•In time step t the volatility sigma becomes,
σ^2 (∆t) =E(S^2 ) -[E(S)]^2 =p(u-1)^2+(1-p) (d-1)^2-(p(u-1)+(1-p)(d-1))^2
σ^2 (∆t) =E(S^2 ) -[E(S)]^2 =p(1-p)(u-1)^2+(1-p) [p(d-1)]^2-2p(1-p)(u-1)(d-1)
=>p(1-p) (u-d)^2=σ^2 (∆t)
=>σ^2 (∆t)=(e^r∆t-d)(u-e^r∆t )
⇒e^r∆t (u+d)-ud-e^2r∆t=σ^2 (∆t)
ignoring higher order powers of t,
(1+r∆t)(u+d)-ud-1-2r∆t =σ^2 (∆t)=> u+d+ur∆t+dr∆t-ud-2r∆t-1=σ^2 (∆t),
let u=1/d,u+1/u-2=σ^2 (∆t) ...(Eqn A)
=>u^2-(σ^2 (∆t)+2)u+1=0 => u=((σ^2 (∆t)+2)+√(σ^4 (∆t)+〖4σ〗^2 (∆t) ))/2=>u=1+σ√∆t => u =e^σ∆t and d=e^(-σ∆t).
•If μ is the stock return in real world then real world probability is, up∗+(1-p∗)d=e^μ∆t=>p∗=(e^μ∆t-1)/(u-d) put this in equation with r-> μ is e^μ∆t (u+d)-ud-e^2r∆t=σ^2 (∆t) shall give same up and down movements as u =e^σ∆t and d=e^(-σ∆t). Therefore, in real world if volatility is σ1 then σ1^2 (∆t) =u+1/u-2 is same as (Eqn A) since u and d are identical in real and risk neutral world.Thus, σ1^2 (∆t)=σ^2 (∆t) =>σ1=σ thus volatility is same as we move from risk neutral to real world.