Black-Scholes Model for Options Pricing
Understanding Black-Scholes model used for pricing European Options
Ever wondered how the Black-Scholes model works? Let’s break it down! 🤔The Black-Scholes model, a cornerstone in options pricing, is your go-to tool for pricing European Call and Put options. Let’s focus on Call Options. The payoff is straightforward — max(ST-K, 0) at expiration (time T). If ST>K, you’re in the money; otherwise, it expires worthless. 💸
Here’s the twist: When buying the option (at time t0), we don’t know ST. Enter Nobel laureates Robert C. Merton and Myron S. Scholes, who gifted us a closed-form formula for pricing European options. 🎁
The Black-Scholes equation for call options is given as below:
C0 = N(d1)*S0 — N(d2)*K*exp{-rt}
Simply put, there are two main terms in this equation, the first, 𝑁(d1)S0 calculates the probability of the option being exercised. This would occur when the underlying price is equal to or greater than the options strike price, K.
The second term, N(d2)*K*exp{-rt}, expresses the inverse probability of the investor not exercising the option.
Breaking down d1: Also, what’s intriguing is breaking down d1. The numerator splits into intrinsic value [ln(S0/K)] and time value [(r + sigma²/2)*T]. Intrinsic value gauges moneyness — proportional to how much S0 exceeds K. Time value has drift (money growth rate) and diffusion (volatility-dependent) terms. The denominator standardizes with sigma and multiplies by sqrt(T) for the time factor. The concept of scaling volatility by sqrt(t) for time t comes from Stochastic Calculus. (Psst, Stochastic Calculus delves into the nitty-gritty — more on that later! 😉)
Applications of Black-Scholes Model: It must be noted that Black-Scholes model is a powerful tool that can be used to price options beyond vanilla European Calls and Puts. For example, with some modification, Black-Scholes model can be converted to Black’s model which can be used to price FX Options.
Also, if Option prices are available in the markets and all the inputs except volatility is unknown, then it can be obtained using Black-Scholes equation by inputting all the parameters in the above equation and solving for volatility using a numerical method. The volatility obtained using such a method is called implied volatility.
Assumptions of Black-Scholes Model: While Black-Scholes model is an extremely useful model for options pricing, there are certain underlying assumptions which must hold true in order for Black-Scholes model to be applicable. These assumptions are given below:
- The condition of no-arbitrage holds true.
- Underlying asset prices are lognormally distributed.
- The risk-free rate is continuous, constant, known, and always available for borrowing or lending.
- Trading is continuous.
- The volatility of the underlying asset is constant and known.
- Markets are frictionless. This means that there are no taxes, no transactions costs and no restrictions on short sales.
- The underlying asset has no cash flow, such as dividends or coupon payments.
It is naturally assumed that the options valued are European options, which can only be exercised at maturity. As a result of this, the model cannot be used to correctly price American options.
