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14 Implied volatility OUTLINE • 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 σ 2 .Ofthese ﬁve quantities, only the asset volatility σ cannot be observed directly. How do we ﬁnd a suitable value for σ?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 the σ that leads to this value. Having found σ ,wemay use the Black–Scholes formula to value other options on the same asset. A σ 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 of σ is described in Chapter 20. We focus here on the case of extracting σ from a European call option quote. An analogous treatment can be given for a put, or, alternatively, the put quote could be converted 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 = 0 131 132 Implied volatility and S = S 0 .) We thus treat the option value as a function of σ only, and, for the rest of this chapter, denote it by C(σ ).Given a quoted value C , our task is to ﬁnd the implied volatility σ that solves C(σ ) = C . Computing the implied volatility requires the solution of a nonlinear equa- tion and hence, from Chapter 13, we may use the bisection method or Newton’s method. We will ﬁnd that it is possible to exploit the special form of the nonlinear equation arising in this context. Since volatility is non-negative, only values σ ∈ [0, ∞) are of interest. Let us look at C(σ ) in the case of large or small volatility. First, as σ →∞,wesee from (8.20) that d 1 →∞and hence N(d 1 ) → 1. Similarly, from (8.21), as σ →∞, d 2 →−∞and hence N(d 2 ) → 0. It follows in (8.19) that lim σ →∞ C(σ ) = S. (14.1) Next, we look at the limit σ → 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 d 1 →∞, N(d 1 ) → 1, d 2 →∞ and N(d 2 ) → 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 d 1 →−∞, N(d 1 ) → 0, d 2 →−∞and N(d 2 ) → 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 + we have d 1 → 0, N(d 1 ) → 1 2 , d 2 → 0 and N(d 2 ) → 1 2 . Hence, C → 1 2 (S − Ee −r(T −t) ) = 0. The three cases are summarized neatly by the formula lim σ →0 + C(σ ) = max(S − Ee −r(T −t) , 0). (14.2) 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.Itfollows 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 condition holds. For further justiﬁcation of this assumption we note from Section 2.6 that if (14.3) is violated then an arbitrage opportunity exists. 14.4 Bisection and Newton 133 For later use, we will calculate the second derivative. Differentiating (10.6) gives ∂ 2 C ∂σ 2 =− S √ T − t √ 2π e − 1 2 d 2 1 d 1 ∂d 1 ∂σ . From (8.20) we have ∂d 1 ∂σ =− log(S/E) +r(T − t) σ 2 √ T − t + 1 2 √ T − t =− log(S/E) + (r − (σ 2 /2))(T − t) σ 2 √ T − t =− d 2 σ and hence ∂ 2 C ∂σ 2 = S √ T − t √ 2π e − 1 2 d 2 1 d 1 d 2 σ = d 1 d 2 σ ∂C ∂σ . (14.4) It follows from (14.4) that ∂C/∂σ is maximum over [0, ∞) at σ = σ , where σ := 2 log S/E +r(T − t) T − t , (14.5) see Exercise 14.1. Moreover, ∂ 2 C/∂σ 2 may be written in the form ∂ 2 C ∂σ 2 = T − t 4σ 3 (σ 4 − σ 4 ) ∂C ∂σ , (14.6) 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 for σ in the form F(σ ) = 0, where F(σ ) := C(σ ) − C .Toapply 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 ) F (σ n ) , (14.7) 134 Implied volatility where F (σ ) = ∂C/∂σ is given by (10.6). Because we know a lot about F,wecan exploit an expansion along the lines of (13.3) that keeps track of the remainder. Using F(σ ) = 0and the Mean Value Theorem, we have σ n+1 − σ = σ n − σ − F(σ n ) − F(σ ) F (σ n ) = σ n − σ − (σ n − σ )F (ξ n ) F (σ n ) , for some ξ n between σ n and σ . Hence, we may write σ n+1 − σ σ n − σ = 1 − F (ξ n ) F (σ n ) . (14.8) We know that F (σ ) is positive and takes its maximum at the point σ in (14.5). Hence, using the starting value σ 0 = σ we must have 0 < F (ξ 0 )<F (σ) in (14.8), so that 0 < σ 1 − σ σ 0 − σ < 1. (14.9) This means that the error in σ 1 is smaller than, but has the same sign as, the error in σ 0 .Toproceed we suppose that σ<σ . Then (14.9) tells us that σ 0 <σ 1 < σ .Now,weknow from (14.6) that F (σ ) < 0 for all σ>σ and we also know that ξ 1 in (14.8) lies between σ 1 and σ .Hence 0 < F (ξ 1 )<F (σ 1 ) and (14.8) gives 0 < σ 2 − σ σ 1 − σ < 1. Continuing this argument gives 0 < σ n+1 − σ σ n − σ < 1, for all n ≥ 0. (14.10) So the error decreases monotonically as n increases. In a similar manner, it can be shown that (14.10) holds in the case where σ> σ , see