Monte Carlo and Black-Scholes share options valuation models
Two widely used methods for valuing share options are the Black-Scholes model and Monte Carlo simulations. A share option is a contract that gives an individual the right to purchase company shares once certain conditions—known as vesting conditions—are met. Upon vesting, the individual can buy the shares at a predetermined strike price, potentially becoming a shareholder.
Valuing share options is important for a variety of reasons, including financial reporting, employee compensation planning, and regulatory compliance. This article explores the key differences between the Black-Scholes and Monte Carlo valuation approaches, providing an overview of their assumptions, strengths, limitations, and practical applications.
Options pricing models for medium sized and high growth companies
Price Bailey value share options once or twice a month for companies. We find that high growth and medium sized businesses tend to be ‘first timers’ in valuing share options. For these groups they can be surprised that complex valuation methods are needed for their options. There are two explanations.
Firstly, the class of share that an option exercises into rarely have the same rights as the existing Ordinary Shares. Those differences create changes in the attractiveness and risk profile of the underlying shares. Commercially buyers and sellers should intuitively expect a different price as a result; tax authorities certainly expect there to be a difference in price between existing Ordinary shares and options over a share with differing rights.
Secondly, the nature of an option means the holder carries the risks of execution and the volatility of a market. An ordinary shareholder also carries these risks, but they do so whilst holding an asset that can be sold in a secondary market for shares. There is no secondary market for options over shares in most medium and high growth businesses. This fundamental trait of an option means that they carry more risk and less reward than most Ordinary shares.
Consequently, convention leads to specific options pricing models being used.
Black–Scholes
So why do I need this?
‘The most common thing I hear is that Black Scholes is only common for big companies.’
Chand Chudasama , Partner, Price Bailey
The Black-Scholes model is a mathematical formula that is most commonly used to calculate the fair value of what is known as a European-style options using a set of inputs. It assumes that the underlying stock price follows a log-normal distribution over time. The model considers factors such as the current share price, the exercise price, the time to expiration, the risk-free rate, the volatility of the stock and dividend yield. By considering these inputs, the Black-Scholes model provides an estimate of the fair value of options.
Black-Scholes may seem over engineered for many businesses but it is an industry standard approach that, when calculated properly using robust inputs, provides the best chance for robust defence of an options price.
The Black-Scholes Formula and what it really means
Black-Scholes Model
\[ C = S N(d_1) – K e^{-rT} N(d_2) \]
\[ d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)t}{\sigma\sqrt{t}} \quad \text{and} \quad d_2 = d_1 – \sigma \sqrt{t} \]
Required inputs
S = Current equity value of the business divided by the number of shares
K = the strike price or exercise price of the Option
r = the Risk-free rate; typically this is a 20 year Treasury bond yield
T = the time remaining until the option expires
σ= the volatility of the share typically based on the movement in stock price from a relevant sample of publicly listed companies
How does Black-Scholes model work?
Essentially, when the broad direction of travel of a stock is known, but the volatility is unknown, we know that the stock price will either go up or down along the way and the journey is not linear. This formula accounts for that volatility. This is also known as Geometric Brownian Motion (GBM).
To understand how Black Scholes determines an options price one has to understand GBM. To understand GBM we find a metaphor of a drunken person walking up a path that goes up a hill tends to help. The broad direction of the drunken person is known – in this case direction is set by the path and the gradient of the hill – most of the steps the person takes are in the right direction but occasionally they stagger a few steps left and right and perhaps even backwards, but they correct. Black-Scholes recreates this for companies on the basis that the direction is known but the pathway to get there can have some trepidation.
This reflects that an Option holder does not get the benefits of a shareholder during the journey to exercise but does carry the risk; therefore, the steeper the hill and the more drunk the person, the greater the difference between an options price and an ordinary share.
Chand Chudasama, Price Bailey, Partner
Top 5 things to remember about using a Black-Scholes model
- Black-Scholes is the industry standard approach to valuing simple options.
- Robust input data is critical.
- A sense check is always needed.
- The sample of comparable companies is critical.
- Understanding GBM helps to understand how the Black-Scholes model works.
Monte Carlo
Black-Scholes can only really be used where the option follows a relatively linear path — for example, where the vesting conditions are straightforward and easy to model. The most common case is where vesting depends solely on a Total Shareholder Return (TSR) target. In these cases, a European-style call option works well.
