Black Scholes Merton Models

Definitions

  • Black-Scholes-Merton models (BSMs) are primarily used to price European-style options and other forms of derivatives.

  • BSMs are also used to calculate Greeks which are foundational measures of risk for a derivative or portfolio of derivatives.

  • BSMs are based on a set of assumptions that, in order to be effective, must be accounted for by modifications including, amongst others, jump diffusion models and transaction fees.

  • The foundational math underlying BSMs are differential equations.

Concrete

  • Concrete’s coverage policies are well modeled by a put spread (a combination of two derivatives). As such, BSMs are well-suited to achieving numerous protocol functions.

  • Put spread example above. Note: Concrete’s policies do not have any payout in between the two strike prices, while traditional put spreads do. This does not fundamentally impact the efficacy/accuracy of this modeling approach.

  • BSMs are used especially heavily for two purposes within Concrete’s risk engine

    • Coverage policy pricing The traditional use case for BSMs is to price derivatives, particularly European style options. Concrete utilizes a modified BSM as an input pricing model for coverage policies.

    • Internal risk management BSMs are used to generate the well known Greeks which are useful measures of risk. They are well-suited for Concrete as coverage policies can be approximately modeled as put option spreads. The Greeks primarily used by Concrete are delta, vega, and gamma.

Models

  • Key assumptions - all are modified and accounted for in Concrete’s specific BSM implementation.

    • European-style options can only be exercised at expiration.

    • No dividends are paid out during the life of the option.

    • Markets are efficient with no transaction costs, and all participants have the same information.

    • The risk-free interest rate and volatility of the underlying asset are known and constant.

    • The returns on the underlying asset are normally distributed.

Basic Black-Scholes-Merton Model

  • The Black-Scholes-Merton model is a mathematical framework employed to determine the theoretical value of European-style financial options. Essentially, it provides a formula for pricing an option based on certain known and estimated variables, including the underlying asset's price, the option's strike price, the time to expiration, and the risk-free interest rate.

  • The model's primary accomplishment is its ability to assign a consistent and rational value to an option, thereby offering clarity in a market where pricing was previously driven by less precise heuristics.

  • The differential equations that define BSMs are as follows: where C(S,t) is the price of the call option with underlying asset price S at time t, X is the option’s strike price, r is the risk-free interest rate, T is the time to expiration, and N(d) denotes the cumulative distribution function of the standard normal distribution.

  • The price of a corresponding put option based on put-call parity is

Greeks

  • Delta

    • Delta, ∆, measures the rate of change of the option value (theoretical) with respect to changes in the underlying asset’s price. It is the first derivative of the option value with respect to the underlying asset’s price.

    • Delta is often used as the ”hedge ratio” - when monitoring option risk, delta is used as the primary indicator. Delta-hedging strategies immunize the position against loss or profit variability occurring due to small movements in the market.

    • Delta ranges from 0 to, 1 for calls in value and 0 to -1 for puts. It reflects the decrease or increase in the option price with respect to a 1 point movement of the underlying asset price. Deep in-the-money options have deltas close to 1 while far out-the-money options have delta values close to 0.

  • Gamma

    • Gamma, Γ, measures the rate of change in the Delta with respect to changes in the underlying price - the second derivative of the value function with respect to the underlying price.

    • When the asset price is near the strike price of the option, gamma is high and approaches 0 for options deep in the money or far out of the money and ultimately approaches zero for deep in-the-money options as well as for far out-the-money options.

    • For effective delta-hedging for a portfolio, gamma of the portfolio must be neutralized to ensure that the hedge will be effective across a wider range of the underlying price movements. All the options which are bought have positive gamma because as the price increases, Gamma also increases, which in turn causes delta to approach 1 from 0 (long call option) and 0 to -1 (long put option). Similarly, inverse is true for short options as for the options which are sold, gamma is negative.

  • Vega

    • Vega measures sensitivity with respect to the implied volatility of the underlying asset. Vega is calculated as the amount of money that the option value will gain or lose per underlying asset as the volatility rises or falls by 1%.

    • Crypto’s volatile nature makes Vega especially relevant to Concrete. For long volatility, Vega is positive and for short volatility, Vega is negative.

Further reading

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