5

This question is on option pricing using Black-Scholes The Black-Scholes pricing problem for European binary call Op tion is given by the partial differeutial equat...

Question

This question is on option pricing using Black-Scholes The Black-Scholes pricing problem for European binary call Op tion is given by the partial differeutial equation and final condition aV 02V OV 5 > E tr5' CtV =0. V(ST) = 952 ds 5 < E V (S,t) is the option value, S is the underlying asset price, is time. T is expiry; is the constant risk-free interest rate, is the constant volatility and E is the strike price. By introducing the following change of variables log { = 29"(T _ t)

This question is on option pricing using Black-Scholes The Black-Scholes pricing problem for European binary call Op tion is given by the partial differeutial equation and final condition aV 02V OV 5 > E tr5' CtV =0. V(ST) = 952 ds 5 < E V (S,t) is the option value, S is the underlying asset price, is time. T is expiry; is the constant risk-free interest rate, is the constant volatility and E is the strike price. By introducing the following change of variables log { = 29"(T _ t), ~(r.t) =V(S.t) . show that the problem can be reduced to 0" ~ O~ {1 I > 0 aw (1.0) = 0r2 Or I <0 where a = 2r/o"_ Show that the term ay can be eliminated by further transformation of variable (1,0) =e-au (1.t) Finally show that the drift term will disappear by putting =I+(a = 1)t' . T =t resulting in the diffusion equation and final condition 0 u 7> 0 u (:. 0) 0-2 3<0 The solution of the above diffusion equation is given in terms of the fundamental solution by (y-=)?[4t ~> u(at) = f(u)e dy: u(:.0) = f(:) = 2V 4u Use this to calculate the option value and show that it is given by V(S.t) r(T-')N(d2)-



Answers

The market demand and market supply functions for a product are given by the following
equations.
QD = 100 – 3P
QS = 10 + 2P
Where, QD
is the quantity demanded, QS
is the quantity supplied and P is the price
of the product
a) Does the market exhibit an excess demand or excess supply at the prices 10 and 20?
Explain. (1 mark)

In this problem. The market demand and markets supply functions for the products are given by the following equations. Right? So here are the equation for QD. Right? So this is actually quantity demanded is represented by QD And quantity supplied is represented by Qs. Okay, So now here it has given QD as 100 -3 b. and second equation is cures. That is quantity supplied. It was true 10 plus two P and P is nothing. But this is a price. Okay? So now that I have already told that QD our security is quantity demanded and QS quantity supplied and P is the price of the product. It could be in dollars. It could be anything, quantity could be anything. Right? And it could be in thousands. It is not given in the problem. But you can take it as in thousands as well. But I'm taking as as the units are not given to us. So that's why I'm taking The same thing which is given to us that it's 100 only. So it will be 100. Okay, So now for the first one, it disaster. Does the market exhibit an excess demand or excess supply at the prize is 10 and 20. Okay. So it is asked uh like this. Right? So, how can we find it, first of all? We need to draw a graph. Right. So for that uh I will solve for 2nd part 1st. Why? Because we need to draw Deep Draft 1st. Right? So here here, second, when it's draw the market on demanding market supply girls for the product on the tiger. Okay, So now, what I can see here is Okay, so you're here are the girls between price and quantity demanded and second uh of says between price and quantity supplied. All right. So now I can make according to this, I have put price 0 10 2030 40 by myself because for 10 and printed has asked that's why I have taken the Difference of 10. Right? You can see and take anything and we will just put this value and like P is the variable, right? So we can put the value of p particular p. and put it in into QD and Q. S equations. So, let us see how the girls can comes out. Right? So, this is the table I have given for QD and Q. S. When we will put it uh for 0 10 2030 and 40. We will get the perspective and using the given in the table in this screen. Yeah. Mhm. Okay, So from here, I can put this value or like 0 10 2030 40. And I'll see the points meeting QD and Q. T. And be right. So here, I can get this point this like this. So here, is that good for price versus quantity demanded? Similarly for quantity supplied, It will be from here like this. Right? So, we are going to put just the points uh Q. S. P. And taking P like zero And here values 10 for quantity supplied, right? So similarly we are doing for all values right? What we have assumed here. Okay, so now what we can do finally we can get this value it as like this. Uh I'm not throwing this is a rough diagram because I don't have that graphing calculator. But yes. Uh it can be 10. Right? I can write it as like this. And here it is 100. Right, Okay, so now this is 70 Right? And you can write your it does this is this is some point. Okay, So it should be at this point should be minus 20 year. Okay, so I can make it a zero. This is somewhere between 30 and 40 is one making it as 30 and 40 like this. Okay, so this is price yeah, what it is us it is us market demand and market supply curves. Right? So this isn't this is quantity jake. So this is complete girl. Right? This isn't dollars. Right? So remember quantity could be in thousands. So the value will change right? Because we have at the end of the answer we have processed multiplied by 1000. Okay, if it is it isn't 1000 spelling. Okay, so this we have got to go right, so this is for part B we can say right, so this is part B. I can write yes as it is said for separately for drawing the demand and supply curves. So you can easily take this, you can see this girl and this girl. Okay. No. Right. It is asked. Okay, now it is us that now we will solve the first part, what it is said that it is said that does the market exhibit excess demand or excess supply at the prices 10 n. Wendy. Okay, Okay. So we will just see the quantities what we are getting here at 10 and 20. Right? So here the QD is 70 right? And us is 30, Similarly for 20, it is, Fury is 40 and 50 is for Q. S. Right? So this implies that for part A. We can right this in place that four price garden, Yeah jake whichever is the larger country that will be having the axis quantity. Right, So 70 -30, that means demand. There is an access demand of 40,000. Thanks uh 70 minus 30 equals to 40 units. Their food. There is and exist demand similarly. What we can see here is for price dollar 20 what we can see here at dollar 20 This was dollar 10 and this is dollar 20. So here this becomes at 40. Mhm. There is an access supply by because 14 minus fifties equals two minus 10 units. Right? That means there is and excess supply supply is more and demand is nice. Right? So this we have found out this is the answer for this problem. Once great. So now moving forward, this is us. That what is what are the equilibrium price and equilibrium quantity off the product, equilibrium quantity and price of the product? Right. So, yeah. So here we can write equilibrium price. So we will just equalize both equations QD and Q S. Right? So here, 100 minus three P equals to 10 plus two P. That will give us prices $ 18. Right? So this is our answer for the first part of it. That is equilibrium price, price will be equilibrium price will be equal to dollar 18. Right, Similarly, equilibrium quantity traded will be we will just put instead of P uh in QD equation we will just put it and we will get here 100 -3 into 18. That will give us 46 s answer. So this is 46 units is you know, equilibrium quantity. So you did late. Okay, so all you can say this is equilibrium quantity of the product. Right? So this is how we solve this problem. I hope your industry thank you for watching

