Greeks

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Basic Concepts of Greeks

The 'Greeks' are quantities that represent the sensitivity of an option's premium. They are used to predict the risk of an option contract.

• Delta - The change in the option premium relative to the change in the price of the underlying currency pair.
• Gamma - Measures the rate of change of Delta (Δ) when the price of the underlying currency pair changes.
• Vega - Measures how the option premium is affected by changes in volatility (σ).
• Theta - Measures the sensitivity of the option premium to the passage of time (τ), or "time decay."
• Rho - Measures the sensitivity of the option premium to changes in interest rates (r).

Delta

Delta $[\Delta = \frac{\partial P}{\partial S}]$ is the change in the option premium relative to the change in the price of the underlying asset.
It represents how much the option value changes when the exchange rate moves by 1 unit.

Example: A 50 call option with 90 days to expiration.

From the perspective of the probability of exercise, Delta represents the likelihood of an ordinary option being exercised at expiration (reflecting the probability of the option being in-the-money at expiration).

• For a long call option, as expiration approaches, the Delta of an ITM option will approach 1, and the Delta of an OTM option will approach 0.
  

• For a short call option, as expiration approaches, the Delta of an ITM option will approach -1, and the Delta of an OTM option will approach 0.
  

Gamma - The Rate of Change of Delta

Gamma $[\Gamma = \frac{\partial \Delta}{\partial S} = \frac{\partial^2 P}{\partial S^2}]$:
Measures the rate of change of Delta (Δ) when the price of the underlying currency pair changes. The question here is: How much will Delta change after a change in the spot price?

Gamma can be positive or negative. A positive Gamma means that the Delta of the option increases as the price of the underlying rises and decreases as the price of the underlying falls. A negative Gamma means that Delta decreases as the price of the underlying rises and increases as the price of the underlying falls.

Generally, all long options have positive Gamma, and all short options have negative Gamma.

Theta - Time Decay

Theta $[\Theta = -\frac{\partial P}{\partial \tau}]$:
Measures the sensitivity of the option premium to the passage of time (τ), or "time decay." As the option contract approaches expiration, time decay accelerates.

• Theta is negative for long calls and long puts, and positive for short calls and short puts.
• For options with the same expiration, ATM options have the largest absolute Theta values. The closer to expiration, the larger the absolute Theta value.
• When the option has a long time to expiration, Theta is small, but it increases rapidly as expiration approaches.
  

Vega - Sensitivity to Volatility

Vega $[v = \frac{\partial P}{\partial \sigma}]$:
Measures how the option premium is affected by changes in volatility (σ). It represents the amount of money gained or lost by the option when volatility increases or decreases by 1%. Vega is especially important in highly volatile markets, as it helps monitor the option premium.

• Buying a vanilla option (call/put) → Vega is positive.
• Selling a vanilla option (call/put) → Vega is negative.

Rho - Sensitivity to Interest Rates

Rho $[\rho = \frac{\partial P}{\partial r}]$:
Measures the sensitivity of the option premium to changes in interest rates (r).

• For call options, Rho is positive, and for put options, Rho is negative. In other words, as interest rates rise, the price of call options increases, while the price of put options decreases.
• Buying a call option is often used as a substitute for buying the underlying stock. Buying stock requires capital, but buying a call option does not. The saved capital can generate interest income, so when interest rates are high, the price of call options rises.
• Interest rates are considered the least important parameter in option pricing models.

Rho and Phi:

Rho measures the sensitivity of the portfolio value to changes in interest rates. Generally, currency options have two Rhos: one for the domestic interest rate and one for the foreign interest rate. It measures the sensitivity of the portfolio value to changes in interest rates. For call options, Rho is typically positive, and for put options, Rho is typically negative.

Rho represents the expected change in the option premium due to a small change in the domestic interest rate. Phi represents the expected change in the option premium due to a small change in the foreign interest rate.
Reference:
The Complete Guide to Option Pricing Formulas - Haug.pdf, P71

Black-Scholes-Merton Greeks

Spot Delta

$$\begin{aligned} \text{Call} \\ \Delta_{\text{call}} &= \frac{\partial c}{\partial S} = e^{(b - r)T} N(d_1) > 0 \\ \text{Put} \\ \Delta_{\text{put}} &= \frac{\partial p}{\partial S} = e^{(b - r)T} [N(d_1) - 1] < 0 \end{aligned}$$

Example: Consider a futures option with six months to expiration. The futures price is 105, the strike price is 100, the risk-free rate is 10% per annum, and the volatility is 36% per annum. Thus, $S = 105$, $X = 100$, $T = 0.5$, $r = 0.1$, $b = 0$, $\sigma = 0.36$.

