Calibration by Maximum Entrop
General theory: PDE approach
The first attempt to extend this approach to a multiperiod setting was made by
Avellaneda et al [4] There, the asset price is postulated to follow dS, = S,u° dt+ S,o° dW, under pe (5.9) dS; = Sefte dt + 510% aw, under P (5 10)
Since measures P and P° are mutually singular, it poses a challenge as H(P|P°) equals infinity The authors in [4] propose a discrete approximation to the entropy distance, with the first-order term referred to as mơ) The key contribution of the paper is the detailed exploration of the solution to the maximization problem, which aligns with the supremum in equation (5.5).
W(S,t) = sup E, |e" T n(o?) ds + > wer™9G (Sr) (5.11) t as the solution of a Hamilton—Jacobi-Bellman (HJB) equation: ow (ert PW aw
— a te °( 5 S$ ser | + ps 35 r — — — rw ae › À¡ô(Œ — T)G;(S) ,ð(‡ — T:)G; (5.12) 5.12 for 9 > 0 and £ < 7, and where ® is the convex conjugate (or Legendre transform) of 7: đ(z) = sup|ứ?z — r(ứ2)] (5.13)
The solution to the calibration problem then consists in finding the minimum of the function F = W(S,t; 2) — 33, AiCi, which follows an equation similar to (5.12)
124 asset price days to expiration
The volatility surface, illustrated in Figure 5.1, was derived by solving equation (5.12), with volatilities constrained between 10% and 20% and an initial estimate of 14.1% The current stock price stands at $1.5, and the surface calibration is based on 25 options referenced in Avellaneda et al [4].
Note that this is entirely equivalent to the solution of (5.8)
The introduction of the HJB equation marked a significant advancement in addressing the calibration problem; however, it faced challenges due to the inherent singularity of the measures This issue was mitigated through an ad hoc discrete approximation of the entropy distance As previously noted, when the equations governing asset price evolution lead to mutually singular measures, the method still facilitates the resolution of the resulting PDE, allowing for the generation of a "volatility surface" graph, exemplified in Figure 5.1.
We should remark that the reason why controlling the volatility was attractive in
The Black-Scholes model's assumption of constant volatility fails to align with actual market observations, as evidenced by the varying implied volatilities of options on the same stock with different strike prices, resulting in the "smile" effect This phenomenon, characterized by a U-shaped graph of implied volatility against strike price, highlights the need for effective volatility management Notably, one significant achievement in this area was the successful reproduction of the smile effect Stochastic volatility models also capture this effect and other empirical characteristics of implied volatilities For an in-depth review of stochastic volatility methodologies and their properties, refer to the work of Fouque et al.
Carmona and Xu [12] proposed that the singularity problem could be addressed by controlling the drift terms 4° and pz, suggesting an expansion of the equations governing asset processes through a stochastic volatility framework In this approach, the asset's volatility Y is driven by a secondary stochastic differential equation, where the drift term can be calibrated while maintaining P ~ P° The Radon-Nikodym derivative of P with respect to P® can be derived in a closed form, resulting in a nonlinear PDE in two spatial dimensions and one time dimension, akin to equation (5.12) While the convexity of the value function guarantees the uniqueness of a minimum, the existence of a solution has not been thoroughly examined These challenges indicate the need for an alternative method to implement the entropy minimization framework, potentially through Monte Carlo simulations, which will be discussed further.
Solution by Monte Carlo simulations
The PDE approach poses significant challenges for stable numerical implementation To address these numerical difficulties, Avellaneda et al [3] proposed a Monte Carlo framework This method focuses on maximizing an entropy measure while imposing specific constraints.
In this article, we shift from enforcing the condition E Gi(Sr,) = C; through equation (5.1) to utilizing Monte Carlo simulations We will generate N scenarios, indexed by j, based on the stochastic differential equation (SDE) that describes the asset's price For each scenario w; and each benchmark contingent claim i, we will assign a discounted payoff G;; = G;(S(w;)) Importantly, we are not restricting the payoffs of claim i to a single time T; Instead, we will implement a discrete equivalent restriction to (5.1).
