One of the questions in Q-Bank was to calculate value of call option using BS model. Do you guys expect to get this in the exam? Are we expected to memorize that horrible formula?
I vaguely remember reading that this year memorization of that formula is not required, but I’d wait on someone else to confirm. Even if memorizing the formula isn’t required, you do need to know the inputs - that much I know.
THE ONLY LOS’s ON BS: Explain the assumptions underlying the Black-Scholes-Merton model and their limitations. Explain how an option price, as represented by the Black-Scholes-Merton model, is affected by each of the input values (the option Greeks).
constant volatility and known (big assumption) continuous risk free rate is known markets have no transaction costs or taxes Underlying doesn’t pay any dividends Options are European
- Price of underlying follows log-normal distribution
Must it be the CONTINUOUS risk-free rate? (Because you can calculate ln(1+i) to get the risk-free rate). Also, only works on European options, not American ones, and should not be used to price bonds.
e^Rcontinuous - 1 = R discrete e^Rcontinuous = Rdiscrete + 1 Rcontinuous = ln(Rdiscrete + 1) Right, shouldn’t be used to price bonds since they have underlying cash flow?
TheAliMan Wrote: ------------------------------------------------------- > constant volatility and known (big assumption) > continuous risk free rate is known > markets have no transaction costs or taxes > Underlying doesn’t pay any dividends > Options are European Nice. Thanks for the refresh.
PS The risk free rate is not only continuous but also CONSTANT (just like volatility); this is also a very tall assumption but not as important since the Rf is a less important input to the BSM then is volatility.
TheAliMan Wrote: > > Right, shouldn’t be used to price bonds since they > have underlying cash flow? how do bonds even get into the picture here to begin with? you can’t use BS to price bonds because the formula is for pricing stock options. and you can’t use a bonzai tree to fix a flat tire either, for identical reasons i suppose
Sorry, I meant options on bonds.
I think we can’t even value options on Bonds using BSM, as they need IRV as constant and known, in the case of Bonds the cash flows are interest rate dependent. Don’t know - I am really starting to mix up my concepts.
TheAliMan Wrote: ------------------------------------------------------- > Sorry, I meant options on bonds. OK. the underlying security in the black-scholes is lognormal, which means it can’t fall below zero but the upside is potentially unlimited. there is no upper bound. a stock fits that description. that is not the case with a bond - you get a series of coupons and a fixed principal payment at the end. even if your discount rate is zero, the maximum you get is the sum of the contractual cash flows. your payoff is capped so its distribution is definitely not lognormal, hence you can’t use BS for an option written on a bond.
nice Mobius Striptease !!!. A good to know concept (not for the exam, but general)
Mobius Striptease Wrote: ------------------------------------------------------- > TheAliMan Wrote: > -------------------------------------------------- > ----- > > Sorry, I meant options on bonds. > > > OK. > the underlying security in the black-scholes is > lognormal, which means it can’t fall below zero > but the upside is potentially unlimited. there is > no upper bound. a stock fits that description. > > that is not the case with a bond - you get a > series of coupons and a fixed principal payment at > the end. even if your discount rate is zero, the > maximum you get is the sum of the contractual cash > flows. your payoff is capped so its distribution > is definitely not lognormal, hence you can’t use > BS for an option written on a bond. This is absolutely incorrect. BS model can easily be extended to price bonds, options on bonds, swaptions, and numerous other fixed income derivatives. The bond price as a funtion of a random interest rate and time can be found as the solution to a partial differential equation, and it happens to be a variant of the Black Scholes PDE.
wyantjs Wrote: ------------------------------------------------------- >> > This is absolutely incorrect. BS model can easily > be extended to price bonds, options on bonds, > swaptions, and numerous other fixed income > derivatives. The bond price as a funtion of a > random interest rate and time can be found as the > solution to a partial differential equation, and > it happens to be a variant of the Black Scholes > PDE. an argument of this sort follows this framework - “black-scholes formula uses concepts from math finance such as risk neutrality principple, similar concepts can be used to price bonds under certain other assumptions, therefore black-scholes can be used to price bonds”. in the end you are telling us nothing. we are talking here about the classical black-scholes formula used to price european calls and puts that takes 5 simple inputs. no, you won’t use it directly to price options on bonds at all.
