Black-Scholes formula

In Pursuit of the Perfect Portfolio: The Stories, Voices, and Key Insights of the Pioneers Who Shaped the Way We Invest

by Andrew W. Lo and Stephen R. Foerster  · 16 Aug 2021  · 542pp  · 145,022 words

-known formulas in all the social sciences. In that same year, Robert C. Merton, a colleague and a friendly rival, published an extension to the Black-Scholes model, adding to the derivatives tool kit. Together, their contributions are often recognized collectively as the Black-Scholes/Merton option-pricing formula. But there’s much

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option pricing formula has been described by mathematician-author Ian Stewart as one of “17 equations that changed the world.”29 But what does the Black-Scholes formula actually tell us? This world-changing namesake formula describes the correct price of a call option, under certain assumptions. However, to fully understand this accomplishment

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underlying stock return, are readily available. The challenge to the implementation of Black-Scholes was a good estimate of volatility. Let’s unpack how the Black-Scholes formula works for pricing a call option. Suppose that IBM is trading for $130 and you could buy a call option on the stock, allowing you

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, and requires specific assumptions, such as constant volatility of the underlying security and a constant interest rate, in order for it to be solved. The Black-Scholes model makes these assumptions in order to solve the pricing puzzle in “closed form” by applying mathematical equations. Thus, a model is an abstraction from reality

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price of a call option will not be exact. Scholes explains that “a model … by definition has an error to it. So, people say the Black-Scholes model doesn’t work, but it depends on the assumptions and how good the assumptions are.”42 Derivatives technology, in contrast to the model, applies mathematical

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powered the explosive growth in the use of derivatives.60 The world of derivatives was dramatically different before and after their 1973 publication containing the Black-Scholes formula. While options on stocks existed in the seventeenth century, before the 1970s, purchasing options in the public mind was considered to be basically the same

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equities as well as indexes such as the S&P 500, the most active U.S. index option.62 Many of the assumptions in the Black-Scholes model, such as zero trading costs and no restrictions on short selling, were originally unrealistic, but the world was starting to change, and commissions were soon

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about to dramatically fall. The Black-Scholes model had an almost immediate impact, hitting the emerging options market in its technological sweet spot. The model helped the exchange to overcome the stigma of

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prices and found that some call options were overvalued by 30–40 percent. As early as 1974, Texas Instruments marketed a handheld calculator with the Black-Scholes model and “hedge ratios” to calculate the number of securities to go long versus short in a hedging portfolio. Scholes lamented, “When I asked [Texas Instruments

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and options are priced, the signals can be interpreted differently. What information is contained in derivatives? Let’s return to the key inputs of the Black-Scholes model. Again suppose we’re interested in a call option, this time on the S&P 500 index. The call option depends on five factors: the

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.”70 This wasn’t a onetime opinion for Buffett. In his 2008 letter, he commented specifically on the Black-Scholes/Merton option pricing model. “The Black-Scholes formula has approached the status of holy writ in finance.… If the formula is applied to extended time periods, however, it can produce absurd results. In

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to pricing a call option that had eluded other researchers. How did Merton figure this out? He later explained, “In addition to naming it the Black-Scholes model,39 my most significant contribution to the model was to show that if you go to shorter and shorter trading intervals, their same dynamic strategy

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assumption of “no arbitrage,” or no riskless profits, one could derive the price of a call option. This model is often referred to as the Black-Scholes model. According to Fischer Black, however, Merton contributed in a significant manner to the development of “their” option-pricing model. “Bob has contributed as much to

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). 35. Interview with authors. 36. Interview with authors. 37. Bernstein (1992). 38. Merton (2014). 39. Merton (1973c) refers to “the Black and Scholes model,” “the Black-Scholes formula,” and simply “Black-Scholes.” This was the first published article to refer to Black-Scholes. However, as Merton (1998, 326n5) described, Merton’s 1970 working

