Exercise 2 point 2

This forum is to discuss the book "the concepts and practice of mathematical finance" by Mark Joshi.

Exercise 2 point 2

Here's a detailed solution.
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ex2pt2_428.zip
mj

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Re: Exercise 2 point 2

Thanks! This is very useful
cggascon

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Re: Exercise 2 point 2

There is something which is not clear to me. In the solution, when it is stated: "substituting our constraint, we get: 1 - 10 alpha" Which constraint is it referring to?
I have just taken this book for self learning without previous real background in finance ( I studied image processing and I am currently working in equities cash ), so I just want to make sure I fully understand the book all the way through.
Thanks
cggascon

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Re: Exercise 2 point 2

110 alpha + beta = 1
mj

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Re: Exercise 2 point 2

Hi.
I hope you can explain some more on this.
So in part I, we have the lower bound is 0 and upper bound is 100/110 which means the initial price must lie some where between 0 and 100/110 and it is defined by the function 1-10*alpha. So what if we make alpha smaller by making Beta > 0. For e.g. alpha = 0.5/110 so that B = 1 - 0.5 = 0.5 and 1 - 10 * (0.5/110) = 105/110 > 100/110 which is our upper bound. So what is the flaw in my argument ?
chituan

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Re: Exercise 2 point 2

well if you increase beta you get a higher upper bound.

But we want the best upper bound, i.e. the lowest one.
mj

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Re: Exercise 2 point 2

It was said that the portfolio had \alpha shares of the security and \beta shares of stock. Given the stock price 100, why the initial value of the portfolio is 100\alpha + \beta ?
wenge

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Re: Exercise 2 point 2

it should say alpha stocks and beta bonds.
mj

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Re: Exercise 2 point 2

Did not want to start a separate section for this. I have two questions - both of them apply equally to all parts of this question so I will just use part 1 as an example:

1) When I am looking at the payoff graph I can say that super-replicating portfolio has to go through points 0 and 110. What I do not understand is why I cannot (and should not) use the alternative upper bound method used in example on p.39 of 2nd edition of book? Using this method the upper boundary would be min(super-replicating portfolio, P), where P is the cost of zero-coupon bond expiring at time of maturity of the option.

2) While my solution matches what was posted I do not use the cost constraint and I do not really understand the importance of having to consider the cost constraint. Is there an example where the constraint I am ignoring would make a difference in finding the super-replicating portfolio?
mikek

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Re: Exercise 2 point 2

the dominating ZCB gives an upper bound but it generally won't be optimal.

The constraint given expresses that the dominating portfolio intersects the pay-off graph. If it doesn't then it can probably be moved down to get a better upper bound.
mj

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Re: Exercise 2 point 2

Thank you for the explanation.
mikek

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Re: Exercise 2 point 2

someone can help?

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The file ./../files/2_3a673eda9b595f6245b658b06c384922 does not exist."
facmello

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Re: Exercise 2 point 2

if you look in the second edition, there should be the full solution.
mj

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Re: Exercise 2 point 2

Hi,

I have a simple question that I couldn't relate to the provided answer.

Where did the 1-10*Alpha came from ? is it from

110*Alpha + Beta = 1 Needed Maturity Price of the portfolio
100*Alpha + Beta Spot ( initial Price ) of the portfolio
-------------------------
Subtracting them from each other

10*alpha = 1

---> 1- 10*Alpha ?

Durazi

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Re: Exercise 2 point 2

Sorry,

To complete my question, as if

110*Alpha + Beta = 1 Maturity
100*Alpha + Beta = 0 Spot

it implies Arbitrage ?!
Durazi

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