# Understanding Black-Scholes Merton closed form formula for Vanilla Call Option

Call (premium) =  S0 N(d1)  -  Ke-r(T-t) N(d2)

To understand this equation we need to start with payoff of call option at maturity.

Max( ST – K, 0)  =>  If at maturity Stock price(ST) ends above Strike(K), Options buyer gets Stock worth ST and pays Strike K for it.

The value of a derivative contract is expected present value of its future cash flows discounted at risk-free rate (under risk-neutral pricing framework). Hence, above equation can be divided into two parts.

1. A) S0 N(d1): Present value of expected Stock price received by the buyer of Option.
2. B) Ke-r(T-t) N(d2): Present value of expected cash flow (Strike) paid by buyer.

Starting with B as it is easy to understand:

B represents expected present value of Strike that is paid to obtain underlying by buyer in case call option matures in the money.

As stated above there are two important terms to consider: present value and expected value.

Present value: As Strike K is constant, so under risk neutral frame-work,  discounting rate of risk-free rate is used to calculate present value. Ke-r(T-t)

Expected value: Strike is only paid when option matures ITM. So to calculate the expected present value of cash-flow, present value mentioned as above needs to be multiplied by the probability of Option ending in the money, which is N(d2).

*Important point: Here is that N(d2) is not actual probability but probability under the risk-neutral framework. To get actual probability all required is to replace risk-free interest rate with actual expected return on the stock.

Coming back to part A:

As similar to discussed above, A is expected present value of Stock received by the buyer. Unlike part B, explanation of constituent terms of A is not straight forward. S0 represents the present value of stock and hence we might think that similar to part B, to get expected value we need to multiply the present value of Stock, S0 with the probability of Option maturing ITM. But in part A Sis multiplied by N(d1). Two probability terms used in part A “N(d1)” and Part B “N(d2)” are different. This is because N(d1) is not a pure probability that option will mature ITM but is: Conditional expected  present value of the unit value of the stock given stock ends in ITM. Which when multiplied by S0 simply gives: Conditional expected present value of the stock given stock ends in ITM.

E[ST/given(ST>K)]e-r(T-t)

Why part A “S0 N(d1)” could not be simplified as part B?

In part B cash-flow in the case of Option maturing ITM was constant “K”. Thus, the two terms present value and probability could be separated. While in the case of part A option buyer receives stock whose value at maturity is not fixed and linked to probability distribution and hence probability term couldn’t be completely separated from expected value.

Conclusion:

S0 N(d1) – Expected present value of what buyer receives as Stock.

Ke-r(T-t) N(d2): Expected present value of what buyer have to pay as Strike.

N(d2): Probability of option maturing ITM (under risk-neutral framework).

N(d1): Conditional expected  present value of a unit value of the stock given stock ends in ITM. And N(d1) is also Delta of an option.

