Option Price and Option Greeks Calculator (Option fair value calculator): Black-Scholes Model

What are the basic option terms I need to know?

Call option: When someone buys a call option for a given quantity, they get the right, but not the obligation, to buy that quantity of the underlying asset at the pre-determined price. The amount of money paid to buy the call option is called the premium, while the pre-determined price is called the strike price.

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Example: If someone buys a call on 125 shares (here, corresponding to 1 lot) with the strike price of ₹7000 on the stock "XYZ", and pays ₹200 for that call, then they get the right to buy 125 shares of the stock at a price of ₹7000 irrespective of whether the price is higher than ₹7000 or not. The total premium paid is equal to ₹200 * 125 = ₹25,000. Should the price fall below ₹7000, then the buyer of the call option merely forefits the premium paid; in other words, the buyer has a fixed, but known maximum loss which is equal to the premium paid of ₹25,000 in this example. The buyer of a call option "breaks even" only if the price of the underlying crosses a certain value that can be calculated as strike price + premium paid; in this example ₹7200. For the seller of a call option, the opposite applies, i.e. the seller of the call option has the obligation to sell the underlying at the strike price even if the spot price of the underlying is larger than the strike price. Intuitively, the call buyer wants the spot price of the underlying to rise while the call seller wants the spot price of the underlying to stay below the strike price. 

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Put option: When someone buys a put option for a given quantity, they get the right, but not the obligation, to sell that quantity of the underlying asset at the pre-determined price. The amount of money paid to buy the call option is called the premium, while the pre-determined price is called the strike price.

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Example: If someone sells a put on 125 shares (here, corresponding to 1 lot) with the strike price of ₹7000 on the stock "XYZ", and pays ₹250 for that call, then they get the right to sell 125 shares of the stock at a price of ₹7000 irrespective of whether the price is lower than ₹7000 or not. The total premium paid is equal to ₹250 * 125 = ₹31,250. Should the price rise above ₹7000, then the buyer of the put option merely forefits the premium paid; in other words, the buyer has a fixed, but known maximum loss which is equal to the premium paid of ₹31,250 in this example. The buyer of a put option "breaks even" only if the price of the underlying crosses a certain value that can be calculated as strike price - premium paid; in this example ₹6750. For the seller of a put option, the opposite applies, i.e. the seller of the put option has the obligation to buy the underlying at the strike price even if the spot price of the underlying is lower than the strike price. Intuitively, the put buyer wants the spot price of the underlying to fall while the put seller wants the spot price of the underlying to be larger than the strike price.

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Delta: Delta is the rate of change in the option price for a unit change in the underlying price, i.e. by how much does the option price change if the underlying increases or decreases by 1 unit, if all other variables such as time, volatility etc are kept constant.

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Gamma: Gamma is the rate of change of delta for a unit change in the underlying price. It tells us by how much the delta will change if the price changes by 1 unit with all other influencing variables like time, volatility, etc are kept constant.

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Vega: Vega is the rate of change of option price for a unit change in volatility. A rapid rise in volatility can lead to large mark-to-market losses on short options.

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Theta: Delta is the rate of change in the option price for a unit change in time, i.e. by how much does the option price change if time changes by 1 unit, e.g. 1 day, if all other variables such as price, volatility, etc are kept constant.

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Rho: Rho is the sensitivity of the option price to the risk free interest rate.

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Implied volatility: The implied volatility is the volatility of an option that is calculated by using option price and other inputs such as underlying price, strike price, days to expiry, and the interest rate as an input and reverse solving for the variable volatility. The calculation involves using an iterative process such as Newton Raphson's method etc.

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What are the formulas used in this options pricing calculator?

This options pricing calculator uses the formulas from the Black-Scholes Model mentioned above.

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What does this option pricing Greeks calculator do?

This calculator uses the inputs: days to expiry, underlying price, option strike price, the risk free interest rate and the implied volatility to calculate the fair price of a call or put option along with the Greeks delta, gamma, vega, theta, and rho. In this calculator, the impact of the dividend yield is neglected.

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What is the impact of changing one or more input variables on the option price and option Greeks?

This calculator has 5 inputs and 6 outputs each for a call and put option. Rather than discussing each possible case, let us look at the most important effects.

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Impact of time on the option price and Greeks

Ceterus paribus, as days to expiry (time) reduces, the option price, delta, and vega reduces accompanied by an increase in theta and gamma. For in-the-money options, the option price would lose its time value and eventually expire at a price equal to or very close to the amount by which the option is in the money. For an out-of-the-money option, the option would expire worthless, i.e. at 0.

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Impact of volatility on the option price and Greeks

Ceterus paribus, as volatility increases, the option price, delta, and theta increases, while gamma and vega decrease. A higher volatility increases the option price irrespective of whether they are at-the-money, in-the-money, or out-of-the-money. Volatility usually rises when uncertainty is high, e.g. before declaration of corporate results, important meetings by central banks, initiation of major investigations or legal cases against the company etc. Once uncertainty passes, the implied volatility usually falls and as a result prices also reduce.  

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Impact of the underlying price on the option price and Greeks

Ceterus paribus, as the underlying price increases, for a call option, the delta and the option price increases, while theta, gamma and vega decrease. For a put option, the delta and option price will decrease as the underlying price increases. For a call and puts options, when the underlying price increases/decreases such that the options become in-the-money, the time value of the option reduces and the intrinsic value of the option increases. For out-of-the-money options, the entire option price is equal to the time value. As the underlying price changes, such that a call or put option becomes further out-of-the-money, the option price reduces.  

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Why is this calculator useful?

This option pricing and option Greeks calculator is useful as it helps calculate the fair price of an option along with the Greeks. Knowing the price for a given implied volatility is useful especially while pricing illiquid options. Knowing the Greeks can help with risk management and the design of option strategies based on risk preferences.

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How do I use this option pricing calculator?

Instructions to use this option pricing calculator are provided above.