Options_2009-05-25_T L.pdf

1. The replicating strategy of the derivative at time 0 is
hS = sinh u - sinh d / s(eu - ed) = 1/2S(0)(1+e-u-d) units of the stock and
hB = eu sinh d - ed sinh u / B(0)er(eu - ed) = -e-r/2B(0)(e-u + e-d) units of the bond.

2. The price of the derivative at time t is
Y(t) = 1/S(T0)c(t, (1-δ)s, S(T0), T) = 1/S(T0)((1-δ)S(t)φ(D1(t)) - S(T0)e-rτφ(D2(t)))
where D1(t) = (1/σ√τ)(ln((1-δ)S(t)/S(T0)) + (r + σ^2/2)τ) and
D2(t) = (1/σ√τ)(ln((1-δ)S(t)/S(T0)) + (r - σ^2/2)τ).

3. The delta of the derivative is positive and does not exceed
ln(1 + a/K) / (σ√(2π(T-t))) at time t < T.

4. (a) The call price equals sφ(d1) - Ke-rτφ(d2) where
d1 = (1/σ√τ)(ln(s/K) + (r + σ^2/2)τ) and d2 = d1 - σ√τ.
(b) The delta of the call equals φ(d1).

5. The price of the derivative at time t is
C(t) = Se^(-rf(T-t))φ(d1) - Ke^(-r(T-t))φ(d2) where
d1 = (1/σ√(T-t))(ln(S/K) + (r - rf + σ^2/2)(T-t)) and
d2 = d1 - σ√(T-t).