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Abstract This article discusses a form of Financial Derivative investment strategy called a ‘Risk Reversal.’ A ‘Risk Reversal’ is also referred to as a ‘Synthetic Long’ in some literature. The article explains the theory, and shows you what risks you are exposed to. It also shows you how to protect your investment once the market has moved in a favorable direction. The theory, the mathematics and a theoretical example are presented. If you would like to contact the author, send a message to . If there are any errors, typos, or defects with this article please make the effort to contact the author. |

A risk reversal is a position in which you simulate the behavior of a long, therefor it is sometimes called a synthetic long. This is an investment strategy that amounts to both buying and selling out-of-money options simultaneously. In this strategy, the investor will first make a market hunch, if that hunch is bullish he will want to go long. However, instead of going long on the stock, he will buy an out of the money call option, and simultaneously sell an out of the money put option. Presumable he will use the money from the sale of the put option to purchase the call option. Then as the stock goes up in price, the call option will be worth more, and the put option will be worth less.

First a quick refresher on buying and selling the call and put options. With options there are basically 4 positions that you can take:

- long the call option - in this case you are buying a contract that gives you the right to purchase an underlying on or before a future date at a predetermined ‘strike ’ price. You give up some money so that you can hold this right.
- short the call option - in this case you are the party that sells a contract which gives someone else the right to purchase (from you) an underlying on or before a future date at the strike price.
- long the put option - in this case you are buying a contract that gives you the right, but not the obligation, to sell an underlying on or before a future date at the strike price.
- short the put option - in this case you are selling to someone else a contract that gives them the right, but not the obligation, to sell an underlying (to you) on or before a future date at the strike price.

Each of the four options mentioned above can be either in or out of the money.

In order to create this position, the investor first must find a stock or commodity that he or she is bullish on. As we study this a bit more, you will notice that it helps if the market is bearish on that commodity because the option pricing tends to be more favorable. Lets now attempt to set up the position. I will choose FSLR, First Solar Corporation (and yes for those of you that read my articles, I frequently choose this stock. The reason is it has a large market cap and high volatility therefore behaving at a boundary. You always want to understand how to apply theory at the boundaries.).

For this risk reversal position, we will somewhat arbitrarily choose an option that is 427 days away from expiry. First looking at the put that we are going to sell: strike price of $ 105.00 , price $ 22.00 , δ = -1.6 and γ = -1.8 (the chance of break-even on the option is estimated at 40%). Since the sale of the put option generated $ 22.00 , that money will be able to purchase a call option with a strike price of $ 145.00 . The call option has a δ = 2.9, γ = 1.3 and costs $ 21.5.

Looking at this portfolio in a tabular form: Underlying Price: 121.15 Days to Expiry: 427

position | cost | value | cash flow | intrinsic value | in-the-money | break even |

short put | (22.00) | (22.00) | - | 0.0 | above $ 105 | $ 83 |

long call | 21.50 | 21.5 | - | 0.0 | above $ 145 | $ 166.5 |

cash in | 0.00 | 0.50 | - | 0.5 | - | - |

Total | (0.5) | 0.0 | 0.5 | 0.5 | - | - |

Now we will just look at several expire situations: Assuming that the option expires with with price at the following:

price at expiry | short put status | long call status | close out value |

75 | (30.00) | 0.00 | (30.00) |

85 | (20.00) | 0.00 | (20.00) |

95 | (10.00) | 0.00 | (10.00) |

105 | 0.00 | 0.00 | 0.00 |

115 | 0.00 | 0.00 | 0.00 |

125 | 0.00 | 0.00 | 0.00 |

135 | 0.00 | 0.00 | 0.00 |

145 | 0.00 | 0.00 | 0.00 |

155 | 0.00 | 10.00 | 10.00 |

165 | 0.00 | 20.00 | 20.00 |

175 | 0.00 | 30.00 | 30.00 |

Notice that as long as the price stays above 105, the strike price of the short put, you will not be negative. The reason is because you used the proceeds from the sale of the short put to fund the long call. As a basis for whether or not this was a good investment, we need to look at the situation if you bought the long position in FSLR on the date in question, and cashed out on the expiry. The following table shows this situation:

