There was an error in the original post. The FinancialDerivative[] function can be used with NonlinearModelFit if the inputs parameters have _?NumericQ. On the post I divided days by 365, because that was needed for Black-Scholes, but not the Mathematica function.
In[]:=
amcall=americanCall[k,days,intrate,σ,div,curprice]
Out[]=
americanCall[k,162,0.0425,σ,0.007908,758.725]
In[]:=
nlm1=NonlinearModelFit[callPrices,amcall,{{σ,0.3}},k]​​nlm1["BestFitParameters"]
NonlinearModelFit
:The step size in the search has become less than the tolerance prescribed by the PrecisionGoal option, but the gradient is larger than the tolerance specified by the AccuracyGoal option. There is a possibility that the method has stalled at a point that is not a local minimum.
Out[]=
FittedModel
americanCall[k,162,3,759.]

Out[]=
{σ0.364905}
In[]:=
Plot[amcall/.nlm1["BestFitParameters"],{k,500,1200},PlotLabel->"Call Price Fit"];​​Show[%,ListPlot[callPrices]]
Out[]=
In[]:=
volam=σ/.nlm1["BestFitParameters"]
Out[]=
0.364905
In[]:=
g4=Plot[{americanCall[k,days,intrate,volam,div,curprice],-americanPut[k,days,intrate,volam,div,curprice]},{k,500,1200}];​​Show[g0,g1,g4,PlotLabel->"Call and Put Fits with AmCall Volatility"]
Out[]=