In this problem. The market demand and markets supply functions for the products are given by the following equations. Right? So here are the equation for QD. Right? So this is actually quantity demanded is represented by QD And quantity supplied is represented by Qs. Okay, So now here it has given QD as 100 -3 b. and second equation is cures. That is quantity supplied. It was true 10 plus two P and P is nothing. But this is a price. Okay? So now that I have already told that QD our security is quantity demanded and QS quantity supplied and P is the price of the product. It could be in dollars. It could be anything, quantity could be anything. Right? And it could be in thousands. It is not given in the problem. But you can take it as in thousands as well. But I'm taking as as the units are not given to us. So that's why I'm taking The same thing which is given to us that it's 100 only. So it will be 100. Okay, So now for the first one, it disaster. Does the market exhibit an excess demand or excess supply at the prize is 10 and 20. Okay. So it is asked uh like this. Right? So, how can we find it, first of all? We need to draw a graph. Right. So for that uh I will solve for 2nd part 1st. Why? Because we need to draw Deep Draft 1st. Right? So here here, second, when it's draw the market on demanding market supply girls for the product on the tiger. Okay, So now, what I can see here is Okay, so you're here are the girls between price and quantity demanded and second uh of says between price and quantity supplied. All right. So now I can make according to this, I have put price 0 10 2030 40 by myself because for 10 and printed has asked that's why I have taken the Difference of 10. Right? You can see and take anything and we will just put this value and like P is the variable, right? So we can put the value of p particular p. and put it in into QD and Q. S equations. So, let us see how the girls can comes out. Right? So, this is the table I have given for QD and Q. S. When we will put it uh for 0 10 2030 and 40. We will get the perspective and using the given in the table in this screen. Yeah. Mhm. Okay, So from here, I can put this value or like 0 10 2030 40. And I'll see the points meeting QD and Q. T. And be right. So here, I can get this point this like this. So here, is that good for price versus quantity demanded? Similarly for quantity supplied, It will be from here like this. Right? So, we are going to put just the points uh Q. S. P. And taking P like zero And here values 10 for quantity supplied, right? So similarly we are doing for all values right? What we have assumed here. Okay, so now what we can do finally we can get this value it as like this. Uh I'm not throwing this is a rough diagram because I don't have that graphing calculator. But yes. Uh it can be 10. Right? I can write it as like this. And here it is 100. Right, Okay, so now this is 70 Right? And you can write your it does this is this is some point. Okay, So it should be at this point should be minus 20 year. Okay, so I can make it a zero. This is somewhere between 30 and 40 is one making it as 30 and 40 like this. Okay, so this is price yeah, what it is us it is us market demand and market supply curves. Right? So this isn't this is quantity jake. So this is complete girl. Right? This isn't dollars. Right? So remember quantity could be in thousands. So the value will change right? Because we have at the end of the answer we have processed multiplied by 1000. Okay, if it is it isn't 1000 spelling. Okay, so this we have got to go right, so this is for part B we can say right, so this is part B. I can write yes as it is said for separately for drawing the demand and supply curves. So you can easily take this, you can see this girl and this girl. Okay. No. Right. It is asked. Okay, now it is us that now we will solve the first part, what it is said that it is said that does the market exhibit excess demand or excess supply at the prices 10 n. Wendy. Okay, Okay. So we will just see the quantities what we are getting here at 10 and 20. Right? So here the QD is 70 right? And us is 30, Similarly for 20, it is, Fury is 40 and 50 is for Q. S. Right? So this implies that for part A. We can right this in place that four price garden, Yeah jake whichever is the larger country that will be having the axis quantity. Right, So 70 -30, that means demand. There is an access demand of 40,000. Thanks uh 70 minus 30 equals to 40 units. Their food. There is and exist demand similarly. What we can see here is for price dollar 20 what we can see here at dollar 20 This was dollar 10 and this is dollar 20. So here this becomes at 40. Mhm. There is an access supply by because 14 minus fifties equals two minus 10 units. Right? That means there is and excess supply supply is more and demand is nice. Right? So this we have found out this is the answer for this problem. Once great. So now moving forward, this is us. That what is what are the equilibrium price and equilibrium quantity off the product, equilibrium quantity and price of the product? Right. So, yeah. So here we can write equilibrium price. So we will just equalize both equations QD and Q S. Right? So here, 100 minus three P equals to 10 plus two P. That will give us prices $ 18. Right? So this is our answer for the first part of it. That is equilibrium price, price will be equilibrium price will be equal to dollar 18. Right, Similarly, equilibrium quantity traded will be we will just put instead of P uh in QD equation we will just put it and we will get here 100 -3 into 18. That will give us 46 s answer. So this is 46 units is you know, equilibrium quantity. So you did late. Okay, so all you can say this is equilibrium quantity of the product. Right? So this is how we solve this problem. I hope your industry thank you for watching

