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-style options 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.
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 proved in the late s that a dynamic revision of a portfolio removes the expected return of the security, thus inventing the risk neutral argument. 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 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.
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. 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 continous revision.
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: 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. 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.
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. 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".
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, , [ citation needed ] transaction costs and taxes Ingersoll, , [ citation needed ] and dividend payout.
As above, the Black—Scholes equation is a partial differential equation , which describes the price of the option over time. 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". The Black—Scholes formula calculates the price of European put 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. 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:. 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 formula can be interpreted by first decomposing a call option into the difference of two binary options: 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 value of the binary call options. These binary options are much less frequently traded than vanilla call options, but are easier to analyze. 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. 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 equivalent martingale probability measure is also called the risk-neutral probability measure. Note that both of these are probabilities in a measure theoretic sense, and neither of these is the true probability of expiring in-the-money under the real probability measure. To calculate the probability under the real "physical" probability measure, additional information is required—the drift term in the physical measure, or equivalently, the market price of risk.
The Feynman—Kac formula says that the solution to this type of PDE, when discounted appropriately, is actually a martingale. Thus the option price is the expected value of the discounted payoff of the option. Computing the option price via this expectation is the risk neutrality approach and can be done without knowledge of PDEs. For the underlying logic see section "risk neutral valuation" under Rational pricing as well as section "Derivatives pricing: They are partial derivatives of the price with respect to the parameter values.
One Greek, "gamma" as well as others not listed here is a partial derivative of another Greek, "delta" in this case. The Greeks are important not only in the mathematical theory of finance, but also for those actively trading. Financial institutions will typically set risk limit values for each of the Greeks that their traders must not exceed. Delta is the most important Greek since this usually confers the largest risk. Many traders will zero their delta at the end of the day if they are speculating and following a delta-neutral hedging approach as defined by Black—Scholes.
The Greeks for Black—Scholes are given in closed form below. They can be obtained by differentiation of the Black—Scholes formula. Note that from the formulae, it is clear that the gamma is the same value for calls and puts and so too is the vega the same value for calls and put options. N' is the standard normal probability density function. In practice, some sensitivities are usually quoted in scaled-down terms, to match the scale of likely changes in the parameters.
For example, rho is often reported divided by 10, 1 basis point rate change , vega by 1 vol point change , and theta by or 1 day decay based on either calendar days or trading days per year. The above model can be extended for variable but deterministic rates and volatilities. The model may also be used to value European options on instruments paying dividends.
In this case, closed-form solutions are available if the dividend is a known proportion of the stock price. American options and options on stocks paying a known cash dividend in the short term, more realistic than a proportional dividend are more difficult to value, and a choice of solution techniques is available for example lattices and grids. For options on indices, it is reasonable to make the simplifying assumption that dividends are paid continuously, and that the dividend amount is proportional to the level of the index.
Under this formulation the arbitrage-free price implied by the Black—Scholes model can be shown to be. It is also possible to extend the Black—Scholes framework to options on instruments paying discrete proportional dividends. This is useful when the option is struck on a single stock. The price of the stock is then modelled as. The problem of finding the price of an American option is related to the optimal stopping problem of finding the time to execute the option. Since the American option can be exercised at any time before the expiration date, the Black—Scholes equation becomes an inequality of the form.
In general this inequality does not have a closed form solution, though an American call with no dividends is equal to a European call and the Roll-Geske-Whaley method provides a solution for an American call with one dividend;   see also Black's approximation. Barone-Adesi and Whaley  is a further approximation formula. Here, the stochastic differential equation which is valid for the value of any derivative is split into two components: With some assumptions, a quadratic equation that approximates the solution for the latter is then obtained.
Bjerksund and Stensland  provide an approximation based on an exercise strategy corresponding to a trigger price. The formula is readily modified for the valuation of a put option, using put—call parity. This approximation is computationally inexpensive and the method is fast, with evidence indicating that the approximation may be more accurate in pricing long dated options than Barone-Adesi and Whaley.
By solving the Black—Scholes differential equation, with for boundary condition the Heaviside function , we end up with the pricing of options that pay one unit above some predefined strike price and nothing below. In fact, the Black—Scholes formula for the price of a vanilla call option or put option can be interpreted by decomposing a call option into an asset-or-nothing call option minus a cash-or-nothing call option, and similarly for a put — the binary options are easier to analyze, and correspond to the two terms in the Black—Scholes formula.
This pays out one unit of cash if the spot is above the strike at maturity. Its value is given by. This pays out one unit of cash if the spot is below the strike at maturity.
This pays out one unit of asset if the spot is above the strike at maturity. This pays out one unit of asset if the spot is below the strike at maturity. Similarly, paying out 1 unit of the foreign currency if the spot at maturity is above or below the strike is exactly like an asset-or nothing call and put respectively. The Black—Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset. The skew matters because it affects the binary considerably more than the regular options.
A binary call option is, at long expirations, similar to a tight call spread using two vanilla options. Thus, the value of a binary call is the negative of the derivative of the price of a vanilla call with respect to strike price:. Skew is typically negative, so the value of a binary call is higher when taking skew into account. Since a binary call is a mathematical derivative of a vanilla call with respect to strike, the price of a binary call has the same shape as the delta of a vanilla call, and the delta of a binary call has the same shape as the gamma of a vanilla call.
The assumptions of the Black—Scholes model are not all empirically valid. In short, while in the Black—Scholes model one can perfectly hedge options by simply Delta hedging , in practice there are many other sources of risk. Results using the Black—Scholes model differ from real world prices because of simplifying assumptions of the model. One significant limitation is that in reality security prices do not follow a strict stationary log-normal process, nor is the risk-free interest actually known and is not constant over time.
The variance has been observed to be non-constant leading to models such as GARCH to model volatility changes. Pricing discrepancies between empirical and the Black—Scholes model have long been observed in options that are far out-of-the-money , corresponding to extreme price changes; such events would be very rare if returns were lognormally distributed, but are observed much more often in practice.
Nevertheless, Black—Scholes pricing is widely used in practice, : Even when the results are not completely accurate, they serve as a first approximation to which adjustments can be made. Basis for more refined models: The Black—Scholes model is robust in that it can be adjusted to deal with some of its failures.
Rather than considering some parameters such as volatility or interest rates as constant, one considers them as variables, and thus added sources of risk. This is reflected in the Greeks the change in option value for a change in these parameters, or equivalently the partial derivatives with respect to these variables , and hedging these Greeks mitigates the risk caused by the non-constant nature of these parameters.
Other defects cannot be mitigated by modifying the model, however, notably tail risk and liquidity risk, and these are instead managed outside the model, chiefly by minimizing these risks and by stress testing.
Solving for volatility over a given set of durations and strike prices, one can construct an implied volatility surface.More...