Black–Scholes model
From the parabolic 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 given the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate).
The main principle behind the model is to hedge the option by buying and selling the underlying asset in a specific way to eliminate risk.
The insights of the model, as exemplified by the Black–Scholes formula, are frequently used by market participants, as distinguished from the actual prices.
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.
Louis Bachelier's thesis[3] in 1900 was the earliest publication to apply Brownian motion to derivative pricing, though his work had little impact for many years and included important limitations for its application to modern markets.
Fischer Black and Myron Scholes demonstrated in 1968 that a dynamic revision of a portfolio removes the expected return of the security, thus inventing the risk neutral argument.
Black and Scholes then attempted to apply the formula to the markets, but incurred financial losses, due to a lack of risk management in their trades.
[11] After three years of efforts, the formula—named in honor of them for making it public—was finally published in 1973 in an article titled "The Pricing of Options and Corporate Liabilities", in the Journal of Political Economy.
[18] Their dynamic hedging strategy led to a partial differential equation which governs the price of the option.
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 with discount factor
Computing the option price via this expectation is the risk neutrality approach and can be done without knowledge of PDEs.
Financial institutions will typically set (risk) limit values for each of the Greeks that their traders must not exceed.
Many traders will zero their delta at the end of the day if they are not speculating on the direction of the market and following a delta-neutral hedging approach as defined by Black–Scholes.
Since the American option can be exercised at any time before the expiration date, the Black–Scholes equation becomes a variational inequality of the form: together with
[29][30] Bjerksund and Stensland[31] provide an approximation based on an exercise strategy corresponding to a trigger price.
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 domestic interest rate, and the rest as above, the following results can be obtained: In the case of a digital call (this is a call FOR/put DOM) paying out one unit of the domestic currency gotten as present value: In the case of a digital put (this is a put FOR/call DOM) paying out one unit of the domestic currency gotten as present value: In the case of a digital call (this is a call FOR/put DOM) paying out one unit of the foreign currency gotten as present value: In the case of a digital put (this is a put FOR/call DOM) paying out one unit of the foreign currency gotten as present value: In the standard Black–Scholes model, one can interpret the premium of the binary option in the risk-neutral world as the expected value = probability of being in-the-money * unit, discounted to the present value.
Market makers adjust for such skewness by, instead of using a single standard deviation for the underlying asset
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 others can be further discussed: Useful approximation: although volatility is not constant, results from the model are often helpful in setting up hedges in the correct proportions to minimize risk.
All other things being equal, an option's theoretical value is a monotonic increasing function of implied volatility.
By computing the implied volatility for traded options with different strikes and maturities, the Black–Scholes model can be tested.
Despite the existence of the volatility smile (and the violation of all the other assumptions of the Black–Scholes model), the Black–Scholes PDE and Black–Scholes formula are still used extensively in practice.
Even when more advanced models are used, traders prefer to think in terms of Black–Scholes implied volatility as it allows them to evaluate and compare options of different maturities, strikes, and so on.
[citation needed] Espen Gaarder Haug and Nassim Nicholas Taleb argue that the Black–Scholes model merely recasts existing widely used models in terms of practically impossible "dynamic hedging" rather than "risk", to make them more compatible with mainstream neoclassical economic theory.
[42] They also assert that Boness in 1964 had already published a formula that is "actually identical" to the Black–Scholes call option pricing equation.
[44] Emanuel Derman and Taleb have also criticized dynamic hedging and state that a number of researchers had put forth similar models prior to Black and Scholes.
[39][46] In his 2008 letter to the shareholders of Berkshire Hathaway, Warren Buffett wrote: "I believe the Black–Scholes formula, even though it is the standard for establishing the dollar liability for options, produces strange results when the long-term variety are being valued...
"[47] British mathematician Ian Stewart, author of the 2012 book entitled In Pursuit of the Unknown: 17 Equations That Changed the World,[48][49] said that Black–Scholes had "underpinned massive economic growth" and the "international financial system was trading derivatives valued at one quadrillion dollars per year" by 2007.
