To many investors, the most appealing feature of the option is that the downside of trading is fully protected while the upside is fully opened. However, what people usually ignore is that the option itself represents a kind of trading strategy and such trading strategy is consistent with the stop loss and keep profit strategy (buy more when make money while cut loss when lose money) which is widely adopted in trading industry. As a result, one can create a synthetic option by trading the underlying with such strategy.

Let's first take a look at an vanilla European option. Instead of having a certain stop loss level, the option is equivalent to holding more underlying when option is more in the money, while hold less underlying when the option moves out of the money. The overall effect of this strategy is that you pay a small cost, but keep the potential of winning an unlimited profit. The cost, as we all know, is the option premium.

Now here comes the question, if option itself is a strategy, there should be some way that we can replicate this strategy with the underlying and cash. And the answer is YES. From the Black-Scholes model introduced in 1973, the year when the whole would of financial engineering starts, the option can be decomposed in to more fundamental assets. The biggest contribution of Black and Scholes is not the elegant close form solution of the European option, instead, it is the dynamic hedging argument that one can replicate an option's payoff by holding a certain amount of underlying and cash. That certain amount of underlying is the well-known option Delta.

In the following part I will use some simple math to show how an option can be replicated with a trading strategy in which we only trade the underlying asset and cash. Let's start with a self-financing portfolio in which we short an option C, borrow ΔS-C of cash, and buy Δ shares of stock S. Initial portfolio value=-C-(ΔS-C)+ΔS=0. On a small time interval t to t+dt, the following relationship holds:

Starting from this relationship, let's first consider an ideal case where there is no financing cost, namely r=0

Where C is the option's value at expiry which is the option's payoff max(S-K,0). A typical payoff a trader will target at is the capital protection style payoff, where K=S0. In this case,

One can also adjust the strike to customize the profit target.

As we can see, the right hand side of the equation is our trading strategy: We hold Δ shares of the underlying S at time t. At time t+dt, the p&l of the strategy is equal to: Δ[S(t+dt)-S(t)]=ΔdS. We then rebalance the portfolio to holding Δ(t+dt) shares of underlying. We keep trading with this strategy all the way to time T which is the target investment horizon. From the formula one can easily capture the main feature of the strategy. For example, if our target is to create a synthetic call option. As the underlying moves against you, Δ will decrease which means you will cut the position as you lose money. While if the price move in favor of you, Δ will increase and you take more profit. This is the fundamental mechanism of an option that protests the downside of the trade.

The overall effect of this trading strategy is quite clear which is shown on the left hand side. max{S(NYSE:T)-S(0),0}-C0 is the option's payoff minus the option premium, which is exactly the profit and lose of buying an option. The following two figures show two different situations based on different simulated stock path. The stock is simulated with simple Geometric Brownian Motion with initial stock price=100, risk free rate=0, volatility=30%, T=1 year. Δ of the trading strategy is calculated with BS model.

Figure 1: If we buy a synthetic call option, when the price goes up we keep the profit

payoff max{S(T)-S0,0} |
47.5088 |

c0 |
11.9235 |

payoff-c0 |
35.5853 |

strategy cumulative p&l at expiry |
36.3248 |

Figure 2: If we buy a synthetic call option, when the price goes down, we cut the position and the downside is protected

payoff max{S(T)-S0,0} |
0 |

c0 |
11.9235 |

payoff-c0 |
-11.9235 |

strategy cumulative p&l at expiry |
-12.0123 |

This conclusion can be generalized to the cases where r is not equal to 0. In the real world there is interest cost in trading known as rollover fee.

where and are the rollover fee of holding the underlying and the option. For example, if we hold Δ of shares at time t, the money we borrow to finance this position is ΔS so that the financing cost (interest) occurring from t to t+dt is ΔSrdt.

One thing needs to be noticed is that Δ(t,S,K) is model based. The most popular models are Black-Scholes, Local Volatility, Stochastic Volatility and Stochastic Local Volatility. Since Stochastic volatility has additional risk factors that cannot be hedged by holding underlying only (usually requires other options), this methodology is better applied to Black-Scholes or Local Volatility model. I personally prefer Local Volatility model because it can also capture the risk from the volatility smile:

The Delta also determines how much premium c0 we are going to pay as the cost to protect the downside risk.

The trading strategy requires the trader to rebalance the portfolio with the updated Δ at a certain frequency. The rebalancing frequency can be customized based on difference investment horizon and transaction costs. Less frequent rebalancing means more replication errors. The best market to implement this strategy is the FX, CFD or futures market where we have huge leverages and the trading costs is relatively low (so that we can rebalance more frequently). In addition, long and short are not restricted. Such strategy can be applied by individuals and institutions who have derivative trading restrictions.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.