C; = ẹ › 0;G:(S(j)) "N 2„PiSu t=1, ,M (5.14) j=l where p; will be the probability (“weight”) of the j-th path We will be minimizing the entropy distance of these path weights with respect to some previous measure q;
In practice, the paths will be generated from our prior distribution, and so we will have q; = 1/N for all 7 In this case, the entropy distance will be
H(pla) = 3p; les (#1) = log N + Šp loạp, j=l 3 =1 (5.15)
The calibration algorithm will then consist of the following steps: e Construct N scenarios w; of the underlying asset price time evolution under our
127 prior measure P9, e build a matrix G;; of the discounted payout of each contingent claim 7 under scenario 7, e minimize 5° p;logp; subject to the conditions NC; = ye PGi for all i 1, ,M
The algorithm generates a vector of calibrated path weights, denoted as p = (p1, p2, , pn) These weights enable the calculation of the price C’ for any contingent claim by taking into account its discounted payouts.
G, under each scenario w;, and taking C" = p- Gj.
An illustrative example . . + -6- 128
This article illustrates the calibration process within the Black-Scholes framework using a specific example We consider a company with a stock price \( S \) governed by the stochastic differential equation \( dS = \mu S dt + \sigma S dW \), where \( W \) represents Brownian motion For this scenario, we assume a zero interest rate, a drift coefficient \( \mu \) of 5%, a volatility \( \sigma \) of 30%, and an initial stock price of 100 Given these parameters, the Black-Scholes model can be employed to directly price options for this stock, allowing us to use the Black-Scholes formula as a benchmark for comparing our calculated option prices.
Our goal is to price twelve options: six calls and six puts with strike prices of 95 and
To effectively calibrate our model, we require benchmark options, specifically a set of eighteen options comprising nine calls and nine puts These options will feature strike prices of 90, 100, and 110, with maturities of 60, 90, and 120 days.
120 days The prices of these options are observed in the market, and, under our assumptions, these would be given by the Black-Scholes formula
We initiated the process by creating 2000 stock price paths based on the risk-neutral dynamics described by the equation dS = rS dt + oS dW Using the specified parameters, we assigned weights to each scenario by minimizing the entropy distance, adhering to 18 constraints outlined earlier Each path was generated employing a first-order scheme.
The constrained minimization problem was formulated as S541 = Sj + Spy At + S;ỉ v At €;j (5.18), where p = r, €; ~ N(O,1), and At was set to 1/365 This process utilized an AMPL program in conjunction with R J Vanderbei’s LOQO solver, with the program code, calib.pro, available in Appendix D For those unfamiliar with the AMPL language, an introduction can be found in reference [31], while LOQO’s technical specifications are detailed in reference [67] The optimization was successfully completed in 13 iterations, requiring only 2 seconds of computational time on an Origin 200 computer equipped with a 270 MHz CPU and 2 Gigabytes of RAM.
The output of the minimization algorithm is a vector of 2000 weights, one for each scenario In order to price an asset, we would use these weights to compute the
| T K | type | BS price || equal weight | %error || Calibrated price | % error
Table 5.1 presents the pricing of various options based on 2,000 scenarios, employing equal weighting for each scenario and entropy-calibrated prices The analysis reveals the average payoffs of the asset across all scenarios, resulting in a comprehensive weighted average that effectively summarizes the potential returns.
The article discusses the "calibrated" price of an asset and contrasts it with a simple average of payoffs, where each scenario is given equal weight without calibration information Table 5.1 presents a comparison of these "calibrated" and "historical" results The first four columns of the table outline the time to maturity in days, the strike price, the type of option (call or put), and the Black-Scholes price for twelve options being analyzed Given the validity of the assumptions behind the Black-Scholes theory, these prices are regarded as accurate, with any deviations identified as errors.
