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

### option hedging remarks

QuoteOriginally posted by: listQuoteOriginally posted by: frenchXWhere is the risk if you buy a call option quoted on an underlying with a constant volatility in a constant interest rate environment (assuming that the premium is calculated with the Black Scholes formula at the right volatility) in a continuous hedging framework in frictionless market ? Have a closer look at the paper of Derman I have posted before, especially the part "The P&L of Hedged Trading strategies"also have a look herehttp://www.stanford.edu/~japrimbs/Publications ... q.pdfthere are several derivation for the Black Scholes equation. constant volatility+continuous hedging means NO RISK.An european vanilla option can be perfectly replicated with bons and stocks in the BS framework.Indeed, the BS option price does not implies any risk. This is 'no free lunch' , 'no arbitrage' price. If you forget about any definition of the price and imagine that we have underlying stock S ( t ; mu, sigma ) , T , K . We define a contract payoff max{ S ( T ) - K , 0 } at T. We have to note paying amount C , C > 0 we can loose C for the set of scenarios S ( T ; mu , sigma, omega ) =< K. This is one of the risk components. Let for example E S ( T ; mu , sigma, omega ) * I { S ( T ; mu , sigma, omega ) < K }. It is average payoff losses at T. If we hold stock its value at T would be S ( T ) that can be less than K for a scenario while for this scenario payoff will be 0. Thus C is lost.OK - here is a really tricky problem (not). Stock is trading at 100. Weknow that when we next observe the stock at time T, it will either double to 200 or halve to 50. What is the price of the 100 strike call on the stock with maturity T ? Assume zero rates, no dividends, no transaction costs.
Last edited by daveangel on August 29th, 2011, 10:00 pm, edited 1 time in total.
knowledge comes, wisdom lingers

TinMan
Posts: 613
Joined: September 21st, 2006, 9:42 am

### option hedging remarks

QuoteOriginally posted by: daveangel OK - here is a really tricky problem (not). Stock is trading at 100. Weknow that when we next observe the stock at time T, it will either double to 200 or halve to 50. What is the price of the 100 strike call on the stock ? Assume zero rates, no dividends, no transaction costs.LOL, just when I thought I was out......... they pull me back in!!

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

### option hedging remarks

QuoteOriginally posted by: TinManQuoteOriginally posted by: daveangel OK - here is a really tricky problem (not). Stock is trading at 100. Weknow that when we next observe the stock at time T, it will either double to 200 or halve to 50. What is the price of the 100 strike call on the stock ? Assume zero rates, no dividends, no transaction costs.LOL, just when I thought I was out......... they pull me back in!!Me too
knowledge comes, wisdom lingers

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

### option hedging remarks

QuoteOriginally posted by: daveangelQuoteOriginally posted by: listQuoteOriginally posted by: frenchXWhere is the risk if you buy a call option quoted on an underlying with a constant volatility in a constant interest rate environment (assuming that the premium is calculated with the Black Scholes formula at the right volatility) in a continuous hedging framework in frictionless market ? Have a closer look at the paper of Derman I have posted before, especially the part "The P&L of Hedged Trading strategies"also have a look herehttp://www.stanford.edu/~japrimbs/Publications ... q.pdfthere are several derivation for the Black Scholes equation. constant volatility+continuous hedging means NO RISK.An european vanilla option can be perfectly replicated with bons and stocks in the BS framework.Indeed, the BS option price does not implies any risk. This is 'no free lunch' , 'no arbitrage' price. If you forget about any definition of the price and imagine that we have underlying stock S ( t ; mu, sigma ) , T , K . We define a contract payoff max{ S ( T ) - K , 0 } at T. We have to note paying amount C , C > 0 we can loose C for the set of scenarios S ( T ; mu , sigma, omega ) =< K. This is one of the risk components. Let for example E S ( T ; mu , sigma, omega ) * I { S ( T ; mu , sigma, omega ) < K }. It is average payoff losses at T. If we hold stock its value at T would be S ( T ) that can be less than K for a scenario while for this scenario payoff will be 0. Thus C is lost.OK - here is a really tricky problem (not). Stock is trading at 100. Weknow that when we next observe the stock at time T, it will either double to 200 or halve to 50. What is the price of the 100 strike call on the stock with maturity T ? Assume zero rates, no dividends, no transaction costs.Your problem was the starting point of my 'alternative' pricing approach. I discussed how this type of options should be priced. You can check it for example inhttp://papers.ssrn.com/sol3/papers.cfm?abstract_id=895679 or http://papers.ssrn.com/sol3/papers.cfm? ... 895679This is the first step of option pricing. The 2nd step is the spot price specification. If the 1st step will be clear for you I can specify the sense of the spot market price. Though I talk about it a few time recently.

