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distribution of option prices under black-scholes
Posted: October 31st, 2011, 11:01 pm
by slslsl
Hello,Suppose I model my stock price S_t using Black_Scholes.Suppose my vanilla call option expires at time T, and at time t S_t is random (eg I'm looking at the process from time 0).Is it possible to derive the distribution of call option prices at time t using the knowledge of the distribution of S_t? Can one work out the moments (analytically) at least? Is there any research on this?
distribution of option prices under black-scholes
Posted: November 1st, 2011, 12:32 am
by bearish
Yes, there is definitely a well defined distribution for the call option price in this case. It's support will be the positive part of the real line; it will be a relatively simple monotonic transformation of the normal distribution; but it will probably not have a nice analytical representation.
distribution of option prices under black-scholes
Posted: November 1st, 2011, 2:25 am
by DavidJN
Robert Merton studied this in some detail a long time ago. Try for example Merton, Robert C., Myron S. Scholes, and Matthew L. Gladstein. "The Returns and Risk of Alternative Call Option Portfolio Investment Strategies." Journal of Business 51 (April 1978): 183-242 and Merton, Robert C., Myron S. Scholes, and Matthew L. Gladstein. "The Returns and Risk of Alternative Put Option Portfolio Investment Strategies." Journal of Business 55 (January 1982): 183-242.
distribution of option prices under black-scholes
Posted: November 1st, 2011, 5:27 am
by frolloos
would it be possible to price (in black-scholes framework) options on the option, and derive using breeden-litzenberger the prob density function?
distribution of option prices under black-scholes
Posted: November 1st, 2011, 6:03 am
by list
QuoteOriginally posted by: slslslHello,Suppose I model my stock price S_t using Black_Scholes.Suppose my vanilla call option expires at time T, and at time t S_t is random (eg I'm looking at the process from time 0).Is it possible to derive the distribution of call option prices at time t using the knowledge of the distribution of S_t? Can one work out the moments (analytically) at least? Is there any research on this?If C ( t , x ; T , K ) is the call option price given S ( t , omega ) = x then C ( t , x ; T , K ) | x = S ( t , omega ) is what you are asking. Then the moments are E C ^ n ( t , x ; T , K ). Whether or not these integrals for different n , K, t , T and for a given distribution S ( t, omega ) be calculated for it is not clear but you can try yourself or might be one did it for some particular cases.
distribution of option prices under black-scholes
Posted: November 1st, 2011, 10:16 am
by ronm
Quotebut it will probably not have a nice analytical representation.Or, may be you can try with approximating the BS option pricing function as 1st or 2nd order approximation?Thanks,