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forward - starting options are path dependent put / call financial contracts characterized by having a strike price expressed in terms of a pre - specified percentage @xmath0 of the asset price taken at a contractually fixed intermediate date @xmath1 $ ] , @xmath2 being the option maturity . the time @xmath3 is known as _ strike - determination time_. the payoff of a forward starting call is therefore @xmath4 these products represent the fundamental component of the so - called cliquets ( see @xcite ) , which are indeed equivalent to a series of forward starting at - the - money options , activated along a sequence of intermediate dates , upon payment of an initial premium . cliquets are often employed to buy protection against downside risk , though preserving an upside potential , for instance in pension plans in order to hedge the guarantees attached to embedded equity linked products . wilmott in @xcite showed that these products are particularly sensitive to the model that one chooses for the dynamics of the underlying s price . in this paper we study a generalization of forward starting options allowing for random strike - determination times . that is to say , we hypothesize that we are making the forward starting option worth , if some future random event happens during the life of the contract with positive probability . the value of the asset at this random time is then compared at maturity with the final value of the underlying , so the call s payoff ( [ fspay ] ) is replaced by @xmath5 for some appropriate random time @xmath6 , where we denoted by @xmath7 . the pricing methodology we will present is an extension of rubinstein s approach ( see @xcite ) , and it can be applied to several existing option models . it is easy to imagine a promising application of this type of contracts in several situations , as we show with the following examples .    * a first example with a speculative goal , could be represented by a call written ( with @xmath8 ) on the manchester united stock ( nyse:*manu * ) , starting the beginning day of the premiere league ( pl ) soccer tournament and ending the closing day of the tournament . the call is triggered by the first time , @xmath6 , the united will eventually lead the tournament with at least three points more than all other teams . if this event happens , then the united will increase its chances to take part to the next champions league ( cl ) tournament , an event which generates substantial additional cash - flows . if , at the end of pl tournament , the united will get the right to play cl , the share value @xmath9 of * manu * at that date will probably be larger than @xmath10 and an investor in the call contract could pocket the difference . * another possible context we may imagine is when a firm expects an adverse market scenario that might cause a downgrading of the firm s rating in the near future . so the firm s owner offers a contract of this type to a smart manager ( hired in the past ) , as an incentive to convince him / her to remain within the firm and to work to improve the firm s position before maturity . as an example , the firm s owner offers this contract to the manager on january , 1st 2015 with maturity @xmath11 year ( with @xmath8 ) . if the event @xmath12 happens , for instance on june , 6th 2015 , then the manager has time until maturity to make the firm s value rise again above @xmath13 and pocket the payoff , otherwise no reward is gained . the manager is therefore interested in assessing the correct current value of the incentive . + in this context , it is also interesting to consider the counterparty credit risk aspects of this type of contract from the point of view of the manager : if the firm ( the counterparty ) does not recover from the adverse market scenario , it might default a short time after the downgrading event or default could even happen before downgrading . this leads to the problem of computing unilateral credit value adjustment ( * cva * ) for this kind of option , since the manager needs to evaluate this further risk of the contract in order to assess correctly the value of the firm s offer . * lastly , it is also possible to consider contracts depending on more than one single random event : for example a cliquet call option could be triggered by the first time the underlying asset reaches a pre - specified upper barrier @xmath14 until the asset reaches , if before maturity @xmath2 , a new barrier level @xmath15 , at that time a second call option with strike @xmath16 and expiring at maturity gets activated . in general , depending on the characteristics of the financial product , one can imagine to construct cliquet type options along some increasing sequence of random times . the literature on classical forward starting options is wide in terms of underlying s dynamics model and evaluation techniques . we shortly review the main contributions , since we will use these results as starting point in our analysis    rubinstein ( 1991 ) ( see @xcite ) was the first to consider the pricing problem of a forward - start option in the black and scholes model . later , valuation of forward