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Fund Advisors director, 189; early life of, 174–75; education of, 174, 176–81; extension of capital asset pricing model and, 180–81; extension to Black-Scholes model published by, 140–41; as financial scientist, 186–87; first options-based mutual fund created by, 187; as Long-Term Capital Management founder, 188; as

Python for Finance

by Yuxing Yan  · 24 Apr 2014  · 408pp  · 85,118 words

as less than two hours, a reader who has no clue about the option theory could price a European call option based on the famous Black-Scholes model. [ 76 ] Chapter 4 In Chapter 5, Introduction to Modules, we will introduce modules formally, and it is the first chapter of a three-chapter block

Transaction Man: The Rise of the Deal and the Decline of the American Dream

by Nicholas Lemann  · 9 Sep 2019  · 354pp  · 118,970 words

, and then priced and traded. By purchasing derivatives, one could protect oneself against the potential losses that a straightforward portfolio of assets inescapably entailed. The Black-Scholes formulas could help determine the price of a derivative in a scientific way, and also the precise mix of assets and derivatives that would most reduce

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in a portfolio. The cause they felt they were serving was reducing the beta, or volatility, of stock and bond holdings. As complicated as the Black-Scholes formula was, the next and final major breakthrough in financial economics was even more complicated. It was invented by Robert C. Merton, a colleague of Scholes

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named Kiyosi Itô (the only previous practical application of whose work was in plotting the trajectories of rockets) that allowed for “dynamic modeling” of the Black-Scholes formula, meaning that all the elements in a portfolio would be constantly recalculated and readjusted as conditions in the markets changed. By this time, the early

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) Binger, Carl bin Laden, Osama Bishop, Amy Bend Bishop, Cortlandt Black, Fischer black Americans, discrimination faced by; in housing; in policing Black Monday Black Panthers Black-Scholes formula Blackstone Blagojevich, Rod Blankenbeckler, Frank Blankfein, Lloyd blitzscaling Bloom, Ron Bloomberg, Michael Bohac, Ben bonds; as fixed-income; high-risk; in WWI Booth, David Bork

The Rise of the Quants: Marschak, Sharpe, Black, Scholes and Merton

by Colin Read  · 16 Jul 2012  · 206pp  · 70,924 words

’s solution and, for that matter, Bachelier’s derivation 70 years earlier, N(d) is the normalized cumulative probability distribution function. Alternative derivations of the Black-Scholes formula Black and Scholes had made a few implicit assumptions in their analysis. First, they assumed that the number of shares outstanding does not change before

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are greater than those predicted by Black-Scholes price estimates based on past volatility, then act as a measure of changing volatility patterns. Extensions The Black-Scholes model must be modified to overcome two of its simplifying assumptions. First, it treats European options that cannot be exercised before expiration, unlike their American counterpart

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, and volatility, but also assumptions on the timing and size of dividends.5 In fact, the Cox, Ross, and Rubinstein model is identical to the Black-Scholes model when dividends are not paid and if there are an infinite number of branch points in the limit between a given time t and t

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offer financial analysts a language to compare and describe option price dynamics. Of course, they all depend on acceptance of the underlying Black-Scholes model. Subsequent to the publication of the Black-Scholes model, but before the many variants that followed, Stephen A. Ross published in 1976 an entirely different approach to pricing called arbitrage pricing

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interests of a particularly eclectic academic, but he also had the confidence to recognize the limitations of his own theories.3 He worried that the Black-Scholes formula would be misapplied if people did not recognize that, in the real world, a stock price could jump much more than anticipated by the Markov

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would shy away from expressing their opinions. And, just as Black remained concerned about the inappropriate use of derivatives markets or the application of the Black-Scholes formula, Scholes is frequently asked to comment on excesses and malaise in modern financial markets. Black had passed away before some of these excesses in financial

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of the primary formula in derivatives markets flashes when derivatives markets capture our national attention. In 1973, on the cusp of the publication of the Black-Scholes formula and the creation of the CBOE, no one could have reasonably imagined that derivatives markets could come to affect us all in incredibly profound ways

