The name indicates that σ is implied by option value data in the market.. Computing the implied volatility requires the solution of a nonlinear tion and hence, from Chapter 13, we may us
Trang 1Implied volatility
O U T L I N E
• the need for implied volatility
• properties of option value as a function of σ
• bisection and Newton for computing the implied volatility
• volatility smiles and frowns
14.1 Motivation
We now put the bisection method and Newton’s method to work on the problem
of computing the implied volatility
14.2 Implied volatility
The Black–Scholes call and put values depend on S, E, r , T − t and σ2 Of thesefive quantities, only the asset volatilityσ cannot be observed directly How do
we find a suitable value forσ? One approach is to extract the volatility from the
observed market data – given a quoted option value, and knowing S, t, E, r and T ,
find theσ that leads to this value Having found σ, we may use the Black–Scholes
formula to value other options on the same asset Aσ computed this way is known
as an implied volatility The name indicates that σ is implied by option value data
in the market
A completely different way to get hold ofσ is described in Chapter 20.
We focus here on the case of extractingσ from a European call option quote An
analogous treatment can be given for a put, or, alternatively, the put quote could beconverted into a call quote via put–call parity (8.23)
14.3 Option value as a function of volatility
We assume that the parameters E, r and T and the asset price S and time t are known (In practice, we will typically be interested in the time-zero case, t = 0
131
Trang 2132 Implied volatility
and S = S0.) We thus treat the option value as a function ofσ only, and, for the
rest of this chapter, denote it by C (σ) Given a quoted value C , our task is to findthe implied volatilityσ that solves C (σ ) = C .
Computing the implied volatility requires the solution of a nonlinear tion and hence, from Chapter 13, we may use the bisection method or Newton’smethod We will find that it is possible to exploit the special form of the nonlinearequation arising in this context
equa-Since volatility is non-negative, only valuesσ ∈ [0, ∞) are of interest Let us
look at C (σ) in the case of large or small volatility First, as σ → ∞, we see from
(8.20) that d1 → ∞ and hence N(d1) → 1 Similarly, from (8.21), as σ → ∞,
d2 → −∞ and hence N(d2) → 0 It follows in (8.19) that
lim
Next, we look at the limitσ → 0+and separate out three cases
Case 1: S − Ee −r(T −t) > 0 In this case log(S/E) + r(T − t) > 0, so as σ → 0+
we have d1→ ∞, N(d1) → 1, d2→ ∞ and N(d2) → 1 Hence, C → S −
Ee −r(T −t).
Case 2: S − Ee −r(T −t) < 0 In this case log(S/E) + r(T − t) < 0, so as σ → 0+ we
have d1→ −∞, N(d1) → 0, d2→ −∞ and N(d2) → 0 Hence, C → 0.
Case 3: S − Ee −r(T −t) = 0 In this case log(S/E) + r(T − t) = 0, so as σ → 0+
Now we recall from Chapter 10 that the derivative of C with respect to σ , that is,
the vega, is given by (10.6) In particular, we know that∂C/∂σ > 0 Since C(σ )
is continuous with a positive first derivative, we conclude that C is monotonic
increasing on [0, ∞) From (14.1) and (14.2), values of C(σ ) must lie between
max(0, S − Ee −r(T −t) ) and S It follows that C(σ) = C has a solution if and
only if
max(S − Ee −r(T −t) , 0) ≤ C < S, (14.3)and if a solution exists it is unique Henceforth, we assume that this conditionholds For further justification of this assumption we note from Section 2.6 that if(14.3) is violated then an arbitrage opportunity exists
Trang 314.4 Bisection and Newton 133For later use, we will calculate the second derivative Differentiating (10.6)gives
√
T − t
= −
log(S/E) + (r − (σ2/2))(T − t)
∂2C
∂σ2 = T − t
4σ3 ( σ4− σ4) ∂C
see Exercise 14.2 The identity (14.6) shows us that C (σ) is convex for σ < σ
and concave for σ > σ This will allow us to get a globally convergent Newton
iteration by suitably choosing the starting value
14.4 Bisection and Newton
We will write our nonlinear equation forσ in the form F (σ) = 0, where F(σ) :=
C (σ) − C To apply the bisection method, we require an interval [σ a , σ b] over
which F (σ) changes sign It follows from (14.1), (14.2) and the monotonicity of
C (σ) that this can be done by fixing K (say K = 0.05) and trying [0, K ], [K, 2K ],
[2K , 3K ],
Newton’s method takes the form
σ n+1= σ n− F (σ n )
Trang 4134 Implied volatility
where F(σ) = ∂C/∂σ is given by (10.6) Because we know a lot about F, we can
exploit an expansion along the lines of (13.3) that keeps track of the remainder
Using F (σ ) = 0 and the Mean Value Theorem, we have
We know that F(σ ) is positive and takes its maximum at the point σ in (14.5).
