11.4 Greeks

The Black-Scholes formula for non-dividend paying underlying assets (11.10) show that there are essentially five parameters, which determine the option price: the current level of the underlying asset $S_t$, the strike price $K$, the continuously compounded risk-free interest rate $r$, the time to expiration $\tau$ and the instantaneous standard deviation $\sigma$ of the underlying. The influence of these parameters on the option price can be investigated by using the quantlet influence from the library finance in XploRe :

dat = influence ()

dat = influence (S,K,r,sigma,tau,carry,opt,pder,v1,ub1,v2,ub2)

displays graphically the influence of the parameters, which enter the Black-Scholes formula, on the option price. The input parameters are specified either interactively or directly.

This quantlet plots the option price against one or two input parameter(s), maintaining all other parameters constant. The sensitivity of the option price related to this parameter(s) gives additional information. This sensitivity can be represented in terms of a number, or indicator, generally referred to as Greek. By knowing numerical values for Greeks, it allows hedge positions using options to be set up. Greeks can be computed and plotted through the quantlet greeks .

dat = greeks ()

dat = greeks (S,K,r,sigma,tau,carry,opt,pder,v1,ub1,v2,ub2)

calculates and plots different sensitivities of the option price. The input parameters are specified either interactively or directly.

When the direct specification of the input parameters is selected, then the first five parameters follow the usual notation: $\tt {S}$ for the current level of the underlying asset, $\tt {K}$ for the strike price, $\tt {r}$ for the continuously compounded risk-free interest rate, $\tt {sigma}$ for the instantaneous standard deviation of underlying asset and $\tt {tau}$ for time to expiration. The parameter $\tt {carry}$ denotes the annualized additional costs as a proportion of the price of the underlying asset. The parameter $\tt {opt}$ is a scalar defining the type of option. It has the value 1 for a call and 0 for a put. The parameter concerned is specified in $\tt {v1}$. The type of sensitivity computed through the quantlet greeks is specified in $\tt {pder}$. It has the value 1 for delta, 2 for gamma, 3 for eta, 4 for delta-k, 5 for vega, 6 for theta and 7 for rho.

The output is a two dimensional plot, which shows the dependence of the option price (when the quantlet influence is used) or of its sensitivity (when the quantlet greeks is used) on the specified parameter. If an additional parameter is specified in $\tt {v2}$, a three dimensional plot with both parameters as explanatory variables is produced.

For graphical representation, the option price is computed within 30 discrete intervals of the explanatory variable(s). This is done mainly in two steps. Firstly, the quantlet asset is used to create a discrete grid of 31 points. To achieve this, the lowest and highest bounds for the parameter(s) are requested. The highest bound must be inputed into $\tt {ub1}$ (and into $\tt {ub2}$ in the case of two exploratory variables). The specified input value of the exploratory variable(s) is considered as the lowest bound. When both quantlets influence and greeks are used interactively, the user can freely decide which bound values to apply. Secondly, the option price is computed for each of the 31 different grid points using the Black-Scholes Formula. The results are presented in a two dimensional, or for two exploratory variables, in a three dimensional plot.

In the following, the sensitivity of the option price with respect to changes in one of the five parameters is analyzed: $\tt {S}$, $\tt {K}$, $\tt {r}$, $\tt {tau}$ and $\tt {sigma}$. Details on these sensitivities can be found in different financial sources, e.g. in the e-book Statistics of Financial Markets, ch. 7.3. .

The following theoretical descriptions are based on Franke et al. (2001), Gibson (1991), Hull (2000), Kwock (1998) and Tompkinks (1994). To demonstrate how the option price and its sensitivity relates to the changes in the parameters above, the quantlets influence and greeks are used.

11.4.1 Delta

The delta ($\Delta$) of a derivative security is defined as the rate of change of its price with respect to the price of the underlying asset. It is the slope of the curve that relates the derivative security price $V$ to the price of the underlying $S$:

$$\Delta=\frac{\partial V}{\partial S}$$

Delta plays a crucial role in portfolio hedging. In the derivation of the Black-Scholes equation a covered call position is maintained by creating a risk-free portfolio, where the writer of a call sells one unit of the call and buys $\Delta$ units of the underlying.

The delta of a call ($\Delta _C$) is always positive, as an increase in the asset price will increase the probability of a positive payoff at expiration resulting in a higher value. On the other hand, there is a negative relationship between the put price and the underlying asset price, as an increase in the asset price, will reduce the put's current exercise value $\left\{K\,exp{(-r\tau)}-S\right\}$ and therefore the put's price will decrease. This explains a negative $\Delta _P$ as given in (11.50).

