Steven R. Dunbar
Department of Mathematics
203 Avery Hall
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

Stochastic Processes and

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Sensitivity, Hedging and the “Greeks”

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Note: These pages are prepared with MathJax. MathJax is an open source JavaScript display engine for mathematics that works in all browsers. See http://mathjax.org for details on supported browsers, accessibility, copy-and-paste, and other features.

_______________________________________________________________________________________________ ### Rating

Mathematically Mature: may contain mathematics beyond calculus with proofs.

_______________________________________________________________________________________________ ### Section Starter Question

Recall when we ﬁrst considered options, see Options..

1. What did we intuitively predict would happen to the value of a call option if the underlying security value increased?
2. What did we intuitively predict would happen to the value of a call option as the time increased to the expiration date?
3. What did we intuitively predict would happen to the value of a call option if the volatility of the underlying security value increased?

_______________________________________________________________________________________________ ### Key Concepts

1. The sensitivity of the Black-Scholes formula to each of the variables and parameters is named, is fairly easily expressed, and has important consequences for hedging investments.
2. The sensitivity of the Black-Scholes formula (or any mathematical model) to its parameters is important for understanding the model and its utility.

__________________________________________________________________________ ### Vocabulary

1. The Delta ($\Delta$) of a ﬁnancial derivative is the rate of change of the value with respect to the value of the underlying security, in symbols
$\Delta =\frac{\partial V}{\partial S}.$

2. The Gamma ($\Gamma$) of a derivative is the sensitivity of $\Delta$ with respect to $S$, in symbols
$\Gamma =\frac{{\partial }^{2}V}{\partial {S}^{2}}.$

3. The Theta ($\Theta$) of a European claim with value function $V\left(S,t\right)$ is
$\Theta =\frac{\partial V}{\partial t}.$

4. The rho ($\rho$) of a derivative security is the rate of change of the value of the derivative security with respect to the interest rate, in symbols
$\rho =\frac{\partial V}{\partial r}.$

5. The Vega ($\Lambda$) of derivative security is the rate of change of value of the derivative with respect to the volatility of the underlying asset, in symbols
$\Lambda =\frac{\partial V}{\partial \sigma }.$

6. Hedging is the attempt to make a portfolio value immune to small changes in the underlying asset value (or its parameters).

__________________________________________________________________________ ### Mathematical Ideas

#### Sensitivity of the Black-Scholes Formula

To start the examination of each of the sensitivities, restate the Black-Scholes formula for the value of a European call option:

$\begin{array}{llll}\hfill {d}_{1}& =\frac{log\left(S∕K\right)+\left(r+{\sigma }^{2}∕2\right)\left(T-t\right)}{\sigma \sqrt{T-t}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {d}_{2}& =\frac{log\left(S∕K\right)+\left(r-{\sigma }^{2}∕2\right)\left(T-t\right)}{\sigma \sqrt{T-t}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

and then

${V}_{C}\left(S,t\right)=S\Phi \left({d}_{1}\right)-K{e}^{-r\left(T-t\right)}\Phi \left({d}_{2}\right).$

Note that ${d}_{2}={d}_{1}-\sigma \sqrt{T-t}$. Figure 1: Value of the call option at various times.

#### Delta

The Delta of a European call option is the rate of change of its value with respect to the underlying security price:

