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From the partial differential equation in the model, known as the Black—Scholes equationone 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 formula led to a boom opțiuni john k hull opțiuni john k hull trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world.

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".

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

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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 formulathat are frequently used by market participants, as distinguished from the actual prices.

The maximum loss for this strategy is realised when, at expiration, the underlying has moved moderately bearishly to the price of the lower strike price. This strategy might be used when the trader believes that there will be a very sharp, downward move and would like to enter the position without paying a lot of premium, as the written puts will offset the cost of the purchased puts. In this case, this strategy can be considered a neutral or bullish play, since the net credit may be kept if the underlying remains at or greater than the upper strike price when the options expire. This position has a complex profile in that the Greeks Vega and Theta affect the profitability of the position differently, depending on whether the underlying spot price is above or below the upper strike. When the underlying's price is at or above the upper strike, the position is short vega the value of the position decreases as volatility increases and opțiuni john k hull theta the value of the position increases as time passes.

These insights include no-arbitrage bounds and risk-neutral pricing thanks to continuous revision. Further, the Black—Scholes equationa 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.

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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.

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