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## Valuation of Options

The value of an option can be estimated using a variety of quantitative techniques based on the concept of risk neutral pricing and using stochastic calculus. In general, standard option valuation models depend on the following factors:

• The current market price of the underlying security
• The strike price of the option, particularly in relation to the current market price of the underlying asset (in the money vs. out of the money)
• The cost of holding a position in the underlying security, including interest and dividends
• The time to expiration together with any restrictions on when exercise may occur, and
• An estimate of the future volatility of the underlying security's price over the life of the option.

### Profit & Loss on Call options

An option contract is profitable, "In-the-money" (ITM) if the market price is more than the contracted strike price. For example, if the contracted strike price is 1000 Fils, and if the market price at expiration is 1400 Fils; the contract buyer profits 400 Fils. An option contract is non-profitable, "Out-of-the-money"(OTM) if the market price is less than the contracted strike price. As in the last example, if the market price declines to 700 Fils while the contracted strike price is 1000 Fils; the contract buyer will not exercise the contract as he will not buy a stock for 1000 Fils while its market price is 700 Fils. An option contract becomes “At-the-money” (ATM) when the market price equals the strike price.

 Option classification Call option contracts In-the-money Market price > strike price Out-of-the-money Market price < strike price At-the-money Market price = strike price

### Pricing fundamentals

An option premium (Ask Price) has two components, its intrinsic value (also called fundamental value) and a time value.

Option Premium = Intrinsic Value + Time Value

The intrinsic value is the value that any given option would have if it were exercised today. An option has intrinsic value if the option is in-the-money. When out-of-the-money, its intrinsic value is zero. The intrinsic value (IV) for an in-the-money option is calculated as the absolute value of the difference between the current price (S) of the underlying and the strike price (K) of the option, floored to zero.

For a call option

IVcall = max {0, S - K}

While, for a put option

IVput = max {0, K - S}

The time value is the residual value of the option after negating its intrinsic value from the option premium. It is basically the risk premium that the seller requires to provide the option buyer with the right to sell/buy the underlying up to the expiration date.

Time value = Premium – Intrinsic value

As the contract draws closer to expiration, the time value erodes and this erosion becomes rapid toward expiration till it reaches zero at the expiration date. Hence, the premium at the expiration date equals only the intrinsic value of the option contract.

### Cox, Ross, Rubinstein  Model

The binomial model breaks down the time to expiration into potentially a very large number of time intervals, or steps.  A tree of stock prices is initially produced working forward from the present to expiration.  At each step it is assumed that the stock price will move up or down by an amount calculated using volatility and time to expiration.  This produces a binomial distribution, or recombining tree, of underlying stock prices. The tree represents all the possible paths that the stock price could take during the life of the option.  At the end of the tree i.e. at expiration of the option, all the terminal option prices for each of the final possible stock prices are known as they simply equal their intrinsic values. ### Corporate Actions

In situations where the precise size of the dividends and the actual time (ex date) can be determined in advance, the option price will incorporate the expected dividends. As far as possible, any adjustments for corporate actions will be made on the last trading day of the options on a cumulative basis.

As the company distributes cash dividends and/or bonus shares, the corresponding number of shares outstanding in the market increases and the share price of the stock decreases to reflect the amount of dividends distributed. Accordingly, the option contract components such as the strike price and contract size are adjusted to reflect the change in stock price as follows:

• New Strike Price: The new strike price shall be arrived at by dividing the old strike price by the adjustment factor (Multiplier M).
• New Contract Size: The new contract size shall be arrived at by multiplying the number of shares by the adjustment factor.

The adjustment factor or the multiplier shall be computed as follows:

Multiplier M = (P / (P – D)) * (1 + S)

Where,

P : is the closing price of the stock one day before the ex-date (cum date)

D : is the cash dividend declared

S : is the stock dividend ratio

New Strike = Old Strike / M

New number of shares = Old number of shares * M

The strike price is rounded off to the nearest tick and the number of shares is rounded off to avoid any fraction value.

For example:

Consider a stock with a price of KD 2.340 paying a dividend of 55 fils and a bonus share of 12%. A call option contract of 10,000 shares expiring in three months and a strike price of 2.340 is assumed to have a price of 196 fils.

The multiplier M is computed as M = 2.340 /(2.340-0.055) * (1+0.12) = 1.147

The revised strike price = Old strike / Multiplier = 2.340/1.147 = 2.041 (rounded to the nearest tick 2.040)

New number of shares = 10,000 * 1.147 = 11,470 shares