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his making. In his defense, any investor interested in hedging risk needs tools to measure and balance risk. To do otherwise would be imprudent. The Black-Scholes formula is one of the best and most intuitive tools for financial risk management to date. The inventors of a useful tool cannot be held responsible

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Long Term Capital Management fiasco. We conclude with the great mind of Robert Merton. This page intentionally left blank 18 The Early Years While the Black-Scholes formula for options pricing remains the contribution most associated with that pair of great minds, the results were motivated behind the scenes by another most intellectually

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skeptical of the static and backward-looking characteristics of the CAPM, and were seeking to create dynamic extensions of it. Merton was convinced that the Black-Scholes formula, which was a special case of Spreckle’s derivation, must be a further special case of a more general and dynamic CAPM. Black had already

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from the dynamic CAPM perspective at a portfolio that is readjusted at each period in time, he could mimic the option returns specified by the Black-Scholes formula by combining positions on the underlying stock with borrowing at the risk-free interest rate. He then 152 The Theory 153 realized the duality of

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clever mathematical construct, Merton was able to relax some of the strongest assumptions that offended those most critical of the Black-Scholes formula. Consequently, he was able to increase the generality of the Black-Scholes formula. He demonstrated that dividends and early calls could also be accommodated by his enveloping portfolio. He managed to show that

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an option for an underlying stock that may exhibit jumps and dividends nonetheless follows the same differential equation defined by Black. Merton demonstrated that the Black-Scholes formula is accurate, even if the option may need to be continuously rebalanced at each point in time to accommodate jumps or ex-dividend repricing. 156

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the CAPM and fundamentals analysis could provide unbiased estimates of stock prices, there was a sudden and pressing need to manage risk and volatility. The Black-Scholes formula filled this void and the CBOE offered the market to do so. This need to manage risk was no longer confined to practitioners of high

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their value from underlying securities have risen geometrically. Many of them are listed on the CBOE. Most of them can be priced according to the Black-Scholes formula, perhaps with some modification. And all of them have allowed moderately sophisticated investors to reduce risk without engaging in the high contracting costs that were

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breaks the dynamic path of the derivatives into a series of steps at various points in time between the valuation date and the expiration date. Black-Scholes model – a model that can determine the price of a European call option based on the assumption that the underlying security follows a geometric Brownian motion

Against the Gods: The Remarkable Story of Risk

by Peter L. Bernstein  · 23 Aug 1996  · 415pp  · 125,089 words

gain less than $2.50. Above 52 3/4, the potential profit is infinite-at least in theory. With all the variables cranked in, the Black-Scholes model indicates that the AT&T option was worth about $2.50 in June 1995 because investors expected AT&T stock to vary within a range

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of a point difference in the case of AT&T. The market clearly expected Microsoft to be more volatile than AT&T. According to the Black-Scholes model, the market expected Microsoft to be exactly twice as volatile as AT&T over the following four months. Microsoft stock is a lot riskier than

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of Political Economy and the CBOE started trading, the hand-held electronic calculator appeared on the scene. Within six months of the publication of the Black-Scholes model, Texas Instruments placed a half-page ad in The Wall Street Journal that proclaimed, "Now you can find the Black-Scholes value using our ... calculator

Solutions Manual - a Primer for the Mathematics of Financial Engineering, Second Edition

by Dan Stefanica  · 24 Mar 2011

. Bonds. 45 2.1 Solutions to Chapter 2 Exercises. . . . . . . . . . . . . . . .. 45 2.2 Supplemental Exercises. . . . . . . . . . . . . . . . . . . . .. 57 2.3 Solutions to Supplemental Exercises. . . . . . . . . . . . . .. 58 3 Probability concepts. Black-Scholes formula. Hedging. 3.1 Solutions to Chapter 3 Exercises. . . . . . . . 3.2 Supplemental Exercises. . . . . . . . . . . . . 3.3 Solutions to Supplemental Exercises. . . . . . Greeks and . . . . . . . .. . . . . . . . .. . . . . . . . .. 63 63 82 83 4

