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Journal of Information, Control and Management Systems, Vol. 3, (2005), No. 1
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OPTION PRICE SENSITIVITY FACTORS AND HEDGING
Zuzana KOZUBÍKOVÁ
Faculty of Management and Informatics
University of Žilina
e-mail: [email protected]
Abstract
In the present paper are described options and factors influencing their price.
The measures of the inluence of individual factors are introduced. These masures
are then applied to set up hedged positions.
Keywords: derivative securities, option price sensitivity, hedging, delta, vega
and gamma neutral portolio
1 INTRODUCTION
Financial instruments are refered to as derivative securities when their value is dependent upon the value of some other underlying assets. To this class of securities belong
forward contracts, financial futures, swaps, options, warrants and also many more complex securities sometimes refered as an exotic derivatives. In last decades the volume
traded on the futures and options markets have grown rapidly. Many derivatives are sold
by financial institutions in the over–the–counter market. Detailed description of most of
this derivative securities can be found for example in [9],[5], [7] or [8].
Options are often present in other assets and liabilities of the firm. Let us consider
a firm financed through an equity issue and a debt issue. The debt issue is composed of
a discount bond. When the bond issue matures the firm is sold out at its market value
and its value is distributed to the equity and bondholders. The bondholders receive their
promised payment and the stockholder receive the residual value of the firm. If the value
of the firm is less than the payment due the bondholders, the firm goes bankrupt. Similar
approach can be applied in pricing the equity of the firm. One can replicate the return
structure of the levered firm by mixing the equity of unlevered firm with the riskless
asset. Also research and development, test marketing, mine surveying and many other
investments which are undertaken prior to the full scale implementation of a project may
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Option price sensitivity factors and hedging
be analyzed as an option on that project.
2 OPTIONS
Now we briefly describe the options. We distinguish two basic types of options.
The call option gives its holder the right but not the obligation to buy the underlying
asset by a certain date for a certain price. The second side of the option contract has
the obligation to buy the contracted asset for a given price when the option is exercised.
Similarly, the put option gives its owner the right to sell the underlying asset by a certain
date for a certain price. In the description of the option is important that it is right not
obligation. So the price of the option is the value of this right.
The price stated in option contract is known as the exercise price os strike price, the
date in contract is known as the expiration date, exercise date or maturity. In connection
with ecercising we distinguish American option which can be exercised at any time up to
the expiration date and European option which can be exercised only on the expiration
date itself.
There are two sides in each option contract. One side is the owner of the option
who has bought the option and we say he has taken the long position in the option. On
the other side is the option writer who sold the option and has taken the short position in
the option. So here are four basic options positions:
1. A long position in the call option
2. A long position in the put option
3. A short position in the call option
4. A short position in the put option
Our aim in this paper is not to describe the options and option positions in details,
but show the the impact of various factors on the option price. Therefore we racall the
option pricing formula stated in [1]:
√
c = SΦ(x) − X(1 + r)−1 Φ(x − σ t)
(1)
where
µ
S
ln
Xr−1
√
x≡
σ t
¶
1 √
+ σ t,
2
(2)
Journal of Information, Control and Management Systems, Vol. 3, (2005), No. 1
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and Φ denotes the distribution function of the standardized normal distribution defined
by formula
Z x
y2
1
√ e− 2 dy.
Φ(x) ≡
2π
−∞
Symbol c denotes the call option price S the current stock price, X the strike price,
t time to maturity, r the interest rate and σ the volatility measured by standard deviation.
Knowing the call option price, we can calculate the put price from the put–call
parity equation (see for example [3] p.42)
c = p + S − Kr−1
where p denotes the put option price.
3 OPTION PRICE SENSITIVITY FACTORS
As we can see from the option pricing formula (1), here are six factors affecting
the option price. To these factors belongs the current price of the underlying asset, the
strike price, the time to expiration, the volatility of the underlying asset price, the risk
free interest rate and the expected cash flows connected with keeping the asset during
the life of the option. Now we briefly describe the measures of the influence of these
factors.
3.1
Delta
The delta of a derivative security we denote ∆ and define 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
illustrates the dependency of the option price on the underlying asset price. It indicates
the amount by which the option premium will change for one–unit change in the price
of the underlying asset.
The delta of the call option we get differentiating the option price c given by (1)
with respect to the underlying asset price. SO we have
∆ =
where x is defined in (2).
∂c
= Φ(x),
∂S
(3)
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Option price sensitivity factors and hedging
(a) Call option
(b) Put option
Figure 1 Variation of the delta with the asset price
The ∆ of the European put option we obtain from the put–call parity as
∆ = Φ(x) − 1.
