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Since market models are models of forward rates, a speci cation of the volatility structure of the forward rates uniquely determines their instantaneous drifts via the no-arbitrage (HJM) condition. We consider a LIBOR market model based on N forward LIBOR rates Fi(t), i = 0, . . . , N 1, where Fi(t) = Fi(t; Ti 1, Ti). Under the Qi probability measure, the measure associated with the Ti-forward measure and the P( , Ti) numeraire (i.e., the price of a bond whose maturity coincides with a maturity of the forward rate) the forward rate is a martingale. Note that Fi(t)P(t, Ti) is the price of a tradable asset, the difference between two discount bonds with notional principal amounts 1/ i. Fi(t)P(t, Ti) = (P(t, Ti 1) P(t, Ti))/ i Since the price of the tradable asset Fi(t)P(t, Ti) divided by the zero-coupon bond numeraire P( , Ti) is a martingale, the ith forward rate evolves by dFi(t) = i(t)Fi(t)dzi(t) In general, we need to know the dynamics of the forward rate Fi(t) under a measure Qk different from Qi, for t min(Tk, Ti 1). We can obtain the dynamics of Fi under the forward-adjusted measure Qk. We assume that each forward rate is driven by a standard Brownian motion zi with a time-dependent lognormal volatility and a time-dependent instantaneous correlation structure E[dzidzj] = ijdt. Under this general framework, the forward dynamics are given by dFi(t) = i(t)Fi(t)dt + i(t)Fi(t)dzi(t) where the drift i is determined by the no-arbitrage condition and depends on the choice of the numeraire (chosen measure). If we take a zero-coupon bond expiring at one of the reset times Tk to be the numeraire, P( , Tk), then the instantaneous drift is:

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It is clear that an individual who has asked to withdraw from a study should not be invited to take part in any primary data collection activity in the future, and systems are needed to put this into effect reliably However, though few studies would deliberately risk irritating a study member in this way, it is less obvious whether that person would choose to receive correspondence from the study team, for example telling them about the latest findings It also seems likely that someone who asks to withdraw from a study would not want to be included in the next stage of any ongoing administrative data collection exercise One example of this is the proposed biennial downloads of National Insurance Contributions and benefits, to which roughly three-quarters of ELSA respondents agree.

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double rate : interest rate double div : dividend yield double strike : strike price double price : stock price double T : time to maturity [out]: double : put price **********************************************************************************/ double BlackScholesModel::calcBSPutPrice(double vol, double rate, double div, double strike, double price, double T) { double prob1; double prob2; double putprice; double d1, d2; d1 = (log(price/strike) + (rate div + (vol)*(vol)/2)*T)/(vol*sqrt(T)); d2 = d1 vol*sqrt(T); prob1 = normalCalc(-d1); prob2 = normalCalc(-d2); putprice = strike*exp(-rate*T)*prob2 price*exp(-div*T)*prob1; return putprice; }

However, this assumption may be wrong; it might be the case that they would not object to future downloads of administrative data as long as it did not involve them personally in any action The appropriate response is even less clear with another administrative data linkage This is the agreement to be flagged on the National Health Service Central Register, which, at some time in the future, will result in information from the person s death certificate being passed to the study team so that their age and cause of death can be identified At the point when an individual withdraws from a study, it is again unclear whether the one-off consent that they gave for this flag, perhaps many years earlier, is necessarily void.