# Option Pricing Model

One of the most common AICPA-approved methods to value private companies with complex capital structures is the Option Pricing Model. The Option Pricing Model (OPM) treats each share class as a call option on the equity value of the entire firm, with exercise prices based on the liquidation preferences of the preferred stock. One notable benefit to using the OPM is that it accounts for the economic rights often seen in venture-capital backed preferred shares, including preferred liquidation preferences and payout seniority. In this method, each share class only has value if the funds available for distribution to shareholders exceed the value of the liquidation preferences at the time of a liquidity event for each of the prior share classes in a company's cap table.

__The OPM and Common Stock__

Using the OPM, the common stock is modeled as a call option that gives its owner the right, but not the obligation, to buy the underlying equity value at a predetermined price. The considered "price" of these common-stock "call options" is based on the value of the entire enterprise at specific equity values ('breakpoints'). Thus, the common stock is considered to be a call option with a claim on the equity at an exercise price equal to the remaining value immediately after all share classes with higher liquidation seniority have liquidated. The Carta OPM uses the Black-Scholes Option Pricing Model.

__OPM Considerations__

The OPM considers the various terms of an enterprise's stockholder agreements that would affect the distributions to each class of equity upon a liquidity event *as of the future liquidation date*, including:

- the level of seniority among securities,
- dividend policy,
- conversion ratios,
- and cash allocations.

__OPM Inputs__

The OPM relies on four inputs:

- the total equity value of the enterprise
- expected time to exit
- the risk-free interest rate as of the valuation date
- the volatility derived from similar publicly traded companies

__OPM Model (Black-Scholes-Merton Formula):__

Where:

S0 = Total Equity Value

X= Equity Breakpoint Value

q = continuously compounded dividend yield

t = time to expiration (% of year)

σ = Volatility

r = risk-free rate

__OPM Example__

Carta is engaged to value ABC, Inc., a small early-stage company that expects to exit in 5 years (t=5). Using a basket of similar public companies, Carta estimates the volatility to be 50%. Additionally, the risk-free rate from the valuation date until the exit date is 1.08% as of the valuation date. Finally, the total equity value of ABC, Inc. is $5,000,000.

*Related: How is total equity value calculated? *The backsolve method

ABC, Inc. Capitalization Table (as of valuation date):

Series Seed Preferred | Common | Common Options | |

Number of shares: | 1,000,000 | 500,000 | 125,000 |

Purchase (or exercise) Price: | $1.00 | - | $0.20 |

Liquidation Multiplier | 1.0x | - | - |

Liquidation Preference | $1,000,000 | - | - |

First, Carta computes the equity breakpoints to determine which share class(es) will have value at various liquidation amounts for the company:

From | To | Description |

$0.00 | $1,000,000.00 | Series Seed Preferred only. |

$1,000,000.00 | $1,100,000.00 | Common shares only. |

$1,100,000.00 | $1,700,000.00 | Common options exercised |

$1,700,000.00 | Infinity | Series Seed Preferred converts to common. |

Next, compute the implied call-option value of each of these breakpoints, using the Black-Scholes-Merton model:$5,000,000 - Total equity value

1.08% - Risk-free rate

50.00% - Volatility

5 years - Time to exit

From | To | Option Value | Incremental Option Value |

$0.00 | $1,000,000.00 | $4,118,100.00 | $881,900.00 |

$1,000,000.00 | $1,100,000.00 | $4,041,170.00 | $76,930.00 |

$1,100,000.00 | $1,700,000.00 | $3,622,827.00 | $418,343.00 |

$1,700,000.00 | Infinity | - | $3,622,827.15 |

- | - | Total: | $5,000,000.00 |

From | To | Series Seed Preferred |
Common Shares |
Common Options |

$0.00 | $1,000,000.00 | 100.00% | 0.00% | 0.00% |

$1,000,000.00 | $1,100,000.00 | 0.00% | 100.00% | 0.00% |

$1,100,000.00 | $1,700,000.00 | 0.00% | 80.00% | 20.00% |

$1,700,000.00 | Infinity | 61.54% | 30.77% | 7.69% |

From | To | Series Seed Preferred |
Common Shares |
Common Options |

$0.00 | $1,000,000.00 | $881,900.00 | $0.00 | $0.00 |

$1,000,000.00 | $1,100,000.00 | $0.00 | $76,930.00 | $0.00 |

$1,100,000.00 | $1,700,000.00 | $0.00 | $334,674.40 | $83,668.60 |

$1,700,000.00 | Infinity | $2,229,487.83 | $1,114,743.91 | $278,595.41 |

Total: | $3,111,387.83 | $1,526,348.31 | $362,263.91 | |

Value per share: | $3.05 |

Thus, the fully-marketable value per new options to be issued equals __$3.05__ per share. However, this assumes that the shares are *fully-marketable* (i.e., able to be easily bought and sold). What should the price be if the shares aren't actually fully-marketable?

Next: Discount for Lack of Marketability

*Note: The figures used above are purely for educational and example purposes. Most capitalization tables are structured quite differently than above, however the Option Pricing Model calculations are done the same.*