Exercise 14.3. Overall, we conclude that with the choice σ 0 = 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, σ 0 = σ is a foolproof starting value for Newton’s method on this particular nonlinear equation. This is therefore our method of choice for computing the implied volatility. Computational example Figure 14.1 illustrates the performance of Newton’s method in the case where S 0 = 3, E = 1, r = 0.05, T = 3 and t = 0. We used 14.5 Implied volatility with real data 135 0 0.5 1 1.5 2 2.2 2.4 2.6 2.8 σ C( σ) starting value iterates 0 2 4 6 8 10 12 14 16 18 10 −15 10 −10 10 −5 10 0 Error Iteration Fig. 14.1. Newton’s method for the implied volatility. Upper picture: iterates. Lower picture: errors. 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 value σ 0 = σ given by (14.5), so monotonic convergence is guaranteed. The up- per picture in Figure 14.1 shows the curve C(σ ) and superimposes the start- ing value (σ 0 , C(σ 0 )) and the subsequent iterates (σ n , C(σ n )). The lower picture plots the size of the error, |σ n − σ |.Wesee that the initial convergence is quite slow, 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 ) = 0isanassumption in the convergence theorem. If F (σ ) were exactly zero then Newton’s method would converge at a rate slower than quadratic – 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 In- ternational Financial Futures and Options Exchange (LIFFE), as reported in the 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 Ex- change. The expiry date for these options was December 2001. 136 Implied volatility Exercise price Option price 5125 475 5225 405 5325 340 5425 280 1 2 5525 226 5625 179 1 2 5725 139 5825 105 5100 5200 5300 5400 5500 5600 5700 5800 5900 0.172 0.174 0.176 0.178 0.18 0.182 0.184 0.186 0.188 0.19 0.192 Exercise price Implied volatility FTSE 100, 22 August , 2001 Current 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. Of course, if the Black–Scholes formula were valid, the volatility would be the same for 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 that start in-the-money than for options starting out-of-the-money. This behaviour is 14.7 Program of Chapter 14 and walkthrough 137 typical for data arising after the stock market crash of October 1987. Pre-crash plots 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 (Manaster and Koehler, 1982). It is also mentioned in (Kwok, 1998). More about implied volatility can be found in (Hull, 2000; Kwok, 1998), for example. The widely reported phenomenon that the implied volatility is not constant as other 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. This should 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 , the σ 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, but none 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 been taken. EXERCISES 14.1. Show that ∂C/∂σ has a unique maximum over [0, ∞) at σ = σ, where σ is deﬁned in (14.5). 14.2. Verify the identity (14.6). 14.3. Suppose σ 0 = σ. Using the fact that F (σ ) > 0 for σ<σ , conﬁrm that (14.10) holds in the case where σ>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,weusech08 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.Weuse a while loop of the form discussed for ch13,witha call to ch10 providing the required vega value. The ﬁnal solution is correct to within 6 × 10 −17 . 138 Implied volatility %CH14 Program for Chapter 14 % % Computes implied volatility for a European call %%%%%%%%%%% parameters %%%%%%%%%% r=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); %%%%%%%%%%%%%%%%%%%%%%%%%%%% %starting value sigmahat = sqrt(2*abs( (log(S/E) + r*T)/T ) ); %%%%%% Newton’s method %%%%% tol = 1e-8; sigma = sigmahat; sigmadiff = 1; k=1; kmax = 100; while (sigmadiff >= tol&k<kmax) [C, Cdelta, Cvega, P, Pdelta, Pvega] = ch10(S,E,r,sigma,tau); increment = (C-C true)/Cvega; sigma = sigma - increment; k=k+1; sigmadiff = abs(increment); end sigma Fig. 14.3. Program of Chapter 14: ch14.m. PROGRAMMING EXERCISES P14.1. Alter ch14 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 the implied volatility as the expiry time varies. Quotes The volatility is the most important and elusive quantity in the theory of derivatives. PAUL WILMOTT (Wilmott, 1998) A smiley implied volatility is the wrong number to put in the wrong formula to obtain the right price. RICCARDO REBONATO (Rebonato, 1999) It is the strong opinion of the author that most traders 14.7 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. SMITH (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. [...]