$$d_1 = \frac{\ln(105 / 100) + (0 + 0.36^2 / 2) \times 0.5}{0.36 \sqrt{0.5}} = 0.3189$$

$$N(d_1) = N(0.3189) = 0.6251$$

$$\Delta_{\text{call}} = e^{(0 - 0.1) \times 0.5} N(d_1) = 0.5946$$

$$\Delta_{\text{put}} = e^{(0 - 0.1) \times 0.5} [N(d_1) - 1] = -0.3566$$

Forward Delta

Forward Delta represents how many units (FOR) of a forward contract a trader needs to buy to hedge a short option position:

$$\phi \mathcal{N}(\phi d_+)$$

Strike From Delta

In some OTC markets, options are quoted in terms of Delta rather than strike price. For example, in the OTC currency options market, it is common to ask for a Delta and expect the salesperson to return a price (in terms of volatility or points) and the strike price given a spot reference. In such cases, it is necessary to find the strike price corresponding to a given Delta. Many option software systems use the Newton-Raphson method or the bisection method to solve this numerically. However, this is not actually necessary. By inverting the cumulative normal distribution function $N^{-1}(\cdot)$, the strike price can be derived analytically from Delta, as described by Wystrup (1999).

For a call option:

$$X_{\text{call}} = S \exp[-N^{-1}(\Delta_{\text{call}} e^{(r - b)T}) \sigma \sqrt{T} + (b + \sigma^2 / 2)T]$$

For a put option:

$$X_{\text{put}} = S \exp[N^{-1}(-\Delta_{\text{put}} e^{(r - b)T}) \sigma \sqrt{T} + (b + \sigma^2 / 2)T]$$

Example: To achieve a Delta of 0.25 for a three-month stock index call option, assume the risk-free rate is 7%, the dividend yield is 3%, the volatility is 50%, and the stock index is trading at 1800. Given parameters: $S = 1800$, $T = 0.25$, $r = 0.07$, $b = 0.07 - 0.03 = 0.04$, $\sigma = 0.5$.

Calculating:

$$N^{-1}(\Delta_{\text{call}} e^{(r - b)T}) = N^{-1}(0.25 e^{(0.07 - 0.04) \times 0.25}) = -0.6686$$

Then, the strike price can be calculated using:

$$X_{\text{call}} = 1800 \times \exp[0.6686 \times 0.5 \sqrt{0.25} + (0.04 + 0.5^2 / 2) \times 0.25] = 2217.0587$$

Thus, to achieve a Delta of 0.25, the strike price should be set to 2217.0587.

Gamma

Gamma is the sensitivity of Delta to small changes in the price of the underlying asset. Gamma is the same for calls and puts:

$$\Gamma_{\text{call, put}} = \frac{\partial^2 c}{\partial S^2} = \frac{\partial^2 p}{\partial S^2} = \frac{n(d_1) e^{(b - r)T}}{S \sigma \sqrt{T}} > 0$$

This is the standard Gamma measure found in most textbooks (e.g., Hull (2005) and Wilmott (2000)). It measures the change in Delta when the underlying asset price changes by one unit.

Example: Consider a stock option with nine months to expiration. The stock price is 55, the strike price is 60, the risk-free rate is 10% per annum, and the volatility is 30% per annum. $S = 55$, $X = 60$, $T = 0.75$.

$$d_1 = \frac{\ln(55 / 60) + (0.1 + 0.3^2 / 2) \times 0.75}{0.3 \sqrt{0.75}} = 0.0837$$

$$n(d_1) = n(0.0837) = \frac{1}{\sqrt{2 \pi}} e^{-0.0837^2 / 2} = 0.3975$$

$$\Gamma_{\text{call, put}} = \frac{0.3975 e^{(0.1 - 0.1) \times 0.75}}{55 \times 0.3 \sqrt{0.75}} = 0.0278$$

Vega

Vega is the sensitivity of the option to small changes in the volatility of the underlying asset. Vega is the same for calls and puts.

$$\text{Vega}_{\text{call, put}} = \frac{\partial^2 c}{\partial \sigma^2} = \frac{\partial^2 p}{\partial \sigma^2} = S e^{(b - r)T} n(d_1) \sqrt{T} > 0$$

Example: Consider an index option with nine months to expiration. The index price is 55, the strike price is 60, the risk-free rate is 10.50% per annum, the dividend yield is 3.55% per annum, and the volatility is 30% per annum. What is the Vega? $S = 55$, $X = 60$, $T = 0.75$, $r = 0.105$, $b = 0.105 - 0.0355 = 0.0695$, $\sigma = 0.3$.