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

### option hedging remarks

QuoteOriginally posted by: frenchXI think you miss the idea of hedging. Sure that if you buy the option and do nothing until expiration then yes the outcome is random and risky. But by hedging, your portfolio is self financing, you don't earn money but you don't loose too, in any scenario. Look at the binomial model, by hedging after one time step to another the variation of your portfolio is the same unconditional that the stocks goes up or down. Read carefully the chapter "how to delta hedge" in the book of Paul.I did not say that it is impossible to construct self-financing portfolio and by using it to provide perfect dynamic hedging. My intention was to illustrate drawback of such construction in Hull book and as far as similar constructions are used in other books it might have sense to check whether or not portfolio constructions are correct. The simple standard way is to test the derivations. We need to use explicate values of each components of the portfolio and check whether or not given portfolio satisfies self-financing definition. If yes then self-financing works if no then one need to try improve it.

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

### option hedging remarks

QuoteOriginally posted by: listQuoteOriginally posted by: daveangelQuoteOriginally posted by: listQuoteOriginally posted by: frenchXWhere is the risk if you buy a call option quoted on an underlying with a constant volatility in a constant interest rate environment (assuming that the premium is ckalculated with the Black Scholes formula at the right volatility) in a continuous hedging framework in frictionless market ? Have a closer look at the paper of Derman I have posted before, especially the part "The P&L of Hedged Trading strategies"also have a look herehttp://www.stanford.edu/~japrimbs/Publications ... q.pdfthere are several derivation for the Black Scholes equation. constant volatility+continuous hedging means NO RISK.An european vanilla option can be perfectly replicated with bons and stocks in the BS framework.Indeed, the BS option price does not implies any risk. This is 'no free lunch' , 'no arbitrage' price. If you forget about any definition of the price and imagine that we have underlying stock S ( t ; mu, sigma ) , T , K . We define a contract payoff max{ S ( T ) - K , 0 } at T. We have to note paying amount C , C > 0 we can loose C for the set of scenarios S ( T ; mu , sigma, omega ) =< K. This is one of the risk components. Let for example E S ( T ; mu , sigma, omega ) * I { S ( T ; mu , sigma, omega ) < K }. It is average payoff losses at T. If we hold stock its value at T would be S ( T ) that can be less than K for a scenario while for this scenario payoff will be 0. Thus C is lost.UOK - here is a really tricky problem (not). Stock is trading at 100. Weknow that when we next observe the stock at time T, it will either double to 200 or halve to 50. What is the price of the 100 strike call on the stock with maturity T ? Assume zero rates, no dividends, no transaction costs.Your problem was the starting point of my 'alternative' pricing approach. I discussed how this type of options should be priced. You can check it for example inhttp://papers.ssrn.com/sol3/papers.cfm?abstract_id=895679 or http://papers.ssrn.com/sol3/papers.cfm? ... 895679This is the first step of option pricing. The 2nd step is the spot price specification. If the 1st step will be clear for you I can specify the sense of the spot market price. Though I talk about it a few time recently.Why don't you just tell us what you think the price of the call is rather than point us at more papers? Do you think the problem is not sufficeintly specified ?
Last edited by daveangel on August 30th, 2011, 10:00 pm, edited 1 time in total.
knowledge comes, wisdom lingers

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

### option hedging remarks

OK - here is a really tricky problem (not). Stock is trading at 100. Weknow that when we next observe the stock at time T, it will either double to 200 or halve to 50. What is the price of the 100 strike call on the stock with maturity T ? Assume zero rates, no dividends, no transaction costs.---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------In economic as well as in finance it is often have used stochastic setting in which a security is assumed to be random. That means for example in continuous case that we know probability that say for arbitrary t we know probability P { a =< S ( t ) < b } for arbitrary t and numbers a , b. In Binomial setting for given coordinate space { 50 , 200 } the stock is given if you know probabilities of the states. If we do not have in continuous or in discrete coordinate space probability distribution then it is measurable function.In your example you do not have a stock as far as you did not define probabilities of the states. For example if at maturity T P_50 = 0.1 and P_200 = 0.9 is the one stock and if P_50 = 0.8 ,P_200 = 0.2 is the other one. In either case these are random variables at T. In other words in your example stock does not define regardless what you wish to ask next.