In Table 5.1, the fifth column displays the predicted prices generated by the historical strategy, while the sixth column indicates the percentage error when these predicted prices are compared to the Black-Scholes prices found in column three.
Column seven of Table 5.1 displays the prices derived from the minimum entropy calibration, which closely align with the Black-Scholes prices found in column five Notably, when rounded to the nearest penny, these prices differ by no more than 2¢ from those in column four, highlighting the effectiveness of this calibration method in comparison to the historical strategy.
130 calibration method, we see that the first had an average absolute difference with respect to the Black-Scholes prices of 1.97%, whereas the second was only 0.24% off
Using just 2,000 scenarios, the minimum entropy method achieves option pricing that is within one or two cents of theoretical values, comfortably fitting within the bid-ask spread Increasing the number of scenarios enhances accuracy significantly; with 10,000 scenarios, the average absolute difference from theoretical prices drops to 0.11%, and with 20,000 scenarios, it further decreases to 0.06% In contrast, the historical strategy shows larger discrepancies, recording differences of 1.40% and 1.20% for the same scenario counts.
The calibration strategy can be effectively utilized even without knowing the specific form of risk-neutral asset dynamics For instance, if paths were generated using a drift of 5% instead of the expected 0%, the calibration process would still be straightforward The algorithm adeptly reweights each path, accommodating the "incorrect" drift In this scenario, with 2,000 generated paths, the average absolute error was impressively low at just 0.15%.
Using 10,000 or 20,000 paths resulted in absolute errors of 0.1% and 0.03%, respectively, closely aligning with previous results In summary, the calibration method demonstrates a performance improvement of approximately ten times over the historical method and remains effective even without knowledge of the risk-neutral asset dynamics.
To quantify the additional information, we can analyze the deviation of the minimum entropy weights for each path from the uniform distribution The entropy H(p|q) reaches its maximum (zero) when the optimal weights align with the uniform distribution, while it attains its minimum (log N) when one path has a calibrated probability of one and all others have zero The authors of [3] propose using d = N!~(?l9)/lo8N as a metric for this deviation from uniformity If the probability vector matches the uniform distribution, this distance measure will approximate N.
As the entropy approached log N, the measure d indicated how well the original scenarios met the constraints In simulations with 10,000 scenarios, when using uy = 5%, the distance between the calibrated path weights and the uniform distribution was d = 9950 Conversely, when paths followed the risk-neutral dynamics, d increased to 9994 This demonstrates that both scenarios provided more information than the uniform distribution, but the calibrated weights offered significantly more information when not generated under risk-neutral conditions.
In this article, we analyze the pricing of Microsoft options based on market data from February 1, 2000 We focus on a selected dataset, excluding options with maturities over 81 days and strikes above $120, while also removing specific options due to numerical challenges in the Monte Carlo entropy maximization algorithm From the remaining options, we selected three with a $105 strike as benchmarks and calibrated them against 13 others We generated 10,000 sample realizations of Microsoft's stock, using a parameter setting of p = r = 5.5%, providing a comprehensive approach to option pricing in this case study.
Once we recovered the maximum entropy weights, we proceeded to price the three options; their prices were estimated to be $1.9368, $4.2435 and $6.3274, for T,
46 and 81 days, respectively The actual market prices, from Table 5.4, are $1.9375,
The first pricing option at $4.125 is competitively priced, while the other two options are slightly higher, at $6.125 and $6.125, which are 12¢ and 20¢ above observed market prices, respectively, reflecting a deviation of about 3% Although these results are not as favorable as those derived from a lognormal diffusion model, the error margins are similar to those observed in the Black-Scholes model when applying the risk-neutral measure for path generation and calculating prices through a simple mean of discounted payoffs.