$\begin{array}{llll}\hfill \Delta & =\frac{\partial {V}_{C}}{\partial S}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\Phi \left({d}_{1}\right)+S{\Phi }^{\prime }\left({d}_{1}\right)\phantom{\rule{1em}{0ex}}\frac{\partial {d}_{1}}{\partial S}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}-Kexp\left(-r\left(T-t\right)\right){\Phi }^{\prime }\left({d}_{2}\right)\phantom{\rule{1em}{0ex}}\frac{\partial {d}_{2}}{\partial S}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\Phi \left({d}_{1}\right)+S\frac{1}{\sqrt{2\pi }}exp\left(-{d}_{1}^{2}∕2\right)\frac{1}{S\sigma \sqrt{T-t}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}-Kexp\left(-r\left(T-t\right)\right)\frac{1}{\sqrt{2\pi }}exp\left(-{d}_{2}^{2}∕2\right)\frac{1}{S\sigma \sqrt{T-t}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\Phi \left({d}_{1}\right)+S\frac{1}{\sqrt{2\pi }}exp\left(-{d}_{1}^{2}∕2\right)\frac{1}{S\sigma \sqrt{T-t}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}-Kexp\left(-r\left(T-t\right)\right)\frac{1}{\sqrt{2\pi }}exp\left(-{\left({d}_{1}-\sigma \sqrt{T-t}\right)}^{2}∕2\right)\frac{1}{S\sigma \sqrt{T-t}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\Phi \left({d}_{1}\right)+\frac{exp\left(-{d}_{1}^{2}∕2\right)}{\sqrt{2\pi }\sigma \sqrt{T-t}}×\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}\left[1-\frac{Kexp\left(-r\left(T-t\right)\right)}{S}exp\left({d}_{1}\sigma \sqrt{T-t}-{\sigma }^{2}\left(T-t\right)∕2\right)\right]\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\Phi \left({d}_{1}\right)+\frac{exp\left(-{d}_{1}^{2}∕2\right)}{\sqrt{2\pi }\sigma \sqrt{T-t}}×\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}\left[1-\frac{Kexp\left(-r\left(T-t\right)\right)}{S}exp\left(log\left(S∕K\right)+\left(r+{\sigma }^{2}∕2\right)\left(T-t\right)-{\sigma }^{2}\left(T-t\right)∕2\right)\right]\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\Phi \left({d}_{1}\right)+\frac{exp\left(-{d}_{1}^{2}∕2\right)}{\sqrt{2\pi }\sigma \sqrt{T-t}}×\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}\left[1-\frac{Kexp\left(-r\left(T-t\right)\right)}{S}exp\left(log\left(S∕K\right)+r\left(T-t\right)\right)\right]\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\Phi \left({d}_{1}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$ Figure 2: Delta of the call option at various times.

Figure 2 is a graph of Delta as a function of $S$ for several values of $t$. Note that since $0<\Phi \left({d}_{1}\right)<1$ (for all reasonable values of ${d}_{1}$), $\Delta >0$, and so the value of a European call option is always increasing as the underlying security value increases. This is precisely as we intuitively predicted when we ﬁrst considered options, see Options.. The increase in security value in $S$ is visible in Figure 1.

#### Delta Hedging

Notice that for any suﬃciently diﬀerentiable function $F\left(S\right)$

$F\left({S}_{1}\right)-F\left({S}_{2}\right)\approx \frac{dF}{dS}\cdot \left({S}_{1}-{S}_{2}\right).$

Therefore, for the Black-Scholes formula for a European call option, using our current notation $\Delta =\partial V∕\partial S$,

$V\left({S}_{1}\right)-V\left({S}_{2}\right)\approx \Delta \cdot \left({S}_{1}-{S}_{2}\right).$

Equivalently for small changes in security price from ${S}_{1}$ to ${S}_{2}$,

$V\left({S}_{1}\right)-\Delta \cdot {S}_{1}\approx V\left({S}_{2}\right)-\Delta \cdot {S}_{2}.$

In ﬁnancial language, we express this relationship as:

“Being long 1 derivative and short $\Delta$ units of the underlying asset is approximately market neutral for small changes in the asset value.”

We say that the sensitivity of the ﬁnancial derivative value with respect to the asset value, denoted $\Delta$, gives the hedge-ratio. The hedge-ratio is the number of short units of the underlying asset which combined with a call option will oﬀset immediate market risk. After a change in the asset value, $\Delta \left(S\right)$ will also change, and so we will need to dynamically adjust the hedge-ratio to keep pace with the changing asset value. Thus $\Delta \left(S\right)$ as a function of $S$ provides a dynamic strategy for hedging against risk.

We have seen this strategy before. In the derivation of Black-Scholes equation, we required that the amount of security in our portfolio, namely $\varphi \left(t\right)$, be chosen so that $\varphi \left(t\right)={V}_{S}$. (See Derivation of the Black-Scholes Equation.) The choice $\varphi \left(t\right)={V}_{S}$ gave us a risk-free portfolio that must change in the same way as a risk-free asset.

#### Gamma: The convexity factor

The Gamma ($\Gamma$) of a derivative is the sensitivity of $\Delta$ with respect to $S$:

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

The concept of Gamma is important when the hedged portfolio cannot be adjusted continuously in time according to $\Delta \left(S\left(t\right)\right)$. If Gamma is small then Delta changes very little with $S$. This means the portfolio requires only infrequent adjustments in the hedge-ratio. However, if Gamma is large, then the hedge-ratio Delta is sensitive to changes in the price of the underlying security.

According to the Black-Scholes formula,

$\Gamma =\frac{1}{S\sqrt{2\pi }\sigma \sqrt{T-t}}exp\left(-{d}_{1}^{2}∕2\right)$

Notice that $\Gamma >0$, so the call option value is always concave-up with respect to $S$. See this in Figure 1.

