Option Strategies
Long Box Spread
Introduction
A long Box Spread is the combination of a bull call spread and a bear put spread. It consist of buying an ITM call at strike $A$, selling an OTM put at strike $A$, buying an ITM put at strike $B < A$, and selling an OTM call at strike $B$, where all of the contracts have the same expiry date. This strategy serves as an delta-neutral arbitration from Option mispricing. Note that it only attains a true profit when the risk-free return is greater than the risk-free interest rate.
Implementation
Follow these steps to implement the long box spread strategy:
- In the
Initialize
initialize
method, set the start date, set the end date, subscribe to the underlying Equity, and create an Option universe. - In the
OnData
on_data
method, select the expiry and strikes of the contracts in the strategy legs. - In the
OnData
on_data
method, select the contracts and place the order.
private Symbol _symbol; public override void Initialize() { SetStartDate(2017, 4, 1); SetEndDate(2017, 4, 30); SetCash(100000); UniverseSettings.Asynchronous = true; var option = AddOption("GOOG", Resolution.Minute); _symbol = option.Symbol; option.SetFilter(universe => universe.IncludeWeeklys().BoxSpread(30, 5)); }
def initialize(self) -> None: self.set_start_date(2017, 4, 1) self.set_end_date(2017, 4, 30) self.set_cash(100000) self.universe_settings.asynchronous = True option = self.add_option("GOOG", Resolution.MINUTE) self._symbol = option.symbol option.set_filter(lambda universe: universe.include_weeklys().box_spread(30, 5))
The BoxSpread
box_spread
filter narrows the universe down to just the four contracts you need to form a box spread.
public override void OnData(Slice slice) { if (Portfolio.Invested) return; // Get the OptionChain if (!slice.OptionChains.TryGetValue(_symbol, out var chain)) { return; } // Select an expiry date and ITM & OTM strike prices var expiry = chain.Max(x => x.Expiry); var contracts = chain.Where(x => x.Expiry == expiry).ToList(); var higherStrike = contracts.Max(x => x.Strike); var lowerStrike = contracts.Min(x => x.Strike);
def on_data(self, slice: Slice) -> None: if self.portfolio.invested: return # Get the OptionChain chain = slice.option_chains.get(self._symbol, None) if not chain: return # Select an expiry date and ITM & OTM strike prices expiry = max([x.expiry for x in chain]) contracts = [x for x in chain if x.expiry == expiry] lower_strike = min([x.strike for x in contracts]) higher_strike = max([x.strike for x in contracts])
Approach A: Call the OptionStrategies.BoxSpread
OptionStrategies.box_spread
method with the details of each leg and then pass the result to the Buy
buy
method.
var boxSpread = OptionStrategies.BoxSpread(_symbol, higherStrike, lowerStrike, expiry); Buy(boxSpread, 1);
box_spread = OptionStrategies.box_spread(self._symbol, higher_strike, lower_strike, expiry) self.buy(box_spread, 1)
Approach B: Create a list of Leg
objects and then call the Combo Market Ordercombo_market_order, Combo Limit Ordercombo_limit_order, or Combo Leg Limit Ordercombo_leg_limit_order method.