When the price of the underlying asset changes, put and call option values move in opposite directions, since $(0\leq\Delta_C\leq1)$ and $(-1\leq\Delta_P\leq0)$. However, the absolute changes in their prices will never exceed those of the underlying asset.

$\Delta _C$ of a European call on a non-dividend paying underlying asset can be easily derived from the Black-Scholes formula (11.10):

$$\Delta_C = \Phi(d_1)+S\frac{1}{\sqrt{2\pi}}exp{\left(-\frac{d^2_1}{2}\right)... ...}{\sqrt{2\pi}}exp{\left(-\frac{d^2_2}{2}\right)}\frac{\partial d_2}{\partial S}$$

$$= \Phi(d_1)+ \frac{1}{\sigma\sqrt{2\pi\tau}} \left[exp{\left(-\frac... ...t(r\tau+ln\frac{S}{K}\right)\right\}} exp{\left(-\frac{d^2_2}{2}\right)}\right]$$

$$= \Phi(d_1)>0$$ (11.49)

The delta of a European put option is then derived from the put-call parity relation:

$$\Delta_P = \frac{\partial P}{\partial S}=\Delta_C-1=\Phi(d_1)-1=-\Phi(-d_1)<0$$ (11.50)

In the following, it is shown through examples how the option price and its delta is calculated and plotted as a function of the underlying asset.

library("finance")

S=230 		; (spot) price of the underlying

K=210 		; exercise price

r=5 		; the annualized risk-free interest rate in % 

sigma=25 		; annualized volatility  in % 

tau=0.5 		; annualized time to expiration

carry=5 		; cost of carry 

opt=1 		; call

v1=1 		; spot price as an explanatory variable

ub1=400 		; highest bound of the spot price

influence(S,K,r,sigma,tau,carry,opt,v1,ub1)



pdr=1 		; delta of the call

greeks(S,K,r,sigma,tau,carry,opt,pdr,v1,ub1)

XLGfindex10.xpl

 

The computation yields a (31x2) dimensional matrix with the prices of the underlying asset in the first column. The second column contains the respective call prices (for the quantlet influence ), or the values of $\Delta _C$ (for the quantlet greeks ).

The two dimensional plot (Figure 11.6) displays a positive relationship between the call price and the underlying, which supports the theoretical results from (11.49). Note, for explanation purposes the underlying ranges from 100 to 400. This is achieved through running the quantlets influence and greeks once again and specifing interactively the parameters with the same values, as in the example above.

Figure 11.7 shows that $\Delta _C$ is an increasing function of $S$. This result is not surprising, since $\frac{\partial \Phi(d_1)}{\partial S}$ is always positive. It follows that the call price is an increasing convex function of the underlying price (see subsection 11.4.3 for further details on convexity).

The fact that the delta changes as the underlying price changes, means that the delta provides only instantaneous information. To remain perfectly risk-free, a hedged position in options may have to be revised continuously. The delta-hedge frequency, depends on the derivative of the delta with respect to the price of the underlying, commonly referred to as the gamma. For detailed explanations and examples on gamma see subsection 11.4.3.

$\parbox{5.5cm}{ \ {\includegraphics[wid... ...ceSa.ps}} \ \caption{\small The price-stock relationship for a European call.}}$ $\parbox{5.5cm}{ {\includegraphics[widt... ...sDeltaSa.ps}} \caption{\small $\Delta_C$\ as a function of the stock price.}}$

In a second example the same procedure is repeated for a put option:

library("finance")
 influence(230,210,5,25,0.5,5,1,1,400) ; put option
 greeks(230,210,5,25,0.5,5,0,1,1,400)  ; delta of the put

The outputs are presented in Figure  (11.8) and Figure  (11.9) respectively. Figure  (11.9) shows that $\Delta _P$ is an increasing function of the asset price $S$, i.e. the put's price decreases at an increasing rate, when the price of the underlying asset increases. In other words, the put's price is a decreasing convex function of the price of the underlying asset.

