Введение
В финансах за покупку или продажу опционов уплачивается или получается цена (премия). В данной статье рассматривается общий расчет этой премии. Более подробную информацию можно найти в следующих источниках: обсуждение математических основ; финансовая инженерия – для практической реализации; а также общие сведения.
Премиальные компоненты
Эта цена может быть разделена на два компонента: внутренняя стоимость и временная стоимость (также называемая "добавочной стоимостью").
Другие факторы, влияющие на премию
Существует множество факторов, влияющих на премию опциона. Эти факторы оказывают влияние на премию опциона с разной степенью интенсивности. Некоторые из этих факторов перечислены ниже:
Price of the underlying: Any fluctuation in the price of the underlying (stock/index/commodity) obviously has the largest effect on premium of an option contract. An increase in the underlying price increases the premium of call option and decreases the premium of put option. Reverse is true when underlying price decreases. Strike price: How far is the strike price from spot also affects option premium. Say, if NIFTY goes from 5000 to 5100 the premium of 5000 strike and of 5100 strike will change a lot compared to a contract with strike of 5500 or 4700. Volatility of underlying: Underlying security is a constantly changing entity. The degree by which its price fluctuates can be termed as volatility. So a share which fluctuates 5% on either side on daily basis is said to have more volatility than e. g. stable blue chip shares whose fluctuation is more benign at 2–3%. Volatility affects calls and puts alike. Higher volatility increases the option premium because of greater risk it brings to the seller. Payment of Dividend: Payment of Dividend does not have direct impact on value of derivatives but it does have indirect impact through stock price. We know that if dividend is paid, stock goes ex dividend therefore price of stock will go down which will result into increase in Put premium and decrease in Call premium. Apart from above, other factors like bond yield (or interest rate) also affect the premium. This is because the money invested by the seller can earn this risk free income in any case and hence while selling option; he has to earn more than this because of higher risk he is taking.
Цена базового актива: Любые колебания цены базового актива (акции/индекса/товара) оказывают, очевидно, наибольшее влияние на премию опционного контракта. Рост цены базового актива увеличивает премию опциона колл, а снижение – премию опциона пут. Обратное верно при снижении цены базового актива.
Price of the underlying: Any fluctuation in the price of the underlying (stock/index/commodity) obviously has the largest effect on premium of an option contract. An increase in the underlying price increases the premium of call option and decreases the premium of put option. Reverse is true when underlying price decreases. Strike price: How far is the strike price from spot also affects option premium. Say, if NIFTY goes from 5000 to 5100 the premium of 5000 strike and of 5100 strike will change a lot compared to a contract with strike of 5500 or 4700. Volatility of underlying: Underlying security is a constantly changing entity. The degree by which its price fluctuates can be termed as volatility. So a share which fluctuates 5% on either side on daily basis is said to have more volatility than e. g. stable blue chip shares whose fluctuation is more benign at 2–3%. Volatility affects calls and puts alike. Higher volatility increases the option premium because of greater risk it brings to the seller. Payment of Dividend: Payment of Dividend does not have direct impact on value of derivatives but it does have indirect impact through stock price. We know that if dividend is paid, stock goes ex dividend therefore price of stock will go down which will result into increase in Put premium and decrease in Call premium. Apart from above, other factors like bond yield (or interest rate) also affect the premium. This is because the money invested by the seller can earn this risk free income in any case and hence while selling option; he has to earn more than this because of higher risk he is taking.
Цена исполнения: Расстояние между ценой исполнения и текущей ценой базового актива также влияет на премию опциона. Например, если NIFTY вырастет с 5000 до 5100, премия опционов с ценой исполнения 5000 и 5100 изменится значительно больше, чем у контракта с ценой исполнения 5500 или 4700.
