Question

Find a similar expression for the delta \(\frac{d}{d S} P^{E}(S)\) of a European put option in the Black-Scholes model.

Short answer

The delta of a European put option in the Black-Scholes model is \( \frac{d}{dS} P^E(S) = -N(-d_1) \).

Step-by-step solution

Recall the Black-Scholes Formula for Put Option

The Black-Scholes formula for the price of a European put option is given by: \[ P^{E}(S) = K e^{-rT} N(-d_2) - S N(-d_1) \] where: \[ d_1 = \frac{\ln(S/K) + (r + \sigma^2 / 2)T}{\sigma \sqrt{T}} \] \[ d_2 = d_1 - \sigma \sqrt{T} \]

Differentiate the Option Price with Respect to S

To find the delta, differentiate \(P^{E}(S)\) with respect to \(S\). Using the chain rule and the properties of the normal distribution, we get: \[ \frac{d}{dS} P^E(S) = - N(-d_1) + S (-d_1') n(-d_1) - N'(-d_1) \] Here, \(N'\) is the derivative of the cumulative distribution function, which equals the probability density function, \(n(-d_1)\).

Simplify the Expression

The term \(N'(-d_1)\) evaluates to \(n(-d_1)\) and use \(d_1'\) simplifies to: \[ d_1' = \frac{1}{S \sigma \sqrt{T}} \] Applying these, we get: \[ \frac{d}{dS} P^E(S) = -N(-d_1) \]

Key concepts

delta of European put option

Delta, often denoted as \(\frac{d}{dS} P^{E}(S)\), measures how the price of a European put option changes with respect to the underlying asset price. In simpler words, it shows the sensitivity of the option's price to a small change in the stock price. To find this, we use the Black-Scholes formula for a European put option and differentiate it concerning the asset price, \(S\).
According to the solution steps, first, we recall the Black-Scholes formula for a European put option:
\[ P^{E}(S) = K e^{-rT} N(-d_2) - S N(-d_1) \]
Here:

• \(K\) is the strike price.
• \(r\) is the risk-free rate.
• \(T\) is the time to maturity.
• \(N(-d_1)\) and \(N(-d_2)\) are the cumulative distribution functions of the standard normal distribution.

\[ d_1 = \frac{\ln(S/K) + (r + \sigma^2 / 2)T}{\sigma \sqrt{T}} \] \[ d_2 = d_1 - \sigma \sqrt{T} \]
Next, by differentiating \(P^{E}(S)\) with respect to \(S\), we get:
\[ \frac{d}{dS} P^{E}(S) = -N(-d_1) \]
This equation shows how sensitive the European put option price is to changes in the stock price.

Black-Scholes formula

The Black-Scholes formula is a crucial component in option pricing. It provides a theoretical estimate for pricing European call and put options. For a European put option, the Black-Scholes formula is:
\[ P^{E}(S) = K e^{-rT} N(-d_2) - S N(-d_1) \]
This formula takes into account several parameters:

• \(S\) is the current stock price.
• \(K\) is the strike price.
• \(T\) is the time until expiration.
• \(r\) is the risk-free interest rate.
• \(\sigma\) (sigma) is the volatility of the stock price.

To break it down further:

• \(N(-d_1)\) and \(N(-d_2)\) are the values from the cumulative distribution function of the standard normal distribution.
• \(d_1\) and \(d_2\) are intermediate variables calculated based on the formula for \(d_1\) and the relationship between \(d_1\) and \(d_2\).

Using these inputs, the formula accounts for both the intrinsic value and the time value of an option. Understanding and using the Black-Scholes formula is key for any student looking to grasp derivative pricing.

option price sensitivity

Option price sensitivity, also termed 'Greeks,' helps us understand how various factors impact the price of an option. Delta, gamma, theta, vega, and rho are the primary Greeks. Focusing on delta, we find how the option price reacts to small movements in the underlying asset price. For the European put option, the delta is:
\[ \frac{d}{dS} P^{E}(S) = -N(-d_1) \]
The negative sign indicates that put option prices move inversely compared to the underlying asset.
Other Greeks provide insight into different sensitivities:

• **Gamma:** Examines the rate of change of delta concerning the underlying price.
• **Theta:** Measures sensitivity to time decay.
• **Vega:** Captures the price sensitivity to volatility changes.
• **Rho:** Looks at sensitivity to the interest rate changes.

By understanding these sensitivities, traders and students can better manage risk and make more informed decisions regarding their option strategies.