$\parbox{5.5cm}{ \includegraphics[width... ...enceSb.ps} \caption{\small The price-stock relationship for a European put.}}$ $\parbox{5.5cm}{ \includegraphics[width... ...eksDeltaSb.ps} \caption{\small $\Delta_P$\ as a function of the stock price.}}$

Both call and put deltas are functions of $S$ and $\tau$. It can be shown that

$$\lim_{\tau\to\infty}\frac{\partial C}{\partial S}=1 \qquad \textrm{for all values of }S.$$

This means that $\Delta _C$ always tends to one, whereas the time to expiration tends to infinity, since there is a higher probability of an increase in the asset price. In addition, the following holds (Kwock; 1998, pp.57):

$$\lim_{\tau\to 0^+}\frac{\partial C}{\partial S}=\left\{\begin{array}{r@{\quad if \quad}l} 1 & S>K\\ \frac{1}{2} & S=K\\ 0 & S<K\end{array}\right.$$

At expiration, delta has different asymptotic limits depending on whether the option is in-the-money $(S>K)$, at-the-money $(S=K)$, or out-of-the-money $(S<K)$. For options deep in-the-money, $\Delta _C$ converges to one. In other words, since the option will be exercised at expiration, the writer of the call should hold the asset to hedge the risk. For deep out-of-the-money, the call will not be exercised and the writer no longer needs to hold the asset. Consequently, $\Delta _C$ will then converge to zero. Hence at expiration, the option will have either a slope of zero (if out-of-the-money) or one (if in-the-money).

It can be surmised that the at-the-money option, which lies in the middle between these extremes, might have a slope of $0.5$. Therefore any time before expiration an out-of-the-money option will have a delta between 0 and $0.5$, and in-the money option will have a delta between $0.5$ and $1$. For example, this can be seen for a call option with six months prior to expiration in Figure 11.7. The "S" shaped curve indicates how the exposure of the call option relative to the underlying asset has a limit loss when the price of the underlying asset falls, and assumes full exposure when the underlying price rises.

Alternative ways to think about delta, as a measure of relative risk of the option to the underlying market, or as the probability of exercise, are explained in Tompkinks (1994, pp. 57-65).

greeks () computes $\Delta _C$ and displays it as a function of $S$ and $\tau$ (Figure 11.10). The input parameters are given interactively. The plot supports the relationship mentioned above.

$\parbox{10cm}{\caption{\small The delta... ...all as a function of the \ stock price $S$\ and of time to expiration $\tau$.}}$

11.4.2 Delta of the Strike

For most options the strike price is fixed, but some option-like securities, such as convertible bonds, can have a variable "strike" price. In this case the price change of the derivative security $V$ with respect to the strike price $K$

$$\Delta_K=\frac{\partial V}{\partial K},$$

may be appropriate. The higher the strike price, the less valuable a call option is, since the strike price represents a higher cost of exercising the call and thereby purchasing the stock. In contrast, the higher the exercise price of a put, the higher its price will be. The Black-Scholes formula clearly confirms these relationships:

$$\frac{\partial C}{\partial K}=-exp{\left(-r\tau\right)}\Phi(d_2)<0,$$ (11.51)

$$\frac{\partial P}{\partial K}=exp{\left(-r\tau\right)}\Phi(-d_2)>0.$$ (11.52)

In the following example, the quantlet influence displays the relationship between the option price and the strike price for a European call (Figure 11.11) and a European put (Figure 11.13). The quantlet greeks is used to plot the respective deltas of the strike (Figure 11.12 and 11.14).

library("finance")
 influence(230,210,5,25,0.5,5,1,2,400) ; call
 greeks(230,200,5,25,0.5,5,1,4,1,400)
 influence(230,210,5,25,0.5,5,0,2,400) ; put
 greeks(230,200,5,25,0.5,5,0,4,1,400)

XLGfindex11.xpl

$\parbox{5.5cm}{ \includegraphics[width... ...nceKa.ps} \caption{\small The price-stock relationship for a European call.}}$ $\parbox{5.5cm}{ \includegraphics[width... ...n{\small $\Delta_{K}$\ of a European call as a function of the strike price.}}$

$\parbox{5cm}{ \includegraphics[width=0... ...enceKb.ps} \caption{\small The price-stock relationship for a European call.}}$ $\parbox{5cm}{ \includegraphics[width=0... ...ure \small $\Delta_{K}$\ of a European put as a function of the strike price.}}$

11.4.3 Gamma

The option gamma ($\Gamma$) is defined as the derivative of delta ($\Delta_V$) with respect to the underlying asset price $S$:

$$\Gamma=\frac{\partial\Delta_V}{\partial S}=\frac{\partial^2 V}{\partial S^2}$$

It represents the change in the curvature of the option at different values of $S$ and is therefore also known as convexity. The increments of gamma are often referred to as the number of deltas that will change when the underlying asset price changes by one tick. By definition, the higher the gamma value is, the more the delta will change when the underlying market price changes.

