The formula for black-scholes with put options is very similar to call options.

We can implement it into code like before.

#The black-scholes formula for put options
def black_scholes_put(S, X, sigma, rf, t):
    d1 = 1/(sigma*t**.5) * (np.log(S/X)  + (rf + sigma **2 /2) * t)
    d2 = d1 - sigma * t**.5
    return  norm.cdf(-d2) * X * np.exp(-rf*t) - norm.cdf(-d1) * S
print(black_scholes_put(50, 55, .08, .03, 1))

Verification of put-call parity can be confirmed.

#We can verify put-call parity

#Call and present value of X
print(black_scholes_call(50, 55, .08, .03, 1) + 55 * np.exp(-.03*1))

#Put and current price of stock
print(black_scholes_put(50, 55, .08, .03, 1) + 50)

Comparison of intrinsic and put value shows how at a certain point the intrinsic value is better than the option value.

#Compute values for the intrinsic value and the black-scholes value
df = pd.DataFrame(list(range(101)), columns = ['Stock Price'])
df['$55 Put Intrinsic Value'] = df['Stock Price'].apply(lambda x: max(55-x, 0))
df['$55 Put Black-Scholes Value'] = df['Stock Price'].apply(lambda x: black_scholes_put(x, 55, .08, .03, 1))
df = df.set_index('Stock Price')

#Plot the value of the option
ax = df.plot(kind='line')
ax.axhline(0, linestyle='--', color='grey')
plt.xlabel("Stock Price")
plt.ylabel("Value")
plt.title("Black Scholes vs. Intrinsic")
plt.show()

ax = df.plot(kind='line')
ax.axhline(0, linestyle='--', color='grey')
plt.xlim([40,70])
plt.xlabel("Stock Price")
plt.ylabel("Value")
plt.title("Black Scholes vs. Intrinsic")
plt.show()

Test with different volatilities.

#Test the values with different volatilities
df = pd.DataFrame(list(range(101)), columns = ['Stock Price'])
df['$55 Put Intrinstic Value'] = df['Stock Price'].apply(lambda x: max(55-x, 0))

for vol in [.08, .14, .20, .26, .32]:
    df[str(int(vol*100))+"% Vol"] = df['Stock Price'].apply(lambda x: black_scholes_put(x, 55, vol, .03, 1))
df = df.set_index('Stock Price')

ax = df.plot(kind='line')
ax.axhline(0, linestyle='--', color='grey')
plt.xlabel("Stock Price")
plt.ylabel("Value")
plt.title("Black-Scholes vs. Volatility")
plt.show()

ax = df.plot(kind='line')
ax.axhline(0, linestyle='--', color='grey')
plt.xlim([40,70])
plt.xlabel("Stock Price")
plt.ylabel("Value")
plt.title("Black-Scholes vs. Volatility")
plt.show()

With higher volatility the option value is greater for puts like calls. Now for the risk-free rate.

#Test the values with different risk-free rates
df = pd.DataFrame(list(range(101)), columns = ['Stock Price'])
df['$55 Put Intrinstic Value'] = df['Stock Price'].apply(lambda x: max(55-x, 0))

for rf in [.03, .06, .09, .12]:
    df[str(int(rf*100))+"% rf"] = df['Stock Price'].apply(lambda x: black_scholes_put(x, 55, .08, rf, 1))
df = df.set_index('Stock Price')

ax = df.plot(kind='line')
ax.axhline(0, linestyle='--', color='grey')
plt.xlabel("Stock Price")
plt.ylabel("Value")
plt.title("Black-Scholes vs. Risk-free Rate")
plt.show()

ax = df.plot(kind='line')
ax.axhline(0, linestyle='--', color='grey')
plt.xlim([40,70])
plt.xlabel("Stock Price")
plt.ylabel("Value")
plt.title("Black-Scholes vs. Risk-free Rate")
plt.show()

The risk-free rate makes the option less valuable. What about the effect of time to maturity?

#Test the values with different time to maturities
df = pd.DataFrame(list(range(101)), columns = ['Stock Price'])
df['$55 Put Intrinstic Value'] = df['Stock Price'].apply(lambda x: max(55-x, 0))

for t in [.25, 1, 2]:
    df["t="+str(t)] = df['Stock Price'].apply(lambda x: black_scholes_put(x, 55, .08, .03, t))
df = df.set_index('Stock Price')

ax = df.plot(kind='line')
ax.axhline(0, linestyle='--', color='grey')
plt.xlabel("Stock Price")
plt.ylabel("Value")
plt.title("Black-Scholes vs. Time to Maturity")
plt.show()

ax = df.plot(kind='line')
ax.axhline(0, linestyle='--', color='grey')
plt.xlim([40,70])
plt.xlabel("Stock Price")
plt.ylabel("Value")
plt.title("Black-Scholes vs. Time to Maturity")
plt.show()