Price of the underlying: Any fluctuation in the price of the underlying (stock/index/commodity) obviously has the largest effect on premium of an option contract. An increase in the underlying price increases the premium of call option and decreases the premium of put option. Reverse is true when underlying price decreases. Strike price: How far is the strike price from spot also affects option premium. Say, if NIFTY goes from 5000 to 5100 the premium of 5000 strike and of 5100 strike will change a lot compared to a contract with strike of 5500 or 4700. Volatility of underlying: Underlying security is a constantly changing entity. The degree by which its price fluctuates can be termed as volatility. So a share which fluctuates 5% on either side on daily basis is said to have more volatility than e. g. stable blue chip shares whose fluctuation is more benign at 2–3%. Volatility affects calls and puts alike. Higher volatility increases the option premium because of greater risk it brings to the seller. Payment of Dividend: Payment of Dividend does not have direct impact on value of derivatives but it does have indirect impact through stock price. We know that if dividend is paid, stock goes ex dividend therefore price of stock will go down which will result into increase in Put premium and decrease in Call premium. Apart from above, other factors like bond yield (or interest rate) also affect the premium. This is because the money invested by the seller can earn this risk free income in any case and hence while selling option; he has to earn more than this because of higher risk he is taking.
Волатильность базового актива: Базовый актив – это постоянно меняющаяся величина. Степень, в которой изменяется его цена, называется волатильностью. Таким образом, акция, колеблющаяся на 5% в любую сторону ежедневно, считается более волатильной, чем, например, стабильные акции «голубых фишек», колебания которых более умеренные – 2–3%. Волатильность влияет как на опционы колл, так и на опционы пут. Более высокая волатильность увеличивает премию опциона из-за повышенного риска для продавца.
Price of the underlying: Any fluctuation in the price of the underlying (stock/index/commodity) obviously has the largest effect on premium of an option contract. An increase in the underlying price increases the premium of call option and decreases the premium of put option. Reverse is true when underlying price decreases. Strike price: How far is the strike price from spot also affects option premium. Say, if NIFTY goes from 5000 to 5100 the premium of 5000 strike and of 5100 strike will change a lot compared to a contract with strike of 5500 or 4700. Volatility of underlying: Underlying security is a constantly changing entity. The degree by which its price fluctuates can be termed as volatility. So a share which fluctuates 5% on either side on daily basis is said to have more volatility than e. g. stable blue chip shares whose fluctuation is more benign at 2–3%. Volatility affects calls and puts alike. Higher volatility increases the option premium because of greater risk it brings to the seller. Payment of Dividend: Payment of Dividend does not have direct impact on value of derivatives but it does have indirect impact through stock price. We know that if dividend is paid, stock goes ex dividend therefore price of stock will go down which will result into increase in Put premium and decrease in Call premium. Apart from above, other factors like bond yield (or interest rate) also affect the premium. This is because the money invested by the seller can earn this risk free income in any case and hence while selling option; he has to earn more than this because of higher risk he is taking.
Выплата дивидендов: Выплата дивидендов не оказывает прямого влияния на стоимость производных финансовых инструментов, но оказывает косвенное влияние через цену акции. Мы знаем, что после выплаты дивидендов акция переходит в состояние «ex-dividend», что приводит к снижению ее цены, увеличению премии опциона пут и снижению премии опциона колл.
Price of the underlying: Any fluctuation in the price of the underlying (stock/index/commodity) obviously has the largest effect on premium of an option contract. An increase in the underlying price increases the premium of call option and decreases the premium of put option. Reverse is true when underlying price decreases. Strike price: How far is the strike price from spot also affects option premium. Say, if NIFTY goes from 5000 to 5100 the premium of 5000 strike and of 5100 strike will change a lot compared to a contract with strike of 5500 or 4700. Volatility of underlying: Underlying security is a constantly changing entity. The degree by which its price fluctuates can be termed as volatility. So a share which fluctuates 5% on either side on daily basis is said to have more volatility than e. g. stable blue chip shares whose fluctuation is more benign at 2–3%. Volatility affects calls and puts alike. Higher volatility increases the option premium because of greater risk it brings to the seller. Payment of Dividend: Payment of Dividend does not have direct impact on value of derivatives but it does have indirect impact through stock price. We know that if dividend is paid, stock goes ex dividend therefore price of stock will go down which will result into increase in Put premium and decrease in Call premium. Apart from above, other factors like bond yield (or interest rate) also affect the premium. This is because the money invested by the seller can earn this risk free income in any case and hence while selling option; he has to earn more than this because of higher risk he is taking.