starting options in stochastic volatility models was addressed by several authors . lucic ( 2003 ) ( see @xcite ) wrote an evaluation formula by employing the asset s price at determination time as numeraire , transforming those options in plain vanilla ones , giving a closed formula for the heston s model . kruse and ngel ( 2005 ) ( see @xcite ) , followed the same approach , but they developed a pricing formula exploiting the distributional properties of the heston s model ; they remarked the importance of such an extension ( w.r.t . the black - scholes environment ) since forward starting options are particularly sensitive to volatility changes . further , guo and hung ( 2008 ) ( see @xcite ) derived quasi - analytic pricing formulas in general stochastic volatility / stochastic interest rates models , presenting jumps in both the asset s price and the volatility . this was important , since , in contrast to the plain vanilla case , the value of a forward - starting option may not always increase with maturity . it depends on the current term structure of interest rates . in the same context , ahlip and rutkowski ( 2009 ) ( see @xcite ) chose a heston s stochastic volatility model and a cir dynamics for the interest rates , with volatility and short rate both correlated with the dynamics of the stock returns . the main result is an analytic formula for the call case , derived by using the probabilistic approach combined with the fourier inversion technique , as developed in carr and madan ( 1999 ) ( see @xcite ) . van haastrecht and pelsser ( 2011 ) , ( @xcite ) , exploiting fourier inversion of characteristic functions , developed a quantitative analysis of the pricing of forward starting options under stochastic volatility and stochastic interest rates ( with ornstein - uhlenbeck dynamics ) , confirming that these not only depend on future smiles , but are also very sensitive to model specifications such as volatility , interest rate and correlation movements , concluding that it is of crucial importance to take all these factors into account for a correct valuation and risk management of these securities . more recently , ramponi ( 2012 ) ( see @xcite ) extended this fourier analysis to regime switching models for the asset price . finally , in 2006 chicago board options exchange introduced options on its implied volatility index ( vix ) , enhancing the interest in forward starting options to forecast future volatility . albanese , lo and mijatovic ( @xcite ) proposed a spectral method to evaluate numerically the joint distribution between the stock price and its realized variance , which gave a way of pricing consistently european , general accrued variance payoffs , forward - starting and vix options .    to evaluate our generalization of fs options to random determination time , we exploit the techniques developed in a number of papers ( refer to @xcite and to @xcite for a general exposition ) to deal with default time in credit derivatives . the main tool we employ is the so called assumption ( h ) , that guarantees that all martingales remain as such when adding the information generated by the default time . in other words , this means that the no - arbitrage structure is not altered by the introduction of the random time . this leads to the so called key lemma , that allows to switch from the global information ( generated by the asset s price and the random time ) to the information generated only by the observable prices .    following the intensity based approach , we give a general expression of the pricing formula either in conditions of independence or not . under independence between the asset s price and the random time , we specialize and explicit our prices in several computable models ( black & scholes , heston , variance gamma ) and we suggest an affine dependency structure for the intensity when the independence assumption is dropped , showing it is still possible to recover a partially easily computable formula .    finally , keeping in mind the second example presented above , we focus on computing the credit value adjustment , due to counterparty risk , required by these products , when in presence of a further default time ( for a general reference see brigo and chourdakis ( 2009 ) , @xcite ) . the paper is organized as follows . section [ sec1a ] gives the set up , in section [ sec2a ] we present a general approach to the evaluation of this kind of product , in section [ sec3a ] we develop some explicitly computable cases under independence . section [ sec4a ] provides some numerical implementation of the derived pricing formulas , finally section [ sec5a ] treats briefly the counterparty risk implied by this product , when a default time for the issuer might occur . we consider a time interval @xmath17 $ ] , with @xmath2 denoting the maturity of the contracts . we assume that the market is made up of a single asset and a bond living in a probability space @xmath18 , with a filtration @xmath19 , made complete and right continuous to satisfy the _ usual hypotheses _ ( see @xcite ) . moreover , we assume we are in absence of arbitrage opportunity and that @xmath20 is a risk neutral measure selected by some criterion .    