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import math |
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from scipy.stats import norm |
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def BSM_withAdjoints(S0, r, y, sig, K, T): |
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#Evaluation |
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sqrtT = math.sqrt(T) |
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df = math.exp(-r * T) |
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F = S0 * math.exp((r - y) * T) |
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std = sig * sqrtT |
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d = math.log(F / K) / std |
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d1, d2 = d + 0.5 * std, d - 0.5 * std |
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nd1, nd2 = norm.cdf(d1), norm.cdf(d2) |
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Call_P = df * (F * nd1 - K * nd2) |
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Put_P = df * (K * (1 - nd2) - F * (1 - nd1)) |
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# Adjoint calculation |
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v_ = 1.0 |
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df_ = v_ * (F * nd1 - K * nd2) |
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F_ = v_ * df * nd1 |
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nd1_ = v_ * df * F |
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K_ = - v_ * df * nd2 |
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nd2_ = - v_ * df * K |
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d2_ = nd2_ * norm.pdf(d2) |
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d1_ = nd1_ * norm.pdf(d1) |
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d_ = d2_ |
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std_ = - 0.5 * d2_ |
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d_ += d1_ |
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std_ += 0.5 * d1_ |
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F_ += d_ / (F * std) |
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K_ -= d_ / (K * std) |
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std_ -= d_ * d / std |
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sig_ = std_ * sqrtT |
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T_ = 0.5 * std_ * sig / sqrtT |
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S0_ = F_ * F / S0 |
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r_ = F_ * T * F if r == 0 else 0 |
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y_ = - F_ * T * F if y == 0 else 0 |
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T_ += F_ * (r - y) * F |
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r_ += - df_ * df * T if r == 0 else 0 |
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T_ += - df_ * df * r |
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return (Call_P,Put_P), (S0_, r_, y_, sig_, K_, T_) |
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def IVFromPremium(C0,S0, r, y, sig, K, T): |
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V0 = sig #initial guess |
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C=100 |
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while abs(C) >= 0.0001: |
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OptionPrice,adjoints= BSM_withAdjoints(S0, r, y, V0, K, T) |
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C1= OptionPrice[0] |
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Cprime = adjoints[3] |
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C = C1 - C0 |
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V0 = V0 - C/Cprime |
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return V0 |
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if __name__ == "__main__": |
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#Example Fwd=22045, Rate & Yeild=0, Vol=0.1352, Strike=22050, DTE=6 (0.016438356) |
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Fwd=22045 |
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Rate=0 |
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Yeild=0 |
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Strike=22050 |
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DTE=0.016438356 #(6 DTE) |
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callPrem = 149.01 |
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initial_guess_vol=0.4 |
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sig= IVFromPremium(callPrem,Fwd, Rate,Yeild, initial_guess_vol, Strike, DTE) |
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print("Implied Vol",sig) |
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OptionPrice,adjoints= BSM_withAdjoints (Fwd, Rate,Yeild, sig, Strike, DTE) |
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print("Call Option Price:", OptionPrice[0]) |
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print("Put Option Price:", OptionPrice[1]) |
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print("S0/Delta:", adjoints[0]) |
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print("r/Rho:", adjoints[1]) |
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print("y/Dividend yeild:", adjoints[2]) |
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print("sig/Vega:", (adjoints[3])) |
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print("K/Strike:", adjoints[4]) |
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print("T/Theta:", (adjoints[5])) |
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