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chocolatemoney
Posts: 322
Joined: October 8th, 2008, 6:50 am

### option hedging remark*

QuoteOriginally posted by: listQuoteOriginally posted by: daveangelQuote On other hand i also tried to present examples that show that say binomial pricing does not always correspond our common sense of the price : to buy calls for the same price which with 99% and 1 % gives you the right to buy underlying of the option. I have asked you over and over again to tell me what price you think this option should trade at if you think 33.3333 is not the right price. Please tell me what it is otherwise please do not comment further.Market price in the example is a rv C ( t , o ) = { 0 , 50 } with probabilities p , 1 ? p. These values come from the definition of market price. The market price depends on the parameter p.An investor who think that the spot market price = premium of the option is 33 1/3 will lose his premium with probability 1-p and will gain 50-33 1/3 with probability p. For different p, say 0.1 or 0.9 risk characteristics of the price 33 1/3 will be different. This is basic risk characteristics of the price 33 1/3. We can calculate mean, variance of return and based on our rules take a decision whether to buy or not this option for 33 1/3 price.Consider more complex example because 2 states at T is a simplest illustration. Let S ( t ) = 2.5 and S ( T ) = { 0.5 , 1 , 2, 4 } with correspondent distribution { 0.1 , 0.3 , 0.2 , 0.4 } , K = 2. I do not know how to present benchmark solution in this case. Other approach defines the market price at t of the call option C ( t , omega ) as a rv taking values { 0 , 1.25 } with probabilities { 0.6 , 0.4}. Here 1.25 comes from the equation ( 4 ? 2 ) / x = 4 / 2If one chose the premium=spot market price equal C ( t ) = 1 then rate of return on the call investment is a rv taking value 0 with prob 0.6 and ( 4 ? 2 )/1.25 = 16/5 with prob 0.4. Expected return is (16/5)*0.4 = 32/25.We can calculate stdv of the return. If we agree with such risk characteristics of the investment we can buy option if no we can determine lower price which risk will be consistent with our strategy. If we have other ?real world? distribution of the stock we get in general other risk characteristics of the investment in call.List,I am just a newbie, but here are my two cents.$1c) The goal of all this financial mathematics stuff is to find the price of the financial products. The point is that when you're saying that the price depends on some scenarios and some probabilities and the investor can fine tune its view of the distribution of the underlying, that does not answer the question. The question is: what is the price of the option? The price is an unique number, expressed in a currency. If it was a real case, with a price quote of 33 1/3 would you: buy the option, sell short it or would you consider the price fair and do nothing?$2c) The beauty of No Arbitrage pricing is that prices come out of a no arbitrage argument. Put in rather simple terms, if you are hedged, you do not have to worry about the value of the underlying at t=Maturity. This point is not evident from your latest post.Cheers.Edit: fixed a grammar error
Last edited by chocolatemoney on September 5th, 2011, 10:00 pm, edited 1 time in total.

list
Topic Author
Posts: 2041
Joined: October 26th, 2005, 2:08 pm

### option hedging remark*

$1c) The goal of all this financial mathematics stuff is to find the price of the financial products. The point is that when you're saying that the price depends on some scenarios and some probabilities and the investor can fine tune its view of the distribution of the underlying, that does not answer the question. The question is: what is the price of the option? The price is an unique number, expressed in a currency. If it was a real case, with a price quote of 33 1/3 would you: buy the option, sell short it or would you consider the price fair and do nothing?/// Of course you right. It is always easier to present a solution of a simpler model. Nevertheless when you see definition of a call option contract t , T , K , S ( ) are given and say find the option price what is the definition of the option price you use.///$2c) The beauty of No Arbitrage pricing is that prices come out of a no arbitrage argument. Put in rather simple terms, if you are hedged, you do not have to worry about the value of the underlying at t=Maturity. This point is not evident from your latest post./// $3c )The beauty is not a primary concern of the finance either in practice nor in theory.$4c) Imagine in theory that you think that your price is perfectly hedged and it occurs that you discovered that you are at risk ? ///