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The Way of Choosing ClothesRunning is either a pleasure or a problemin the vocation.To begin with, it is a way of touristing.However, you have adifficult in adjusting to run in your holiday.If you want to wait for eveningto run, it is especially hard.You may feel so tired at that time and be eagerto have a hot bath or have a drink in the wineshop.It may have dangerous torun in the night because of unknowing condition of the road.Running in themorning is the best way to solve problem.For all this, you may always meetaccidental things.One of my best cheerful experiences of running is in theWeston Sue wares MEL of Englandin the night.My wife and I drive the car a day and I feel so tired.The hotelwhere we live is in the seaside.It is a huge Victorian architecture and towersstands everywhere.To make me happy is that there is two miles plank road withthe light shining.Before going out I wear the sport clothes.It is a warmevening and then the smell of sea flutters in the sky.Suddenly, I get recoveryand then begin running relax and unsurely.A Scottish set up a clinic in theWeston Sue wares MEL He is also taking exercise and we are running together fora while.
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22. MixEdura says:
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while the nations parks, books and afterward going on country towns are a significant charm for ecotourists and those similar to African lifestyle, for several site visitors it is actually the nations okay tropical climate and loosening.
The Way of Choosing ClothesRunning is either a pleasure or a problemin the vocation.To begin with, it is a way of touristing.However, you have adifficult in adjusting to run in your holiday.If you want to wait for eveningto run, it is especially hard.You may feel so tired at that time and be eagerto have a hot bath or have a drink in the wineshop.It may have dangerous torun in the night because of unknowing condition of the road.Running in themorning is the best way to solve problem.For all this, you may always meetaccidental things.One of my best cheerful experiences of running is in theWeston Sue wares MEL of Englandin the night.My wife and I drive the car a day and I feel so tired.The hotelwhere we live is in the seaside.It is a huge Victorian architecture and towersstands everywhere.To make me happy is that there is two miles plank road withthe light shining.Before going out I wear the sport clothes.It is a warmevening and then the smell of sea flutters in the sky.Suddenly, I get recoveryand then begin running relax and unsurely.A Scottish set up a clinic in theWeston Sue wares MEL He is also taking exercise and we are running together fora while.
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though the nations parks, books and also in the works country towns are a significant fascination for ecotourists and those taking into consideration African lifestyle, for several site visitors it is actually the nations normal tropical climate and loosening.
The Way of Choosing ClothesRunning is either a pleasure or a problemin the vocation.To begin with, it is a way of touristing.However, you have adifficult in adjusting to run in your holiday.If you want to wait for eveningto run, it is especially hard.You may feel so tired at that time and be eagerto have a hot bath or have a drink in the wineshop.It may have dangerous torun in the night because of unknowing condition of the road.Running in themorning is the best way to solve problem.For all this, you may always meetaccidental things.One of my best cheerful experiences of running is in theWeston Sue wares MEL of Englandin the night.My wife and I drive the car a day and I feel so tired.The hotelwhere we live is in the seaside.It is a huge Victorian architecture and towersstands everywhere.To make me happy is that there is two miles plank road withthe light shining.Before going out I wear the sport clothes.It is a warmevening and then the smell of sea flutters in the sky.Suddenly, I get recoveryand then begin running relax and unsurely.A Scottish set up a clinic in theWeston Sue wares MEL He is also taking exercise and we are running together fora while.
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while the nations parks, books and next stirring country towns are a significant charm for ecotourists and those taking into account African lifestyle, for several site visitors it is actually the nations enjoyable tropical climate and loosening.
The Way of Choosing ClothesRunning is either a pleasure or a problemin the vocation.To begin with, it is a way of touristing.However, you have adifficult in adjusting to run in your holiday.If you want to wait for eveningto run, it is especially hard.You may feel so tired at that time and be eagerto have a hot bath or have a drink in the wineshop.It may have dangerous torun in the night because of unknowing condition of the road.Running in themorning is the best way to solve problem.For all this, you may always meetaccidental things.One of my best cheerful experiences of running is in theWeston Sue wares MEL of Englandin the night.My wife and I drive the car a day and I feel so tired.The hotelwhere we live is in the seaside.It is a huge Victorian architecture and towersstands everywhere.To make me happy is that there is two miles plank road withthe light shining.Before going out I wear the sport clothes.It is a warmevening and then the smell of sea flutters in the sky.Suddenly, I get recoveryand then begin running relax and unsurely.A Scottish set up a clinic in theWeston Sue wares MEL He is also taking exercise and we are running together fora while.
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25. MixEdura says:
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though the nations parks, books and along with in the works country towns are a significant appeal for ecotourists and those later than African lifestyle, for several site visitors it is actually the nations pleasing tropical climate and loosening.
The Way of Choosing ClothesRunning is either a pleasure or a problemin the vocation.To begin with, it is a way of touristing.However, you have adifficult in adjusting to run in your holiday.If you want to wait for eveningto run, it is especially hard.You may feel so tired at that time and be eagerto have a hot bath or have a drink in the wineshop.It may have dangerous torun in the night because of unknowing condition of the road.Running in themorning is the best way to solve problem.For all this, you may always meetaccidental things.One of my best cheerful experiences of running is in theWeston Sue wares MEL of Englandin the night.My wife and I drive the car a day and I feel so tired.The hotelwhere we live is in the seaside.It is a huge Victorian architecture and towersstands everywhere.To make me happy is that there is two miles plank road withthe light shining.Before going out I wear the sport clothes.It is a warmevening and then the smell of sea flutters in the sky.Suddenly, I get recoveryand then begin running relax and unsurely.A Scottish set up a clinic in theWeston Sue wares MEL He is also taking exercise and we are running together fora while.
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The player must battle Skynet enemies from the upcoming film as well as new enemies specifically designed for the game.Red Faction: GuerrillaIt is an upcoming third person shooter video game in development by Volition, Inc.and published by THQ.The game is set to be released June 2, 2009 for the Xbox 360, PlayStation 3, and Microsoft Windows platforms.The game is the third installment in the Red Faction series.In the gameplay of Red Faction: Guerrilla, the protagonist, Alec Mason, will be able to run, jump, hijack vehicles, and use weapons to traverse across the open world of Mars and fight the EDF Earth Defense Force.Weapons in the game will include: machine gun turrets, rocket turrets, remote charges, a sledgehammer, machine guns, and many others.PrototypePrototype is an upcoming sandbox style action game in development by Radical Entertainment and based on an original IP developed in house.The game is slated for release on June 9th, 2009 for the Xbox 360, Playstation 3 and Microsoft Windows.

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