## Questions regarding Chp 3 and Chp 5

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

### Questions regarding Chp 3 and Chp 5

Hi,

I was wondering if anyone can help me with the following questions. Thanks in advance.

Chp3

Page 45. (Last sentence of 3.1.1 Pricing in a one-step tree by hedging) It has been shown, using the hedging argument, that the price of the option is 5. We then have the sentence "This implies that our option is worth 5 and that 5 is the only arbitrage-free price". I agree that 5 is indeed the price of the option. In fact, the law of one price guarantees that this price is unique. Hence we have shown uniquess of the price. However, where exactly has it been shown (in this section 3.1.1) that an option price of 5 is in fact arbitrage-free? How do we even know there exists a price for the option which is arbitrage free? Do we need the risk-neutral valuation to show that this is indeed the case?

Page 45. (Under 3.1.2 Risk-neutral valuation) It talks about how the hedging argument is independent of probabilities:
"...The argument remains valid regardless of what the probability of an up-jump is". It then goes on to say a probability of 1 for an up jump is not valid. Isn't this a bit contradictory: on one hand it says the argument works regardless of what probability is taken and the next statement says it will not work when the probability is 1. Is there an assumption here that a probability of 1 implies the certain event; that is, if the probability of an up jump is 1 then the only event that can occur is that of the up jump - there is no down jump.

Page 48. (Last paragraph of section 3.1.2) It says "Thus the risk-neutral price gives a lower bound on the set of arbitrage -free prices, and the hedging argument gives an upper bound". I was wondering, how is this deduction exactly obvious given the previous paragraphs?

Page 72 (Exercise 3.9). I was wondering if the following argument is valid. Since there is no riskless bond this is the same as saying the risk-free rate is r=0. Then use risk neutral valuation to get the price of A: 0.5*(110-100)=5.

Chp 5

Page 108 (the formula between formula 5.19 and 5.20). Here the bound is C|y-z|^3. I think this was deduced by applying thm D1 (page 470) where we have k+1 = 3. That is, we need the function f to be 3 times differentiable. However, thm 5.1 (page 110) says f is only twice differentiable. This effectively means the proof has used stronger assumptions than the statement in the thm. Is there a more rigorous proof that requires f to be only 2 times differentiable?

Page 116 (first sentence in section 5.8 ) "before proceeding to the solution, we prove that the solution to the equation is, in fact, the unique arbitrage-free price for the option". My question is similar to the first question I asked. I agree uniqueness (and also the price being well - defined) have definitely been shown in this section. However, which sentence shows that the price of the option is in fact arbitrage-free?

Chp 3 and 5

Page 61 and 112 (Stock follows GBM). In chp 3, section 3.7.1, it was argued (via CLT) that the stock has a log normal distribution where the first paramter is \mu * T. It was defined on page 60, that \mu is called the drift. However, in chp 5, it was shown (via SDE) that this parameter is in fact (\mu - 0.5 * \sigma^2 ) * T [formula 5.46]. On page 111, it was also said the \mu in this term stands for drift. Am I to interpret that \mu [first one] = (\mu [second one] - 0.5 * \sigma^2 )? Which \mu then refers to the "drift"?

Many Thanks.
harrychamp

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