Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
Stochastic Processes and
Advanced Mathematical Finance
Sensitivity, Hedging and the “Greeks”
Mathematically Mature: may contain mathematics beyond calculus with proofs.
Recall when we ﬁrst considered options, see Options..
To start the examination of each of the sensitivities, restate the Black-Scholes formula for the value of a European call option:
Note that .
The Delta of a European call option is the rate of change of its value with respect to the underlying security price:
Figure 2 is a graph of Delta as a function of for several values of . Note that since (for all reasonable values of ), , 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 is visible in Figure 1.
Notice that for any suﬃciently diﬀerentiable function
Therefore, for the Black-Scholes formula for a European call option, using our current notation ,
Equivalently for small changes in security price from to ,
In ﬁnancial language, we express this relationship as:
“Being long 1 derivative and short 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 , 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, will also change, and so we will need to dynamically adjust the hedge-ratio to keep pace with the changing asset value. Thus as a function of 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 , be chosen so that . (See Derivation of the Black-Scholes Equation.) The choice gave us a risk-free portfolio that must change in the same way as a risk-free asset.
The Gamma () of a derivative is the sensitivity of with respect to :
The concept of Gamma is important when the hedged portfolio cannot be adjusted continuously in time according to . If Gamma is small then Delta changes very little with . 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,
Notice that , so the call option value is always concave-up with respect to . See this in Figure 1.
The Theta () of a European claim with value function is
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 , 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,
Note that 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
Given the parameters , and , and any 4 of , , , and the remaining quantity is implicitly determined.
The 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,
so is always positive. An increase in the risk-free interest rate means a corresponding increase in the derivative value.
The Vega () 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 , , and ; referring to Vega by the corresponding Greek letter name.) For a European call option on a non-dividend-paying stock,
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].
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.
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.
For given parameter values, the Black-Scholes-Merton call option “greeks” Delta and Gamma are sampled at a speciﬁed array of times and at a speciﬁed 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 values and an array of values, but it is not vectorized for arrays in the parameters , , , and . 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 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 , taking logarithms of , 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 for the speciﬁed array of times in two side-by-side subplots in a single plotting frame.
R script for Black-Scholes call option greeks Delta and Gamma..
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Last modiﬁed: Processed from LATEX source on August 17, 2016