SERVING THE QUANTITATIVE FINANCE COMMUNITY

Kielers
Posts: 31
Joined: September 30th, 2010, 5:00 pm

### option hedging remarks

list, you are correct the differential of the function P is given by Equation 2, however, ask yourself what use is this equation without knowledge of the future? Equation 1 is the P&L process given by the replicating portfolio, this is a deterministic process (under BS) and is therefore very useful. Peter Carr has written about this, have a look at his website.

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

### option hedging remarks

There are two different problems. 1st if we try to derive BS eq we need a correct derivation. I might be wrong but the step that finite difference of the hedge portfolio is a part of any BS derivation. 2nd let the underlying of the BS eq is mathematically correct. What does BS's interpretation of the option price state? If Buyer and seller will use BS price then there is no arbitrage exist in option price. It looks like recommendation than a formal definition of the price. It reminds a recommendation let chess white player plays such and such and black player plays such and such than in say t = 30 steps they arrive at the peace end. I think that everyone here understand that any option price is risky for either option gambler and therefore risk is a component of the option price. This only fact was missed by B&S. Let BS price is agreed to use by the people who involved in the business. The value of the risk for either counterparty is then the only one what should be added to the broad picture.

bearish
Posts: 5897
Joined: February 3rd, 2011, 2:19 pm

### option hedging remarks

Why do you say things like: "I might be wrong...", "I will appreciate any remark", "I will also thankfull if you will point my errors too", etc.? What you are obviously trying to say is that we are all idiots and only you understand anything about financial mathematics. Whenever anybody tries to point (out?) your errors, you do not come across as "thankfull".This would not be much of an issue if you only commented in your own threads (although OT would be a better forum), but you also routinely pollute other legitimate threads with your nonsense (like the current "Modeling Spot Exchange Rate with Stochastic Interest Rates" thread in the Technical Forum). Sorry, I know I promised to try to ignore you, but that was yesterday.

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

### option hedging remarks

B&S replaced real underlying dS = mu S dt + sigma S dw for heuristic underlying with drift riskfree 'r'. This underlying does not exist and they stated that if options with given fixed sigma will be priced as no existed risk neutral model stock then there is no arbitrage. This is essence of the option pricing.As you often state about my errors correct me my view without emotions. I say that 1st we define market price of a call. It is a function of scenario: for S( T ) = < K option price is 0, i.e. no one will pay for option which outcome equal 0. For a scenario for which S ( T ) > K option price is define such that rate of return on the real underlying and option should be equal. Actually this rule is often used in finance. Spot market price is a number which specified by the market surplus and demand. If we wish we choose the BS number, other choice it can be conditional expectation with respect to observations up to t or some more complex statistics. For given distribution of the underlying we can calculate risk characteristics attached to the chosen number-price.

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

### option hedging remarks

@List: i wrote the attached document for myself some while ago. it shows in detail why Black-Scholes-Derivation /replicating self-financing portfolios is correct. I used a general numeraire, but think of N as the money market account with constant interest rate if you like. doesn't make a difference. if the attached file doesn't make it clear to you, i don't really know what will..

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

### option hedging remarks

QuoteOriginally posted by: frolloos@List: i wrote the attached document for myself some while ago. it shows in detail why Black-Scholes-Derivation /replicating self-financing portfolios is correct. I used a general numeraire, but think of N as the money market account with constant interest rate if you like. doesn't make a difference. if the attached file doesn't make it clear to you, i don't really know what will..Can you state what these formulas prove? If we open a math book and the theorem statement is removed. We read only proof and we could not justify anything without the theorem's statement. Theorem statement might be correct or incorrect and in either case we verify what is stated with the proof itself. Without the statement we could not say anything.

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

### option hedging remarks

QuoteOriginally posted by: frolloos and the frustration is?So......, how's the thread working out for you so far?Don't give up yet, try pointing out the utter stupidity of the statement ...'option price is define such that rate of return on the real underlying and option should be equal.'Or what about missing the point of BS by light years in therefore risk is a component of the option price. This only fact was missed by B&S.

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

### option hedging remarks

Quote So......, how's the thread working out for you so far?well it seems that we have moved on from forwards to options. a step in the right direction.
knowledge comes, wisdom lingers

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

### option hedging remarks

QuoteOriginally posted by: TinManQuoteOriginally posted by: frolloos and the frustration is?So......, how's the thread working out for you so far?point taken, and i am joining the ranks of those feeling listless - pun intended.

frenchX
Posts: 5911
Joined: March 29th, 2010, 6:54 pm

### option hedging remarks

That's my last attempt on this thread. this document is very clear mathematically about the derivation of Black Scholes using self financing portfoliohttp://www.ntu.edu.sg/home/nprivault/MA5182/chapter5.pdf

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

### option hedging remarks

The problem of definition of the option price and whether exists or not self-financing portfolio are related to each other but not the same we talk about definition or constructing option price. In any BS price construction it is never stated before derivation what we call call-option price. Let us assume for now that all mathematics is perfectly correct. They use some formulas and arrive at the solution of the BS equations which should by definition signify the call option price. Correct way to this construction is to say in advance that we call the option price the function = BSE solution and next to show that this function satisfies a PDE ( BSE ) and provides hedging for a portfolio ( we call this portfolio self-financing). It looks two approaches are equivalent but not actually not completely. In benchmark approach when definition appears at the end one can justify definition by the help of its derivation. Say because it provides perfect hedge as usually can be read. If we use other approach then such arguments fail because it does not exclude other possibility and other definition can posess other valuable properties that are better corresponds to real market.Ignoring technical details the only difference that should be add to option price is its risk characteristics. For given T , K they are : risk to loss premium. average losses, average profit, and correspondent to this parameters volatilities. I have a feeling that any trader think about such characteristics before buying options regardless whether he use BS price or its adjustment.And the only one conclusion it follows from alternative approach in options pricing. Buying options is more risky than it presented by BS pricing.This outlook of the problem based on assumption that BS mathematics is completely perfect.

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

### option hedging remarks

About self-financing. If we use explicit form of the call option price and substitute this solution into portfolio used by Hull in his derivation of the BSE it very easy to see that chosen portfolio does not self-financing because the change of its value over short interval lost the term which is difference between above formulas 1, 2 we discussed above. And it is only the unique way to justify self financing condition of the portfolio used in BS derivation. Just use explicit form of the BSE solution and substitute it into the portfolio formula and check the self-financing definition.

frenchX
Posts: 5911
Joined: March 29th, 2010, 6:54 pm

### option hedging remarks

Where 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.
Last edited by frenchX on August 29th, 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: 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.

frenchX
Posts: 5911
Joined: March 29th, 2010, 6:54 pm

### option hedging remarks

I 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.

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