¨
Orebro University
School of Business and Economics Statistics, advanced level thesis, 15hp Supervisor: Farrukh Javed
Examiner: Nicklas Pettersson Autumn 2016
Calibration of European Call options
with time varying volatility
A Bayesian and frequentist analysis
Hugo Sj¨oqvist 1992-07-19
Abstract
In this thesis we calibrated, analyzed and compared different methods to esti-mate the Black & Scholes model for options pricing with time varying volatility on the Swedish OMXS30 market. The thesis consisted of the volatility being estimated with the GARCH(1,1) and the Stochastic Volatility model, via both the frequentist and the Bayesian approaches. For the Bayesian approach, we set the priors as; normal and exponential for the GARCH(1,1) model and normal, adjusted beta and Inverse-Gamma for the parameters of the stochastic volatility. The result proved that the different methods outperformed each other depending on the length and frequent change of the stock price’s volatility and that further studies are needed to decide the most optimal method.
Acknowledgements
First and foremost, I would like to thank my supervisor Farrukh Javed for helping and guiding me throughout the whole thesis - without his knowledge I would have been lost more than once. I would also like to extend my thanks to my examiner, Nicklas Pettersson, for notably improving my thesis with his feedback. Also, a big thanks to Magnus Wiktorsson at Lund University for sharing his data at such a short notice. Finally, I would like to thank my older brother, Axel. Not just for helping me with the grammatical parts and structure of the thesis, but also for unknowingly being there and listening to my unreasonable complaining throughout this work.
Contents
1 Introduction 1
1.1 Purpose . . . 1
1.2 Outline . . . 1
2 Background 1 2.1 The Brownian motion . . . 1
2.2 The Stock price dynamics . . . 3
2.3 Option trading . . . 4
2.3.1 Black & Scholes model . . . 4
2.4 The Volatility σ . . . 5
2.4.1 Historical and implied volatility . . . 5
2.5 Volatility smiles: Going beyond the standard B&S model . . . 6
2.5.1 Volatility following the GARCH model . . . 7
2.5.2 Volatility following the SV model . . . 8
2.6 The Bayesian theory . . . 8
3 Theory & Models 10 3.1 Alternative B&S approach: Time determining volatility . . . 10
3.1.1 GARCH with a Bayesian approach . . . 10
3.2 Alternative B&S approach: Stochastic volatility . . . 12
3.2.1 SV with a Bayesian approach . . . 12
4 Data 14 5 Results and analysis 15 5.1 The GARCH(1,1) model results . . . 16
5.2 SV results . . . 16
5.3 Periodical analyses . . . 16
5.3.1 The GARCH(1,1) models . . . 17
5.3.2 The stochastic volatility models . . . 18
6 Discussion and conclusions 20 7 Appendix 21 7.1 Proof of the Black Scholes formula . . . 21
7.2 The Kalman Filter . . . 22
1
Introduction
In 1900 Louis Bachelier found that street corner traders put a price on their products based on future probabilistic events which later became his prime research interest. However, his dissertation was not received well in the academic community (Lebon 2000) and his idea remained untouched for the next five decades until the mathemati-cian Edward Thorpe applied Bachelier’s work with a mathematical way of analyzing wagering. The breakthrough was not achieved until the early 1970s when Fisher Black and Myron Scholes utilized Bachelier’s theory on pricing options which is nowadays known under the famous Black & Scholes model for option pricing. Their proposed method was further modified by Robert Merton. This work finally appeared in the Journal of Finance in 1972. In recognition for their efforts concerning pricing options they were awarded the Swedish National Bank’s Prize in Economic Sciences in Memory of Alfred Nobel in 1997 (Hull 2005, p. 281).
1.1
Purpose
The purpose of this thesis is to analyze different estimation methods for the volatility to be utilized in the Black & Scholes model. The main goal is to compare and evaluate these methods and with that contribute to the existing research in this field.
1.2
Outline
This thesis will start with a mathematical background explanation of the Black & Scholes model for options pricing including the theories behind volatility. In chapter 3, the different models that will be used to estimate the volatility will be explained, followed up by chapter 4 with a short review of the data used in the thesis. Chapter 5 will display and analyze the results of the estimated methods, both with a general estimate for the whole time period and sensitivity analyzes for different, shorter time periods. The thesis is then concluded in chapter 6 where we discuss the results and come to a conclusion.
2
Background
This section will explain and show the theory behind the Black & Scholes model and also give an explanation of stocks, options trading and finally the volatility for the GARCH(1,1) model and the stochastic volatility model.
2.1
The Brownian motion
The Black & Scholes model is based on the Brownian motion (also referred to as the Wiener process). The Brownian motion is assumed to be a Markov time stochastic
process with a mean change of 0 and a variance rate of 1.
For it to work, assume the variable z to follow a Brownian motion if the following two properties are fulfilled:
Property 1. For a small period of time ∆t the change in ∆z is
∆z = √∆t (1)
where follows a standard normal distribution, ∼ N (0, 1).
