ON THE USE OF NUMERAIRES IN OPTION PRICING by Simon ...

whereσ Z (t) =σ S + Σ p (t, T 1 ), (21)and where W 1 is a Q T 1Wiener process.Under the assumptions above the volatility σ Z is deterministic, thus guaranteeingthat Z has a lognormal distribution. We can in fact writedZ t = Z t kσ Z (t)k dV 1t ,where V 1 is a scalar Q T 1Wiener process. We may thus use a small variation ofthe Black-Scholes formula to obtain the final pricing resultProposition 5.1 The price, at t, of the convertible bond is given by the formulaΠ (t; X )=S t N[d 1 ] ¡ p(t, T 1 )N[d 2 ]+p(t, T 1 ),whered 1 =µ µ 1Stp ln+ 1 σ2 (t, T 0 ) p(t, T 1 ) 2 σ2 (t, T 0 ) ,d 2 = d 1 ¡ p σ 2 (t, T 0 ),σ 2 (t, T 0 ) =Z T0σ Z (t) = σ S +tkσ Z (u)k 2 du,Z T1tσ f (t, s)ds6 Employee stock ownership plans6.1 Institutional setupIn employee stock ownership plans (ESOP) it is common to include an optionof essentially the following form: The holder has the right to buy a stock at theminimum between its price in 6 months and in 1 year minus a rebate (say 15%).The exercise is one year.6.2 Mathematical modelIn a more general setting the ESOP is a contingent claim X ,tobepayedoutat time T 1 ,oftheformX = S T ¡ β min [S T1 ,S T0 ] , (22)sointheconcretecaseabovewewouldhaveβ =0.85, T 0 =1/2 andT 1 =1.The problem is to price X at some time t · T 0 , and to this end we assume a16