... means to compute option values 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 value E(X ) = a and variance var(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... intervals 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 expected value The idea of using pseudo-random number generators to compute estimates of expected values was touched on in Chapter 4 Here we pull these two threads together and introduce the Monte Carlo approach to valuing an option As we will see in... highly relevant is (Hammersley and Handscombe, 1964), whilst a short and very accessible modern perspective is given by (Madras, 2002) Monte Carlo, pseudo-random number generation and other simulation issues are treated in detail in (Ripley, 1987) Boyle’s classic 1977 paper (Boyle, 1977), which won the Journal of Financial Economics’ All-Star Paper Award 2002, introduced Monte Carlo for option valuation... describe and implement, and, as we will see in Chapters 18 and 19, has the advantage that it is readily adapted to a range of non-European options for which no analytical formula is available In particular, the binomial method provides the simplest means to value American options In studying the method, we revisit two ideas, discrete asset price models and risk neutrality 16.2 Method The binomial method uses... estimates is O(1/ M) and hence, √ after dividing the difference by h, we expect an overall error of O(1/(h M)) for (15.8) This is unfortunate: we want to make h small to get a good derivative approximation in (15.7), but doing so forces us to take even more samples than the basic Monte Carlo option value strategy would need Another way to view the difﬁculty is to note that in order to satisfy ourselves... (M−1) i=1 (Vi − a M ) M The output provides an approximate option price a M and an approximate 95% conﬁdence interval (15.5) Computational example We now use the Monte Carlo method to value a European call option, so (S(T )) = max(S(T ) − E, 0) We will use the Black–Scholes formula (8.19) to compute the exact value and then see how 15.4 Monte Carlo for Greeks 145 Option value approximation 100.3 100.2 100.1... 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 approximating E(X ) to the problem of approximating E(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... must operate in order to replicate the option The Monte Carlo approach can be used to compute approximate partial derivatives; although it must be handled 146 Monte Carlo method with care and may prove expensive We focus here on the case of computing the time-zero delta of a European-style option with payoff (S(T )), but the principles apply generally A simple Taylor series expansion shows that the... of √ the 1/ M’ makes higher accuracy extremely costly To reduce the error to, say, 10−4 would take of the order of 108 samples, and to reduce it to 10−6 would take of the order of 1012 samples; see Exercise 15.4 ♦ 15.3 Monte Carlo for option valuation We are now in a position to use Monte Carlo for option valuation We consider a European-style option with payoff that is some function of the asset price... for the corresponding Monte Carlo option value approximation We also experimented with the corresponding algorithm that uses different pseudo-random numbers for the two options The results were much worse; for the M values used here, no digits of accuracy were recorded, and the standard deviations were around 50 000 times larger ♦ 15.5 Notes and references There are many texts that discuss general Monte . highly relevant is (Hammersley and Handscombe, 1964), whilst a short and very accessible modern perspective is given by (Madras, 2002). Monte Carlo, pseudo-random number generation and other simulation. number, not a random variable, so p is either in I or outside it, and it is meaningless to speak of the probability of p lying in I (the Bayesians, on the other hand, consider p a random variable. two threads together and introduce the Monte Carlo approach to valuing an option. As we will see in Chapter 19, this provides a powerful means to compute option values in cases where no analytical
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