before formalizing mathematically the new contract previously described , we need to recall briefly the evaluation of classical forward - start options .    denoting by @xmath21 the discounting process , the price of a forward - start call with percentage @xmath0 and strike - determination time @xmath22 is given by @xmath23,\qquad 0\le t\le   t.\ ] ] in market models where the price of a plain vanilla call option is represented by a deterministic homogeneous function of degree @xmath24 in two spatial variables @xmath25 , that is @xmath26 applied to the given pair @xmath27 , the price of a classical ( call ) forward - start option is @xmath28|\cf_t]={\mathbf{e}}^p[b(t , u ) call(u , s_u , s_u\alpha , t)|\cf_t]\\ \nonumber\!\!\!&=&\!\!\!{\mathbf{e}}^p[b(t , u)call(u , \gamma 1 , \gamma \alpha , t)\big |_{\gamma = s_u } |\cf_t]={\mathbf{e}}^p[b(t , u)(\gamma call(u , 1 , \alpha , t))\big |_{\gamma = s_u } |\cf_t]\\ \label{hom}\!\!\!&=&\!\!\ ! { \mathbf{e}}^p[b(t , u)s_u call(u,1,\alpha , t ) |\cf_t].\end{aligned}\ ] ] scale - invariant models ( see @xcite ) certainly hold this property . furthermore , if the call price is a deterministic function of the current price of the underlying and it does not include any additional stochastic factor , we may conclude that the price of the forward - start option verifies @xmath29 = { \mathbf{e}}^p[b(t , u)s_u|\cf_t ] call(u,1,\alpha , t ) = s_t \ call(u,1,\alpha , t),\ ] ] where in the last equality we used the martingale property of the discounted asset price . notice that in such a case the portfolio having @xmath30 units of the stock  replicates " the @xmath31-value of the forward - start option . sometimes an analogous result may be achieved also in presence of stochastic factors via change of numeraire arriving at a formula of the type @xmath32 = s_t{\mathbf{e}}^{\bar p}[call(u,1,\alpha , t)|\cf_t]\ ] ] for some proper equivalent measure @xmath33 . we want to generalize these contracts allowing the strike - determination time to be random , thus we need to consider a market model ( and hence a probability space ) where the asset price @xmath34 lives together with a random time @xmath6 . we therefore assume that there exists a @xmath35algebra @xmath36 containing @xmath37 and a filtration @xmath38 , satisfying the usual hypotheses , that makes @xmath39 and @xmath40 adapted processes ( that is to say that @xmath6 is stopping time w.r.t . @xmath41 ) and that we may extend the probability @xmath20 to a new probability measure @xmath42 on the measure space @xmath43 . hence , referring to the previous notation , we have @xmath44 for all @xmath45 and from now on we will assume @xmath46 where @xmath47 . we recall that the filtration stopped at the stopping time @xmath6 will be denoted @xmath48 and it means @xmath49 besides we introduce the classical and fundamental hypothesis    * ( h)*every @xmath50martingale remains a @xmath51martingale . this assumption , known as the h - hypothesis , is widely used in the credit risk literature ( see e.g. @xcite and references therein ) . with this hypothesis , we may conclude that @xmath52 remains in general a @xmath53semimartingale and @xmath54 a martingale under the extended probability @xmath42 . so we can replace the arbitrage free pricing formula ( [ deterministic ] ) by the more general expression @xmath55.\ ] ]    from now on , this product will be called a * random time forward starting * option and we will refer to it as * rtfs * options . notice that the payoff remains @xmath56measurable . in what follows we treat only the call case as , under hypothesis ( h ) , a natural extension of the put - call parity holds for these products @xmath57- { \mathbf{e}}^{q}[b(t , t)(\alpha s_{\tau\wedge t}- s_t)^+|\cal g_t ] = s_t - \alpha s_{\tau\wedge t } b(t , \tau\wedge t).\ ] ]    lastly we remark that for @xmath8 , these options are worth something only if the random occurrence happens before maturity , in a sense we could say they are triggered by the random time . in this section we want to understand how to make formula ( [ random ] ) more explicitly computable .    as a general consideration , let us notice that @xmath58\\ \nonumber & = & { \mathbf{e}}^{q}[b(t , t)(s_t-\alpha s_{\tau})^+\mathbf 1_{\{0 < \tau\le t\}}|\cal g_t]+(1-\alpha ) { \mathbf{e}}^{q}[b(t , t)s_t \mathbf 1_{\ { \tau > t\}}|\cal g_t]\\ \nonumber & = & { \mathbf{e}}^{q}[b(t , t)(s_t-\alpha s_{\tau})^+|\cal g_t]\mathbf 1_{\{0 < \tau\le t\}}+ { \mathbf{e}}^{q}[b(t , t)(s_t-\alpha s_{\tau})^+\mathbf 1_{\{t < \tau\le t\}}|\cal g_t]\\ & + & ( 1-\alpha)s_t\mathbf 1_{\{t<\tau\}}- ( 1-\alpha){\mathbf{e}}^{q}[b(t , t)s_t\mathbf 1_{\ { t < \tau\le t\}})|\cal g_t].\end{aligned}\ ] ] having written @xmath59 as @xmath60 .    