Hence, using the starting value σ0= σ we must have 0 < F(ξ0) < F( σ ) in
(14.8), so that
0< σ1− σ
This means that the error inσ1 is smaller than, but has the same sign as, the error
in σ0 To proceed we suppose thatσ < σ Then (14.9) tells us thatσ0< σ1<
σ Now, we know from (14.6) that F(σ) < 0 for all σ > σ and we also know
thatξ1 in (14.8) lies betweenσ1 andσ Hence 0< F(ξ1) < F(σ1) and (14.8)
So the error decreases monotonically as n increases.
In a similar manner, it can be shown that (14.10) holds in the case whereσ >
σ , see Exercise 14.3 Overall, we conclude that with the choiceσ0 = σ the error
will always decrease monotonically as n increases It follows that the error must
tend to zero, and the theory from Chapter 13 then shows that convergence must
be quadratic Hence, σ0= σ is a foolproof starting value for Newton’s method
on this particular nonlinear equation This is therefore our method of choice forcomputing the implied volatility
Computational example Figure 14.1 illustrates the performance of Newton’s
method in the case where S0= 3, E = 1, r = 0.05, T = 3 and t = 0 We used
Trang 514.5 Implied volatility with real data 135
σ = 0.15 in order to compute the Black–Scholes value for C, and then applied
Newton’s method to see how quickly σ could be found We took the starting
valueσ0= σ given by (14.5), so monotonic convergence is guaranteed The per picture in Figure 14.1 shows the curve C (σ ) and superimposes the start-
up-ing value(σ0, C(σ0)) and the subsequent iterates (σ n , C(σ n )) The lower picture
plots the size of the error,|σn − σ | We see that the initial convergence is quiteslow, but ultimately the characteristic second order behaviour emerges The slow
initial decrease in the error may be caused by the fact that F(σ ) is close to zero (Recall that F(σ ) = 0 is an assumption in the convergence theorem If F(σ )
were exactly zero then Newton’s method would converge at a rate slower thanquadratic – see (Ortega and Rheinboldt, 1970), for example) ♦
14.5 Implied volatility with real data
We now look at the implied volatility for call options traded on the London ternational Financial Futures and Options Exchange (LIFFE), as reported in the
In-Financial Times on Wednesday, 22 August 2001 The data is for the FTSE 100
index, which is an average of 100 equity shares quoted on the London Stock change The expiry date for these options was December 2001
Trang 6Current asset price
Fig 14.2 Implied volatility against exercise price for some FTSE 100 index data.
The initial asset price (on 22 August 2001) was 5420.3 We took values of r =
0.05 for the interest rate and T = 4/12 for the duration of the option Figure 14.2
shows the implied volatility computed for the eight different exercise prices Ofcourse, if the Black–Scholes formula were valid, the volatility would be the samefor each exercise price We see that in this example the implied volatility varies
by around 10% We also note that the implied volatility is higher for options thatstart in-the-money than for options starting out-of-the-money This behaviour is
Trang 714.7 Program of Chapter 14 and walkthrough 137typical for data arising after the stock market crash of October 1987 Pre-crashplots of implied volatility against exercise price would often produce a convex
smile shape; more recent data tends to produce more of a frown.
14.6 Notes and references
The convergence analysis for Newton’s method is based on the article (Manasterand Koehler, 1982) It is also mentioned in (Kwok, 1998) More about impliedvolatility can be found in (Hull, 2000; Kwok, 1998), for example
The widely reported phenomenon that the implied volatility is not constant asother parameters are varied does, of course, imply that the Black–Scholes formu-las fail to describe perfectly the option values that arise in the marketplace Thisshould be no surprise, given that the theory is based on a number of simplifying as-sumptions Despite the disparities, the Black–Scholes theory, and the insights that
it provides, continue to be regarded highly by both academics and market traders.Indeed, it is common for option values to be quoted in terms of ‘vol’; rather than
giving C , theσ such that C (σ ) = C in the Black–Scholes formula is used to
describe the value
Many attempts have been made to ‘fix’ the nonconstant volatility discrepancy
in the Black–Scholes theory A few of these have met with some success, butnone lead to the simple formulas and clean interpretations of the original work.Chapter 17 of (Hull, 2000) gives a good overview of the directions that have beentaken
E X E R C I S E S
14.1. Show that ∂C/∂σ has a unique maximum over [0, ∞) at σ = σ ,
whereσ is defined in (14.5).
14.2. Verify the identity (14.6).
14.3. Suppose σ0= σ Using the fact that F(σ) > 0 for σ < σ , confirm
that (14.10) holds in the case whereσ > σ
14.7 Program of Chapter 14 and walkthrough
In ch14, listed in Figure 14.3, we implement Newton’s method for implied volatility of a European call After setting up r,S,E,T and tau, we use ch08 from Chapter 8 to compute the call value, C_true, corresponding to a volatility of sigma_true=0.3 Our task is then to recover the volatility that produces the call value C_true We use a while loop of the form discussed for ch13, with a call to ch10 providing the required vega value The final solution is correct to within 6 × 10−17.