The gamma of a long European call and a long European put on a non-dividend paying underlying asset is

$$\Gamma_C=\Gamma_P=\frac{\partial\Delta_C}{\Delta S}=\frac{\partial^2 C}{\partial S^2}=\frac{1}{S\sigma\sqrt{\tau}} \Phi^\prime(d_1)>0,$$ (11.53)

whereas, for a dividend paying asset it is

$$\Gamma_C=\Gamma_P=\frac{\partial\Delta_C}{\Delta S}=\frac{\partia... ...rac{exp{\left(-\frac{d^2_1}{2}\right)}}{S\sigma\sqrt{\tau}} \Phi^\prime(d_1)>0,$$ (11.54)

with $d_1$ defined as in equation (11.10) and $\Phi^\prime(d_1)$ defined as

$$\Phi^\prime(d_1)=\frac{1}{2\sqrt{\pi}}exp{\left(-\frac{d^2_1}{2}\right)}.$$ (11.55)

A positive gamma as in (11.53) and (11.54) means that changes in the amount of delta have the same direction as changes in the underlying market. This explains why for any European vanilla call or put option, the curves of the option price functions are convex with respect to the asset price (Figure 11.7 and 11.9).

With gamma being positive, the buyers of the options gain from movements in the price of the underlying assets. For this reason, holders of the options are often referred to as being "long gamma". For sellers of the options, the gamma exposure is exactly opposite to that of buyers of options. Those who sells options can be hurt when gamma is high and the underlying market price moves. Therefore they are often referred to as "short gamma".

When the curvature of the option value is small, its gamma has a low value. A low value of gamma implies that delta changes slowly with the asset price and so adjustments required to keep a portfolio delta neutral can be made less frequently. When gamma is high, in absolute terms, delta is highly sensitive to the price of the underlying asset. It is then quite risky to leave a delta-neutral portfolio unchanged for any remaining length of time.

Intuitively, gamma jointly measures how close the current market is to the option strike price and how close the option is to expiration (Tompkinks; 1994, pp. 67). The closer the market price is to the strike price and the closer the option is to expiration, the higher the gamma will be. This is illustrated in Figure (11.15) for a European call option with $\tt S=230$, $\tt K=200$, $\tt r=5\%$, $\tt\sigma=25\%$, $\tt\tau=0.5$ and $\tt cost\; of \; carry=5\%$. The computation is done interactively by using the quantlet greeks ().

$\parbox{11cm}{\caption{\small Gamma of ... ... (put) as a function of the \ stock price $S$\ and time to expiration $\tau$.}}$

When the option is at-the-money ($S=K=230$) with one minute remaining until expiration, it has the highest possible gamma value of $1.0$. The reason for this, is that if the underlying asset price moves the tiniest increment up, the option will then be in-the-money with a delta of $1.0$ (Figure 11.15). If on the other hand, the underlying asset price falls by an infinitesimally small amount, the option will then be out-of-the-money with a delta of zero.

More moving away from this extreme situation, where the option is at-the-money at expiration, then the lower the gearing effect of the option will be, hence the lower the gamma. The larger the difference between the current underlying asset price and the options's strike price, the less is the time value of the option and the lower the gamma. Additionally, an option with more time remaining until expiration will have a lower gamma (Figure 11.15).

So far the speed (delta) and acceleration (gamma) features of options over the underlying market price have been examined. In the following the other factors are discussed, the most important being volatility.

11.4.4 Vega

The change in the option price with respect to the change in implied volatility (section 11.5) is called vega. It is a measure of the option exposure to changes in implied volatility within the option market (Tompkinks; 1994, pp. 69). The vega of a European vanilla call $\Lambda_C$ and put $\Lambda_C$ on non-dividend paying underlying asset can be derived from the Black-Scholes formula:

$$\Lambda_C = \frac{\partial C}{\partial \sigma}= S\Phi^\prime(d_1)\frac{\parti... ...} -Kexp{\left(-r\tau\right)}\Phi^\prime(d_2)\frac{\partial d_2}{\partial\sigma}$$

$$= S\sqrt{\tau}\Phi^\prime(d_1)>0$$ (11.56)

$$\Lambda_P = \frac{\partial P}{\partial \sigma}=\frac{\partial C}{\partial \si... ...partial}{\partial \sigma}\left\{Kexp{\left(-r\tau\right)}-S\right\}=\Lambda_C>0$$ (11.57)

with $d_1$ and $\Phi^\prime(d_1)$ given in 11.10 and 11.55 respectively.