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

### option hedging remarks

QuoteOriginally posted by: listOK - here is a really tricky problem (not). Stock is trading at 100. Weknow that when we next observe the stock at time T, it will either double to 200 or halve to 50. What is the price of the 100 strike call on the stock with maturity T ? Assume zero rates, no dividends, no transaction costs.---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------In economic as well as in finance it is often have used stochastic setting in which a security is assumed to be random. That means for example in continuous case that we know probability that say for arbitrary t we know probability P { a =< S ( t ) < b } for arbitrary t and numbers a , b. In Binomial setting for given coordinate space { 50 , 200 } the stock is given if you know probabilities of the states. If we do not have in continuous or in discrete coordinate space probability distribution then it is measurable function.In your example you do not have a stock as far as you did not define probabilities of the states. For example if at maturity T P_50 = 0.1 and P_200 = 0.9 is the one stock and if P_50 = 0.8 ,P_200 = 0.2 is the other one. In either case these are random variables at T. In other words in your example stock does not define regardless what you wish to ask next.so you can't price the option ?
knowledge comes, wisdom lingers

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

### option hedging remarks

Quote so you can't price the option ?Though your underlying does not defined in stochastic setting nevertheless if we consider your level of definition of the underlying market object we can define option premium in a similar way. For the scenario that leads to 50 value of the call option is 0, and for the scenario that brings 200 the premium is the solution of the equation [ S ( T ) - K ] - x / x = [ S ( T ) - S ( t ) ] / S ( t ) which represent equality of the rate of return for stock and its call option. This equation can be reduced to [200 - 100] / x = 200/100 or x = 50. Thus call option price admits two values { 0 , 50 }. If we assignto stock values at T probabilities p , q = 1 - p these probabilities will be assigned to the corresponding values of the call option. This scheme with given distribution S ( T ) gives us possibility to calculate any risk characteristics of the option premium for one step economy with arbitrary finite or continuous coordinate space for S ( T ). The equation that was used follows from the equality principle. We call two investments are equal on [ 0 , T ] if the promised equal rate of return for any point of time [ 0 , T ] in the case of continuous time and for each leg/step on [ 0 , T ] for discrete time.This principle more accurate presents notion of equality than PV concept. It is easy to present a numeric deterministic example of two stocks defined for dates 0 < t < T which are not equal in the sense of the equality principle but having equal PVs. As far as rates of return will be different the PV concept does not exclude arbitrage while the equality principle does exclude arbitrage.

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

### option hedging remarks

QuoteOriginally posted by: listQuote so you can't price the option ?Though your underlying does not defined in stochastic setting nevertheless if we consider your level of definition of the underlying market object we can define option premium in a similar way. For the scenario that leads to 50 value of the call option is 0, and for the scenario that brings 200 the premium is the solution of the equation [ S ( T ) - K ] - x / x = [ S ( T ) - S ( t ) ] / S ( t ) which represent equality of the rate of return for stock and its call option. This equation can be reduced to [200 - 100] / x = 200/100 or x = 50. Thus call option price admits two values { 0 , 50 }. If we assignto stock values at T probabilities p , q = 1 - p these probabilities will be assigned to the corresponding values of the call option. This scheme with given distribution S ( T ) gives us possibility to calculate any risk characteristics of the option premium for one step economy with arbitrary finite or continuous coordinate space for S ( T ). The equation that was used follows from the equality principle. We call two investments are equal on [ 0 , T ] if the promised equal rate of return for any point of time [ 0 , T ] in the case of continuous time and for each leg/step on [ 0 , T ] for discrete time.This principle more accurate presents notion of equality than PV concept. It is easy to present a numeric deterministic example of two stocks defined for dates 0 < t < T which are not equal in the sense of the equality principle but having equal PVs. As far as rates of return will be different the PV concept does not exclude arbitrage while the equality principle does exclude arbitrage.but what is the price ? is it something between 0 and 50 ?
knowledge comes, wisdom lingers