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heat equation. In fact , u(x , t) is the fundamental solution of the heat equation , and is used in the PDE deri飞ration of the Black-Scholes formula for pricing European plain vanilla options. Also , note that u(x , t) is the same as the density function of a normal variable with mean

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) 且as probability 去 of occurring Formally, the discrete probability function P : S • [0 , 1] is $76mil + C3 B 3 十 C4 B4 . and C$ (II) Probability concepts. Black-Scholes formula. Greeks and Hedging. P(x , y) (2.17) The system (2.17) has solution B 3 = $3.25mil and B 4 = -5.75mil. We conclude

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((X 三 t+s)n(x 主 t)) P(X 三 t) e 一 α (t+ s) _,"" If ε'-'0 工 e P(X 三 t + s) Problem 6: Use the Black-Scholes formula to price both a put and a call option wit且 strike 45 expiring in six months on an underlying asset with spot price 50 and

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volatility 20% paying dividends continuously at 2% , if interest rates are constant at 6%. Solution: Input for the Black-Scholes formula: P(X 三 t) S = 50; K = 45; T - t = 0.5;σ= 0.2; q = 0.02; r = 0.06. 一讪-甲- l f(x)g(川三 (l

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the same maturity if and only if q 三飞 where r is the constant risk free rate. Use the Put-Call parity, and then use the Black-Scholes formula to prove this result. For a non-dividend-paying asset , i.e. , for q = 0 , we find that Solution: For at-th e-money options

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call (i.e. , with S = K) is C- P = e-q(T-t) N( -d1 ) δC 歹歹工 vega(C). Therefore , Volga(P)z 73 Alternatively, the Black-Scholes formulas for at-the-money options can be written as Note that θd 1 3.1. SOLUTIONS TO CHAPTER 3 EXERCISES Se-q(T-t) - K

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, i.e. , show that θ2C 一一=-=δ K2 > O. 一 Se-q(T-t) N'(d 1 ) - Ke-r(T-t) N'(d 2 ). By differentiating the Black-Scholes formula Se-q(T一忖T(d 1 ) - Ke-r(T-t) N(d 2 ) w山w Se-q(T一叫鞋一 Ke-r(T一川也)在一 e-r(T一切(d2

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) _e-r(T-t) N(d 2 ) , (坐 -252} 飞 θKθK } - σK j2作 (T - t) e-r(T-t) e二Ji>O 口 Solution: The input in the Black-Scholes formula for the Gamma of the call is S = K = 50 ， σ= 0.3 , r = 0.05 , q = O. For T = 1/24 (assuming a 30

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higher. If you have a long position in either put or call options you are essentially "10日g volatility" . (ii) The i即ut in the Black-Scholes formula for the Gamma of the call is S = K = 50 ， σ= 0.3 ， γ = q = 0. For T = 1/24 , T = 1/4, and T

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put option must satisfy the following no-arbitrage condition: K e- rT - Se- qT S P < K e- rT . (3.15) 78 CHAPTER 3. PROBABILITY. BLACK-SCHOLES FORMULA. Solution: One way to prove these bounds on the prices of European options is by using the Put-Call parity, i.e , P 十 Se- qT

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(0) 二 20 is the spot price of the underlyi吨 asset and the value P(O) = 4.9273 of the put option is obtained using the Black-Scholes formula. (ii) The Delta of the put option position is -1000N( -d l ) = -803. (Here and in the rest of the problem , the values of Delta

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衍 = 旷♂沪叫 叮 气(← J山e 一 工矿 M (1 一 ν e o- 2M - O 1) , c ap er wo hu 4EU Taylor's formula and Taylor series. ATM approximation of Black-Scholes formulas. 5.1 Solutions to Chapter 5 Exercises Problem 1: Show that the cubic Taylor approxi口lation of vfτx around 0 IS "十 Z 自 1 +王一旦+主3 2