When the volatility rises, the option's set of favorable outcomes will also rise. As a result, the chances are higher for the option to be either deeper in-the-money or deeper out-of-the-money at expiration. Since the option bears no downside risk, there is no penalty when the option expires deeper out-of-the-money, but a higher payoff when it expires deeper in-the-money. Due to this antisymmetric payoff structure, the vegas for long options are positive, i.e. for the option buyer, the exposure to changes in implied volatility is positive and consequently the vega is positive. By symmetry, the option writer benefits from a decrease in implied volatility and therefore has vega negative exposure.

Using interactive menus in the quantlet greeks (), the vega of a European call option is presented as a function of the underlying asset and of time to expiration (Figure 11.16). For at-the-money options, the longer the time to expiration, the higher the sensitivity of the option to the changes in volatility, and hence the higher the vega. In other words, whereas gamma of an at-the-money option increases as the expiration date approaches (see Figure 11.15), the reverse is true for vega.

$\parbox{11cm}{\caption{\small The vega ... ...l (put) as a function of the stock price $S$\ and time to expiration $\tau$.}}$

11.4.5 Eta

Eta ($\eta$) of a derivative security defines the elasticity of its price $V$ with respect to the underlying price $S$:

$$\eta=\left(\frac{\partial V}{\partial S}\right)\left(\frac{S}{V}\right)$$

This elasticity parameter measures the percentage change in security price for a unit percentage change in the asset price. The elasticity of a European vanilla call ($\eta _C$) on non-dividend paying underlying asset is found to be:

$$\eta_C = \left(\frac{\partial C}{\partial S}\right)\left(\frac{S}{C}\right)= \frac{S\Phi(d_1)}{S\Phi(d_1)-Kexp{\left(-r\tau\right)}\Phi(d_2)}>1$$ (11.58)

Equation (11.58) implies that a call option is riskier than the underlying asset in terms of change in percentage. It can be shown that the elasticity is high when the asset price is low (out-of-the-money), and it decreases monotonously with the price of the underlying asset (Kwock; 1998, pp. 57). For sufficiently large values of $S$, $\eta _C$ converges to one, as $C$ approaches $S$ when $S$ tends to infinity. This relationship can be seen in Figure (11.17), which plots the elasticity of a European call from the following example:

library("finance")
 greeks(230,210,5,25,0.5,5,1,3,2,400) ; call

$\parbox{8cm}{ \caption{\small $\eta_C$\ as a function of the stock \ price.}}$

The elasticity of a European put price ($\eta _C$) on non-dividend paying underlying asset is:

$$\eta_P = \left(\frac{\partial P}{\partial S}\right)\left(\frac{S}{C}\right)=\frac{-S\Phi(-d_1)}{Kexp{\left(-r\tau\right)}\Phi(-d_2)-S\Phi(-d_1)}$$ (11.59)

Equation (11.59) shows that the absolute value of the put elasticity can be less or greater than one. Therefore, a European put option may or may not be riskier than the underlying asset in terms of change in percentage.

For both put and call options, their elasticities increase in absolute value when the corresponding options become more out-of-the-money and move closer to expiration (Kwock; 1998, pp. 57).

11.4.6 Theta

Theta ($\Theta$) of a derivative security is defined as the rate of change of its price $V$ with respect to time $t$ with all other factors remaining constant:

$$\Theta=\frac{\partial V}{\partial t}=-\frac{\partial V}{\partial \tau}$$

It is sometimes referred to as the 'time decay' of security. The derivative of the call price as determined by the Black-Scholes model in (11.10) with respect to time shows that

$$\Theta_C=\frac{\partial C}{\partial t}=-\frac{\partial C}{\partia... ...c{S\sigma}{2\sqrt{\tau}}\Phi^\prime(d_1)-rKexp{\left(-r\tau\right)}\Phi(d_2)<0.$$ (11.60)

Using then the put-call parity relation 11.18, $\Theta_P$ is derived as

$$\Theta_P=\frac{\partial P}{\partial t}=-\frac{S\sigma}{2\sqrt{\tau}}\Phi^\prime(d_1)+rKexp{\left(-r\tau\right)}\Phi(-d_2).$$ (11.61)

The longer the time to expiration of a European call option, the higher its price $(\frac{\partial C}{\partial \tau}>0$). The reason for this, is that by prolonging the time interval until the option's expiration, simultaneously increases the chances of the option either ending in-the-money at expiration and making a profit, or ending out-of-the-money and expiring worthless. This also holds for an American call on a non-dividend paying underlying asset, which will not be exercised before expiration.