Probability of Feasibility - -. 2-2
Problem Formulation -. -. 5+58+250- 136
The question that we will address in this section is “what is the probability that the problem will be feasible, given that we generate N simulations”?
As we will see, finding this probability requires knowledge of the joint distribution of (G,(S), ,Gaz(S)), which can be quite complicated for two reasons: the function
G is generally nonlinear, and the joint distribution of (S7,, ,57,,.) is usually not available in closed form?
We will therefore concentrate on a special case: the calibration of a Vasicek [68] model for interest rates to the observed prices of zero-coupon bonds This problem,
2In this last statement, K is the number of payoff dates So far we have not restricted each benchmark asset 1 to have just one single payoff date T;
The Vasicek model is a widely recognized short-rate model, and zero-coupon bonds serve as the standard instruments for calibrating all interest rate models This article aims to establish a lower bound for the feasibility probability associated with this problem.
The equation for the short rate therefore will be dr; = a(b — re) dt+o dW, (5.22)
Note that this is a special case of the Ornstein-Uhlenbeck process We generalize the model by assuming a and 6 are themselves functions of time: dr, = ag ( by _ Tr) đt +ơ dW, (5.23)
To effectively calibrate our model, it is essential to have a number of parameters that matches the number of assets involved However, similar to the standard Vasicek model, the short rate (r) is characterized by a Gaussian distribution.
Since {r;¿} is a Gaussian process, {X;} is Gaussian too We will write ứx for its mean and o% for its variance
As for the zero-coupon bond payoff, it is given by
Now we see that G; = e*% = Y,, where Y, is a lognormally distributed random variable
We aim to determine the probability that a set of points (C1, ,Cyw) lies within the convex hull of a given set Y, which consists of N realizations of a lognormal random variable Y.
Feasibility Condition -.- +2 -2
To determine the feasibility of observed prices being within the convex hull of simulated prices, we analyze the two-dimensional case We focus on the vector of observed prices (C1, C2) and assess its position relative to the convex hull formed by the simulated prices (Y1, Y2), , (Yj, Yk) This approach allows us to establish a probability for a sufficient condition that ensures the feasibility of the observed prices within the defined regions.
These regions have an intuitive meaning: if, for a given w;, we have (Y?, ¥?) € Qu, this means that the discounted payouts of both assets are larger than C, and C2 under
3There is an abuse of notation here, as Y is taken to be the set of realizations of the process Y;
In Figure 5.4, the quadrants Q1-Q4 illustrate a scenario where only three realizations are necessary for feasibility The regions are centered around the coordinates (C, Ca), highlighting the significance of these specific realizations in the analysis.
If we establish one realization (Y1, Y2) in each of the four regions, our benchmark price vector (C1, C2) will fall within the convex hull formed by these four points.
The likelihood of satisfying the feasibility condition is likely underestimated, as demonstrated by examples where feasibility is achieved with just three simulations (Y‘, Y@, Y@)) In M dimensions, there are 2^M orthants, yet only M+1 strategically selected simulations are needed to encompass (C1, C2) within the convex hull of the Y's.
In evaluating our probability bound, it is important to note that as M increases, the expression 2” >> M + 1 suggests that the number of simulations, M + 1, becomes unrealistic for achieving feasibility in Af-dimensional scenarios.
Our algorithm calculates the probability of all orthants being filled, as well as the likelihood of fewer than 2 of them being filled Users can target a specific number of simulations, n, to assess feasibility, although only n = 2M ensures guaranteed results.
Probability of Feasibility - 22
Calculating the probability of events occurring in specific quadrants can be challenging due to the dependence between the jointly normal variables (X1, X2) This dependence complicates the computation of P{(Yi, Yo) ∈ Qi} However, a relevant lemma allows us to analyze G ∈ convY concerning the X variables, simplifying the handling of dependence issues due to their Gaussian distribution.