Again, you are wrong. You really shouldn’t speak of topics in which you are clearly ignorant on. It is often the case that we assume interest rates follow a very similar process to the GBM process used to model stock prices. CIR, Vasicek, etc… are all very similar. Since bonds are functions of interest rates, they are by definition interest rate derivatives. The BS model was developed under a general setting that pertains to any derivative security whose random underlying follows the GBM process, and is also a function of time. The “classical black-scholes formula” is simply the solution to the general PDE. Slight variations of the PDE often give slight variations of the solution. Hence, I am telling you something, and a very important something at that. I am telling you that your claim that the BS model cannot be used to price bonds is WRONG.
wyantjs Wrote: ------------------------------------------------------- > Again, you are wrong. You really shouldn’t speak > of topics in which you are clearly ignorant on. > It is often the case that we assume interest rates > follow a very similar process to the GBM process > used to model stock prices. CIR, Vasicek, etc… > are all very similar. Since bonds are functions > of interest rates, they are by definition interest > rate derivatives. The BS model was developed > under a general setting that pertains to any > derivative security whose random underlying > follows the GBM process, and is also a function of > time. The “classical black-scholes formula” is > simply the solution to the general PDE. Slight > variations of the PDE often give slight variations > of the solution. Hence, I am telling you > something, and a very important something at that. > I am telling you that your claim that the BS > model cannot be used to price bonds is WRONG. Do you need me to draw you the difference between “pricing a bond” and “pricing an option written on a bond”? using the Black-Scholes formula, i can price an option written on the level of your ignorance if i want to, assuming it is lognormal. seems like a safe assumption cause your upside might be unlimited. an option written on a bond has the bond as the underlying, and the distribution of the bond price is bounded from above. hence it is not lognormal. plain and simple
No, you don’t. But maybe I should explain to you that a bond essentially is a derivative whose underlying is an interest rate. Therefore, when pricing a stock option, we model the stock price. When pricing a bond, we model the interest rate, not the bond price…plain and simple. I can argue with you on this all day long. I solve PDE’s daily, and do so for the pricing of bonds regularly.
the discussion is about pricing an option written on a bond. the bond price is the underlying here, and i can model it in hundreds of ways, depending on the specification and parameters of the problem, and my freakin mood for the day. i can model it as an option on the interest rates, or i can model the firm value without modeling exogenous default rates, i can do whatever i want depending on the problem constraints - the point is that the bond price is not lognormally distributed, and you can’t value the option with the bond price as the underlying by applying black-scholes formula directly. i suspect you know very well what i’m talking about but you are acting cuckoo by talking about semi-related crap. in any case, your good buddy Frank Fabozzi will enlighten you that “there are three assumptions underlying the Black Scholes model that limit its use in pricing options on bonds. First, the probability distribution for the prices assumed by the Black-Scholes option pricing model permits some probability - no matter how small - that the price can take any positive value. For a bond, however, we know that the price cannot exceed the sum of the coupon payments plus the maturity value. Thus, unlike stock prices, bond prices have a maximum value.” seems pretty much verbatim what i said above, which you qualified as “absolutely incorrect” and justified by “i solve PDEs daily” http://books.google.com/books?id=slhRH3ufmecC&pg=PA193&lpg=PA193&dq=can+black+scholes+be+used+to+price+bonds&source=bl&ots=1u-NCR_Gdc&sig=fHg-36yUMR6r9DOZYwsbay-nPMQ&hl=en&ei=iFQDSvv7BIrAM6T8kNgE&sa=X&oi=book_result&ct=result&resnum=8#PPA193,M1 i believe i’m done whoopping your ass on the forum for the day. it’s all educational and in good fun