The variation of the delta of a call option and a put option with the asset price is
shown in figure 1. From the graph we can observe that delta values for the call option
lies between 0 and 1 and the option premium is most sensible if the asset price is close
to the strike price.
3.2
Theta
The theta of derivative security Θ is the rate of change of the value of the derivative
security with respect to the time with all else remaining the same. For a European call
option on a non–dividend paying stock we have
Θ =
√
∂c
SΦ0 (x)σ
√
= −
− rX e−rt Φ(x − σ t).
∂t
2 t
(4)
Journal of Information, Control and Management Systems, Vol. 3, (2005), No. 1
(a) Variation of the theta as a function of the
asset price
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(b) Variation of the theta as a function of the
time to expiration for different asset prices
(S = 30 dashed line, S = 40 solid line and
S = 20 dotted line)
Figure 2 Variation of the theta of the call option
where x is defined in (2) and Φ0 (x) is the density of the standard normal distribution
given by formula
x2
1
Φ0 (x) = ϕ(x) = √ e− 2 .
2π
For the European put option we have
√
SΦ0 (x)σ
√
Θ = −
+ rX e−rt Φ(σ t − x).
2 t
Theta measures the amount at which the time value of the option decays as time
elapses.How the theta behaves as the time elapses and as the price changes is illustrated
on the figure 2.
3.3
Gamma
The gamma Γ of the option measures the change in its delta for one–unit change in
the price of the underlying asset. The hedging strategies based on the creating the delta–
neutral portfolios will hold only for small changes in the asset price. The gamma gives
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Option price sensitivity factors and hedging
(a) Variation of the gamma as a function of the
asset price
(b) Variation of the gamma as a function of
the time to expiration for different asset prices
(S = 28 dashed line, S = 30 solid line and
S = 32 dotted line)
Figure 3 Variation of the gamma of the call option
the option some convexity. If gamma is large, delta is highly sensitive on the underlying
asset price changes and it is quit risky if the delta–neutral portfolio is left unchanged for
any length of time.
To find an explicit expression for the Γ we derive the equation (3) with respect to
the price of the underlying asset. So we get
Γ =
∂∆
1
√ Φ0 (x),
=
∂S
Sσ t
(5)
where x is defined in (2).
Figure 3 shows how gamma changes as expiry approaches as well when asset price
changes. The gamma tends to zero as time approaches the maturity, but it is very large
if the current price is equal to the strike price.
Journal of Information, Control and Management Systems, Vol. 3, (2005), No. 1
3.4
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Vega
The vega of an option denoted Λ measures its sensitivity on the volatility of the
underlying asset. To express the vega we have to take derivative of the the Black–Scholes
formula (1) with respect to the volatility σ. We get
Λ =
√
∂c
= S tΦ0 (x)
∂σ
(6)
where x is defined in (2).
The vega of an option is always positive. Therefore as the volatility rises the option
premium, whether a put or call, will also rise.
3.5
Rho
The rho, ρ of an option is the rate of change in its price with respect to the change in
the interest rate. For a European call option we can evaluate the ρ as the partial derivative
of the option price given by (1) with respect to the interest rate r. We obtain
√
(7)
ρ = Xt e−rt Φ(x − σ t)
where x is defined in (2).Similarly, for the European put option we have
√
ρ = −Xt e−rt Φ(σ t − x).
4 HEDGING METHODS
4.1
Delta hedging
The first step in the hedging strategies is to make the portfolio immune to small
changes in the price of the underlying asset. Response to this changes is mesured by the
∆ of the option. Therefore this startegy is also called the delta hedging.
We recall the delta of the underlying asset is by definition equal to 1. The aim of the
delta hedging is to create the portfolio of options and underlaying asset with delta equal
to zero. Such position is refered as delta neutral. From equation (3) follows we can set
up the delta neutral portfolio as portfolio that contains:
1 call option sold
Φ(x) units of the underlying asset.
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Option price sensitivity factors and hedging
Similarly the short position in the put option can be secured by selling short Φ(x) − 1
shares of stock on each put option sold.
It is important to emphasize that the investor’s position is hedged only for a relatively short period of time. That means in practice the hedge must be periodically
rebalanced. The strategy how to make the portfolio delta–neutral was presented in [6].
4.2
Gamma neutral portfolio
We refer the portfolio of options and underlying assets as gamma neutral if its
gamma has zero value. The only way one can change the gamma of such portfolio is
adding some traded options to the portfolio.
