Calendar Stock Option Spreads Call options in spread

In this type of spread we initiate positions in options which expire on different dates. For example, say we sell a 50 call expiring in a month, and buy a 50 call expiring in 2 months. We know this will be a debit spread—net we will have to pay because the longer-dated call will have a higher price.

Let's assume:

The premia were determined using the Black-Scholes model with a constant volatility of 30%, a 1% risk free rate, and a $49 stock price. Implementing the spread gives us a cash flow of $1.27 - $1.98 = -$0.71 today.

Let's calculate the profit on the spread when the shorter-dated option expires (1 month), assuming we sell the longer-dated option on that date. Also assume volatility stays constant. We assume we sell the longer-dated call at the Black-Scholes value of the call.

To do this let's load a function that will calculate the Black-Scholes call option value into a Python session.

Load required libraries:

import numpy as np
from scipy.stats import norm

def black_scholes_call(stock, strike, risk_free, vol, time):
    d1 = (np.log(stock / strike) + (risk_free + vol * vol / 2) * time) / (vol * np.sqrt(time))
    d2 = d1 - vol * np.sqrt(time)
    call = stock * norm.cdf(d1) - strike * np.exp(-risk_free * time) * norm.cdf(d2)
    return(round(call, 2))

We can now create a function to value our calendar spread, and apply it to various underlying stock prices using a list comprehension:

def calendar_spread_1(stock, vol):
    profit_call_bought = black_scholes_call(stock, 50, .01, vol, 1/12) - 1.98
    profit_call_sold = 1.27 - max(stock - 50, 0)
    total_profit = profit_call_bought + profit_call_sold
    return(round(total_profit, 2))

profit_1_same_vol = [calendar_spread_1(x, 0.3) for x in range(30, 70)]

Plot the spread profit:

import plotly.express as px

fig = px.line(x = range(30, 70), y = profit_1_same_vol, template='plotly_dark', title="Profit on Calendar Spread (Constant Volatility)").update_layout(
    xaxis_title="Stock Price", yaxis_title="Spread Profit"
)

fig.add_hline(y=0)

fig.write_html("./figs/profit_same_vol.html")

Note this resembles the profit on a butterfly, except the transition to the max loss has curvature.

This is how the calendar spread is defined in textbooks, however it is not particularly realistic. Stock volatility is not constant, and will likely change. In the next few plots we'll look at the spread profit when volatility increases and decreases. Last, we'll look at the most realistic scenario: volatility decreases of the stock price increases, and volatility increases if the stock price decreases (known as the leverage effect).

Flip the charts about the x axis.