However, once the vesting conditions become more complex — such as including multiple hurdles, performance gateways, or branching dependencies — Black-Scholes becomes less useful. In those scenarios, the Monte Carlo method is typically more appropriate. Back to the metaphor, this is like changing the single drunken person scenario into one where thousands of different drunken people, walking or running up thousands of different hills with thousands of different pathways and gradients.
As valuers, we often follow a simple rule of thumb:
- If the option is straightforward and linear — use Black-Scholes and a sense check.
- If it’s anything more complicated — use Monte Carlo and a sense check.
How does Monte Carlo work?
Monte Carlo simulations are based on the idea that you can’t calculate the perfect option value using a single deterministic formula. Instead, you run the same valuation model thousands (sometimes hundreds of thousands) of times — each time introducing slight randomness into the input assumptions. These randomised “paths” simulate the potential real-world outcomes of the stock price and vesting conditions. Once all the outcomes are generated, the average result is used as the estimated option value.
Each simulation calculates a “payoff” — essentially, how much the option would be worth if the stock followed that specific path. The final valuation is the average of all those payoffs, discounted back to present value to reflect the time value of money.
While the method might sound grand, the core concept is relatively simple: simulate many possible futures, then average them.
Practical limitations
Although Monte Carlo is widely used, it’s surprisingly opaque in practice. Most valuers rely on variations of the same Excel-based model — yet valuers rarely share the raw excel workbook. Auditors rarely receive the underlying workings and typically have to rebuild their own models to test the results, which can be time-consuming and inefficient.
There’s also a particular challenge around volatility. Monte Carlo require a volatility input to simulate how unpredictable the shares might be. In theory, this can be taken from comparable listed companies. In practice, however, those comparables are often poor. A £20m freight-forwarding business in the UK has little in common with a £300m pan-European logistics company with its own truck fleet, FX exposure, and different operational risks — yet the volatility input is often borrowed from the latter.
This creates a risk of distortion, particularly when the underlying business model is not well aligned with the listed peer group. While averaging 100,000 simulations might mathematically smooth out the issue, poor quality inputs can still undermine the credibility of the valuation. That’s why digging into comparables and volatility assumptions is so important — and often overlooked.
Top 5 things to remember about using a Monte Carlo model
- Monte Carlo models are excellent for complex options.
- They run thousands of tests.
- Robustness on the inputs is key.
- A sense check of the answer is always needed.
- Valuers rarely share their raw workings.
Comparison of Black-Scholes and Monte Carlo Methods
| Category | Black-Scholes Model | Monte Carlo Simulation |
| Typical Use Case | Best suited for simple, linear options – e.g. where the vesting condition is straightforward, such as total shareholder return. | Used for complex options – e.g. with performance gateways, branching vesting structures, or non-linear conditions. |
| Core Inputs | • Current stock price (S) • Strike price (K) • Time to maturity (T) • Risk-free interest rate (r) • Volatility (σ) • Dividend yield (q) (optional) |
• Same base inputs as Black-Scholes • Additional modelling of complex vesting conditions • Random variables to simulate possible market and vesting paths • Discount rate for present value calculations |
| Methodology | A single mathematical formula produces a fair value based on fixed inputs. Based on Geometric Brownian Motion (GBM). | Runs thousands (or hundreds of thousands) of simulations to model random outcomes. Each simulation generates a “payoff” which is averaged and discounted to today’s value. |
| Assumptions | • Efficient markets • Constant risk-free rate • Log-normal distribution of returns • No taxes or transaction costs • No dividends (in original form) |
• Underlying variables (e.g. stock price, volatility) follow normal or log-normal distributions • Quality of valuation depends on relevance of input data and comparables |
| Strengths | • Quick to run • Transparent and well-understood • Widely accepted for reporting |
• Can handle complex vesting structures and non-linear payouts • Flexible – allows for tailored modelling of real-world behaviour • More reflective of uncertain market conditions |
| Limitations | • Can be overly simplistic for real-world scenarios • Often relies on public company comparables that may not match private business risk • Outputs can feel theoretical or “synthetic” |
• More resource-intensive to build and test • Less transparent (especially when modelling workbooks aren’t shared) • Volatility assumptions can be flawed if comparables are poor |
| Best Practice Tip | Use for simple options, but sense-check against a discounted cash flow (DCF) where possible. | Ensure robust justification of input assumptions, especially for volatility and peer comparables. Ask tough questions when auditing or reviewing. |
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We always recommend that you seek advice from a suitably qualified adviser before taking any action. The information in this article only serves as a guide and no responsibility for loss occasioned by any person acting or refraining from action as a result of this material can be accepted by the authors or the firm.