Calculating:

$$d_1 = \frac{\ln(55 / 60) + (0.0695 + 0.3^2 / 2) \times 0.75}{0.3 \sqrt{0.75}} = -0.0044$$

$$n(d_1) = n(-0.0044) = 0.3989$$

$$\text{Vega}_{\text{call, put}} = 55 e^{(0.0695 - 0.105) \times 0.75} \times 0.3989 \sqrt{0.75} = 18.5027$$

To convert this to Vega for a 1% change in volatility, we divide Vega by 100. Thus, if volatility changes from 30% to 31%, the option value will increase by approximately 0.1850.

Theta

Theta is the sensitivity of the option to small changes in the time to expiration. As time to expiration decreases, Theta is typically expressed as the negative of the partial derivative with respect to time.

Call:

$$\begin{aligned} \Theta_{\text{call}} &= -\frac{\partial c}{\partial T} = -\frac{S e^{(b - r)T} n(d_1) \sigma}{2 \sqrt{T}} - (b - r) S e^{(b - r)T} N(d_1) \\ &- r X e^{-rT} N(d_2) \leq \geq 0 \end{aligned}$$

Put:

$$\begin{aligned} \Theta_{\text{put}} &= -\frac{\partial p}{\partial T} = -\frac{S e^{(b - r)T} n(d_1) \sigma}{2 \sqrt{T}} + (b - r) S e^{(b - r)T} N(-d_1) \\ &+ r X e^{-rT} N(-d_2) \leq \geq 0 \end{aligned}$$

Example: Consider a European put option on a stock index currently priced at 430. The strike price is 405, the time to expiration is one month, the risk-free rate is 7% per annum, the dividend yield is 5% per annum, and the volatility is 20% per annum. $S = 430$, $X = 405$, $T = 0.0833$, $r = 0.07$, $b = 0.07 - 0.05 = 0.02$, $\sigma = 0.2$.

Calculating:

$$d_1 = \frac{\ln(430 / 405) + (0.02 + 0.2^2 / 2) \times 0.0833}{0.2 \sqrt{0.0833}} = 1.0952$$

$$d_2 = 1.0952 - 0.2 \sqrt{0.0833} = 1.0375$$

$$n(d_1) = n(1.0952) = \frac{1}{\sqrt{2 \pi}} e^{-1.0952^2 / 2} = 0.2190$$

$$N(-d_1) = N(-1.0952) = 0.1367 \quad N(-d_2) = N(-1.0375) = 0.1498$$

$$\begin{aligned} \Theta_{\text{put}} &= \frac{-430 e^{(0.02 - 0.07) \times 0.0833} n(d_1) \times 0.2}{2 \sqrt{0.0833}} \\ &+ (0.02 - 0.07) \times 430 e^{(0.02 - 0.07) \times 0.0833} N(-d_1) \\ &+ 0.07 \times 405 e^{-0.07 \times 0.0833} N(-d_2) = -31.1924 \end{aligned}$$

Thus, the daily time decay (Theta) is -31.1924 / 365 = -0.0855.

Rho

Rho is the sensitivity of the option to small changes in the risk-free interest rate.

Call:

$$\rho_{\text{call}} = \frac{\partial c}{\partial r} = T X e^{-rT} N(d_2) > 0$$

Put:

$$\rho_{\text{put}} = \frac{\partial p}{\partial r} = -T X e^{-rT} N(-d_2) < 0$$

Example: Consider a European call option on a stock currently priced at 72. The strike price is 75, the time to expiration is one year, the risk-free rate is 9% per annum, and the volatility is 19% per annum. Thus, $S = 72$, $X = 75$, $T = 1$, $r = 0.09$, $b = 0.09$, $\sigma = 0.19$.

Calculating:

$$d_2 = \frac{\ln(72 / 75) + (0.09 - 0.19^2 / 2) \times 1}{0.19 \sqrt{1}} = 0.1638$$

$$N(d_2) = N(0.1638) = 0.5651$$

$$\rho_{\text{call}} = 1 \times 75 e^{-0.09 \times 1} N(d_2) = 38.7325$$

If the risk-free rate changes from 9% to 10%, the call option price will increase by approximately 0.3873.