#### Theta: The time decay factor

The Theta ($\Theta$) of a European claim with value function $V\left(S,t\right)$ is

$\Theta =\frac{\partial V}{\partial t}.$

Note that this deﬁnition is the rate of change with respect to the real (or calendar) time, some other authors deﬁne the rate of change with respect to the time-to-expiration $T-t$, so be careful when reading.

The Theta of a claim is sometimes refereed to as the time decay of the claim. For a European call option on a non-dividend-paying stock,

$\Theta =-\frac{S\cdot \sigma }{2\sqrt{T-t}}\cdot \frac{exp\left(-{d}_{1}^{2}∕2\right)}{\sqrt{2\pi }}-rKexp\left(-r\left(T-t\right)\right)\Phi \left({d}_{2}\right).$

Note that $\Theta$ for a European call option is negative, so the value of a European call option is decreasing as a function of time, conﬁrming what we intuitively deduced before. See this in Figure 1.

Theta does not act like a hedging parameter as do Delta and Gamma. Although there is uncertainty about the future stock price, there is no uncertainty about the passage of time. It does not make sense to hedge against the passage of time on an option.

Note that the Black-Scholes partial diﬀerential equation can now be written as

$\Theta +rS\Delta +\frac{1}{2}{\sigma }^{2}{S}^{2}\Gamma =rV.$

Given the parameters $r$, and ${\sigma }^{2}$, and any 4 of $\Theta$, $\Delta$, $\Gamma$, $S$ and $V$ the remaining quantity is implicitly determined.

#### Rho: The interest rate factor

The rho ($\rho$) of a derivative security is the rate of change of the value of the derivative security with respect to the interest rate. It measures the sensitivity of the value of the derivative security to interest rates. For a European call option on a non-dividend paying stock,

$\rho =K\left(T-t\right)exp\left(-r\left(T-t\right)\right)\Phi \left({d}_{2}\right)$

so $\rho$ is always positive. An increase in the risk-free interest rate means a corresponding increase in the derivative value.

#### Vega: The volatility factor

The Vega ($\Lambda$) of a derivative security is the rate of change of value of the derivative with respect to the volatility of the underlying asset. (Some authors denote Vega by variously $\lambda$, $\kappa$, and $\sigma$; referring to Vega by the corresponding Greek letter name.) For a European call option on a non-dividend-paying stock,

$\Lambda =S\sqrt{T-t}\frac{exp\left(-{d}_{1}^{2}∕2\right)}{\sqrt{2\pi }}$

so the Vega is always positive. An increase in the volatility will lead to a corresponding increase in the call option value. These formulas implicitly assume that the price of an option with variable volatility (which we have not derived, we explicitly assumed volatility was a constant!) is the same as the price of an option with constant volatility. To a reasonable approximation this seems to be the case, for more details and references, see [2, page 316].

#### Hedging in Practice

It would be wrong to give the impression that traders continuously balance their portfolios to maintain Delta neutrality, Gamma neutrality, Vega neutrality, and so on as would be suggested by the continuous mathematical formulas presented above. In practice, transaction costs make frequent balancing expensive. Rather than trying to eliminate all risks, an option trader usually concentrates on assessing risks and deciding whether they are acceptable. Traders tend to use Delta, Gamma, and Vega measures to quantify the diﬀerent aspects of risk in their portfolios.

#### Sources

The material in this section is adapted from ‘Quantitative modeling of Derivative Securities by Marco Avellaneda, and Peter Laurence, Chapman and Hall, Boca Raton, 2000, pages 44–56,; and Options, Futures, and other Derivative Securities second edition, by John C. Hull, Prentice Hall, 1993, pages 298–318.

_______________________________________________________________________________________________ ### Algorithms, Scripts, Simulations

#### Algorithm

For given parameter values, the Black-Scholes-Merton call option “greeks” Delta and Gamma are sampled at a speciﬁed $m×1$ array of times and at a speciﬁed $1×n$ array of security prices using vectorization and broadcasting. The result can be plotted as functions of the security price as done in the text. The calculation is vectorized for an array of $S$ values and an array of $t$ values, but it is not vectorized for arrays in the parameters $K$, $r$, $T$, and $\sigma$. This approach is taken to illustrate the use of vectorization and broadcasting for eﬃcient evaluation of an array of solution values from a complicated formula.

In particular, the calculation of ${d}_{1}$ uses broadcasting, also called binary singleton expansion, recycling, single-instruction multiple data, threading, or replication.