// Select the call and put contracts var itmCall = contracts.Single(x => x.Strike == lowerStrike && x.Right == OptionRight.Call); var otmCall = contracts.Single(x => x.Strike == higherStrike && x.Right == OptionRight.Call); var itmPut = contracts.Single(x => x.Strike == higherStrike && x.Right == OptionRight.Put); var otmPut = contracts.Single(x => x.Strike == lowerStrike && x.Right == OptionRight.Put); var legs = new List<Leg>() { Leg.Create(itmCall.Symbol, 1), Leg.Create(itmPut.Symbol, 1), Leg.Create(otmCall.Symbol, -1), Leg.Create(otmPut.Symbol, -1), }; ComboMarketOrder(legs, 1);
# Select the call and put contracts itm_call = next(filter(lambda x: x.right == OptionRight.CALL and x.strike == lower_strike, contracts)) otm_call = next(filter(lambda x: x.right == OptionRight.CALL and x.strike == higher_strike, contracts)) itm_put = next(filter(lambda x: x.right == OptionRight.PUT and x.strike == higher_strike, contracts)) otm_put = next(filter(lambda x: x.right == OptionRight.PUT and x.strike == lower_strike, contracts)) legs = [ Leg.create(itm_call.symbol, 1), Leg.create(itm_put.symbol, 1), Leg.create(otm_call.symbol, -1), Leg.create(otm_put.symbol, -1), ] self.combo_market_order(legs, 1)
Strategy Payoff
This is a fixed payoff, delta-neutral strategy. The payoff is
$$ \begin{array}{rcll} C_T^{ITM} & = & (S_T - K_{-})^{+}\\ C_T^{OTM} & = & (S_T - K_{+})^{+}\\ P_T^{ITM} & = & (K_{+} - S_T)^{+}\\ P_T^{OTM} & = & (K_{-} - S_T)^{+}\\ Payoff_T & = & (C_T^{ITM} - C_{T_0}^{ITM} + P_T^{ITM} - P_{T_0}^{ITM} - C_T^{OTM} + C_{T_0}^{OTM} - P_T^{OTM} + P_{T_0}^{OTM})\times m - fee\\ & = & (K_{+} - K_{-} + C_{T_0}^{OTM} + P_{T_0}^{OTM} - C_{T_0}^{ITM} - P_{T_0}^{ITM})\times m - fee \end{array} $$ $$ \begin{array}{rcll} \textrm{where} & C_T^{ITM} & = & \textrm{ITM Call value at time T}\\ & C_T^{OTM} & = & \textrm{OTM Call value at time T}\\ & P_T^{ITM} & = & \textrm{ITM Put value at time T}\\ & P_T^{OTM} & = & \textrm{OTM Put value at time T}\\ & S_T & = & \textrm{Underlying asset price at time T}\\ & K_{+} & = & \textrm{Higher strike price}\\ & K_{-} & = & \textrm{Lower strike price}\\ & Payoff_{T_1} & = & \textrm{Payout total at time T1}\\ & C_{T_0}^{ITM} & = & \textrm{ITM Call price when the trade opened (debit paid)}\\ & C_{T_0}^{OTM} & = & \textrm{OTM Call price when the trade opened (credit received)}\\ & P_{T_0}^{ITM} & = & \textrm{ITM Put price when the trade opened (debit paid)}\\ & P_{T_0}^{OTM} & = & \textrm{OTM Put price when the trade opened (credit received)}\\ & m & = & \textrm{Contract multiplier}\\ & T & = & \textrm{Time of expiration} \end{array} $$The following chart shows the payoff at expiration:
![long box spread strategy payoff](https://cdn.quantconnect.com/i/tu/box-spread-payoff.png)
The payoff is only dependent on the strike price and the initial asset prices.
If the Option is American Option, there is a risk of early assignment on the contracts you sell.
Example
The following table shows the price details of the assets in the algorithm:
Asset | Price ($) | Strike ($) |
---|---|---|
ITM Call | 27.30 | 810.00 |
ITM Put | 28.00 | 857.50 |
OTM Call | 1.05 | 857.50 |
OTM Put | 1.50 | 810.00 |
Underlying Equity at expiration | 843.25 | - |
Therefore, the payoff is
$$ \begin{array}{rcll} Payoff_T & = & (K_{+} - K_{-} + C_0^{OTM} + P_0^{OTM} - C_0^{ITM} - P_0^{ITM})\times m - fee\\ & = & (857.50 - 810.00 + 1.05 + 1.50 - 27.30 - 28.00)\times100 - 1.00\times4\\ & = & -529.00\\ \end{array} $$So, the strategy loses $529.
The following algorithm implements a long box spread Option strategy: