Can a portfolio value consisting of longing a delta shares of stocks and shorting a call option greater than strike price?

Question

While trying to implement Black-Scholes delta hedging for a European call option using Python, I came across the following phenomena:

Given a portfolio consisting of longing a delta shares of stocks and shorting a call option, its value can never be greater than the strike price of the call option.

Is this true? If yes, can this be proven?

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For reference, the following is my Python code: 

# evolve stock prices under GBM SDE solution in N steps

# BS parameters
S0 = 120
K = 100
r = 0.05
d = 0
sigma = 0.2
T = 1

# number of discretization steps
N = 50

stock_prices = np.ndarray(shape = (50))
stock_prices[0] = S0

num_rows, num_cols = 5, 5
num_graphs = num_rows * num_cols

_, ax = plt.subplots(num_rows, num_cols, figsize = (15,8))

for j in range(num_graphs):
    for i in range(1, N):
        stock_prices[i] = GBM_formula(stock_prices[i-1], K, r, d, sigma, T)

    ax[j // num_cols, j % num_cols].plot(stock_prices, label = 'Stock Prices')

    # Black-Scholes hedging strategy
    # hedging simulator
    # A delta-neutral portfolio (from option's seller point of view) consists of longing delta shares of stocks and shorting a call option.

    len_of_stock_prices = len(stock_prices)
    portfolio = [0] * len_of_stock_prices
    for i in range(len_of_stock_prices):
        portfolio[i] = Greeks(stock_prices[i], K, r, d, sigma, T).delta() * stock_prices[i] - Option(stock_prices[i], K, r, d, sigma, T).european_call()

    ax[j // num_cols, j % num_cols].plot(portfolio, label = 'Portfolio value')
    ax[j // num_cols, j % num_cols].legend()

The GBM_formula scripts can be found at my Github https://github.com/hongwai1920/Implement-Option-Pricing-Model-using-Python/blob/master/scripts/GBM.py. Same goes to Option and Greek https://github.com/hongwai1920/Implement-Option-Pricing-Model-using-Python/blob/master/scripts/Option.py

The following contain 20 plots of stock prices and the corresponding portfolio values.

Answer 1

When you look at the Black Scholes formula it seems straightforward: The price of an option is \begin{equation} \mathrm C(\mathrm S,\mathrm t)= \mathrm N(\mathrm d_1)\mathrm S - \mathrm N(\mathrm d_2) \mathrm K \mathrm e^{-rt} \label{eq:1} \end{equation}

Your Delta is \begin{equation} \mathrm \Delta(\mathrm S,\mathrm t)= \mathrm N(\mathrm d_1) \label{eq:2} \end{equation}

So your portfolio value is \begin{equation} S\Delta(\mathrm S,\mathrm t)\mathrm - C(\mathrm S,\mathrm t)\mathrm = \mathrm N(\mathrm d_2) \mathrm K \mathrm e^{-rt} \label{eq:3} \end{equation}

Which is smaller than K as one term is a CDF and the other the exponential of a negative number, both smaller than 1.