Let us suppose we have a delta neutral portfolio with gamma equal to Γ and let
the gamma of options traded is ΓT . If we add to our portfolio ωT of traded option, the
gamma of our portfolio is then changed to ωT ΓT + Γ. Our aim is to get portfolio with
zero gamma. Putting the portfolio gamma equal zero and solving the equation
0 = ωT ΓT + Γ.
for ωT we have ωt = − ΓΓT .
In this process we have to keep in mind that including the traded option into the
portfolio changes the delta of the position. Therefore we have to rebalance the position
in the underlying asset to maintain the delta neutrality.
EXAMPLE
Suppose that a portfolio is delta neutral and its gamma is −2.500. Let the delta and
gamma of a traded call option are 0.45 and 1.25 respectively. We can make the portfolio
gamma neutral if we include a long position in 2.500
1.25 = 2.00 traded call options. However, the delta of our portfolio is then changed to the value of 2.000 × 0.45 = 0.900.
That means 0.900 units of the underlying asset must be sold to keep the delta neutrality.
4.3
Vega neutral portfolio
We refer the portfolio as vega neutral if its vega has zero value. Since the vega
of the underlying assets has zero value the vega of portfolio can be changed only by
incorporating the traded options into the portfolio. Let us denote the portfolio vegaas
Journal of Information, Control and Management Systems, Vol. 3, (2005), No. 1
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Λ and the vega of options traded as ΛT We repeate the consideration from the previous
paragraph and we get the amount of the options which makes he portfolio vega neutral.
is − ΛΛT .
Portfolio which is gamma neutral must not be in general vega neutral and vice versa.
Therefore, if we want the portfolio to be both gamma and vega neutral, we need to add
to the portfolio at least two traded option with given underlying asset.
EXAMPLE
Let us consider a delta neutral portfolio, its gamma let be −4 000 and vega is
−6 000. Let the gamma of traded option is 0.2, its vega 3.000 and delta 0.5. The port6 000
= 2 000 traded options. But
folio can be vega neutralized including a position in 3.000
the delta of portfolio increase to 0.5 × 2 000 = 1 000 and that require selling 1 unit
of underlying asset to maintain the delta neutrality. The gamma of the portfolio is then
reduced to −4 000 − 0.2 × 2 000 = −3 600.
To make the portfolio both, the gamma and vega neutral, let us suppose there is
traded a second option with gamma 0.4, vega 2.000 and delta 0.4. If ω1 and ω2 denote
the amount of options included into the portfolio, they has to be solution of the system
of equations
−4 000 + 0.2ω1 + 0.4ω2 = 0
−3 000 +
3ω1 +
2ω2 = 0.
Solving the previous system we have ω1 = −7 000 and ω2 = 13 500. It means that we
need to sell 7 000 of the first traded options and buy 13 500 of the second traded options.
The delta of the portfolio is changed to the value −7 000 × 0.5 + 13 500 × 0.4 = 1 900.
Hence 1 900 units of the underlying asset have to be sold to maintain the delta neutrality
of the portfolio.
5 CONCLUSION
We have shown the measures of the option price sensitivity. The neutralisation of
the portfolio with respect to these factors provides protection of the positions in the options. Throughout it would be wrong to conclude that in practice traders are continually
rebalancing their porfolios to keep them delta, gamma and vega neutral. The transaction
costs can make this strategy too expansive.
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Option price sensitivity factors and hedging
The delta, gamma and vega are used to quantify the different aspects of risk in the
option portfolio. Traders then consider the downside risk of the porfolio and only if it is
unacceptable they take an appropriate position in options and undelying assets.
REFERENCES
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Political Economy, 81 (May–June 1973), pp. 637–659.
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1991.
[3] Cox,J., Rubinstein,M.:Option Markets, Prentice Hall, Englewood Cliffs, New Jersy
1985.
[4] Elton,E.J., Gruber,M.J.:Modern Portfolio Theory and Investment Analysis, Wiley,
New York 1987.
[5] Hull,J.:Options, Futures and Other Derivative Securities, Prentice Hall, Englewood
Cliffs, New Jersy 2002.
[6] Kozubı́ková,Z.: Management of the Financial Risk in the Relation Between the Assets and Liabilities, Journal of Information, Control and Management Systems, Vol.
2 (2004), No.2, pp.173–180.
[7] Pinda,Ľ.:Deriváty cenných papierov, (In Slovak), IURA EDITION, Bratislava 2001.
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Acknowledgement
This paper has been supported by grant VEGA 1/2630/05.