for the first term there is not much to say , there @xmath61 and it behaves as a call price with strike price @xmath62 , which is completely known at time @xmath31 . the third term represents the guaranteed payoff of this contract , so we have to study the remaining terms . hypothesis ( h ) is going to help us for the last one . we know that @xmath63 is a martingale under @xmath36 and that the event @xmath64 ( since for any @xmath65 , @xmath66 ) , so conditioning internally w.r.t . @xmath67 , we obtain @xmath68={\mathbf{e}}^{q}[{\mathbf{e}}^q(b(t , t)s_t\mathbf 1_{\ { t < \tau\le t\}}|\cg_\tau)|\cal g_t]\\ & = & { \mathbf{e}}^{q}[{\mathbf{e}}^{q}(b(t , t)s_t|\cal g_\tau)\mathbf 1_{\ { t < \tau\le t\}}|\cal g_t]={\mathbf{e}}^{q}(b(t,\tau)s_\tau\mathbf 1_{\ { t < \tau\le t\}}|\cal g_t],\end{aligned}\ ] ] where we used the optional sampling theorem , assuming enough integrability of the asset price process , as @xmath69 almost surely . the second term may be rewritten as @xmath70={\mathbf{e}}^{q}[{\mathbf{e}}^q(b(t , t)(s_t-\alpha s_{\tau})^+\mathbf 1_{\ { t < \tau\le t\}}|\cg_\tau)|\cal g_t]\\ & = & { \mathbf{e}}^{q}[b(t,\tau){\mathbf{e}}^{q}(b(\tau , t)(s_t-\alpha s_{\tau})^+|\cal g_\tau)\mathbf 1_{\ { t < \tau\le t\}}|\cal g_t]={\mathbf{e}}^q[b(t,\tau ) c(\tau,\tau , t)\mathbf 1_{\ { t < \tau\le t\}}|\cal g_t],\end{aligned}\ ] ] with @xmath71 defined w.r.t . @xmath72 as in ( [ deterministic ] ) , where we extended the definition to include also @xmath73 .    to proceed with our evaluation we have two possibilities . either @xmath74 , i.e. @xmath75 is an @xmath50stopping time , or not . in the first case , some computable situations may occur . for instance , if @xmath76 is a hitting time ( choosing the barrier level @xmath77 ) , we have @xmath78\mathbf 1_{\{0 < \tau \le t\}}+ { \mathbf{e}}^{q}[b(t , t)(s_t-\alpha h)^+\mathbf 1_{\{t < \tau \le t\}}|\cf_t]\\ & + & ( 1-\alpha)s_t\mathbf 1_{\{t<\tau\}}- ( 1-\alpha){\mathbf{e}}^{q}[b(t , t)s_t\mathbf 1_{\ { t < \tau\le t\}})|\cf_t].\end{aligned}\ ] ] summarizing @xmath79+(1-\alpha)s_t\mathbf 1_{\{t<\tau\}},\end{aligned}\ ] ] where  barrier price " denotes the price of a barrier call option with strike price @xmath80 which is activated as soon as the threshold @xmath81 is reached by the asset price . if this event does not happen before maturity @xmath2 , at maturity the contract is not going to pay zero to the holder as in standard barrier contracts , but @xmath82-percent of the terminal asset price @xmath9 , since the first three pieces of the formula are zero . therefore , if @xmath83 , this random starting forward option differs from a plain vanilla barrier option by a specific non negative value , never exceeding @xmath84 . other situations may be more or less complex to evaluate , depending upon the definition of the stopping time @xmath6 .    in general , since the observable is the asset s price process , it would be interesting to be able to write the pricing formula in terms of @xmath85 , rather than @xmath86 . for that we have the following key lemma ( see @xcite , @xcite ) . [ key ] for any integrable @xmath87measurable r.v . @xmath88 , the following equality holds @xmath89=q(\tau > t|\cg_t)\frac{{\mathbf{e}}^q\big[\mathbf 1_{\{\tau > t\}}y|\cf_t\big]}{q(\tau > t|\cf_t)}.\ ] ]    applying this lemma to the second and fourth term of ( [ general ] ) , respectively with @xmath90 and remembering that @xmath91 is @xmath51measurable , we obtain @xmath92&=&\mathbf 1_{\{\tau > t\ } } \frac{{\mathbf{e}}^q[b(t,\tau)c(\tau,\tau , t)\mathbf 1_{\{t < \tau\le t\}}|\cf_t]}{q(\tau > t|\cf_t)}\\ \label{key2 } { \mathbf{e}}^q[b(t,\tau)s_\tau\mathbf 1_{\{t < \tau\le t\}}|\cal g_t]&=&\mathbf 1_{\{\tau > t\ } } \frac{{\mathbf{e}}^q[b(t,\tau)s_\tau\mathbf 1_{\{t < \tau\le t\}}|\cf_t]}{q(\tau > t|\cf_t)}.\end{aligned}\ ] ] the above may be rewritten following the hazard process approach . let us denote the conditional distribution of the default time @xmath6 given @xmath85 as @xmath93 and let us notice that for @xmath94 , @xmath95 . in order to apply the so called intensity based approach , we need to assume that @xmath96 for all @xmath97 ( which automatically excludes that @xmath98 ) to well define the so called risk or hazard process @xmath99 with this notation , we rewrite ( [ key1 ] ) and ( [ key2 ] ) as @xmath100=(1-h_t){\mathbf{e}}^q[b(t,\tau)c(\tau,\tau , t)(h_t - h_t)\mathrm e^{\gamma_t}|\cf_t]\\ \label{key4 } & & { \mathbf{e}}^q[b(t,\tau)s_\tau\mathbf 1_{\{t < \tau\le t\}}|\cal g_t=(1-h_t){\mathbf{e}}^q[b(t,\tau)s_\tau(h_t - h_t)\mathrm e^{\gamma_t}|\cf_t],\end{aligned}\ ] ] but by proposition 5.1.1 ( ii ) page 147 of @xcite these conditional expectations may be written as @xmath101&=&{\mathbf{e}}^q[\int_t^t b(t , u)c(u , u , t ) df_u|\cf_t]\\ \label{key6 } { \mathbf{e}}^q[b(t,\tau)s_\tau(h_t - h_t)|\cf_t]&=&{\mathbf{e}}^q[\int_t^t b(t , u)s_udf_u|\cf_t].\end{aligned}\ ] ] if we have the additional hypotheses @xmath102 for all @xmath31 and    ( ind ) : :    @xmath103 and @xmath104 are independent ,    then @xmath75 is independent of @xmath85 , hypothesis ( h ) is automatically satisfied and @xmath105 is deterministic , so the previous formula becomes @xmath106={\mathbf{e}}^q[\int_t^t b(t , u)c(u , u , t ) df_u|\cf_t]\\ \nonumber&=&\int_t^t{\mathbf{e}}^q [ { \mathbf{e}}^q(b(t , t)(s_t-\alpha s_u)^+|\cf_u)|\cf_t]df_u=\int_t^t{\mathbf{e}}^q [ b(t , t)(s_t-\alpha s_u)^+|\cf_t ] df_u\\ & = & \int_t^tc(t , u , t)df_u\end{aligned}\ ] ] and the other is treated similarly . we can summarize the above results in the following    [ gener1 ] with the above notation , we have    1 .   