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(* Module/function to compute box option values. Inputs:
p0 - current price of underlying instrument
v - volatility of underlying instrument (per year)
p1 - box price lower limit
p2 - box price upper limit
t1 - time to start of option, in hours
t2 - time to end of option, in hours
Output: probability of hitting the box
showit := Module[{},
Export["/tmp/math.jpg",%, ImageSize->{800,600}]; Run["display /tmp/math.jpg&"]]
(* Source: *)
boxvalue[p0_, v_, p1_, p2_, t1_, t2_] :=
(* the pdf of Log[price] at t1; chance that Log[price]=x *)
pdflp[x_] =
PDF[NormalDistribution[Log[p0], Sqrt[t1]*v/Sqrt[365.2425*24]]][x];
(* the anti-cdf of the max of Log[price]-Log[price[t1]] between t1 and t2;
chance this value is more than x, for x > 0 *)
cdfmaxlp[x_] = 1-Erf[x/(v*Sqrt[t2-t1]/Sqrt[24*365.2425/Sqrt[2]])];
(* 3 cases: price[t1] < p1, between p1 and p2, or price[t1] > p2 *)
(* these are silly global variables I added at the last second + I know
this is bad programming practise *)
upandin = NIntegrate[pdflp[x]*cdfmaxlp[Log[p1]-x],{x,-Infinity,Log[p1]}];
hitleftedge = NIntegrate[pdflp[x]*1,{x,Log[p1],Log[p2]}];
downandin = NIntegrate[pdflp[x]*cdfmaxlp[x-Log[p2]],{x,Log[p2],Infinity}];
(* returns several useful quantities:
chance of hit
value of $1000 hit/miss options [2 quants]
psuedodelta of hit/miss options [2 quants]
psuedotheta of hit/miss options [2 quants]
psuedovega of hit/miss options [2 quants]
quants[p0_, v_, p1_, p2_, t1_, t2_] :=
Module[{p, hv, mv, deltah, thetah, vegah, deltahigh, deltalow, thetahigh,
thetalow, vegahigh, vegalow, deltam, thetam, vegam},
p = boxvalue[p0, v, p1, p2, t1, t2];
hv = 1000/p;
mv = 1000/(1-p);
deltahigh = boxvalue[p0+1/20000, v, p1, p2, t1, t2];
deltalow = boxvalue[p0-1/20000, v, p1, p2, t1, t2];
thetahigh = boxvalue[p0, v, p1, p2, t1+1/120, t2+1/120];
thetalow = boxvalue[p0, v, p1, p2, t1-1/120, t2-1/120];
vegahigh = boxvalue[p0, v+1/200, p1, p2, t1, t2];
vegalow = boxvalue[p0, v-1/200, p1, p2, t1, t2];
deltah = 1000/deltahigh - 1000/deltalow;
deltam = 1000/(1-deltahigh) - 1000/(1-deltalow);
thetah = 1000/thetahigh - 1000/thetalow;
thetam = 1000/(1-thetahigh) - 1000/(1-thetalow);
vegah = 1000/vegahigh - 1000/vegalow;
vegam = 1000/(1-vegahigh) - 1000/(1-vegalow);
{p, hv, mv, deltah, deltam, thetah, thetam, vegah, vegam}
(* Binary options:
Binary options are a limiting case of box options, but the formulas are
p0 - current price of underlying instrument
v - volatility of underlying instrument (per year)
s - strike price of binary option
e - time to option expiration, in hours
Output: probability binary call will be in money
bincallvalue[p0_, v_, s_, e_] =
1-CDF[NormalDistribution[Log[p0],Sqrt[e/(365+2425/10000)/24]*v], Log[s]]
bincallvalue2[p0_, v_, s_, e_] =
1-CDF[NormalDistribution[Log[p0],v*Sqrt[e]], Log[s]]
(* Barrier options:
Another limiting case of box options; doesn't offer these, but close
barriervalue[p0_, v_, s_, e_] =
(1 - Erf[(-Log[p0] + Log[s])/(2*Sqrt[e]*v)])/2