Vanna

$$\frac{\partial^2 v}{\partial \sigma \partial x} = -e^{-r_f \tau} n(d_+) \frac{d_-}{\sigma}$$

Vanna is sometimes referred to as $dvega/dspot$. It reflects how Vega changes as the underlying asset price changes. Trader's Vanna assumes a 1% relative change in the underlying asset price. The origin of the term "Vanna" is unclear. It is suspected to have originated from an article by Tim Owens in the 1990s in Risk Magazine, where he asked, "Want to lose a lot of money?" and explained how failing to hedge second-order Greeks like Vanna and Volga could lead to losses.

Volga

$$\frac{\partial^2 v}{\partial \sigma^2} = x e^{-r_f \tau} \sqrt{\tau} n(d_+) \frac{d_+ d_-}{\sigma}$$

Volga is sometimes referred to as $\upsilon omma$ or $\upsilon olgamma$ or $dvega/dvol$. Volga reflects how Vega changes as volatility changes. Trader's Volga assumes a 1% absolute change in volatility.

Numerical Greeks (Sensitivity Calculation Using Numerical Methods)

So far, we have only discussed analytical Greeks. A commonly used alternative is numerical Greeks, also known as finite difference approximations. The main advantage of numerical Greeks is that their calculation is model-independent. As long as we have an accurate model to calculate the derivative's value, finite difference approximations can provide the Greek values we need.

First-Order Greeks

The first-order partial derivative $\frac{\partial f(x)}{\partial x}$ can be approximated using the central difference method:

$$\frac{c(S+\Delta S,X,T,r,b,\sigma)-c(S-\Delta S,X,T,r,b,\sigma)}{2\Delta S}$$

For time derivatives, since we know the direction of time movement, it is more accurate to use backward differences (one-sided finite difference). That is:

$$\Theta\approx\frac{c(S,X,T,r,b,\sigma)-c(S,X,T-\Delta T,r,b,\sigma)}{\Delta T}$$

Numerical Greeks have several advantages over analytical Greeks. For example, if we have a sticky delta volatility smile, we can adjust the volatility accordingly when calculating numerical Delta. (When the volatility curve changes with the underlying asset price, we call it a sticky delta volatility curve; in other words, the volatility for a given strike price moves with the underlying asset price.)

$$\Delta_{\text{call}}\approx\frac{c(S+\Delta S,X,T,r,b,\sigma_{1})-c(S-\Delta S,X,T,r,b,\sigma_{2})}{2\Delta S}$$

Additionally, numerical Greeks are model-independent, whereas the analytical Greeks discussed above are specific to the BSM model.

Second-Order Greeks

For speed and other third-order derivatives $\frac{\partial^3f(x)}{\partial x^3}$, we can use the following approximation:

$$\text{Speed}\approx\frac{1}{\Delta S^{3}}[c(S+2\Delta S,\ldots)-3c(S+\Delta S,\ldots)+3c(S,\ldots)-c(S-\Delta S,\ldots)]$$

Mixed Greeks

For mixed derivatives $\frac{\partial f(x,y)}{\partial x\partial y}$, such as:

$$\begin{aligned} \text{DdeltaDvol}&\approx\frac{1}{4\Delta S\Delta\sigma}\\ &\times[c(S+\Delta S,\ldots,\sigma+\Delta\sigma)-c(S+\Delta S,\ldots,\sigma-\Delta\sigma)\\ &-c(S-\Delta S,\ldots,\sigma+\Delta\sigma)+c(S-\Delta S,\ldots,\sigma-\Delta\sigma)] \end{aligned}$$

DdeltaDvol and charm can be obtained through numerical calculations. For DdeltaDvol, it is usually divided by 100 to get the "correct" representation, i.e., for a 1% change in volatility.

Third-Order Mixed Greeks

For Greeks like DgammaDvol, we need to calculate third-order mixed derivatives $\frac{\partial^3f(x,y)}{\partial x^2\partial y}$:

$$\begin{aligned} \text{DgammaDvol}& \approx\frac{1}{2\Delta\sigma\Delta S^{2}} \\ &\times[c(S+\Delta S,\ldots,\sigma+\Delta\sigma)-2c(S,\ldots,\sigma+\Delta\sigma) \\ &+c(S-\Delta S,\ldots,\sigma+\Delta\sigma)-c(S+\Delta S,\ldots,\sigma-\Delta\sigma) \\ &+2c(S,\ldots,\sigma-\Delta\sigma)-c(S-\Delta S,\ldots,\sigma-\Delta\sigma)] \end{aligned}$$

For DgammaDvol, it is also usually divided by 100 to get the "correct" representation.