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

### option hedging remarks

QuoteOriginally posted by: listQuote so you can't price the option ?For the scenario that leads to 50 value of the call option is 0, and for the scenario that brings 200 the premium is the solution of the equation [ S ( T ) - K ] - x / x = [ S ( T ) - S ( t ) ] / S ( t ) which represent equality of the rate of return for stock and its call option.Why have you set these things equal? They will not be since the option is a leveraged version of the underlying so at most you can say it is a multiple (k) of underlying in every scenario under a single time step in a complete market (this renders the rest of the post nonsense).This gives you another way to get the price from two equations:1) [(200-100)-x]/x = k(200-100)/1002) [0-x]/x = k(50-100)/100Solving these gives you a unique value for x.

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

### option hedging remarks

but what is the price ? is it something between 0 and 50 ?-------------------------------------------------------------------------------example 1if the real probability of 50 = 1 and prob of 200 = 0. Then with prob 1 call option equal to 0example 2if the real probability of 50 = 0 and prob of 200 = 1. Then with prob 1 call option equal to 50example 3if the real probability of 50 is close to 1 say 0.999 and prob of 200 = 0.001 . Then with prob 0. 999 call option will be equal to 0. This 3 example shows that to say something about the price we need to know the stock distribution. If you do not know it we could not say anything about call option price on theoretical level.

TinMan
Posts: 613
Joined: September 21st, 2006, 9:42 am

### option hedging remarks

QuoteOriginally posted by: listbut what is the price ? is it something between 0 and 50 ?-------------------------------------------------------------------------------example 1if the real probability of 50 = 1 and prob of 200 = 0. Then with prob 1 call option equal to 0example 2if the real probability of 50 = 0 and prob of 200 = 1. Then with prob 1 call option equal to 50example 3if the real probability of 50 is close to 1 say 0.999 and prob of 200 = 0.001 . Then with prob 0. 999 call option will be equal to 0. This 3 example shows that to say something about the price we need to know the stock distribution. If you do not know it we could not say anything about call option price on theoretical level.Hmmmm....... if only there was some way to take this unknown distribution out of the problem.Why can't someone come up with a way of doing that?Maybe we could award some kind of prize....
Last edited by TinMan on August 30th, 2011, 10:00 pm, edited 1 time in total.

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

### option hedging remarks

QuoteOriginally posted by: ACDQuoteOriginally posted by: listQuote so you can't price the option ?For the scenario that leads to 50 value of the call option is 0, and for the scenario that brings 200 the premium is the solution of the equation [ S ( T ) - K ] - x / x = [ S ( T ) - S ( t ) ] / S ( t ) which represent equality of the rate of return for stock and its call option.Why have you set these things equal? They will not be since the option is a leveraged version of the underlying so at most you can say it is a multiple (k) of underlying in every scenario under a single time step in a complete market (this renders the rest of the post nonsense).This gives you another way to get the price from two equations:1) [(200-100)-x]/x = k(200-100)/1002) [0-x]/x = k(50-100)/100Solving these gives you a unique value for x.I used the principle that states that two investments are equal fro t to T when rates of return are equal. Because stock depends on parameter omega we apply this principle for each omega. These are only two scenarios associated with outcome S ( T ) = 50 and S ( T ) = 200.In general if stock takes b ( 1 ) < b ( 2 ) < ... values at T with probabilities p(1) , p(2) , ... then market call option price at t is a random variable taking values that can be found from the eq [ b ( j ) - K ] I ( b ( j ) > K ) / x = b ( j ) / S ( t ) Now if you tell me for example that you admits risk to lose premium say 5% I can define minimum premium if all values are defined numerically. If you will tell me that you wish to have average profit say 5% I could define your minimum premium. There other risk characteristics could be on demand for example average profit/ average loss = 3/1 We also can combine such single simple risk characteristics.

bearish
Posts: 5790
Joined: February 3rd, 2011, 2:19 pm

### option hedging remarks

Wow! I think we just hit rock bottom. Guess you never know what will happen if you take your eyes off a thread for a few hours around here...