Question

Consider a self-financing Markovian portfolio (in continuous time) containing various derivatives of the single underlying asset in the Black-Scholes model. Denote the value (pricing function) of the portfolio by \(P(t, s)\). Show that the following relation must hold between the various greeks of \(P\). $$ \Theta_{P}+r s \Delta_{P}+\frac{1}{2} \sigma^{2} s^{2} \Lambda_{P}^{\prime}=r P $$ Hint: Use Exercise \(7.4\).

Short answer

Question: Show that the relation between the greeks of a self-financing Markovian portfolio in the Black-Scholes model holds as follows: $$ \Theta_{P}+r s \Delta_{P}+\frac{1}{2} \sigma^{2} s^{2} \Lambda_{P}^{\prime}=r P $$ Answer: By recalling the Black-Scholes equation and the definitions of the greeks (Theta, Delta, and Gamma), we rewrote the equation in terms of the greeks, and then rearranged the equation to obtain the desired relation. Hence, the given relation between the greeks of the portfolio holds in the Black-Scholes model.

Step-by-step solution

Recall the Black-Scholes equation and derivative definitions

The Black-Scholes equation is given as: $$ \frac{\partial P}{\partial t} + r s \frac{\partial P}{\partial s} + \frac{1}{2} \sigma^2 s^2 \frac{\partial^2 P}{\partial s^2} - r P = 0 $$ The greeks are defined as follows: - Theta (\(\Theta_P\)): Represents the sensitivity of the portfolio value to time, i.e., it is the partial derivative of the portfolio value with respect to time. $$ \Theta_P = \frac{\partial P}{\partial t} $$ - Delta (\(\Delta_P\)): Represents the sensitivity of the portfolio value to the change in the underlying asset price, i.e., it is the first partial derivative of the portfolio value with respect to the underlying asset price. $$ \Delta_P = \frac{\partial P}{\partial s} $$ - Gamma (\(\Lambda_P'\)): Represents the second-order sensitivity of the portfolio value to changes in the underlying asset price, i.e., it is the second partial derivative of the portfolio value with respect to the underlying asset price. $$ \Lambda_P' = \frac{\partial^2 P}{\partial s^2} $$ Now, we can use these definitions to rewrite the Black-Scholes equation in terms of the greeks.

Rewrite the Black-Scholes equation in terms of greeks

Substitute the definitions of Theta, Delta, and Gamma into the Black-Scholes equation: $$ \Theta_{P}+r s \Delta_{P}+\frac{1}{2} \sigma^{2} s^{2} \Lambda_{P}^{\prime} - r P = 0 $$ To show the relation given in the exercise, we need to prove that: $$ \Theta_{P}+r s \Delta_{P}+\frac{1}{2} \sigma^{2} s^{2} \Lambda_{P}^{\prime} = r P $$

Show that the given greek relation holds

Since $$ \Theta_{P}+r s \Delta_{P}+\frac{1}{2} \sigma^{2} s^{2} \Lambda_{P}^{\prime} - r P = 0 $$ we can rearrange the equation and get: $$ \Theta_{P}+r s \Delta_{P}+\frac{1}{2} \sigma^{2} s^{2} \Lambda_{P}^{\prime} = r P $$ Thus, we have shown that the given relation between the greeks of the portfolio holds.

Key concepts

Theta

Theta, often symbolized by \(\Theta\), represents the rate of decline in the value of a portfolio or a derivative with respect to time, keeping all other factors constant. In simpler terms, it measures how sensitive the price of a portfolio is to the passage of time.
This is important because options typically lose value as they approach expiration, a phenomenon known as "time decay."
The mathematical representation of Theta is:

• \(\Theta_{P} = \frac{\partial P}{\partial t}\)

For a trader, understanding Theta allows better prediction of how an option's price might change over time.
It is critical to factor in Theta when constructing a strategy that requires timing, such as an options spread.
Each day that passes decreases the extrinsic value of an option, particularly when the option is at-the-money. This is why Theta is especially significant as expiration nears.

Delta

Delta, denoted as \(\Delta\), is a measure that indicates how much the price of an option is expected to move per one-point movement in the price of the underlying asset.
This concept helps in gauging the direction sensitivity of the option's price in relation to the asset's price.
Technically, Delta is the first partial derivative of the option price with respect to the underlying asset's price:

• \(\Delta_{P} = \frac{\partial P}{\partial s}\)

Delta values generally range between 0 and 1 for call options, and between 0 and -1 for put options.
A Delta of 0.5 indicates a 50% probability that the option will end up in-the-money at expiry.

Practical Implications of Delta

Delta is useful for creating a hedged portfolio, or "delta-hedging," where the goal is to offset or neutralize the directional risk.
For instance, a Delta of 0.8 means an option price is likely to gain 0.8 units for each unit increase in the underlying price, allowing investors to adjust positions accordingly.

Gamma

Gamma, represented as \(\Lambda_P'\), measures the rate of change of Delta with respect to changes in the underlying asset's price.
In simple words, it tells us how the Delta itself will change as the underlying asset's price changes.
Gamma is the second partial derivative of the option's price with respect to the underlying asset's price:

• \(\Lambda_P' = \frac{\partial^2 P}{\partial s^2}\)

Gamma is a crucial greek for understanding how stable or unstable an option's Delta is, particularly in scenarios where large price movements are expected.
A high Gamma indicates that the Delta can change significantly, improving predictiveness of the hedge.

Role of Gamma in Option Strategies

Gamma helps in crafting more effective option strategies as it provides an insight into how an option's sensitivity (Delta) can evolve with market movements.
Traders must often adjust their portfolios to account for Gamma, making it helpful in managing portfolio sensitivity to ensure it's aligned with a desired level of risk tolerance.
As options move closer to being in-the-money or out-of-the-money, Gamma affects how much attention traders need to pay to their Delta adjustments.