Calculating Delta Using Numerical Methods

raw delta = (v₁ – v₂) / (2 × bump)

Here:

• raw delta

  • Calculated using the finite difference method:
    raw delta = (v₁ – v₂) / (2 × bump)
  • Here, v₁ and v₂ are the option prices after bumping the spot up and down, respectively, and bump is the unit change in the spot.

• premium adjusted delta (adjusted delta)

  • Takes into account the impact of the option premium on sensitivity.
  • For foreign exchange options, it is usually necessary to deduct the proportion of the option price relative to the spot.

• ccy2 delta

  • Generally refers to the delta denominated in the foreign currency.
  • Defined as:
    ccy2 delta = raw delta

• ccy1 delta

  • Refers to the delta denominated in the domestic currency, expressed as:
    ccy1 delta = raw delta – v₀/S
  • Where v₀ is the option price, and S is the current spot exchange rate.

How to Calculate Forward Delta for American Options?

Some option pricing models, such as the BAW or PDE models for American options, do not include "forward" as a direct input parameter in their calculations. Therefore, it is not possible to directly compute the forward delta by altering the forward. Below, we present a commonly used method to obtain the forward delta for foreign exchange American options through the "bump and reprice" approach. The key insight is to recognize that:

$$\text{Forward} = S \times \exp\left[(r_d - r_f) \times (T - t)\right]$$

If your pricing program only provides $S$, $r_d$, and $r_f$, the forward is not explicitly passed as a separate input parameter, but it can be calculated using the above formula.

1. Basic Idea

Suppose you want to calculate the forward delta, which is the sensitivity of the option price to the forward, denoted as:

$$\Delta_F = \frac{\partial V}{\partial F}$$

Since the relationship between the forward and the spot is:

$$F = S \times \exp\left[(r_d - r_f) \times (T - t)\right]$$

Using the chain rule, we have:

$$\Delta_F = \frac{\partial V}{\partial S} \times \frac{\partial S}{\partial F} = \frac{\Delta_S}{\exp\left[(r_d - r_f) \times (T - t)\right]}$$

where $\Delta_S = \frac{\partial V}{\partial S}$ is the traditional spot delta.

2. Bump and Reprice Calculation Method

Using the bump and reprice method, you can follow these steps:

1. Calculate the Base Price
   Use the current $S$, $r_d$, and $r_f$ to price the option, obtaining the option price $V(S)$.

2. Determine the Bump Size
   Assume you want to bump the forward by $\Delta F$. As mentioned earlier, since the relationship between $F$ and $S$ is:

   $$\Delta F = \exp\left[(r_d - r_f) \times (T - t)\right] \times \Delta S$$

   Therefore, to bump the forward by $\Delta F$, you need to increase $S$ by:

   $$\Delta S = \frac{\Delta F}{\exp\left[(r_d - r_f) \times (T - t)\right]}$$

3. Calculate the Bumped Price
   Let the new spot be $S_{\text{bumped}} = S + \Delta S$, then re-run the FDM model with the new spot to obtain the bumped price $V(S_{\text{bumped}})$.

4. Calculate the Spot Delta

   $$\Delta_S \approx \frac{V(S_{\text{bumped}}) - V(S)}{\Delta S}$$

5. Convert to Forward Delta
   According to the chain rule, the forward delta is:

   $$\Delta_F \approx \frac{\Delta_S}{\exp\left[(r_d - r_f) \times (T - t)\right]}$$

   Or equivalently, directly use the price change from bumping the forward:

   $$\Delta_F \approx \frac{V(S_{\text{bumped}}) - V(S)}{\Delta F}$$

   Both methods are consistent under the conversion relationship between $\Delta S$ and $\Delta F$. The key is to ensure that the bump corresponds to the change in the forward, rather than simply applying an absolute bump to $S$.

6. Summary

• Due to the relationship between the foreign exchange forward and spot, $F = S \times \exp\left[(r_d - r_f)(T - t)\right]$, when you want to calculate the forward delta using the bump and reprice method, you can determine the corresponding spot bump based on the forward bump.

• The specific method is: set the forward bump $\Delta F$, calculate the corresponding $\Delta S$, re-price to obtain $V(S_{\text{bumped}})$, derive the spot delta, and then divide by $\exp\left[(r_d - r_f)(T - t)\right]$ to get the forward delta.

• Alternatively, you can directly express the bump value in terms of the price change:

  $$\Delta_F \approx \frac{V(S + \Delta S) - V(S)}{\Delta F}$$

  This way, even if the program parameters do not explicitly include the forward, you can still derive the forward delta through the spot bump process.