From the stochastic volatility processes we now want to construct the appro- priate stock price processes. Carr, Geman, Madan and Yor use two different approaches which differs in terms of the filtration in which the martingale condition is based on. The first approach prohibits arbitrage based only on the stock price, by assuming that investors can only condition trades on the level of the stock price. The second approach instead assumes that trades can be conditioned also on the level of the L´evy process and the time as defined after the time change. This second approach therefore is arbitrage free not just from the point of view of stock based arbitrage, but also from that one based on the level of the L´evy process and the new clock. This sec- ond approach though more appealing theoretically has the drawback that the variables which determine the stock price process are less easily observable than the stock price itself.
To implement the first approach, Carr, Madan, Geman and Yor deter- mine the risk neutral distribution for the stock price at each future time as a the exponential of the VGSV process and of the GCMYSV process, normalized to reflect the initial term structure of forward prices. Note that here the exponential is just the ordinary exponential and not a stochastic exponential as it will be in the second approach. By doing this the model is spot-forward arbitrage free; furthermore arbitrage of calendar spreads of option is also not allowed since it is based only on the stock price. Variance gamma and CGMY models modifications realized with this approach are de- fined VGSA and CGMYSA, where the letters “SA” remembers that they are free only from stock price based arbitrage. To write the formal expressions of the characteristic functions, let’s define S(t) the stock price at time t, r the constant continuously compounded interest rate and qthe constant continu- ously compounded dividend yield. Consider the class of stochastic volatility L´evy processes, Z(t) from equation (3.13), then we have and we can define the stock price at time t as
S(t) = S(0)exp[(r−q)t+Z(t)
E[exp(Z(t))] Noting that
Carr, Madan, Geman and Yor show that the characteristic function for the logarithm of the stock price at time t for the generic stochastic volatility L´evy process is given by
E[exp(iuln(S(t)))] = = exp{iu[ln(S(0)) + (r−q)t]} × φ(−iψX(u), t, y(0);k, η, λ)
φ(−iψX(−i), t, y(0);k, η, λ)iu
In the particular case of the VGSA model, the characteristic function for the logarithm of the stock price at time t is given by
exp{iu[ln(S(0)) + (r−q)t]} × φ(−iψV G(u; 1, G, M), t, C;k, η, λ)
φ(−iψV G(−i; 1, G, M), t, C;k, η, λ)iu while for the CGMYSA model, the characteristic function for the logarithm of the stock price at time t can be written as
exp{iu[ln(S(0)) + (r−q)t]}
× φ(−iψCGM Y(u; 1, G, M, Yp, Yn, ζ), t, C;k, η, λ) φ(−iψCGM Y(−i; 1, G, M, Yp, Yn, ζ), t, C;k, η, λ)iu
The second approach is realized by compensating the pure jump processes VGSV and CGMYSV to form martingales. These martingales are then exponentiated to yield martingale candidates for forward prices. The au- thors name VGSAM and CGMYSAM the models created with this approach, where the letter “M” stays for martingale. Without entering in the detail of the derivation of the results22, we can say that for generic stochastic volatility L´evy process, the stock price can be define by
S(t) =S(0) exp[(r−q)t] exp{X[Y(t)]−Y(t)ψX(−i)} where it can proved that
exp{X[Y(t)]−Y(t)ψX(−i)}
22For more details see Peter Carr, H´elyette Geman, Dilip B. Madan and Marc Yor,
“Stochastic Volatility for L´evy Processes”, Mathematical Finance, Vol. 13, No. 3, July 2003, page 359-360.
is a martingale. In this case the characteristic function for the logarithm of the stock price at time t is given by
E[exp(iuln(S(t)))] = = exp{iu[ln(S(0)) + (r−q)t]} ×φ(−iψX(u)−uψX(−i), t, y(0);k, η, λ) Moreover for the VGSAM, the characteristic function for the logarithm of the stock price at time t is given by
exp{iu[ln(S(0)) + (r−q)t]}
×φ[−iψV G(u,1, G, M)−uψV G(−i,1, G, M), t, C;k, η, λ]
while for the CGMYSAM the characteristic function for the logarithm of the stock price at time t is
exp{iu[ln(S(0)) + (r−q)t]} ×φ[−iψCGM Y(u,1, G, M, Yp, Yn, ζ)+
−uψCGM Y(−i,1, G, M, Yp, Yn, ζ), t, C;k, η, λ]
The stochastic processes obtained using this second approach are appealing from a theoretical but not from practical point of view because they are based on information which is not available in the market. VGSAM and CGMYSAM models are in fact martingales with respect to the enlarged fil- tration, which includes information from the driving L´evy process and knowl- edge about the subordinator given by the time integrated Cox, Ingersoll and Ross process. If these two processes cannot be separately ascertained from a time series of prices, then serious problems in terms of relevance of the associated martingale condition. Geman, Madan and Yor23 provides condi- tions under which the two processes can be determined from a time series of underlying prices. However even if the two processes can be determined from a time series, most likely the rich dynamics of the option price cannot be adequately captured by a martingale which reflects movements only in two processes. Hence this martingale condition based on a filtration which cannot be observed is not really interesting from a practical perspective.
The attention is therefore focused on the models obtained with the first approach: VGSA and CGMYSA, which provide a better empirical perfor- mance than the martingale ones. VGSA and CGMYSA require only the abil- ity to observe stock prices and so they generate risk neutral price processes
23H´elyette Geman, Dilip B. Madan and Marc Yor, “Stochastic Volatility, Jumps and
whose expectation is consistent with the initial term structure of forward prices, but they do not require that the forward prices are martingales with respect to the filtration generated by the L´evy process and the subordinator.