Trang 8r = 0.03; S = 2; E = 2; T = 3; tau = T; sigma true = 0.3;
[C true, Cdelta, P, Pdelta] = ch08(S,E,r,sigma true,tau);
while (sigmadiff >= tol & k < kmax)
[C, Cdelta, Cvega, P, Pdelta, Pvega] = ch10(S,E,r,sigma,tau);
P14.1 Alterch14 to deal with a put option
P14.2 Acquire some real option data, either electronically or via a newspaper,
and create a figure like Figure 14.2 If possible, investigate the behaviour of theimplied volatility as the expiry time varies
Quotes
The volatility is the most important and elusive quantity
in the theory of derivatives.
P A U L W I L M O T T (Wilmott, 1998)
A smiley implied volatility is the wrong number
to put in the wrong formula
to obtain the right price.
R I C C A R D O R E B O N A T O (Rebonato, 1999)
It is the strong opinion of the author that most traders
Trang 914.7 Program of Chapter 14 and walkthrough 139 will gain an improved performance by concentrating their efforts
on a better prediction of the volatility input into a Black–Scholes type model
rather than introducing other pricing techniques.
A L H S M I T H (Smith, 1986)
In those days, before the publication of the Black–Scholes option-pricing formula, warrants were often grossly mispriced Thorpe soon developed a computer program
to identify such opportunities; its deployment was so successful that,
by 1970, both Thorpe and Kassouf had abandoned academe for greener pastures.
JAMES CASE, reviewing the book (Bass, 1999) in Society for Industrial and Applied
Mathematics (SIAM) News, Jan/Feb, 2001.
Trang 11• Monte Carlo for option valuation
• Monte Carlo for Greeks
15.1 Motivation
Chapter 12 showed that valuing an option can be regarded as computing an pected value The idea of using pseudo-random number generators to computeestimates of expected values was touched on in Chapter 4 Here we pull these twothreads together and introduce the Monte Carlo approach to valuing an option
ex-As we will see in Chapter 19, this provides a powerful means to compute optionvalues in cases where no analytical formulas are available
15.2 Monte Carlo
To begin, we consider the case of a general random variable X , whose expected
valueE(X) = a and variancevar(X) = b2 are not known Suppose
• we are interested in computing an approximation to a (and possibly b), and
• we are able to take independent samples of X using a pseudo-random number generator.
We know from Table 4.2 that computing the average of a large number of samples
can give a good approximation to the mean Hence, if we let X1, X2, , X M denote independent random variables with the same distribution as X then we
Trang 12142 Monte Carlo method
to be a good approximation to a We say that an approximation toE(X) is biased if it has the same expected value as X It is easily shown that a M in(15.1) is unbiased; see Exercise 15.1 To estimate the variance, sincevar(X) :=
un-E((X − E(X))2), an obvious choice is (M
By the Central Limit Theorem,M
i=1X i behaves like anN(Ma, Mb2) random
variable, so
a M − a is approximatelyN
0, b2M
We could also say that aM − a is approximately anN(0, 1) random variable scaled
by b /√M This suggests that sampling a M for large M should give an tion to a that is correct to O (1/√M ).
approxima-We can make this argument more quantitative by using the idea of a confidenceinterval that was introduced in Section 6.5 If we had equality in (15.3) then, from(6.15),
The ratio b /√M appearing in (15.4) is often refered to as the standard error
Re-placing the unknown b by the approximation bM we see that the unknown expected
value a lies in the interval
error, we also compute the variance approximation b2M in (15.2) Having bM allows
us to compute the confidence interval (15.5) (or indeed, a confidence interval forsome other percentage, such as 99%; see Exercise 6.8)
Trang 13Fig 15.1 Monte Carlo approximations toE(e Z ), where Z ∼ N(0, 1) Crosses
are the approximations, vertical lines give computed 95% confidence intervals Horizontal dashed line is at heightE(e Z ) =√e.
There are two key features to note
(i) The size of the confidence interval shrinks like the inverse square root of the number
of samples To reduce the ‘error’ by a factor of 10 requires a hundredfold increase in the sample size This is a severe limitation that typically makes it impossible to get very high accuracy from a Monte Carlo approximation.
(ii) The size of the confidence interval is directly proportional to the standard deviation, that is the square root of the variance, of the random variable under consideration In practice, it is highly desirable to transform the problem of approximatingE(X) to the
problem of approximatingE(Y ) where Y is another random variable that has the same mean as X but a smaller variance This idea, known as variance reduction, forms a
vital part of practical Monte Carlo algorithms The two most popular approaches are covered in Chapters 21 and 22.
Computational example In Figure 15.1 we give results from a Monte Carlo
simulation of E(e Z ), where Z ∼N(0, 1) In this case we can work out
analyt-ically that E(e Z ) =√e; see Exercise 15.3 We used 13 different sample sizes,
M= 25, 26, 27, , 217 For each sample size, the picture plots the computed
mean, a M, with a cross and gives the computed 95% confidence interval as a
vertical line, often called an error bar Note that both axes have logarithmic
scales The exact mean,√
e, is represented as a dashed line We see that as M
in-creases the computed mean generally becomes more accurate and the confidence