$\frac {\partial C}{\partial \tau }$ for a European call and a European put is computed in XploRe , by specifying the input parameters interactively. The plots are shown in Figure (11.18) and (11.19). The underlying price $S$ ranges from 150 to 400. The values of the other input parameters are $\tt {K=210}$, $\tt {r=5\%\;p.a.}$, $\tt {\sigma=25\%\; p.a.}$, $\tt {\tau=0.5\;years}$, $\tt {cost\; of\;carry=5\%}$.

$\parbox{5.5cm}{ \ \includegraphics[widt... ...al C}{\partial \tau}$\ for a European \ call as a function of the stock price}}$ $\parbox{5.5cm}{ \ \includegraphics[widt... ...ial P}{\partial \tau}$\ for a European \ put as a function of the stock price}}$

The positive $\frac {\partial C}{\partial \tau }$ for a European call, is consistent with the result in (11.60), as greeks considers $\frac {\partial C}{\partial \tau }$ as theta, instead of $\frac{\partial C}{\partial t}$.

The theta of a European call (Figure 11.18) has its greatest absolute value when the call option is at-the-money, as it may become in-the-money or out-of-the-money soon thereafter. It has a small absolute value when the option is sufficiently out-of-the-money, as it will be highly unlikely for the option to become in-the-money later on. It tends asymptotically to $-rKexp{\left(-r\tau\right)}$ when the asset price is sufficiently high.

The theta of a European put (Figure 11.19) can be any sign, depending on the relative magnitudes of the two terms, which have opposite signs in (11.61). When the European put is deep in-the-money, S assumes a small value, so that $\Phi(-d_2)$ tends to one. The second term $rKexp{\left(-r\tau\right)}\Phi(-d_2)$ is then greater than the first $\frac{S\sigma}{2\sqrt{\tau}}\Phi^\prime(d_1)$. In this case, $\frac {\partial C}{\partial \tau }$ is negative and the theta as defined in (11.61) is positive. When the option is at-the-money or out-of-the-money, $\frac {\partial C}{\partial \tau }$ is typically positive and hence the theta of a European put is negative, as the longer the time to expiration, the higher the chances of positive outcomes.

Note, the ambiguous relationship between a put's price and time to expiration does not hold for American options. An American call or put will always show a positive relationship between its price and time to expiration, which corresponds to a negative theta. When the time to expiration is prolonged, an American option has therefore the additional right to be exercised in the prolonged time interval and consequently has a higher value.

11.4.7 Rho

The Rho ($\rho$) of a derivative security is defined as the rate of change of derivative price $V$ with respect to the interest rate $r$:

$$\rho=\frac{\partial V}{\partial r}$$

The rho of a European call $\rho _C$ and the rho of a European put $\rho _P$ for non-dividend paying underlying asset is derived from the Black-Scholes formulae (11.10) and (11.11), respectively

$$\rho_C= \tau Kexp{\left(-r\tau\right)}\Phi(d_2)>0 \quad\textrm{for $r>0$\ and $K>0$},$$ (11.62)

and from the put-call parity relation

$$\rho_p= -\tau Kexp{\left(-r\tau\right)}\Phi(-d_2)<0 \quad\textrm{for $r>0$\ and $K>0$}.$$ (11.63)

A higher interest rate decreases the present value of the cost of exercising the European call at expiration (which is a similar effect to the strike price decreasing) and so increases the call price. The reverse effect holds for the put price. Signs of the rho for call and put prices in (11.62) and (11.63) confirm this effect.

The following example calculates $\rho _C$ and $\rho _P$ and plots them against the underlying asset price $S$ (Figure 11.20 and 11.21).

library("finance")
 greeks(230,210,5,25,0.5,5,1,7,1,400) ; call
 greeks(230,210,5,25,0.5,5,0,7,1,400) ; put

$\parbox{5.5cm}{ \ \includegraphics[widt... ...on{\small $\rho_C$\ as a function of the \ price of the underlying asset $S$.}}$ $\parbox{5.5cm}{ \includegraphics[width... ...ion{\small $\rho_P$\ as a function of the price of the underlying asset $S$.}}$