Update black_scholes.py

Add vanna and vomma

fypy/pricing/analytical/black_scholes.py

@@ -147,6 +147,46 @@ def black76_vega(F: float,
     d_1 = (np.log(F / K) + 0.5 * vol_st ** 2) / vol_st
     return disc * F * norm.pdf(d_1) * np.sqrt(T)
 
+def black76_vanna(F: float,
+                  K: Union[float, np.ndarray],
+                  vol: Union[float, np.ndarray],
+                  disc: float,
+                  T: float) -> Union[float, np.ndarray]:
+    """
+    Vanna(s) for strike(s)
+    :param F: float, forward price
+    :param K: float or array, the Strike(s)
+    :param vol: float or array, the Volatility(ies) ... if float, all strikes get same vol, else a vol smile
+    :param disc: float, the discount factor, e.g. 0.99
+    :param T: float, time to maturity of option
+    :return: float or np.ndarray, same shape as strikes
+    """
+    vol_st = vol * np.sqrt(T)
+    d_1 = (np.log(F / K) + 0.5 * vol_st ** 2) / vol_st
+    d_2 = d_1 - vol_st
+    return -disc * norm.pdf(d_1) * d_2 / vol
+
+
+def black76_vomma(F: float,
+                  K: Union[float, np.ndarray],
+                  vol: Union[float, np.ndarray],
+                  disc: float,
+                  T: float) -> Union[float, np.ndarray]:
+    """
+    Vomma(s) for strike(s)
+    :param F: float, forward price
+    :param K: float or array, the Strike(s)
+    :param vol: float or array, the Volatility(ies) ... if float, all strikes get same vol, else a vol smile
+    :param disc: float, the discount factor, e.g. 0.99
+    :param T: float, time to maturity of option
+    :return: float or np.ndarray, same shape as strikes
+    """
+    sqt = np.sqrt(T)
+    vol_st = vol * sqt
+    d_1 = (np.log(F / K) + 0.5 * vol_st ** 2) / vol_st
+    d_2 = d_1 - vol_st
+    vega = disc * F * norm.pdf(d_1) * sqt
+    return vega * d_1 * d_2 / vol
 
 def black76_delta(F: float,
                   K: Union[float, np.ndarray],