purchase price | price at expiry | unrealized gain (loss) | |

121.15 | 75.00 | (46.15) | |

121.15 | 85.00 | (36.15) | |

121.15 | 95.00 | (26.15) | |

121.15 | 105.00 | (16.15) | |

121.15 | 115.00 | (6.15) | |

121.15 | 125.00 | 3.85 | |

121.15 | 135.00 | 13.85 | |

121.15 | 145.00 | 23.85 | |

121.15 | 155.00 | 33.85 | |

121.15 | 165.00 | 43.85 | |

121.15 | 176.00 | 53.85 | |

price at expiry | long value | synthetic long value | |

75.00 | (46.15) | (30.00) | |

85.00 | (36.15) | (20.00) | |

95.00 | (26.15) | (10.00) | |

105.00 | (16.15) | 0.00 | |

115.00 | (6.15) | 0.00 | |

125.00 | 3.85 | 0.00 | |

135.00 | 13.85 | 0.00 | |

145.00 | 23.85 | 0.00 | |

155.00 | 33.85 | 10.00 | |

165.00 | 43.85 | 20.00 | |

176.00 | 53.85 | 30.00 | |

From the above table, you can see that your down side risk is significantly reduced. However, your upside reward is also reduced. However, that is not the whole story. There is another side to this, and that is leverage, which can be thought of as amplification.

The long position in FSLR cost real cash, in fact it cost 121.15 dollars per share. The synthetic option does not cost you any capital. Therefore, one can do a study about leverage. The synthetic option has very high leverage because it does not cost any capital, where as the long position in FSLR is costing you 121.15 dollars per share.

Now in reality, the option contract may not ever reach expiry. The reason is because the trader holding the contract may attempt to sell the long call option and or buy back the short put option prior to expiry. The reason for buying back the short put may be to minimize risk. Another reason, may simply to behave as a savvy investor by selling high. Understanding and managing the position requires some guide lines for how to proceed. Rules:

- Market Value of the Portfolio - At any instant in time, the market value of the portfolio is the sale value of the long call option - cost of buying back the short put option. If this is a positive number, then there are unrealized gains to be had.
- Negative Market Value - If the market value of this portfolio is negative, the intrinsic value may still be 0.0. The reason for that is because the options may still both be out of the money.
- reducing risk - If the value of the short put drops significantly, it may be cost effective to buy it back. Buying it back would mean that there is no downside risk.
- both the call and put are liquid asset - there is still a market, and these assets are liquid.
- Black-Scholes modeling - The value of the two options the short put and the long call are going to fundamentally be based on the Black-Scholes options pricing model.

So how does one proceed? Using the Black-Scholes terms will help us identify some things. Creating the nomenclature first:

- - the cost to buy back the put option that we sold.
- - the value of the call option we are holding.
- - the sensitivity of the put option’s value to the change in the underlying stock price.
- - the sensitivity of the call option’s value to a change in the underlying stock price.
- - the sensitivity of delta on the put to a change in the underlying stock price.
- - the sensitivity of delta on the call to a change in the underlying stock price.
- - the sensitivity of the call option value to the passage of time.
- - the sensitivity of the put option value to the passage of time.

from the Article Delta Neutral Hedging ( Vol 1 No. 1)

ignoring θ for the time being, we can construct an expression for the value of the portfolio.

Plugging in and simplifying (several steps in one)

Now solving this equation generally as a function of Stock Price, and Time to Expiry would be very complex, and far beyond the scope of this article. However intuitively, for very small changes in the price of the underlying, and assuming that there is no change in the price of the option, δ or γ we can see that the value of the portfolio will go up if the underlying goes up. Also, we can easily see that if the price of the underlying goes up any significant amount, we could by back the put option we short sold.

Now placing the values from our portfolio into the equation:

Now with this very simplified Option Pricing Model, we will calculate the value of the portfolio as a function of price, assuming that we are still a long way from expiry.

price at expiry | ΔP | portfolio intrinsic value | portfolio market (swag est. V _{portfolio}) |

115.00 | -6.15 | 0.00 | (6.48) |

125.00 | 3.85 | 0.00 | 2.98 |

135.00 | 13.85 | 0.00 | 10.79 |

A risk reversal strategy or ‘Synthetic Long’ is a technique in which an investor takes a market bias position by buying and selling financial derivatives to simulate the behavior of going long on a position. The benefits of the strategy include being highly leveraged, and reducing your down side risk if you are able to ride the position to the end. In a long position, your portfolio’s value will drop dollar for dollar with every drop in the underlying stock’s price (this drop in value would be an unrealized loss if you don’t sell). Creation of the risk-reversal or synthetic long position, the intrinsic value of your stock position will not decrease until the stock has crossed the strike price of the short put.

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