Call Delta Portfolio Value Bound

Article Quant Q&A · Author: Idonknow

Summary

The document asks whether a position long delta shares and short one European call can be worth more than the call’s strike under the Black–Scholes model. The proposed portfolio value is calculated as the stock price multiplied by the call delta, less the call price. The question arises from simulations of stock paths and portfolio values.

Using the Black–Scholes expressions, the answer identifies delta as the cumulative normal probability at d1 and the call price as that delta-weighted stock value less the discounted strike weighted by the cumulative normal probability at d2. Subtracting the call value from the delta stock position leaves the discounted strike multiplied by the d2 probability. Since that probability is below one and the discount factor is below one for a positive interest rate and positive elapsed time, the result is below the strike. This is a model-based argument; it assumes the stated Black–Scholes setup and does not address other option models, dividend complications, or transaction costs.

Key ideas

In Black–Scholes, call delta is the cumulative normal probability evaluated at d1.
Subtracting the call price from the delta stock position yields the discounted strike weighted by the d2 probability.
Under the stated positive-rate setup, both multipliers are below one, so the portfolio value is below the strike.
The argument relies on Black–Scholes assumptions.

Tags

Options Derivatives pricing Volatility

Full text

# evolve stock prices under GBM SDE solution in N steps


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












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?

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 by SachaTheBrave (score 1, accepted)

https://quant.stackexchange.com/a/54627

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.

Why Delta Shares Minus a Call Option Stay Below the Strike

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