Property 2. The value for the difference (∆) in z between two short intervals of time (∆t) are independent.
Then it follows from the first property that ∆z follows a normal distribution with: E(∆z) = 0
σ∆z =
√ ∆t σ∆z2 = ∆t
Now consider the change in the value of z for a long period of T , i.e. z(T ) − z(0). It can be implemented as PN
i=1∆zi for the small intervals of the length ∆t, where N = ∆tT .
Thus, z(T ) − z(0) = N X i=1 i √ ∆t (2) where i ∼ N (0, 1) ∀i
It then follows again from the first property that z(T ) − z(0) is normally distributed, with E[z(T ) − z(0)] = 0 σ2[z(T )−z(0)]= N ∆t = T σ[z(T )−z(0)] = √ T
A drift rate (dz ) is the expected change per time unit for a stochastic process with a variance rate that equals the variance per unit time. Previously we assumed a Brow-nian motion with a drift rate of 0 and a variance rate of 1.
For a variable x the generalized Wiener process can be written as
dx = a dt + b dz (3)
where a and b are constants.
the condition that b dz = 0 we have that dx = a dt, dx dt = a
Integrating the equation with respect to time t yields x = x0+ at
where x0 is the value of x at time 0.
2.2
The Stock price dynamics
In the previous section it was assumed that the expected drift rate a was constant. However, in stock price dynamics the drift rate is changing over the course of time. It can be replaced with the expected drift divided by the stock price (i.e. the expected return) since assuming a constant attribute of this factor is more acceptable. Assume St to be the stock price at the time t, then for a constant parameter µ, the expected
drift rate can be assumed to be µSt. The expected increase in St for an interval of
time ∆t is then µS∆t. µ is the parameter of the expected rate of return for the stock. If the stock price’s volatility is 0, the model above implies that the difference for the stock price is
∆S = µSt ∆t
Then one can assume that
∆S St
∼ N (µ∆t, σp∆t) (4)
Assume that ∆t → 0, then
dS = µSt dt
dS St
= µ dt Integrating between the time 0 and T will yield
Where ST and S0 are the stock prices at time T respectively 0.
Now assume that the change of the stock price’s standard deviation in the time-period ∆t is proportional to the stock price which will lead to the following model
dS = µSt dt + σSt dz
i.e.
dS St
= µ dt + σ dz (6)
where σ is the volatility of the stock price and µ is the expected rate of return.
2.3
Option trading
Option trading is frequently used with stocks. According to (Iacus 2011), ”An option is a contract that gives the right to sell or buy a particular financial product at a given price on a predetermined date” (Iacus 2011, p. 2). In other words, an option is not necessarily something materialistic, but rather a right to sell or buy a particular product at a decided, specific date.
There are generally three main types of options: Asian, American and European. The Asian Option has a payoff that depends of the average price of the underlying asset during a specified time-period. The American Option has an attribute so one can exercise it during any time of its life (Hull 2005, p. 741). The European Option is limited so that one can only access it at the end of its life (Hull 2005, p. 748). It should be noted that the names has nothing to do with the place of the market, but rather the typology of the contract (Iacus 2011, p.7).
Example 1. Consider that a person buy 50 shares of a specific stock for a strike price of $100 in a European market. The current stock price is $76 with a maturity date of 4 months (recall that for the European market, the buyer can not sell the share before the contract expires), the price of an option to purchase one share is $10. The initial investment is then 50 × $10 = $500. With a stock price less than $100, the person will not choose to exercise the contract. If the stock price is above $100 the option can be sold. A stock price valued $105 will give a profit of ($105 - $100) × 50 = $250 on the initial $500 investment. With a starting stock price of $76, the profit would have been a bit over (105-76)/76 = 38 %.
2.3.1 Black & Scholes model
In order to price the option, one needs a mathematical formulation. One such formu-lation is the Black & Scholes (1972). Consider the following model for the expected return for a European call option - the amount of money one gets for selling an option
(the full mathematical explanation can be seen in Appendix 6.1):
CBS = E[max(ST − K, 0)]
= e−rT[E(ST)N (d1) − KN (d2)] (7)
Notations can be seen below. Here we assume that ST to be lognormal and that
E(ST) = S0erT with the standard deviation of ln(ST) = σ
√
T . Then we can rewrite the CBS as:
CBS = e−rT[S0erTN (d1) − KN (d2)] = S0N (d1) − Ke−rTN (d2) (8)
So the Black & Scholes Pricing Formulas for the European Option (Hull 2005, p. 295): CBS = S0N (d1) − Ke−rTN (d2) (9) PBS = Ke−rTN (−d2) − S0N (−d1) (10) where d1 = ln ˆ E(ST ) K + r+σ22 T σ√T and d2 = ln ˆ E(ST ) K −r+σ22 T σ√T = d1− σ √ T .