In this problem. The market demand and markets supply functions for the products are given by the following equations. Right? So here are the equation for QD. Right? So this is actually quantity demanded is represented by QD And quantity supplied is represented by Qs. Okay, So now here it has given QD as 100 -3 b. and second equation is cures. That is quantity supplied. It was true 10 plus two P and P is nothing. But this is a price. Okay? So now that I have already told that QD our security is quantity demanded and QS quantity supplied and P is the price of the product. It could be in dollars. It could be anything, quantity could be anything. Right? And it could be in thousands. It is not given in the problem. But you can take it as in thousands as well. But I'm taking as as the units are not given to us. So that's why I'm taking The same thing which is given to us that it's 100 only. So it will be 100. Okay, So now for the first one, it disaster. Does the market exhibit an excess demand or excess supply at the prize is 10 and 20. Okay. So it is asked uh like this. Right? So, how can we find it, first of all? We need to draw a graph. Right. So for that uh I will solve for 2nd part 1st. Why? Because we need to draw Deep Draft 1st. Right? So here here, second, when it's draw the market on demanding market supply girls for the product on the tiger. Okay, So now, what I can see here is Okay, so you're here are the girls between price and quantity demanded and second uh of says between price and quantity supplied. All right. So now I can make according to this, I have put price 0 10 2030 40 by myself because for 10 and printed has asked that's why I have taken the Difference of 10. Right? You can see and take anything and we will just put this value and like P is the variable, right? So we can put the value of p particular p. and put it in into QD and Q. S equations. So, let us see how the girls can comes out. Right? So, this is the table I have given for QD and Q. S. When we will put it uh for 0 10 2030 and 40. We will get the perspective and using the given in the table in this screen. Yeah. Mhm. Okay, So from here, I can put this value or like 0 10 2030 40. And I'll see the points meeting QD and Q. T. And be right. So here, I can get this point this like this. So here, is that good for price versus quantity demanded? Similarly for quantity supplied, It will be from here like this. Right? So, we are going to put just the points uh Q. S. P. And taking P like zero And here values 10 for quantity supplied, right? So similarly we are doing for all values right? What we have assumed here. Okay, so now what we can do finally we can get this value it as like this. Uh I'm not throwing this is a rough diagram because I don't have that graphing calculator. But yes. Uh it can be 10. Right? I can write it as like this. And here it is 100. Right, Okay, so now this is 70 Right? And you can write your it does this is this is some point. Okay, So it should be at this point should be minus 20 year. Okay, so I can make it a zero. This is somewhere between 30 and 40 is one making it as 30 and 40 like this. Okay, so this is price yeah, what it is us it is us market demand and market supply curves. Right? So this isn't this is quantity jake. So this is complete girl. Right? This isn't dollars. Right? So remember quantity could be in thousands. So the value will change right? Because we have at the end of the answer we have processed multiplied by 1000. Okay, if it is it isn't 1000 spelling. Okay, so this we have got to go right, so this is for part B we can say right, so this is part B. I can write yes as it is said for separately for drawing the demand and supply curves. So you can easily take this, you can see this girl and this girl. Okay. No. Right. It is asked. Okay, now it is us that now we will solve the first part, what it is said that it is said that does the market exhibit excess demand or excess supply at the prices 10 n. Wendy. Okay, Okay. So we will just see the quantities what we are getting here at 10 and 20. Right? So here the QD is 70 right? And us is 30, Similarly for 20, it is, Fury is 40 and 50 is for Q. S. Right? So this implies that for part A. We can right this in place that four price garden, Yeah jake whichever is the larger country that will be having the axis quantity. Right, So 70 -30, that means demand. There is an access demand of 40,000. Thanks uh 70 minus 30 equals to 40 units. Their food. There is and exist demand similarly. What we can see here is for price dollar 20 what we can see here at dollar 20 This was dollar 10 and this is dollar 20. So here this becomes at 40. Mhm. There is an access supply by because 14 minus fifties equals two minus 10 units. Right? That means there is and excess supply supply is more and demand is nice. Right? So this we have found out this is the answer for this problem. Once great. So now moving forward, this is us. That what is what are the equilibrium price and equilibrium quantity off the product, equilibrium quantity and price of the product? Right. So, yeah. So here we can write equilibrium price. So we will just equalize both equations QD and Q S. Right? So here, 100 minus three P equals to 10 plus two P. That will give us prices $ 18. Right? So this is our answer for the first part of it. That is equilibrium price, price will be equilibrium price will be equal to dollar 18. Right, Similarly, equilibrium quantity traded will be we will just put instead of P uh in QD equation we will just put it and we will get here 100 -3 into 18. That will give us 46 s answer. So this is 46 units is you know, equilibrium quantity. So you did late. Okay, so all you can say this is equilibrium quantity of the product. Right? So this is how we solve this problem. I hope your industry thank you for watching

