Black-Scholes Plots

Dennis SilvermanMathematical Physics 212  BDepartment of Physics and AstronomyU . C .  Irvine

<< Statistics`NormalDistribution`

?CDF

?NormalDistribution

pdfn[y_] = PDF[NormalDistribution[0, 1], y]

Plot[normalint[y], {y, -3, 3}]

nd[d_] := CDF[NormalDistribution[0, 1], d]

nd[0]

Plot[nd[y], {y, -3, 3}]

Interest rate :

r = 0.05/365

v = Sqrt[.3*2*r]

(v^2) is the variance rate of the square of the relative stock
       price fluctuations,  
    assuming a random walk Gaussian distribution withthe variance spreading proportional to the time .   Here we 
      take(v^2)/2  =  0.3 r,  
    or a v of about 1  % fluctuation per day .

v^2/(2*r)

We take the maturity time tmat of 90 days :

tmat = 90

d1 := 
      (Log[xc] + (r + v^2/2) (tmat - t))/
        (v *Sqrt[tmat - t])

d2 := 
      (Log[xc] + (r - v^2/2) (tmat - t))/
        (v*Sqrt[tmat - t])

Scale the call option price ω by the strike price c 
      and call it therelative call value wc = ω/c .

Scale the stock price x by the strike price c and call it 
      the relativestock value xc = x/c .

The scaled solution to the Black - Scholes equation is then

wc := xc*nd[d1] - Exp[-r (tmat - t)] nd[d2]

wc

timetable := Table[wc, {t, 0, tmat, 29.99}]

t =.

\(bs := 
      Plot[Evaluate[timetable], {xc, 0.9, 1.1}, 
		
        AxesLabel -> {"Stock/Strike", ""}, 
        PlotRange -> {{0.9, 1.1}, {0.0, 0.12}}, 
		GridLines -> Automatic, 
        
	PlotLabel -> "Black-Scholes Call/Strike for t from 0 to 90"]
        \)

bs