under hypothesis @xmath107 the price of a * rtfs * call option is given by @xmath108\mathbf 1_{\{0 < \tau\le t\}}+(1-\alpha)s_t\mathbf 1_{\{\tau > t\}}\\ & + & \mathbf 1_{\{\tau > t\}}{\mathbf{e}}^q[\int_t^t b(t , u)c(u , u , t)\mathrm e^{-(\gamma_u-\gamma_t)}d\gamma_u|\cf_t]\\ & -&(1-\alpha)\mathbf 1_{\{\tau > t\ } } { \mathbf{e}}^q[\int_t^t b(t , u)s_u \mathrm e^{-(\gamma_u- \gamma_t)}d\gamma_u|\cf_t]\end{aligned}\ ] ] 2 .   under hypothesis @xmath109 the price of a * rtfs * call option is given by @xmath110\mathbf 1_{\{0 < \tau\le t\}}\\ & + & \mathbf 1_{\{\tau > t\}}\big [ \int_t^tc(t , u , t)\mathrm e^{-(\gamma_u- \gamma_t)}d\gamma_u+(1-\alpha)s_t\mathrm e^{-(\gamma_t-\gamma_t ) } \big ] \end{aligned}\ ] ]    part ( a ) is only the assembling of the various pieces previously presented . as for part ( b ) , because of independence , @xmath111 is deterministic and it gets pulled out of the expectation . since @xmath42 is a martingale measure , the integrands of the last two terms verify the martingale property w.r.t . @xmath85 under @xmath42 .    [ rem1 ] if we assume that @xmath112 is absolutely continuous w.r.t . the lebesgue measure , that is to say @xmath113 for some @xmath50adapted and non negative process @xmath114 called intensity ( see @xcite ) , then also @xmath115 is absolutely continuous and if we denote by @xmath116 its density , necessarily we have @xmath117    in this case we have    1 .   under hypothesis @xmath107 the price of a * rtfs * call option is given by @xmath118\mathbf 1_{\{0 < \tau\le t\}}+(1-\alpha)s_t\mathbf 1_{\{\tau > t\}}\\ \nonumber & + & \mathbf 1_{\{\tau > t\}}{\mathbf{e}}^q[\int_t^t b(t , u)c(u , u , t ) \lambda_u\mathrm e^{-\int_t^u\lambda_s ds}du|\cf_t]\\   \nonumber & -&(1-\alpha)\mathbf 1_{\{\tau > t\ } } { \mathbf{e}}^q[\int_t^t b(t , u)s_u \lambda_u\mathrm e^{-\int_t^u\lambda_s ds}du|\cf_t]\end{aligned}\ ] ] 2 .   under hypothesis @xmath109 the price of a * rtfs * call option is given by @xmath119\mathbf 1_{\{0 < \tau\le t\}}\\ \nonumber&+&\mathbf 1_{\{\tau > t\}}\big [ \int_t^tc(t , u , t ) \lambda_u\mathrm e^{-\int_t^u\lambda_s ds}du+(1-\alpha)s_t\mathrm e^{-\int_t^t\lambda_s ds } \big ] \end{aligned}\ ] ]    the above formulas , either in presence of independence or not , rely on the knowledge of the model and of the conditional distribution of the default time . hence , deriving an explicit or implementable formula for the pricing will depend heavily on the modelling choices we are going to make .    of course , the independent case is much easier than the other and we focus on it in the next section , to show that there are some interesting treatable cases . we will always take @xmath8 . when we no longer assume independence of @xmath75 and @xmath34 , we need to resort to formula ( [ hazard1 ] ) and a reasonable choice is to make the hazard rate depend also on the price process . as a first attempt one may try an affine model , so we decide that @xmath120 where @xmath121 are deterministic , positive and bounded functions , while @xmath122 is a positive @xmath51adapted process independent of @xmath85 , @xmath123 .    by using the optional projection theorem ( see @xcite theorem vi.57 ) , we may arrive at the pricing formula @xmath124\mathbf 1_{\{0 < \tau\le t\}}\\ \!\!\!&+&\!\!\!\int_t^t a(u)call(u,1,\alpha , t){\mathbf{e}}^q [   b(t , u)s_u^2\mathrm e^{-\int_t^u   a(s ) s_sds}|\cf_t]{\mathbf{e}}^q[\mathrm e^{-\int_t^u b(s ) z_sds}]du\\ \!\!\!&+&\!\!\!\int_t^t b(u)call(u,1,\alpha , t){\mathbf{e}}^q [    b(t , u)s_u\mathrm e^{-\int_t^u   ( a(s ) s_sds}|\cf_t]{\mathbf{e}}^q[z_u\mathrm e^{-\int_t^u   b(s ) z_sds}]du\\ \!\!\!&+&\!\!\!(1-\alpha){\mathbf{e}}^q[b(t , t)s_t\mathrm e^{-\int_t^t a(s ) s_sds}|\cf_t]{\mathbf{e } } [ \mathrm e^{-\int_t^t b(s ) z_sds}]\big \},\end{aligned}\ ] ] where all the above expectations are of the same type . if not explicitly computable , the expectations can be evaluated via monte carlo or transform methods , once a grid of points @xmath125 in order to approximate the time integrals has been fixed . more precisely , one applies the monte carlo approach to the inner expectations by generating @xmath126 independent paths of the processes @xmath34 and @xmath122 at the grid points . we now show some examples where the above formulas for the rtfs options can be explicitly computed , of course the first case to analyze is the black and scholes model . * pricing in the black & scholes market *    in the classical bs market with constant risk - free rate @xmath127 and volatility @xmath128 , the price of a plain vanilla european call with maturity @xmath2 and strike @xmath129 will be denoted by @xmath130 , while the price of of a forward - start option with maturity @xmath2 , strike - determination time @xmath3 and percentage @xmath8 ( see @xcite ) is simply @xmath131 where @xmath132 @xmath133 hence in formula ( [ fs_pricebs ] ) we have @xmath134 . let us notice that @xmath135 is always true .    henceforth , under ( ind ) , the price of a ( * rtfs * ) option is given by @xmath136 the valuation of ( [ gfspricebs ] ) can be found in closed form in some simple cases . if @xmath137 , @xmath138 under @xmath42 , then for the second term we obtain @xmath139,\ ] ] where @xmath140 and it is clear that @xmath141 for all values of the parameters . the above integrals specialize depending on the choice of the parameters . if @xmath142 we have three cases . if @xmath143 then these integrals can be explicitly computed exploiting integration by parts and the gaussian density , obtaining ( keeping in mind that @xmath144 ) @xmath145.