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Phunfactory
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Moments in a Stochastic Volatility Model with Jumps

April 21st, 2015, 12:56 pm

Hello, currently I'm working on a stochastic volatility model with jumps. The dynamics are the following:[$] dV_t = \kappa_V (\bar{V} - V_t) dt + \sigma_V \sqrt{q_t} dW^V_t[$][$] dq_t = \kappa_q (\bar{q} - q_t) dt + \sigma_q \sqrt{q_t} dW^q_t + Z_q dN^q_t[$],where [$]dW^VdW^q = \rho [$], Z_q is a exponential distributed variable with mean [$] \mu_q [$] and [$] N^q [$] is a compound poisson process with intensity [$]l0[$].I'm interested in the moments of [$] V_t[$] . I calculate the first moment of [$] V_t[$] under the assumption that [$]\sigma_V \sqrt{q_t} dW^V_t[$] is a martingale and, hence, get the usual first moment of a CIR process: [$] \mathbb{E}V_t = V_0 e^{-\kappa_V t} + \bar{V}(1-e^{-\kappa_V t})[$]However, if I do a MC simulation for a set of model parameters and compare the analytic first moment with the simulated first moment the discripancy is hugh. For example:I get [$] \mathbb{E}V_t = 1 [$] analytically and [$] \mathbb{E}V_t = 2.0303[$] via MC simulation using the following parameters: kappa_V = 1.4515; V_bar = 1; sigma_V = 1.3394; kappa_q = 2.0224; q_bar = 1; sigma_q = 1.3394; l0 = 0.1073; mu_q = 16.4938; rho = 0.944; V_0 = 1;Where is my error? Could it be that the jump size mu_q so big that the assumption that [$]\sigma_V \sqrt{q_t} dW^V_t[$] is a martingale does not hold anymore?Do anyone know how to calculate the analytic moment of $V_t$ nonetheless?Grettings
Last edited by Phunfactory on April 20th, 2015, 10:00 pm, edited 1 time in total.
 
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Orbit
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Joined: October 14th, 2003, 5:34 pm

Moments in a Stochastic Volatility Model with Jumps

April 21st, 2015, 2:04 pm

Wait, so [$]V[$] is the volatility process and [$]q[$] is the equity process?First, I think you meant [$]dW^VdW^q=\rho dt[$] but that's no big deal.Next, I think you need to "balance" your jumps in the drift part of the equity returns, right? Otherwise you would have a systematic bias emerge and there would be no risk-neutral martingale. It would drift offtrack like the S.S. Minnow on a three hour tour.
 
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Phunfactory
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Moments in a Stochastic Volatility Model with Jumps

April 21st, 2015, 2:56 pm

You where right with[$] dW^VdW^q= \rho dt [$]I know that I don't have any jump compensation in the drift of the dynamics of [$]q_t[$] and this is done - sofar - on purpose. I'am looking on a SVVJ model, but the stock does not matter for my problem. Therefore I didn't mention it. As you have observed [$] V[$] is by design no martingal, because of the drift. However, the process is mean reverting with long-term-level [$] \bar{V}[$], which is reasonable for a volatility process, which I try to model.
 
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Orbit
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Moments in a Stochastic Volatility Model with Jumps

April 21st, 2015, 4:05 pm

No I meant the equity process, doesn't the lack of drift compensation account for the issue you're seeing?
 
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Alan
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Moments in a Stochastic Volatility Model with Jumps

April 21st, 2015, 5:16 pm

QuoteOriginally posted by: PhunfactoryDo anyone know how to calculate the analytic moment of $V_t$ nonetheless?Yes, you can confirm/refute your guess for E[V_t] by using the joint char. function of (V(t),q(t))
 
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agnoatto
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Moments in a Stochastic Volatility Model with Jumps

May 4th, 2015, 10:27 am

Hi,I hope I am not misunderstanding what you are doing, but the dynamics of your instantaneous variance process are not those of a square root process: in the diffusion term of V you should have [$]\sqrt{V_t}dW^V_t[$] in place of [$]\sqrt{q_t}dW^V_t[$]. If you are Euler-discretizing the dynamics as you gave them it does not surprises me that you can not match the results.Finally, if [$]q_t[$] is your asset price process, you should also have [$]\sqrt{V_t}dW^q_t[$]Anyway, hoping I am not misunderstanding.Cheers,
Prof Alessandro Gnoatto, PhD
Department of Economics
University of Verona
Via Cantarane 24 - 37129 Verona - Italy
37129, Verona, Italy
 
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Alan
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Moments in a Stochastic Volatility Model with Jumps

May 4th, 2015, 12:30 pm

See the OP's Apr 21 comment: the asset price SDE has simply not been written and the volatility model is a two component model as written. The full model would have three components and the full generator is so-called affine under some non-correlation assumptions not stated. Anyway, that's my take on what's been posted so far.
Last edited by Alan on May 3rd, 2015, 10:00 pm, edited 1 time in total.
 
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kermittfrog
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Moments in a Stochastic Volatility Model with Jumps

May 8th, 2015, 1:10 pm

QuoteOriginally posted by: AlanQuoteOriginally posted by: PhunfactoryDo anyone know how to calculate the analytic moment of $V_t$ nonetheless?Yes, you can confirm/refute your guess for E[V_t] by using the joint char. function of (V(t),q(t))I second Alan on this. You will have to come up with the moment generating function. Another point: is V supposed to be your vol process? If so: it seems to be possible that this process can become negative, no?