Lemma 5.1 Let f be a continuous, monotonically increasing function If (C,, ,Car) ws in the convex hull of ((w#', Lee YY), g=l, ,N, then (fF(Ci), ,f(Car)) as in the convex hull of (f(Y2"), , f(¥@)); N > M
Proof Since f(C) = f((Ci, -,Car)) = (f(C1), f(Car)), it is sufficient to prove this Lemma for one dimension
We note first that the strict monotonicity property is equivalent to the following property:
Let A= {x:2>2,} and B= {x: x < x2} Then, the convex hull formed by x; an f (z1), and similarly, for all z € B, f(z) < f(x)
Now, consider a point € € H, the convex hull of (1,, 22): € € H=> € € ANE € B
As we have seen in the previous paragraph, this implies that f (€) > f(z.) and ƒ() < f(x2) Therefore, we conclude that f(€) € conv (f(z1), f(z2)) O
The Lemma demonstrates that a point € within the convex hull formed by two points x1 and x2 cannot surpass or fall behind the values f(z1) or f(z2) when a continuous, monotonically increasing function f is applied Notably, the logarithm function serves as an example of such a continuous and monotonically increasing function Specifically, when we define C; as log C;, Lemma 5.1 establishes that the probability P{(C1, C2) € convY} is equivalent to the probability P{(C1, C2) € conv.X}.
To compute the lower bound on the probability of feasibility, we must first determine the method for generating our scenarios Given that the distribution of (X1, , Xar) is jointly normal, we aim to position the point (C1, C0) within the convex hull of these variables Therefore, we select the parameters a and b in equation (5.23) to ensure that jx = (C1, C0) This setup allows us to proceed with our proof.
Theorem 5.2 Assume we are calibrating a Vasicek model to the observed prices of
M zero-coupon bonds are analyzed, with specific parameters a and b selected to establish px = (Ci, sa Tự) The feasibility probability P of the Monte Carlo scheme, after N realizations, is defined by a specific lower bound.
9 ng k Tự 1 N—k—n+— -—ry pec SG) Ge) tt++ -+ne, The procedure that we will implement is based on the
146 idea that stock B has no options traded on it, but it is highly correlated with stock
A, which has some liquid options that we can calibrate our model to
We will generate path realizations for assets A and B with a specified correlation p, assigning probability weights to these paths Under the probability measure Q established by these weights, it is ensured that E2G,;(S4) equals CA, where i ranges from 1 to M, representing the stocks involved.
In evaluating benchmark options, it is essential to determine if the defined probability measure Q satisfies the condition E°G,(S7) = C,, for k = 1, ,A, where these options pertain to the pricing of stock B that we aim to analyze.
We chose stock A to be Cisco Systems; Microsoft Corporation was our stock B
We will take our initial date to be February 1, 2000; Microsoft was then trading at
To effectively price options on Microsoft while calibrating our path weights for accurate Cisco options pricing, we need a strong correlation between the returns of both companies Analysis reveals a correlation of 50.73% in their log returns for the year leading up to February 1, 2000 While this correlation appears significant, it remains uncertain whether it is adequate for pricing Microsoft options based on Cisco option prices.
Choosing the volatilities 04 and og for our simulations is complex, as various methods exist to identify volatility parameters in log-normal asset pricing models, including historical and implied volatilities However, no single method proves entirely satisfactory For instance, we estimated the historical volatilities of Cisco and Microsoft to be 46.42% and 40.75%, respectively, during the period from February 1, 1999, to February 1, 2000.
1, 1999 to February 1 2000, these were 45.85% and 43.63%, respectively
The implied volatilities of the nineteen Cisco options analyzed varied between 69% and 38% We selected a volatility of 74% to effectively price the component, as outlined in Chapter 9 of Campbell et al [11].
We analyzed 147 options with expirations up to 172 days, averaging the implied volatilities of three options at that maturity, resulting in a value of 46.1% This figure closely aligns with Cisco's historical volatilities, which were 46.42% over the past year and 45.85% over the last three months.
