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Valuation of options

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Title: Valuation of options  
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Subject: Finite difference methods for option pricing, Mathematical finance, Options (finance), Trinomial tree, Derivative (finance)
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Valuation of options

Further information: Option: Model implementation

In finance, a price (premium) is paid or received for purchasing or selling options. This price can be split into two components.

These are:

  • Intrinsic value
  • Time value

Intrinsic value

The intrinsic value is the difference between the underlying spot price and the strike price, to the extent that this is in favor of the option holder. For a call option, the option is in-the-money if the underlying spot price is higher than the strike price; then the intrinsic value is the underlying price minus the strike price. For a put option, the option is in-the-money if the strike price is higher than the underlying spot price; then the intrinsic value is the strike price minus the underlying spot price. Otherwise the intrinsic value is zero.

For example, when a DJI call (bullish/long) option is 18,000 and the underlying DJI Index is priced at $18,050 then there is a $50 advantage even if the option were to expire today. This $50 is the intrinsic value of the option.

In summary, intrinsic value:

= current stock price – strike price (call option)
= strike price – current stock price (put option)

Time value

The option premium is always greater than the intrinsic value. This extra money is for the risk which the option writer/seller is undertaking. This is called the Time value.

Time value is the amount the option trader is paying for a contract above its intrinsic value, with the belief that prior to expiration the contract value will increase because of a favourable change in the price of the underlying asset. The longer the length of time until the expiry of the contract, the greater the time value. So,

Time value = option premium – intrinsic value

Other factors affecting premium

There are many factors which affect option premium. These factors affect the premium of the option with varying intensity. Some of these factors are listed here:

  • Price of the underlying: Any fluctuation in the price of the underlying (stock/index/commodity) obviously has the largest impact 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 has an impact on 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.
  • Time till expiry: Lesser the time to expiry, option premium follows the intrinsic value more closely. On the expiry date Time Value approaches zero.
  • 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 let’s say a 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.

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.

Pricing models

Because the values of option contracts depend on a number of different variables in addition to the value of the underlying asset, they are complex to value. There are many pricing models in use, although all essentially incorporate the concepts of rational pricing, moneyness, option time value and put-call parity.

Amongst the most common models are:

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