\end{aligned}\ ] ] when @xmath146 ( hence @xmath147 ) , then the above formulas reduce to @xmath148\end{aligned}\ ] ] and for @xmath149 we lose the gaussian integral and we arrive at @xmath150\\ \label{case3a2 } \!\!\!\!\!\!a_2(t , t)\!\!\ ! & = & \!\!\ ! \frac \lambda{\lambda\!-\ ! \big \{\mathrm e^{-r(t\!-t)}\cn ( c_2\sqrt{t\!\!-\!t})\!- \!\mathrm e^{-\lambda(t\!-t ) } \big [ \frac 12 + \!\frac{c_2}{\sqrt{2\lambda\!-\!c_1 ^ 2}}\ ! \int^{\sqrt{(2\lambda\!- c_1 ^ 2)(t\!-t)}}_0\!\ ! \frac{\mathrm e^{\frac{z^2}2}}{\sqrt{2\pi}}dz\big ] \big \}.\end{aligned}\ ] ] if instead condition @xmath151 holds , we arrive in any case to an explicit formula with @xmath152 the same as in ( [ case1a1 ] ) and @xmath153 given by @xmath154.\ ] ]    * ( the merton market model ) *    the above results can be extended with moderate computational effort to the merton jump - diffusion market model ( see e.g. @xcite ) @xmath155 where @xmath156 is a brownian motion , @xmath157 a poisson process with jump - arrival intensity @xmath158 , @xmath159 all independent of each other , @xmath160 - 1 $ ] is the jump - size expectation , @xmath128 is the volatility and @xmath127 is the risk - free rate .    under ( ind ) , for @xmath8 and under the martingale @xmath42 , if @xmath161 , with @xmath162 , the price of a ( * rtfs * ) option is given by @xmath163 where @xmath164 , @xmath165 and @xmath166 are given as in ( [ a1a2bs ] ) for appropriate sets of parameters varying with @xmath167 . the explicit formula is available from the authors , upon request . * pricing with fourier transform : the variance gamma model *    here we assume that the price dynamics is of the form @xmath168 , where @xmath169 is a lvy process . fourier transform is a well established computational tool for derivative pricing , first introduced in @xcite and then further developed by several authors ( e.g. see @xcite or @xcite and the references therein ) . different ways of applying this technique lead to different representation formulas for the risk neutral price of a plain vanilla call option , we choose to consider @xmath170 for @xmath171 and @xmath172 , with @xmath173 $ ] the generalized characteristic function of @xmath174 . because of the independence of the increments , @xmath175 does not depend on @xmath52 , so the model is scale - invariant and , applying ( [ scale ] ) , we may conclude that the price of the standard forward starting option with determination time @xmath3 is @xmath176 by inserting this fourier representation of the forward starting call price into the formulas of proposition 1.1 or remark [ rem1 ] , we get a double integral representation of our rtfs call price .    as usual , under hypothesis ( ind ) , setting @xmath8 , we obtain @xmath177\mathbf 1_{\{0 < \tau\le t\ } } \\   & + & \mathbf 1_{\{\tau > t\ } }   \frac{s_t}{2 \pi } \int_{\mathrm{i } \nu - \infty}^{\mathrm{i } \nu + \infty } \frac 1{\mathrm{i } z - z^2 } \big [ \int_t^t \mathrm{e}^{- r(t - u)(1 + \mathrm{i } z ) } \phi_{x_t- x_u}(-z ) \lambda_u\mathrm e^{-\int_t^u\lambda_s ds}du \big ] dz.\end{aligned}\ ] ] in some cases the inner integral can be solved in closed form . as a matter of fact , if we choose a variance gamma model for the price dynamics @xmath178 with @xmath156 a standard brownian motion and @xmath179 a gamma process independent of @xmath156 , with parameters 1 and @xmath180 and @xmath181 ( see @xcite ) , we have @xmath182 where @xmath183 , with @xmath184 and @xmath185 , denotes the principal complex logarithm . taking also ( for ease of computation ) a constant hazard rate @xmath186 , we may conclude @xmath187\mathbf 1_{\{0 < \tau\le t\ } }   \nonumber \\ \!\!\ ! & + & \!\!\!\mathbf 1_{\{\tau > t\ } }   \frac{s_t}{2 \pi}\lambda \mathrm{e}^{-\lambda(t - t ) } \int_{\mathrm{i } \nu - \infty}^{\mathrm{i } \nu + \infty}\frac 1{\mathrm{i } z - z^2}\ ; \frac { 1 -   \mathrm{e}^{-[r ( 1+\mathrm{i } z)+\frac 1\mu\ln ( 1-\mathrm{i}b\mu z + c^2 \mu \frac{z^2}2 ) - \lambda](t- t)}}{r ( 1+\mathrm{i } z)+\frac 1\mu\ln ( 1-\mathrm{i}b\mu z + c^2 \mu \frac{z^2}2 ) - \lambda}dz,\end{aligned}\ ] ] leading to a computable formula . it is reasonable that the same arguments might be extended to tempered stable distributions as well as to the carr - geman - madan - yor model ( see @xcite ) . * pricing in the heston stochastic volatility market . *    assuming independence , we can arrive to an explicit formula also in the heston stochastic volatility model , given ( under the risk neutral measure ) by @xmath188 where @xmath127 is a constant interest rate . here @xmath85 has to be the natural filtration generated by the couple @xmath189 , which is jointly a bidimensional markov process , hence the final pricing formula will depend only upon the initial values of both @xmath34 and @xmath128 .    as before , we start from the expression @xmath190\mathbf 1_{\{0 < \tau\le t\}}\\ & + & \mathbf 1_{\{\tau > t\}}\big [ \!\int_t^t\!