To accurately price options on Microsoft stock, we opted to use a calculated average of the one-year and three-month volatilities, resulting in a figure of 42% This decision was made independently of the implied volatilities for Microsoft options; however, it aligns well with the implied volatilities observed in the longest maturity options.
The choice of drifts is much less crucial, since the minimum entropy algorithm will correct for it We chose to model both as a4 = up =r = 5.5%
Table 5.3 shows the sixteen Cisco call options that we calibrated our paths to
Due to numerical challenges, we were unable to calibrate the originally intended nineteen options, leading us to consider discarding the three least liquid options to potentially simplify our calibration problem.
The effectiveness of using the path weights derived from minimizing the entropy measure for pricing Cisco options raises a critical question: how accurately can we apply these same weights to price Microsoft options? Regrettably, the findings indicate that this approach does not yield satisfactory results for Microsoft options pricing.
Table 5.4 presents 25 Microsoft options prices alongside their calibrated counterparts, revealing a significant discrepancy in performance The calibrated prices fall short of market expectations, particularly for options that are deeply out of the money, highlighting the inadequacy of the calibration method Notably, Microsoft was trading at $98.0625 on February 1, 2000.
This situation is actually not all that surprising The model described in equations
9We can gain a rough notion of the liquidity for each option from the unofficial volume figures published by the Wall Street Journal for each option contract
Table 5.3: February 1, 2000: sixteen benchmark options for Cisco Systems Cisco’s stock price was $109.9375
(5.29)-(5.30) can be re written as dSÀ = dSP =
Siu, dt + SAo, dWA SPup dt + SZứg [V1 ~ p2 dW2" + o dW?]
In our analysis, we observe that the components W4 and W/4” are orthogonal, as highlighted in equation (5.32) Notably, there exists a randomness component in Microsoft's stock that our methodology does not account for, represented by the term (,/1 — p? dWA~) Given that p is approximately 50% in this scenario, the unmodelled randomness is as impactful as the modelled components, leading to a considerable error due to its omission in our analysis.
These considerations allow us to hope that perhaps as p increases, we may get better results than those of Table 5.4 We will discuss an approach to test this hypothesis next
Table 5.4: February 1, 2000: Calibrated and observed prices for 25 Microsoft options Microsoft’s stock price was $98.0625
High correlation comparisons -. -. 151 5.43 Pricing options on an index - - - - {so 156 5.5 Conclusion 2.2.2.0 ee ee ee ee 159
To accurately price a set of options on a specific asset, it is crucial to achieve a high correlation between that asset and the index This high correlation is most feasible when the asset is a significant constituent of the index.
Large capitalization assets, such as those in the Dow Jones Global Indexes, exhibit a high correlation with the index due to their capitalization-weighted nature For instance, Du Pont Corporation represented approximately 30%-40% of the High Rating Basic Materials Index, demonstrating an 80% return correlation with the index from 1997 to 2000 This indicates that companies dominating their respective indexes can serve as effective proxies for the entire index in calibration efforts.
To test our hypothesis, we will create an artificial index that assigns significant capitalization to Cisco Systems, simulating the behavior of a "dominating company." Our calibration will focus on options traded for this stock Previously, we modeled Cisco's stock price, determining a volatility of 46.1% Detailed information about Cisco's benchmark options can be found in Table 5.3 on page 149, and Figure 5.5 illustrates Cisco's stock performance.
Oracle Corporation will be added to the index, joining Cisco to form a weighted average, with Cisco initially holding a significantly larger capitalization, thereby dominating the index.
19T he prices have been normalized to 100 to allow comparison with the Dow Jones Global Indexes
| Fm F | t 5 oT { q Ì ‘ j yk T tot a Ỷ } $ ` + | t ị qt | Ị H q Ỉ , v
The stock performance of Cisco and Oracle is illustrated in Figure 5.5, with prices normalized to 100 as of January 1, 1992 To compare our prices exclusively to Cisco, we will first establish benchmark prices by calibrating our paths to options on both Cisco and Oracle.
