hedging purposes. The best known example is the delta - hedging. On the basis of Rho- sensitivity, for example, can be determined as an option portfolio to be hedged against changes in interest rate refinancing.

The diagram opposite shows the curves are plotted with European call values ​​of different residual maturity. These do not overlap, and the higher, the longer the remaining term. The lowest, kinked curve is the intrinsic value of the option depending on the current base rate today.

Problems

Despite various shortcomings, the Black -Scholes model in the financial world is often applied.

In the Black-Scholes model, the volatility σ is assumed to be constant. All ex -post calculations of standard deviations of returns, however, show that the volatility over time is not constant.

In addition, the model includes the simplifying assumption that returns are normally distributed. The normal distribution has little weight on their ends, making the arrival of extreme events too little account can be taken.

Market prices of options also show that their implied volatilities may arise at any given time depending on the moneyness different values. This results in the so-called volatility smile. In addition, at a given time depends implied volatility of options on the same underlying instrument of their remaining term from ( time structure of volatility). Both observations are not consistent with the model assumption of a single volatility.

Advanced models in which the volatility is assumed to be decreasing function of the stock price, such as the CEV model, deliver better results.

Derivation

The Black- Scholes model can be interpreted as a limiting case of time and value discrete binomial model according to Cox, Ross and Rubinstein by the trading intervals are set shorter.

And remove controls. The share price returns in the discrete model are binomial. They converge to normal distribution. The share prices are then normally distributed logarithmically in any time. In general, a step number of 100 is sufficient, with the restriction of exotic options or option sensitivities.