To the market, demand and supply. So are Y axis has always price and our X axis is quantity. So let's start with um quantity demanded Q. D. Is equal to 100 -3 peak. Okay. Yeah. Mhm. So when price is equal to zero our demand R. Q. D. is going to be 100. Okay Yeah so let's start what's he do? Okay over here. Okay so then when price is say Just plug in 10 and 20? Yeah 10 20. Okay so when the price is 10 If we plug 10 in for p here We get that you d when p equals 10 Is 100 -30 so it's 70. So again this is 100 here this is going to be 70 and then Q. D. one p equals 20 Is going to be 100 -60 because three times 20 is 60 so that's 40 so 40 there we go. So now we've got three points we can draw this and it's not going to be perfect but we don't need it to be here we go. Cool. Okay so now quantity supplied Our line is 10-plus 2 p. So we can start in the same place say when P is zero When he is zero. R Quantity supplied is 10 zero. Okay so then when p equals 10 Q. S. Is equal to 10-plus 2 times 20 so 30. Okay that's got to be somewhere up here. Okay okay and then sorry that spot is not right uh silly me when prices 10 so along the Y axis when prices 10 the supply is which is a quantity, I'll move this a little bit 30 right there, that's better. Okay so then when the price equals 20 the quantity supplied is 10 plus two times 2010 plus 40 which is 50. So when the price is 20 we got 50 50. There we go. Okay so what do we have to figure out Um does the market exhibit excess demand or excess supply at prices too? And I mean 10 and 20 sorry so our first is going to be at Price Equal to 10. It looks like Um quantity supplied as we found here is equal to 30 and quantity demanded, Connect demanded is equal to 70. So in this case demand quantity demanded greatly exceeds quantity, supplies supplied. So quantity demanded is an excess. In our 2nd point At P equals 20 Our quantity supplied is 50 and our quantity demanded is 40. So in this case quantity supplied is an excess compared to quantity. Don't know why I wrote a five. Their quantity demanded. And if we look at this just a little bit further past the scope of the problem will find that this is our equilibrium point somewhere here. So this obeys the law of um the law of supply and demand right? As when the prices are lower there is higher demand. And when the prices are higher there is um less demand as you can see on this side and the opposite is true for supply. So when there's a lower price, the supplies lower, when there is a higher price of the good, the supply is higher. So that's how you solve this problem graphically, and this is how we find our answers again here, quantity supplied as an access and your quantity a mandate is in excess.


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