Table 5.5 presents the eight Oracle options used for model calibration, with an average implied volatility of 77% The drift is set at the risk-free interest rate of 5.5%, and the estimated correlation between Cisco and Oracle over the past six months is 75%.
We will simulate 10,000 sample realizations of Cisco and Oracle stock to construct a new index based solely on these two stocks in the "new economy" sector Each day, the index is determined by the stock prices of Cisco and Oracle, resulting in three simulated prices: one for Cisco, one for Oracle, and one for the index itself.
On February 1, 2000, Oracle Corporation's stock was trading at $50.50, leading to the establishment of eight benchmark options By calibrating our model to accurately reflect the prices of all 24 benchmark options, we can utilize the path weights derived from the two determining stocks to price options on the index itself These index option prices will serve as the benchmark for comparing any other assigned prices for these options.
In practice, it is often challenging to find benchmark options for every constituent of an index, especially when there are numerous components Additionally, calibrating to options on all index constituents may not be feasible Therefore, it is important to analyze the significance of a single company's weight within the index to determine if we can accurately approximate the index option prices by relying solely on the options of that particular stock.
To illustrate the concept, we can assign equal weight to each company in the index as of February 1, 2000, and generate 10,000 simulations of each company's stock price By calculating the index value daily, we can determine the correlation between Cisco's stock value and the index, which stands at 73.2%, aligning closely with our expectations Furthermore, by calibrating these 10,000 simulations, we can replicate the prices of all 24 benchmark companies Given that both Cisco and Oracle hold equal weight on the first day and exhibit a correlation of 75%, we anticipate that the correlation between the resulting index and each stock will also be approximately 75%.
With 153 path weights established, we can now proceed to price the 36 options detailed in Table 5.6, which outlines their expiration times, strike prices, and corresponding values in the first three columns.
Analyzing the impact of calibrating our paths solely to the 16 Cisco benchmark options reveals significant insights, as illustrated in Table 5.6 Despite a mean percentage error of 60%, the findings indicate that certain options, especially those deeply in the money (set at $100), are significantly overpriced Furthermore, options with expiration periods of 18 and 46 days demonstrate more favorable pricing compared to those with extended maturities.
Increasing Cisco's share in the index from 50% to 95% leads to improved results, as evidenced by the comparison of errors presented in Table 5.7.
Table 5.7 highlights that an initial capitalization of 80-90% or a correlation between the stock and the index exceeding 95% is essential for minimizing errors For instance, a 0.96% pricing error in 18-day options, achieved with a 95% weight for Cisco, results in a mere 5¢ difference between benchmark and realized prices Given that the bid-ask spread for trading options typically starts at 6.25¢ and can exceed 12.5¢, this discrepancy is not particularly concerning Additionally, a 3.78% error with Cisco at 85% capitalization translates to less than a 20¢ difference.
All things considered, however, it still seems that this calibration algorithm re- quires too large a correlation between the calibrated assets and the index to be appli-
2This may be due in part to the fact that we have fewer options for longer maturities: just one Oracle option at 137 days and three Cisco options at 172 days
According to Hull [38], the exchange-set limits on bid-ask spreads vary based on the option's price: 25¢ for options under 50¢, 50¢ for those priced between 50¢ and $10, 75¢ for options between $10 and $20, and $1 for options over $20 However, Cox and Rubinstein [16] highlight that CBOE statistics show bid-ask spreads are generally lower than 25¢.
Table 5.6: Index options: Cisco is 50% on February 1, 2000 Fourth column are prices calibrated only to Cisco options
Cisco’s initial | Correlation | Median | Median Median capitalization | Index—Cisco | error error TF | error T