\!c_h(t , u , t , s_t , \sigma_t ) \lambda_u\mathrm e^{-\int_t^u\lambda_s ds}du+(1\!-\alpha)s_t\mathrm e^{-\int_t^t\lambda_s ds } \big ] .\end{aligned}\ ] ] by employing the results in @xcite ( or @xcite ) , we can find an explicit formula for the forward start option . indeed by virtue of the scale invariant property verified also by this model , we have @xmath191 where @xmath192 is the standard price of a call option in the heston model ( for details see for instance @xcite lemma 2.1 ) and @xmath193 and @xmath194 is a modified bessel function of the first kind .    the previous method can be adapted to cover also the case of affine stochastic interest rate market model proposed in @xcite , where the interest rate follows a hull and white model and the stochastic volatility is a ornstein - uhlenbeck process . in this section we report the results of the numerical implementation of some of the models presented before : the black & scholes , the variance gamma and the heston models , with a random time @xmath195 with parameter @xmath162 . in all experiments we set @xmath8 , @xmath196 and we assume the independent hypothesis ( ind ) . analytical or closed - form prices , eventually requiring numerical quadrature , are compared with standard monte carlo approximations of the arbitrage free pricing formula ( [ random ] ) . sample paths of the underlying dynamics are evaluated at the random time @xmath6 and the corresponding payoffs are collected to estimate ( [ random ] ) . in our experiments we simulated @xmath197 samples / paths and remark [ rem1 ] can be easily designed . ] .    adaptive gauss - lobatto quadrature was used to approximate representation integrals ( vg and heston models ) , which required the evaluation of extended fourier transforms involving multivalued functions , such as the complex logarithm . to avoid the well - known numerical instabilities due to a wrong formulation of the characteristic function , we used the so - called _ rotation count algorithm _ by kahl and jckel ( see e.g. @xcite ) . all routines were coded and implemented in matlab , version 8.0 ( r2012b ) on an intel core i7 2.40 ghz machine running under windows 7 with 8 gb physical memory . for each model we report the closed - form ( cf ) and the mc prices for different values of @xmath186 , fixing the other model parameters . in parenthesis we report the absolute error w.r.t . the cf price , followed by the corresponding length of the @xmath198 confidence interval . without loss of generality we set @xmath199 , @xmath200 and @xmath201 .    in the black & scholes model the rtfs price is given by formulas ( [ gfspricebs])-([case4a2 ] ) . exact simulations of the geometrical brownian motion with @xmath202 at the random time @xmath6 are used to estimate the pricing formulas . .*black & scholes * model for different values of @xmath186 . [ cols="^,^,^,^",options="header " , ]     lastly , we choose the random time @xmath195 with varying @xmath203 in order to compare at time @xmath196 , the rtfs price and the price of a fs option with determination time @xmath204 . in other words we compare @xmath205 with @xmath206 , see figure ( [ figall1 ] ) . in all the instances prices decrease as @xmath3 increases , moreover we notice that : i ) for @xmath207 ( i.e. @xmath208 ) fs and rtfs prices both tend to the price of an atm plain vanilla call ; ii ) for @xmath209 ( i.e. @xmath210 ) the fs value trivially goes to @xmath211 , while the value of a rtfs option stays positive since @xmath212 .    . [ figall1],width=680,height=226 ] in this section we are interested in computing the credit value adjustment ( cva ) for a rtfs option , when one of the parties is subject to a possibly additional default time . the interest lies in the fact that those products would be otc and knowing that the cva is computable might be convenient . we first recall what the cva is for an additive cashflow . let us denote by @xmath213 the discounted value of the cashflow between time @xmath31 and @xmath2 ; this value has to be additive , which means that for times @xmath214 it holds @xmath215 all european contingent claims with integrable , @xmath56measurable payoff @xmath216 , trivially verify ( [ additive ] ) since @xmath217 . we call net present value of @xmath218 @xmath219 we now look into unilateral counterparty credit risk . let us suppose that the cashflow is between two parties @xmath220 ( buyer ) and the counterparty @xmath221 , that is subject to default , with default time @xmath222 . then , the discounted value of the cashflow has to be adjusted by subtracting from the defaultless case the quantity @xmath223 where @xmath224 represents a deterministic recovery rate ( see @xcite ) .    in the case of classical forward start options with fixed strike - determination time @xmath3 , it is possible to compute directly the cva . we denote by @xmath225 the filtration generated by the process @xmath226 and by @xmath227 . under independence between @xmath228 and @xmath225 , taking for granted that @xmath54 remains a martingale in the enlarged filtration @xmath229 w.r.t . a risk neutral probability measure @xmath230 which extends the risk neutral probability @xmath20 , we have @xmath231 where the key lemma and the same notation as in section 1 were used , under the assumption that @xmath232 .    