The calculation relies on using the rules for calculation and handling of inﬁnity and NaN (Not a Number) which come from divisions by $0$, taking logarithms of $0$, and negative numbers and calculating the normal cdf at inﬁnity and negative inﬁnity. The plotting routines will not plot a NaN which accounts for the gaps or omissions.

The scripts plot Delta and Gamma as functions of $S$ for the speciﬁed $m×1$ array of times in two side-by-side subplots in a single plotting frame.

#### Scripts

Geogebra
R
1m <- 6
2n <- 61
3S0 <- 70
4S1 <- 130
5K <- 100
6r <- 0.12
7T <- 1
8sigma <- 0.1
9
10time <- seq(T, 0, length = m)
11S <- seq(S0, S1, length = n)
12
13numerd1 <- outer(((r + sigma^2/2) * (T - time)), log(S/K), "+")
14d1 <- numerd1/(sigma * sqrt(T - time))
15Delta <- pnorm(d1)
16
17factor1 <- 1/(sqrt(2 * pi) * sigma * outer(sqrt(T - time), S, "*"))
18factor2 <- exp(-d1^2/2)
19Gamma <- factor1 * factor2
20
21old.par <- par(mfrow = c(1, 2))
22matplot(S, t(Delta), type = "l")
23matplot(S, t(Gamma), type = "l")
24par(old.par)
Octave
1m = 6;
2n = 61;
3S0 = 70;
4S1 = 130;
5K = 100;
6r = 0.12;
7T = 1.0;
8sigma = 0.10;
9
10time = transpose(linspace(T, 0, m) );
11S = linspace(S0,S1,n);
12
13d1 = (log(S/K) + ( (r + sigma^2/2)*(T-time)))./(sigma*sqrt(T-time));
14Delta = normcdf(d1);
15
16factor1 = 1./bsxfun( @times, sqrt(2*pi)*sigma*sqrt(T-time), S);
17factor2 = exp(-d1.^2/2);
18Gamma = factor1 .* factor2;
19
20subplot(1,2,1)
21plot(S,Delta)
22title("Delta")
23subplot(1,2,2)
24plot(S,Gamma)
25title("Gamma")
Perl
1use PDL::NiceSlice;
2use PDL::Constants qw(PI);
3
4sub pnorm {
5    my ( $x,$sigma, $mu ) = @_; 6$sigma = 1 unless defined($sigma); 7$mu    = 0 unless defined($mu); 8 9 return 0.5 * ( 1 + erf( ($x - $mu ) / ( sqrt(2) *$sigma ) ) );
10}
11$m = 6; 12$n = 61;
13$S0 = 70; 14$S1 = 130;
15$K = 100; 16$r = 0.12;
17$T = 1.0; 18$sigma = 0.10;
19
20$time = zeroes($m)->xlinvals($T,0.0); 21$S = zeroes($n)->xlinvals($S0,$S1); 22 23$logSoverK = log($S/$K);
24$n12 = (($r + $sigma**2/2)*($T-$time)); 25$numerd1 = $logSoverK +$n12->transpose;
26$d1 =$numerd1/( ($sigma*sqrt($T-$time))->transpose); 27$Delta = pnorm($d1); 28 29$denom1 = (sqrt(2*PI)*$sigma*sqrt($T-$time))->transpose; 30$factor1 = 1/($denom1*$S);
31$factor2 = exp(-$d1**2/2);
32$Gamma =$factor1 * $factor2; 33 34# file output to use with external plotting programming 35# such as gnuplot, R, octave, etc. 36# Start gnuplot, then from gnuplot prompt 37# set key off # avoid cluttering plots 38# set multiplot layout 1,2 39# plot for [n=2:7] greeks.dat using 1:(column(n)) with lines 40# plot for [n=9:13] greeks.dat using 1:(column(n)) with lines 41## note that column 8 is NaN, because division by 0 42# unset multiplot 43 44open( F, ">greeks.dat" ) || die "cannot write:$! ";
45wcols $S,$Delta, \$Gamma, *F;
46close(F);
SciPy
1import scipy
2
3m = 6
4n = 61
5S0 = 70
6S1 = 130
7K = 100
8r = 0.12
9T = 1.0
10sigma = 0.10
11
12time = scipy.linspace(T, 0.0, m)
13S = scipy.linspace(S0, S1, n)
14
15logSoverK = scipy.log(S / K)
16n12 = (r + sigma ** 2 / 2) * (T - time)
17numerd1 = logSoverK[scipy.newaxis, :] + n12[:, scipy.newaxis]
18d1 = numerd1 / (sigma * scipy.sqrt(T - time)[:, scipy.newaxis])
19
20from scipy.stats import norm
21Delta = norm.cdf(d1)
22
23denom1 = (scipy.sqrt(2 * scipy.pi) * sigma * scipy.sqrt(T - time))[:,
24        scipy.newaxis]
25factor1 = 1 / (denom1 * S)
26factor2 = scipy.exp(-d1 ** 2 / 2)
27Gamma = factor1 * factor2
28
29# file output to use with external plotting programming
30# such as gnuplot, R, octave, etc.
31# Start gnuplot, then from gnuplot prompt
32#    set key off # avoid cluttering plots
33#    set multiplot layout 1,2
34#    plot for [n=2:7] greeks.dat using 1:(column(n)) with lines
35#    plot for [n=9:13] greeks.dat using 1:(column(n)) with lines
36##     note that column 8 is NaN, because division by 0
37#    unset multiplot
38
39scipy.savetxt(greeks.dat, scipy.column_stack((scipy.transpose(S),
40              scipy.transpose(Delta), scipy.transpose(Gamma))),
41              fmt=%4.3f)

