Question

The price S(u), 0 􏰃 u 􏰃 t, of the share is driven by a geometric Brownian motion: S(u) = Seμu+σW(u). A proportional dividend on this share is paid continuously at rate q > 0 and is reinvested in the share. The continuously compounded interest rate is r. Compute the no-arbitrage price of a derivative with expiration time t and payoff function

R(t) = [S(t/3)S(2t/3)S(t)]1/3 .

Answer 1

Consider the Brownian motion given by

$S(u)=Se^{μu+σW(u) }$ $0≤u≤t ……(a)$

Payoff function is given by

$R(t)=[S(\frac{t}{3})S(\frac{2t}{3})S(t)^{\frac{1}{3}} ……(b)$

where t is the expiration time.

For this first find the value of $S \space at \space u=0,t,(\frac{t}{3}),(\frac{2t}{3})$

$At \space u=0$

equation(a) becomes

$S(0)=Se^{0+σW(0)}$

$S(0)=Se^{σW(0) } ……(c)$

at

 u=t

$S(t)=Se^{μt+σW(t) } ……(d)$

$at u=\frac{t}{3}$

$S(\frac{t}{3})=Se^{μ(\frac{t}{3})+σW(\frac{t}{3}) } ……(e)$

$at u=\frac{2t}{3}$

$S(\frac{2t}{3})=Se^{μ(\frac{2t}{3})+σW(\frac{2t}{3}) } ……(f)$

Put the values from equation(c) ,equation(d) ,equation(e) and equation(f)in equation (b)

$R(t)=[S(\frac{t}{3})S(\frac{2t}{3})S(t)]^{\frac{1}{3}}$

$R(t) =[(Se^{μ(\frac{t}{3})+σW(\frac{t}{3})})(Se^{μ(\frac{2t}{3})+σW(\frac{2t}{3})})(Se^{μt+σW(t)})]^\frac{1}{3}$

$=[(S^{3}e^{μ(\frac{t}{3})+σW(\frac{t}{3})})(e^{μ(\frac{2t}{3})+σW(\frac{2t}{3})})(e^{μt+σW(t)})]^\frac{1}{3}$

$=S[(e^{μ(\frac{t}{3})+σW(\frac{t}{3})})(e^{μ(\frac{2t}{3})+σW(\frac{2t}{3})})(e^{μt+σW(t)})]^\frac{1}{3}$

$=S[ e^{μ}(\frac{t}{3}+\frac{2t}{3}+t)eσW(\frac{t}{3}+\frac{2t}{3}+t) ]^{\frac{1}{3}}$

$=S[e^{μ(2t)}e^{σW(2t)}]^\frac{1}{3}$

$R(t)=S[e^{μ(2t)+σW(2t)}]^\frac{1}{3 } ……(g)$

Now we find no-arbitrage price of a derivative .

differentiate equation(g) with respect to t.

$R'(t)=\frac{S}{3}[e^{μ(2t)+σW(2t)}]^{−\frac23}(e^{μ(2t)+σW(2t})(2μ+2σW)$

$=\frac{S}{3}[e^{μ(2t)+σW(2t})]^\frac{1}{3} (2μ+2σW)$

$R'(t)=\frac{2S}{3}[e^{μ(2t)+σW(2t)}]^\frac{1}{3} (μ+σW)$