Помимо вышеперечисленного, на премию влияют и другие факторы, такие как доходность облигаций (или процентная ставка). Это связано с тем, что средства, вложенные продавцом, в любом случае могут приносить гарантированный доход, и, следовательно, при продаже опциона он должен получать доход выше этого уровня из-за повышенного риска.
Price of the underlying: Any fluctuation in the price of the underlying (stock/index/commodity) obviously has the largest effect on premium of an option contract. An increase in the underlying price increases the premium of call option and decreases the premium of put option. Reverse is true when underlying price decreases. Strike price: How far is the strike price from spot also affects option premium. Say, if NIFTY goes from 5000 to 5100 the premium of 5000 strike and of 5100 strike will change a lot compared to a contract with strike of 5500 or 4700. Volatility of underlying: Underlying security is a constantly changing entity. The degree by which its price fluctuates can be termed as volatility. So a share which fluctuates 5% on either side on daily basis is said to have more volatility than e. g. stable blue chip shares whose fluctuation is more benign at 2–3%. Volatility affects calls and puts alike. Higher volatility increases the option premium because of greater risk it brings to the seller. Payment of Dividend: Payment of Dividend does not have direct impact on value of derivatives but it does have indirect impact through stock price. We know that if dividend is paid, stock goes ex dividend therefore price of stock will go down which will result into increase in Put premium and decrease in Call premium. Apart from above, other factors like bond yield (or interest rate) also affect the premium. This is because the money invested by the seller can earn this risk free income in any case and hence while selling option; he has to earn more than this because of higher risk he is taking.
После кризиса
После финансового кризиса 2007–2008 годов в оценку были включены соображения кредитного риска контрагента, в то время как ранее для дисконтирования выплат использовалась безрисковая ставка. В области ценообразования опционов произошло три основных изменения:
For discounting, the overnight indexed swap (OIS) curve is typically used for the "risk free rate", as opposed to LIBOR as previously (LIBOR is due to be phased out by the end of 2021, with replacements including SOFR and TONAR); see Relatedly, the "Multi curve framework" is now standard in the valuation of interest rate derivatives and for fixed income analysis more generally. As mentioned, option pricing models must consider the volatility surface, and the numerics will then require a zeroth calibration step, such that observed prices are returned before new prices and / or "greeks" can be calculated. To do so, banks will apply local or stochastic volatility models, such as Heston mentioned above (or less common, implied trees). The risk neutral value, no matter how determined, is adjusted for the impact of counterparty credit risk via a credit valuation adjustment, or CVA, as well as various of the other XVA which may also be appended.
Для дисконтирования обычно используется кривая овернайт индексированных свопов (OIS) в качестве "безрисковой ставки", в отличие от ранее использовавшейся LIBOR (LIBOR планируется к полному прекращению использования к концу 2021 года, а в качестве замены рассматриваются SOFR и TONAR); см. Также, "многокривая модель" (Multi curve framework) стала стандартной практикой в оценке процентных деривативов и в анализе фондового рынка в целом. Как упоминалось, модели ценообразования опционов должны учитывать поверхность волатильности, и численные методы потребуют первоначального этапа калибровки, чтобы воспроизвести наблюдаемые рыночные цены перед расчетом новых цен и/или "греков". Для этого банки применяют локальные или стохастические модели волатильности, такие как упомянутая выше модель Хестона (или менее распространенные, например, деревья подразумеваемой волатильности). Нейтральная к риску стоимость, независимо от метода ее определения, корректируется на влияние кредитного риска контрагента посредством кредитной корректировки (CVA), а также различных других XVA, которые также могут быть добавлены.
For discounting, the overnight indexed swap (OIS) curve is typically used for the "risk free rate", as opposed to LIBOR as previously (LIBOR is due to be phased out by the end of 2021, with replacements including SOFR and TONAR); see Relatedly, the "Multi curve framework" is now standard in the valuation of interest rate derivatives and for fixed income analysis more generally. As mentioned, option pricing models must consider the volatility surface, and the numerics will then require a zeroth calibration step, such that observed prices are returned before new prices and / or "greeks" can be calculated. To do so, banks will apply local or stochastic volatility models, such as Heston mentioned above (or less common, implied trees). The risk neutral value, no matter how determined, is adjusted for the impact of counterparty credit risk via a credit valuation adjustment, or CVA, as well as various of the other XVA which may also be appended.