After watching one of the Wolfram Finance Platform videos from Michael Kelly I found out how to write the Black Scholes equation in Mathematica, in terms of N the CDF of the Normal Distribution and the two auxiliary equations. My code has been converted to InputForm, it'll be easier to read as StandardForm.
cdfDist[x_] := CDF[NormalDistribution[], x];
auxEqu1[s_, sigma_, k_, t_, r_, q_] := (Log[s/k] + t*(r - q))/(sigma*Sqrt[t]) + (sigma*Sqrt[t])/2;
auxEqu2[s_, sigma_, k_, t_, r_, q_] := (Log[s/k] + t*(r - q))/(sigma*Sqrt[t]) - (sigma*Sqrt[t])/2;
And finally,
BlackScholesCall[s_, k_, v_, r_, q_, t_] :=
s E^(-q t) cdfDist[auxEqu1[s, v, k, t, r, q]] - k E^(-r t) cdfDist[auxEqu2[s, v, k, t, r, q]];
BlackScholesPut[s_, k_, v_, r_, q_, t_] :=
k E^(-r t) cdfDist[-auxEqu2[s, v, k, t, r, q]] - s E^(-q t) cdfDist[-auxEqu1[s, v, k, t, r, q]];
I then followed Michael's instructions to compute a first-order differential analysis of the Black Scholes Theta Call and Put, i.e. the total derivative of the BlackScholesCall and Put with respect to the following rules:
SetAttributes[{s, k, q}, Constant];
stochasticRules = {Dt[r, {t}] -> Subscript[alpha, r], Dt[sigma, {t}] -> Subscript[alpha, sigma]};
And finally evaluating:
StochasticCallTheta[s_, k_, sigma_, r_, q_, t_] =
-Evaluate[Dt[BlackScholesCall[s, k, sigma, r, q, t], {t}]] /. stochasticRules;
StochasticPutTheta[s_, k_, sigma_, r_, q_, t_] =
-Evaluate[Dt[BlackScholesPut[s, k, sigma, r, q, t], {t}]] /. stochasticRules;
Simplify[StochasticCallTheta[s, k, sigma, r, q, t]]
However, the final output is in terms of s, k, sigma, r, q and t. When I'm speaking with people in the Financial Engineering community they expect to see the output in terms of the two auxiliary equations I defined above (auxEqu1, auxEqu2). I looked through the Documentation to see if there was a "simplify and substitute" function, but couldn't find one. This is essentially want I want to do in Mathematica syntax
Simplify[StochasticCallTheta //. Thread[{ (Log[s/k] + t*(r - q))/(sigma*Sqrt[t]) +
(sigma*Sqrt[t])/2, (Log[s/k] + t*(r - q))/(sigma*Sqrt[t]) -
(sigma*Sqrt[t])/2} ->{auxiliary1,auxiliary2}]
But the ReplaceRepeated doesn't work like that because it only sees patterns. I don't mind having to write two or three lines to persuade Mathematica into transforming the output of StochasticCallTheta in terms of the auxiliary equations, but I just can't think of how to even start.