__________________________________________________________________________ ### Problems to Work for Understanding

1. How can a short position in 1,000 call options be made Delta neutral when the Delta of each option is $0.7$?
2. Calculate the Delta of an at-the-money 6-month European call option on a non-dividend paying stock, when the risk-free interest rate is 10% per year (compounded continuously) and the stock price volatility is 25% per year.
3. Use the put-call parity relationship to derive the relationship between
1. the Delta of a European call option and the Delta of a European put option,
2. the Gamma of a European call option and the Gamma of a European put option,
3. the Vega of a European call option and a European put option, and
4. the Theta of a European call option and a European put option.
1. Derive the expression for $\Gamma$ for a European call option.
2. For a particular scripting language of your choice, modify the script to draw a graph of $\Gamma$ versus $S$ for $K=50$, $r=0.10$, $\sigma =0.25$, $T-t=0.25$.
3. For a particular scripting language of your choice, modify the script to draw a graph of $\Gamma$ versus $t$ for a call option on an at-the-money stock, with $K=50$, $r=0.10$, $\sigma =0.25$, $T-t=0.25$.
4. For a particular scripting language of your choice, modify the script to draw the graph of $\Gamma$ versus $S$ and $t$ for a European call option with $K=50$, $r=0.10$, $\sigma =0.25$, $T-t=0.25$.
5. Comparing the graph of $\Gamma$ versus $S$ and $t$ with the graph of ${V}_{C}$ versus $S$ and $t$ in of Solution the Black Scholes Equation, explain the shape and values of the $\Gamma$ graph. This only requires an understanding of calculus, not ﬁnancial concepts.
1. Derive the expression for $\Theta$ for a European call option, as given in the notes.
2. For a particular scripting language of your choice, modify the script to draw a graph of $\Theta$ versus $S$ for $K=50$, $r=0.10$, $\sigma =0.25$, $T-t=0.25$.
3. For a particular scripting language of your choice, modify the script to draw a graph of $\Theta$ versus $t$ for an at-the-money stock, with $K=50$, $r=0.10$, $\sigma =0.25$, $T=0.25$.
1. Derive the expression for $\rho$ for a European call option as given in this section.
2. For a particular scripting language of your choice, modify the script to draw a graph of $\rho$ versus $S$ for $K=50$, $r=0.10$, $\sigma =0.25$, $T-t=0.25$.
1. Derive the expression for $\Lambda$ for a European call option as given in this section.
2. For a particular scripting language of your choice, modify the script to draw a graph of $\Lambda$ versus $S$ for $K=50$, $r=0.10$, $\sigma =0.25$, $T-t=0.25$.
4. For a particular scripting language of your choice, modify the script to create a function within that language that will evaluate the call option greeks Delta and Gamma at a time and security value for given parameters.

__________________________________________________________________________ ### References

   Marco Allavenada and Peter Laurence. Quantitative Modeling of Derivative Securities. Chapman and Hall, 2000. HG 6024 A3A93 2000.

   John C. Hull. Options, Futures, and other Derivative Securities. Prentice-Hall, second edition, 1993. economics, ﬁnance, HG 6024 A3H85.

__________________________________________________________________________ 1. Stock Option Greeks. video on the meaning and interpretation of the rates of change of stock options with respect to parameters.

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