to extend the previous formula to the case of the rtfs options we need to consider two random times : the strike - determination time @xmath75 and the counterparty default time @xmath222 . hence we need to extend the original filtration @xmath228 with both @xmath233 and @xmath234 , obtaining @xmath235 accordingly , we assume there exists an extension of the risk neutral probability @xmath20 to a common probability space @xmath236 that we still denote by @xmath42 and we set the corresponding key assumption    * ( hc)*every @xmath85 martingale remains a @xmath72 martingale . taking @xmath8 for simplicity , we have that the price of a defaultable rtfs call option is @xmath237.\ ] ] so , in presence of a recovery rate @xmath238 , we obtain that the cva for this product has to be @xmath239= ( 1\!-\!r ) { \mathbf{e}}^{q}[\mathbf 1_{\{t<\tau_c\le t\}}b(t , t)(s_t - s_{\tau\wedge t})^+|\cal g_t].\end{aligned}\ ] ] we may decompose the expectation as @xmath240\\ \!\!\!&=&\!\!\!{\mathbf{e}}^{q}[\mathbf 1_{\{t<\tau_c\le t\}}\mathbf 1_{\{\tau > t\}}b(t , t)(s_t\!-\!s_{\tau\wedge t})^+|\cal g_t]\!+{\mathbf{e}}^{q}[\mathbf 1_{\{t<\tau_c\le t\}}\mathbf 1_{\{\tau\le t\}}b(t , t)(s_t\!-\!s_{\tau\wedge t})^+|\cal g_t]\\ \!\!\!&=&\!\!\!{\mathbf{e}}^{q}[\mathbf 1_{\{t<\tau_c\le t\}}\mathbf 1_{\{t<\tau\le t\}}b(t , t)(s_t - s_{\tau})^+|\cal g_t ] + \mathbf 1_{\{\tau\le t\}}{\mathbf{e}}^{q}[\mathbf 1_{\{t<\tau_c\le t\}}b(t , t)(s_t - s_{\tau})^+|\cal g_t],\end{aligned}\ ] ] where the last equality is justified by the fact that the first term in the first passage is equal to 0 . in the second equality the crucial term is the first , as the second reduces to the cva of a standard forward start option , for which formula ( [ cvafs ] ) applies . we may handle the first term under condition of independence between @xmath222 and @xmath241 . denoting by @xmath242 , we have @xmath243 and applying the key lemma to @xmath72 and @xmath244 we obtain @xmath245\\ & = & q(\tau_c > t|\cg_t)\frac{{\mathbf{e}}^{q}[\mathbf 1_{\{t<\tau_c\le t\}}\mathbf 1_{\{t < \tau\le t\}}b(t , t)(s_t - s_{\tau})^+|\cf^h_t]}{q(\tau_c > t|\cf^h_t)}.\end{aligned}\ ] ] the independence of @xmath246 from the remaining factors gives @xmath245\\ & = & q(\tau_c > t|\cg_t)\frac { q(t < \tau_c\le t)}{q(\tau_c > t)}{\mathbf{e}}^{q}[\mathbf 1_{\{t < \tau\le t\}}b(t , t)(s_t - s_{\tau})^+|\cf^h_t]\end{aligned}\ ] ] and if @xmath247 , the above reduces to @xmath248=q(t < \tau_c\le t ) { \mathbf{e}}^{q}[\mathbf 1_{\{t < \tau\le t\}}b(t , t)(s_t - s_{\tau})^+|\cf^h_t]\ ] ] which is the weighted value of a rtfs option , exactly as it happened with the standard fs option . summarizing we have    under hypothesis ( hc ) and the condition that @xmath249 , if @xmath222 is independent of @xmath241 , then the cva of a defaultable rtfs call option of price @xmath250 , with recovery rate @xmath251 $ ] , is given by @xmath252 . lastly we remark that if the two random times coincide , @xmath253 , then we may conclude that @xmath254 which implies that the defaultable price is @xmath255 . the authors thank prof . c. chiarella for useful suggestions at an early stage of this work . d. brigo , k .chourdakis , _ counterparty risk for credit default swaps : impact of default volatility and spread correlation _ , international journal of theoretical and applied finance , vol . 12 , no . 7 , 1007 - 1029 , ( 2009 ) . g. h. cheang , c. chiarella , _ a modern view on merton s jump - diffusion model _ , in `` stochastic processes , finance and control : a festschrift in honor of robert j. elliott '' , samuel n. cohen , dilip madan , tak kuen siu and hailiang yang ( eds . ) , pg 217 - 234 , world scientific , ( 2012 ) . a. van haastrecht , a. pelsser _ accounting for stochastic interest rates , stochastic volatility and a general correlation structure in the valuation of forward starting options _ , journal of futures markets , vol . 31 , no . 2 , 103 - 125 ( 2011 ) . a. ramponi , _ on fourier transform methods for regime - switching jump - diffusions and the pricing of forward starting options _ , international journal of theoretical and applied finance , vol . 15 , no . 5 , ( 2012 )
we introduce a natural generalization of the forward - starting options , first discussed by m. rubinstein ( @xcite ) . 
 the main feature of the contract presented here is that the strike - determination time is not fixed ex - ante , but allowed to be random , usually related to the occurrence of some event , either of financial nature or not . 
 we will call these options * random time forward starting ( rtfs)*.    we show that , under an appropriate  martingale preserving " hypothesis , we can exhibit arbitrage free prices , which can be explicitly computed in many classical market models , at least under independence between the random time and the assets prices . 
 practical implementations of the pricing methodologies are also provided . 
 finally a credit value adjustment formula for these otc options is computed for the unilateral counterparty credit risk .    * keywords * : random times , forward - starting options , cva .    * jel classification * : g13