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Black-Scholes Option Pricing Formula
In their 1973 paper, The Pricing of Options and Corporate Liabilities, Fischer Black and Myron Scholes published an option valuation 歡迎登陸 Real Options Valuation 網站 formula that today is known as the Black-Scholes model. It has become the standard method of pricing options.
The Black-Scholes formula calculates the price of a call option to be:
|C = price of the call option|
|S = price of the underlying stock|
|X = option exercise 歡迎登陸 Real Options Valuation 網站 price|
|r = risk-free interest rate|
|T = current time until expiration|
|N() = area under the normal curve|
|d1 = [ ln(S/X) + (r + σ 2 /2) 歡迎登陸 Real Options Valuation 網站 歡迎登陸 Real Options Valuation 網站 歡迎登陸 Real Options Valuation 網站 T ] / σ T 1/2|
|d2 = d1 - σ T 1/2|
Put-call parity requires that:
Then the price of a put option is:
The Black-Scholes model assumes that the option can be exercised only at expiration. It requires that both the risk-free rate and the 歡迎登陸 Real Options Valuation 網站 volatility of the underlying stock price remain constant over the period of analysis. The model also assumes that the underlying stock does not pay dividends; adjustments can be made to correct for such distributions. For example, the present value of estimated dividends can be deducted from the stock price in the model.
Warrants are call options issued by a corporation. They tend to have longer durations than do exchange-traded call options. Warrants can be valued by the Black-Scholes model, but some modifications must be made to the parameters.
When warrants are exercised, the company typically issues new shares at the exercise price to fill the order. The resulting increase in shares outstanding dilutes the share value. If there 歡迎登陸 Real Options Valuation 網站 歡迎登陸 Real Options Valuation 網站 were n shares outstanding, and m warrants are exercised, α 歡迎登陸 Real Options Valuation 網站 represents the percentage of the value of the firm that is represented by the warrants, where
When using the Black-Scholes model to value the warrants, it is worthwhile to use total amounts instead of per share amounts in order to better account for the dilution. The current share price S becomes the enterprise value (less debt) to be acquired by the warrant holders. The exercise price is the total warrant exercise amount, adjusted for the fact that in paying cash to the firm to exercise the warrants, the warrant holders in effect are paying a portion of the cash, α, to themselves.
The inputs to the Black-Scholes model for both 歡迎登陸 Real Options Valuation 網站 option pricing and warrant pricing are outlined in the following 歡迎登陸 Real Options Valuation 網站 table.
Black-Scholes Parameters for Pricing Options and Warrants
|Input Parameter||Option Pricing||Warrant Pricing|
|S||current share price||α V, where V is enterprise value minus debt.|
|X||exercise price per share||total warrant exercise 歡迎登陸 Real Options Valuation 網站 amount multiplied|
by (1 - α).
|T||current time to expiration||average T for warrants|
|r||interest rate||interest rate|
|σ||standard deviation of stock return||standard deviation for returns on enterprise value, including warrants|
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real option valuation
Dear Sir, as well as I understood the value of option comprices intrinsic value and time value. The intrinsic value 歡迎登陸 Real Options Valuation 網站 is difference between asset price and exercise price.
In the technical article related real option valuation there was example with the following initial data:
Asset Value (Pa) = $90m
Exercise price (Pe) = $140m
Exercise date (t) = Four years 歡迎登陸 Real Options Valuation 網站 歡迎登陸 Real Options Valuation 網站
Risk free rate (r) = 5%
Volatility (s) = 40%
as the result of call option calculation 20.8
So, intrinsic value = 60, and the time value 20.8-60=-39.2
How it could be explained taking that option value is always 歡迎登陸 Real Options Valuation 網站 positive?
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The intrinsic value is the amount you would pay for an option if you could 歡迎登陸 Real Options Valuation 網站 exercise it now.
On your figures, the intrinsic value is 歡迎登陸 Real Options Valuation 網站 zero. Nobody would pay 140 for something worth 90 :-).
(Had the asset value been 140 and the exercise price had been 90, then the intrinsic value would be 140 – 90 = 50. Obviously the value of the option would be completely different and not be 20.8.)
Real Option Valuation 1.0
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