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Joined: July 22nd, 2015, 2:12 pm

Stochastic Volatility

July 20th, 2017, 3:43 pm

Never have looked before at the pricing details. In my understanding stochastic volatility should imply stochastic BS pricing. In other words regardless of a particular distribution of the [$]\sigma ( * , \omega )[$] any path of the sigma leads to unique BS price of the option. Hence I thought that option price under a SV model should be a statistics of
 [$]C ( t , S ( t ) ; T , K ; \sigma ( * , \omega )[$]    (1)
ie it should be mean of (1) or something like adjustment of this mean. Note that in this case we should arrive at explicit market risk implied by the SV model. Such risk will represent overvalued / undervalued option price with respect to real world realization of the [$]\sigma ( * , \omega )[$] 
I have looked at some papers and discovered somewhat different approaches. I will thankful for comments or a reference with primary idea of option valuation under SV.
In particular I looked through 1 ch Gatheral book and found  his derivation of the 'BSE' with reference on (1998) Wilmott paper. The derivation somewhat confused me. a) we assume that market is underlying with SV, option contract and bond. There is no other assets. In derivation it was used undefined asset [$] V_1 [$] for hedged portfolio. It was shown that option pricing eq does not depend on this auxiliary asset, ie it's existence does not actually needed for pricing but it is used to eliminate stock risk term dS and volatility risk term dv in derivation of the 'BSE'. How can one use undefined function to eliminate some terms? It might be I missed here something? 
 
frolloos
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Re: Stochastic Volatility

July 20th, 2017, 3:55 pm

You're going very fast. At this rate we'll be done with stochastic local volatility before the end of this week.
 
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Re: Stochastic Volatility

July 20th, 2017, 4:05 pm

You're going very fast. At this rate we'll be done with stochastic local volatility before the end of this week.
To slow down a little bit we can start with the simplest SV model. Let random variable [$]\sigma ( \omega )[$] takes two values a and b with known probabilities [$] p_a , p_b , p_a + p_b = 1[$] . How does option price should be defined. Such example will provide a basis for option price definition with SV. If we fail to present formal definition in this simplest case it looks like everything done would be formally close to nonsense. 
 
frolloos
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Joined: September 27th, 2007, 5:29 pm
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Re: Stochastic Volatility

July 20th, 2017, 4:19 pm

Yes! So how would you define the option price? With a bit of luck you will re-derive the mixing formula.
 
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Re: Stochastic Volatility

July 20th, 2017, 5:23 pm

Yes! So how would you define the option price? With a bit of luck you will re-derive the mixing formula.
My approach to definition of the option price does not depends whether volatility is random or not. I think that 0-order price of a traded asset can be the break even value for the rates of  return regardless whether it a security or a derivative.