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in the previous exercise. IPBS，r=O俨0 一凡pprox，r=O，q=O 1= 0.002604 = 0.26%. 自nd C jz and therefore From the Black-Scholes formula , we - P 自 σS1/.!.-e qT+ 2 PBS ,r=O ,q=O = 5.968592 , Usi吨 the P 十 Se- qT JZehedSM-frT) (ii) From the

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Black-Scholes formula , we find that (iii) for ATM call and put options do not satis有T the Put-Call parity: C 用 σ Sy ;1r we obtain that

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' t' θγ7 一一一: 一 vega …一 δC _ -τ一;。 。σ θC θT Denote by C (S , K , T， σ， r) the value of the call option obtained from the Black-Scholes formula. (ii) The forward and central difference approximations ~f and ~e for ~， and the central difference approximation f e for fare ~f -= C(S 十 dS， K

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= 一 6.1. SOLUTIONS TO CHAPTER 6 EXERCISES 147 Solution: (i) Usi吨 the formulas for the Greeks of a plain vanilla call option derived from the Black-Scholes formula , we find that ~ = 0.839523; f = 0.086191; vega = 0.01501; P = 7.045377; 8 = -1. 394068. (ii) The Black-Scholes value of the call

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) θC 一 θt +' 1 叮叮 θ2C δC ';a 2 S 2 一丁 +(γ - q)S一 2- ~ 8υθS - rC - 0, where C = C (S , t) is given by the Black-Scholes formula. Solution: Although direct computation can be used to show this result , we will use the version of the Black-Scholes PDE invol飞ring the Greeks

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) e-T- O. σ Se-q(T-t) _ d~ ~ v ~~G /~ ., e-~ ， 2 、/2作 (T … t) and substitute for C the value given by the Black-Scholes formula , i.e. , C - Se-旷-t) N(d 1 ) - Ke-r(T-t) N(d 2 ). Then , @+jd付 + (r …仙一 γc qSe-q(T-t) N(d

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the call option corresponding to a spot price S + dS of the underlying asset. (ii) Compute the Delta and Gamma of the call using the Black-Scholes formula , and the approximation errors I~e - ~I and Ife q. Note that these approximation errors stop improving , or even worsen , as dS becomes too small

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deteriorated very quickly. To explain this phenomenon , denote the exact value 2 of Delta by ~exaet. Note that the value of ~ is given by the Black-Scholes formula , i. e. , ~ = ~BS = e- qT N(d 1 ). This value is computed using a numerical approximation of N(d 1 ) that is accurate within 7

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0日ly know that (6.37) I~BS - ~exaetl < 10- 6 . When computing the 且nite difference approximation ~e ， we use a numerical estimation of the Black-Scholes formula to compute C(S + dS) and C (S - dS) which once again involves the numerical approximation of the cumulative density of the standard normal variable

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price fell. From this reasoning, using nifty mathematics after making some simplifying assumptions, they derived an exact formula for the price of options – the famous Black–Scholes formula – which allowed a price to be calculated using a handful of parameters: the asset price, the strike, the option maturity, the volatility and yield of

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was published in May 1973, a month after the opening of the Chicago Board Options Exchange. To compute the value of an option using the Black-Scholes formula all you needed, in addition to the strike price, the current price, and the duration of the option, was the interest rate on government bonds

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of these areas, the key was the development of mathematical methods to price risk. Almost all of these methods relied, to some extent, on the Black-Scholes formula and the bell curve. Simply by invoking the ghost of Louis Bachelier, it was possible to take much of the danger out of finance. Or

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. The price of the option ought to be the same as whatever it cost to construct an investment portfolio that achieved the same end. The Black-Scholes formula enabled the rapid pricing of options and paved the way for explosive growth in derivatives markets. Greek academics have even used it to work out

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idea that the price of the option ought to be the same as the cost of constructing a perfect hedge for the underlying asset. The Black-Scholes formula, which coincided with the computerization of trading, enabled the rapid pricing of options and paved the way for huge growth in derivatives markets.7 At

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