chocolatemoney
Posts: 322
Joined: October 8th, 2008, 6:50 am

QuoteOriginally posted by: list$4c) Imagine in theory that you think that your price is perfectly hedged and it occurs that you discovered that you are at risk ? ///So, you don't buy ACD explanation in this thread because you don't buy the no arbitrage / hedging / replication argument he made ?? daveangel Posts: 17031 Joined: October 20th, 2003, 4:05 pm ### option hedging remark* Quote So, you don't buy ACD explanation in this thread because you don't buy the no arbitrage / hedging / replication argument he made ?? yeah - but not sure if he is a seller or not ! knowledge comes, wisdom lingers list Topic Author Posts: 2041 Joined: October 26th, 2005, 2:08 pm ### option hedging remark* chocolatemoney, I think that along with perfectly hedged BS-type solution there always exists another one. This is C ( t , x ) = 0. No one either buyer or seller of an option can lose anything and hence it is also no-arbitrage price. If one agrees then the next significant future open problem is to identify the situation when 0 is the better or worse than BS price. Of course this is not a simple problem as far as the primary tool ?no-arbitrage? could not be applied for identification. One of the most significant advantages of the 0-solution is zero risk. I am not sure and you can correct me that BS price is somewhat risky. I head that sometimes not at every day people can lose their premium? bearish Posts: 5693 Joined: February 3rd, 2011, 2:19 pm ### option hedging remark* I'm a buyer! list Topic Author Posts: 2041 Joined: October 26th, 2005, 2:08 pm ### option hedging remark* QuoteOriginally posted by: bearishI'm a buyer!It was joke, something like divide by 0. Sorry. It is a good way to stop in time. chocolatemoney Posts: 322 Joined: October 8th, 2008, 6:50 am ### option hedging remark* List,bearish is a buyer, and I would be buying calls for free as well.The reason is this: if the underlying ends above the strike, I make money; otherwise, I lose nothing. It's a free bet and I'd like to have as many free-calls at possible.I am afraid the market does not work like you think. If the price is zero, anybody would be buying. So the price would increase. This means that zero is not a "fair" price for the option. A trader, market maker, investment bank might for some reason refuse to trade a specific contract. In this case, the contract would have no bid or ask price. But, nevertheless, the option would still have a fair value (or fair price) and that would not be zero, maybe 0.0000000000000000000000000000001 but not zero.The fact that the option price is also called "premium" is misleading, now that I think about it. The price of the option is not really a "premium over something" that could be zero if people's not willing to pay such a "premium".Cheers.furthermore, a price has to be unique - just like at the grocery store. An apple does not go for a vector a real valued prices. An apple is sold for$0.20 or whatever.

chocolatemoney
Posts: 322
Joined: October 8th, 2008, 6:50 am

### option hedging remark*

oh, thanks god.

list
Topic Author
Posts: 2041
Joined: October 26th, 2005, 2:08 pm

### option hedging remark*

QuoteOriginally posted by: chocolatemoneyoh, thanks god.you have the definition of a call option contract t , T , K , S ( ). What is the definition of the option price you use. We have call payoff and we say let us find the price of the call option what does it mean without writing some formulas?
Last edited by list on September 5th, 2011, 10:00 pm, edited 1 time in total.

bearish
Posts: 5693
Joined: February 3rd, 2011, 2:19 pm

### option hedging remark*

A European call option is completely characterized by its payoff function at maturity (taken to be time T): max(S(T)-K,0). There is no role to play for quantities at time t, for any t<T. Separately, the economy in which the call option lives is potentially complicated, but we have been trying to keep things simple by pinning down the behavior of the stock price process to assume that it is a single step binomial, with the interest rate equal to zero, and with the stock not paying any dividends over the period that we care about. The last piece of the puzzle is an assumption that we can buy and sell the stock at the stated prices, with no transaction costs. In this world (trivial as it may be), there is not much more to say.

chocolatemoney
Posts: 322
Joined: October 8th, 2008, 6:50 am

### option hedging remark*

List, by the way, is this your book?

daveangel
Posts: 17031
Joined: October 20th, 2003, 4:05 pm

### option hedging remark*

QuoteOriginally posted by: chocolatemoneyList, by the way, is this your book? out who the publisher is and then google them
knowledge comes, wisdom lingers

frolloos
Posts: 1621
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### option hedging remark*

ACD
Posts: 107
Joined: April 19th, 2004, 8:09 am

### option hedging remark*

QuoteOriginally posted by: listchocolatemoney, I think that along with perfectly hedged BS-type solution there always exists another one. This is C ( t , x ) = 0. No one either buyer or seller of an option can lose anything and hence it is also no-arbitrage price.I know I asked you this already, but do you know what arbitrage is? From your statement here the answer seems to be no.Let me give you 2 alternative definitions for arbitrage, and I will show that your price allows for both of them. The principle of arbitrage is that it's not possible to make a return above the risk-free rate without taking risk. In daveangel's example we've defined the risk free rate to be zero so this means a return above zero in this setting. So to formalise the definition (these are the two definitions of arbitrage I am aware of):Definition 1: For no arbitrage to exist there does not exist a portfolio which has a present value of 0 and a future value that is greater than zero with probability 1. Let's call this the strong form.To show your price violates this under the assumptions let's consider a portfolio where we buy 1 option, sell 2/3 of S and use the proceeds of this sale to buy 66 2/3 worth of bond. At time t this portfolio is valueless using your option price:x * C(t) + y* S(t) + z * B(t) = 1 * 0 - 2 / 3 * 100 + 66 2 / 3 = 0Now let's consider the value of the portfolio at time T: x * C(T) + y* S(T) + z * B(T)The two scenarios give us the following results:1 * 100 - 2 / 3 * 200 + 66 2 / 2 = 33 1/31 * 0 - 2 / 3 * 50 + 66 2 / 2 = 33 1/3The portfolio becomes worth 33 1/3 regardless of what happens, therefore with probability 1 I get a guaranteed 33 1/3 with no investment. The strong form of arbitrage exists.Definition 2: For no arbitrage to exists there does not exist a portfolio which has a present value of 0, which cannot take a negative value in the future and can have a positive value with probability > 0. Let's call this the weak form.You option price C(t)=0, violates this. To see this we just look at the two possible outcomes:S(T) = 50 => C(T)=0S(T) = 200 => C(T)=100This violates the definition and so it allows for the weak form of arbitrage.QuoteOriginally posted by: listchocolatemoney, One of the most significant advantages of the 0-solution is zero risk How is it zero risk? If I sell an option for zero I have a non-zero probability of having to pay out 100 at expiry.
Last edited by ACD on September 6th, 2011, 10:00 pm, edited 1 time in total.