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blackscholes.py

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+'''
+Name : Alex Habegger
+Link : https://miamioh.instructure.com/courses/160636/files/folder/Project?preview=23048333
+Goal: Experiment with Black Scholes in Python
+'''
+
+from scipy import stats
+from scipy.stats import norm
+import math
+
+'''
+The Black-Scholes model makes certain assumptions:
+ - No dividends are paid out during the life of the option.
+ - Markets are random (i.e., market movements cannot be predicted).
+ - There are no transaction costs in buying the option.
+ - The risk-free rate and volatility of the underlying asset are known and constant.
+ - The returns on the underlying asset are log-normally distributed.
+ - The option is European and can only be exercised at expiration.
+'''
+def black_scholes(s, k, t, v, rf, div, cp):
+    '''
+    Price an option using the  Black-Scholes model
+
+    :param s: initial stock price
+    :param k: strike price
+    :param t: expiration time
+    :param v: volatility
+    :param rf: risk-free rate
+    :param div: dividend
+    :param cp: +1 / -1 for call/put
+    :return: option price
+    '''
+
+    d1 = (math.log(s/k)+(rf-div+0.5*math.pow(v,2))*t)/(v*math.sqrt(t))
+    d2 = d1 - v*math.sqrt(t)
+    optprice = cp*s*math.exp(-div*t)*stats.norm.cdf(cp*d1) - cp*k*math.exp(-rf*t)*stats.norm.cdf(cp*d2)
+
+    return optprice
+
+''' HELPER FUNCTIONS '''
+
+def d1(S,K,T,r,sigma):
+    return(math.log(S/K)+(r+sigma**2/2.)*T)/(sigma*math.sqrt(T))
+
+def d2(S,K,T,r,sigma):
+    return d1(S,K,T,r,sigma)-sigma*math.sqrt(T)
+
+
+''' CALLS '''
+
+def call_price(s, k, t, v, rf, div):
+    return black_scholes(s, k, t, v, rf, div, 1)
+
+''' THE GREEKS (CALLS) '''
+
+def call_delta(S, K, T, r, sigma):
+    '''
+    S = spot price of an asset
+    K = strike price
+    r = risk-free interest rate
+    t = time to maturity
+    σ = volatility of the asset
+    '''
+    return norm.cdf(d1(S, K, T, r, sigma))
+
+def call_gamma(S, K, T, r, sigma):
+    return norm.pdf(d1(S, K, T, r, sigma)) / (S * sigma * math.sqrt(T))
+
+def call_vega(S, K, T, r, sigma):
+    return 0.01 * (S * norm.pdf(d1(S, K, T, r, sigma)) * math.sqrt(T))
+
+def call_theta(S, K, T, r, sigma):
+    return 0.01 * (-(S * norm.pdf(d1(S, K, T, r, sigma)) * sigma) / (2 * math.sqrt(T)) - r * K * math.exp(-r * T) * norm.cdf(
+        d2(S, K, T, r, sigma)))
+
+def call_rho(S, K, T, r, sigma):
+    return 0.01 * (K * T * math.exp(-r * T) * norm.cdf(d2(S, K, T, r, sigma)))
+
+def call_implied_volatility(Price, S, K, T, r):
+    sigma = 0.001
+    while sigma < 1:
+        Price_implied = S * \
+            norm.cdf(d1(S, K, T, r, sigma))-K*math.exp(-r*T) * \
+            norm.cdf(d2(S, K, T, r, sigma))
+        if Price-(Price_implied) < 0.001:
+            return sigma
+        sigma += 0.001
+    return "Not Found"
+
+
+''' PUTS '''
+
+def put_price(s, k, t, v, rf, div):
+    return black_scholes(s, k, t, v, rf, div, -1)
+
+''' THE GREEKS (PUTS) '''
+
+def put_delta(S, K, T, r, sigma):
+    return -norm.cdf(-d1(S, K, T, r, sigma))
+
+def put_gamma(S, K, T, r, sigma):
+    return norm.pdf(d1(S, K, T, r, sigma)) / (S * sigma * math.sqrt(T))
+
+def put_vega(S, K, T, r, sigma):
+    return 0.01 * (S * norm.pdf(d1(S, K, T, r, sigma)) * math.sqrt(T))
+
+def put_theta(S, K, T, r, sigma):
+    return 0.01 * (-(S * norm.pdf(d1(S, K, T, r, sigma)) * sigma) / (2 * math.sqrt(T)) + r * K * math.exp(-r * T) * norm.cdf(
+        -d2(S, K, T, r, sigma)))
+
+def put_rho(S, K, T, r, sigma):
+    return 0.01 * (-K * T * math.exp(-r * T) * norm.cdf(-d2(S, K, T, r, sigma)))
+
+def put_implied_volatility(Price, S, K, T, r):
+    sigma = 0.001
+    while sigma < 1:
+        Price_implied = K*math.exp(-r*T)-S+bs_call(S, K, T, r, sigma)
+        if Price-(Price_implied) < 0.001:
+            return sigma
+        sigma += 0.001
+    return "Not Found"
+
+
+if __name__ == "__main__":
+    test1 = black_scholes(100.0, 100.0, 1.0, 0.3, 0.03, 0.0, -1)
+    test2 = call_price(100.0, 100.0, 1.0, 0.3, 0.03, 0.0)
+    test3 = put_price(100.0, 100.0, 1.0, 0.3, 0.03, 0.0)
+
+    print("Num : {0}".format(test1))
+    print("Num : {0}".format(test2))
+    print("Num : {0}".format(test3))
+