Calculating Implied Volatility Using the Black-Scholes Model – Bitcoin Options Example

I have previously discussed Implied Volatility (IV) in past posts. The Black-Scholes model is one of the most widely used tools for options pricing. It enables traders to determine the theoretical value of an option based on variables such as the underlying asset price, strike price, time to maturity, risk-free rate, and implied volatility (IV). When it comes to Bitcoin options, solving for implied volatility can provide valuable insights into market expectations of price fluctuations.

In this post, we will walk through the process of using the Black-Scholes formula to calculate implied volatility for Bitcoin options, as well as its significance in cryptocurrency trading. Previously, I mentioned we would apply these calculations to the VIX index. However, instead, we will perform a real-world calculation for a Bitcoin option with 30 days to maturity.

What is Implied Volatility?

Implied volatility (IV) measures the market’s expectation of future volatility for the underlying asset, which in this case is Bitcoin. It is derived from the market price of an option. Unlike historical volatility, which analyses past price movements, IV represents the market’s forecast of price fluctuations.

The Black-Scholes Model: A Quick Recap

The Black-Scholes formula for a European call option is as follows:

$$C = S \cdot N(d_1) – K \cdot e^{-rT} \cdot N(d_2)$$

Where:

• $C$: Option price
• $S$: Current price of the underlying asset (Bitcoin, in this case)
• $K$: Strike price
• $T$: Time to maturity (in years)
• $r$: Risk-free rate
• $N(x)$ Cumulative distribution function of the standard normal distribution
• $d_1 = \frac{\ln(S / K) + (r + \frac{1}{2} \sigma^2) T}{\sigma \sqrt{T}}$
• $d_2 = d_1 – \sigma \sqrt{T}$
• $\sigma$: Implied volatility (the variable we solve for)

The challenge in solving for IV lies in the fact that it is not explicitly given in the formula. Instead, a numerical approach, such as Newton-Raphson iteration, must be employed to find the value of $\sigma$ that matches the market price of the option.

Steps to Solve for Implied Volatility

Here is how to solve for implied volatility for Bitcoin options using the Black-Scholes model:

1. Input Parameters:
   • Bitcoin price ($S$): The current price of Bitcoin (e.g., $93,491).
   • Strike price ($K$): The price at which the option can be exercised (e.g., $100,000).
   • Time to maturity ($T$): The number of days until expiration, expressed in years (e.g., 30 days = 0.0822 years).
   • Risk-free rate ($r$): Typically taken from the U.S. Treasury yield or an equivalent benchmark (e.g., 4% = 0.04).
   • Market price of the option ($C$): The observed market price of the option (e.g., $2,500).
2. Numerical Approach:
   Use an iterative method, such as Newton-Raphson, to estimate implied volatility:
   • Start with an initial guess for $\sigma$, typically 0.2 (20%).
   • Calculate the theoretical option price using the Black-Scholes formula.
   • Adjust iteratively to minimise the difference between the theoretical price and the market price.
3. Python Implementation:

import math

from scipy.stats import norm

# Black-Scholes Call Option Price

def black_scholes_call(S, K, T, r, sigma):

    d1 = (math.log(S / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * math.sqrt(T))

    d2 = d1 – sigma * math.sqrt(T)

    call_price = S * norm.cdf(d1) – K * math.exp(-r * T) * norm.cdf(d2)

    return call_price

# Implied Volatility Solver

def implied_volatility(S, K, T, r, market_price, tol=1e-6, max_iter=100):

    sigma = 0.5  # Initial guess for IV

    for i in range(max_iter):

        price = black_scholes_call(S, K, T, r, sigma)

        d1 = (math.log(S / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * math.sqrt(T))

        vega = S * norm.pdf(d1) * math.sqrt(T)

        price_diff = market_price – price

        if abs(price_diff) < tol:

            return sigma

        sigma += price_diff / vega

    raise ValueError(“Implied volatility did not converge”)

# Parameters

S = 93491  # Current Bitcoin price in USD

K = 100000  # Strike price in USD

T = 30 / 365  # Time to maturity in years

r = 0.04  # Risk-free rate

market_price = 2500  # Market price of the option in USD

# Calculate Implied Volatility

iv = implied_volatility(S, K, T, r, market_price)

print(f”Implied Volatility: {iv:.2%}”)

 

Why Is Implied Volatility Important in Bitcoin Options?

1. Market Sentiment: High IV suggests traders expect significant price swings in Bitcoin, while low IV indicates more stable conditions.
2. Options Pricing: IV directly influences the premium of an option, helping traders evaluate whether an option is overpriced or underpriced.
3. Risk Management: Understanding IV is crucial for constructing hedging strategies and managing risk effectively.

Conclusion

Implied volatility is a vital metric in Bitcoin options trading, and solving for it using the Black-Scholes model involves an iterative approach. The real-world calculation above shows an IV of 52.73%, indicating substantial market expectations for Bitcoin price movements over the next 30 days. This serves as a forward-looking measure of volatility, suggesting that traders anticipate Bitcoin’s price to fluctuate by ±52.73% during this period.

Cryptocurrency markets are inherently volatile, and implied volatility offers a valuable perspective for gauging future uncertainty. By mastering IV calculations, traders can gain a competitive advantage in this fast-paced market.

Caio Marchesani