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54) Let us denote the rate changes at time t for the time period s s + ] in the upstate and the downstate of the rst random shock and the upstate and the downstate of the second random shock as 0 1 0 1 V (t s) C V (t s) C Y1(t s) = B @ 1 A Y2(t s) = B@ 2 A U1 (t s) U2 (t s) respectively. 55) Pr(Y1 = V1 Y2 = V2 ) = q00(t) Pr(Y1 = V1 Y2 = U2 ) = q01 (t) Pr(Y1 = U1 Y2 = V2 ) = q10(t) Pr(Y1 = U1 Y2 = U2) = q00 (t): We obtain the well known discrete version of the Heath-Jarrow-Morton model 15]. It is easy to see that this is a fourth-nomial model with u0 = V1 + V2 u1 = V1 + U2 u2 = U1 + V2 u3 = U1 + U2 : Finally, we gives an arti cial example to show how to compute a risk-neutral probability measure for a given trinomial model.
Consequently, we look for the risk-neutral probability measure Q such that Q(t) = Q is also independent of t 2. 49) The model we just obtained is the well known Ho-Lee model 16]. This model has a very appealing feature: recombining. In a recombining binomial model, the security prices are determined only by the number of upstates and the number of downstates that have occured in the past and are independent of the order of those states occured. If a model is recombining, the number of states increases linearly instead of exponentially.
All the paths (with probability one) of W (t) are continuous. It is easy to see that for any Wiener process W (t), 1 (W (t) ; t) is a standard Wiener process. Thus, from now on we always denote a standard Wiener process as W (t) and alternative Wiener processes are written in the form t + W (t). Sometime we need to deal with a Wiener process starting at a point, say x, away from zero. A standard Wiener process in this case is x + W (t). We call it the standard Wiener process starting at x and denote it as W (t) W (0) = x: As an application of Wiener processes, we consider a market with a riskfree bond and a risky security over the period 0 T ].
Lecture Notes in Mathematical Finance by S.Lin.