N (d1) and N (d2) are the cumulative probabilities from a standard normal
distribu-tion for d1 respectively d2. CBS is the Value of Call and PBS is the Value of Put –
i.e. the economic gain one would get by selling and buying the options respectively at the time of its maturity T with the Black Scholes model. S0 indicates the price of the
stock at the time zero, K is the ”Strike price” (or ”Exercise price”) – the price agreed on in the contract (Hull 2005, p. 6). r is the risk-free interest rate corresponding to the life of the option (Hull 2005, p. 101). Finally we have that σ is the stock-price volatility, i.e. the measure of uncertainty for the returns provided by the stock.
2.4
The Volatility σ
In the original Black & Scholes model, the stock price volatility σt is assumed to be
constant over time. Many approaches have been made to try to counter the constant volatility assumption, the most basic approaches will be explained in this section. 2.4.1 Historical and implied volatility
Generally there are two kind of ways to estimate the constant volatility. The first approach is to use historical data and calculate its variance. Have n + 1 as the number of observations, Si the stock price at the end of interval i (i = 0, 1, 2, ..., n). Let
ui = ln
Si
Si−1
. Then the standard deviation s for the stock price is given by
s = s
Pn
i=1(ui− ¯u)2
We know from previous sections that ui can be assumed to follow a normal distribution
with the standard deviation σ√t. i.e. s is simply an estimate of σ√t such that
ˆ σhist = qPn i=1(ui−¯u)2 n−1 √ t = s √ t (12)
with ˆσhist as the estimated volatility on historical data (Hull 2005, p. 286-287).
The implied volatility is more frequently used in practice by stock traders. The implied volatility is the volatility estimated by option prices observed in the market (Hull 2005, p. 300).
Assume the basic Black & Scholes model
CBS = S0N (d1) − Ke−rTN (d2) (13)
Since N (d1) ∼ N (0, 1) and N (d2) ∼ N (0, 1) we have that
N (d1) = 1 √ 2π Z d1 −∞ e−t2dt = 1 2 1 + erf d√1 2 (14) N (d2) = 1 √ 2π Z d2 −∞ e−t2dt = 1 2 1 + erf d√2 2 (15) with erf as the error function. Therefore
CBS = S0 1 2 1 + erf d√1 2 − Ke−rT1 2 1 + erf d√2 2 (16) To solve for σ might be a bit cumbersome, if at all possible. Another approach to solve it is a reject/accept search procedure. Try out different values of σ (the rest constant) until one find the most optimum value that corresponds for the market’s call option. In 1988 Brenner and Subrahmanyam proposed a formula to approximate the implied volatility (Olga et al. 2007)
σimp≈ r 2π T C S (17)
Where T is the time till maturity, C the value of call and S the underlying price.
2.5
Volatility smiles: Going beyond the standard B&S model
In the standard B&S model, volatility was considered as constant. Research (Engle 1982) has proven that assumption to be false. Volatility smiles (also known as volatility smirks) are a special graphical relationship between the call option prices (based on implied volatility) CBS, the strike price K and the time to maturity T . The implied
volatility has different values for a given time of maturity and it also changes with the strike price in a non-linear way. Sometimes it can appear as a graphically u-shaped relationship between the three variables, as can be seen in Figure 1.
Figure 1: A graphical example of the volatility smile (Sitter 2009) 2.5.1 Volatility following the GARCH model
A historical volatility type is to consider the volatility to follow a time-series. GARCH models provide one way of capturing the time series structure. GARCH stands for Generalized Autoregressive Conditional Heteroscedasticity, it was originally proposed by Engle (1982) and generalized by Bollerslev (1985) in a way to include lagged variance in the ARCH model, which will give a basic GARCH(p, q) model as
σt2 = α0+ q X i=1 αi2t−i+ p X i=1 βiσt−i2 (18)
with q as the order of the ARCH terms 2 given by a basic equation yt = x0tb + t, p
the lag length of σ2. α
0, αi and βi are constants.
The GARCH-process follows a discrete time-series while the stock price for the Black & Scholes model has been written as a continuous. Recall the previous section about the Brownian motion, one can rewrite it as a geometric Brownian motion
∆S = µS ∆t + σS√∆t (19)
Following the same steps as in the previous section, one can see that∆SS ∼ N (µ∆t, σ√∆t). The GARCH(1,1) shows that the model is sufficient in capturing the volatility (Hansen & Lunde 2005).
2.5.2 Volatility following the SV model
Another way to predict the historical volatility is to use the Stochastic Volatility (SV) model by assuming that y = (y1, y2, ..., yn)0 is a vector with an expected value of zero.
The main assumption in SV modeling is that each ythas its own independent variance
which can relax the assumption of homoscedasticity. One assumes its logarithm to follow an AR(1) process, which makes it similar but notably different from the GARCH model. It should be noted that the major difference between the GARCH(1,1) and SV model is that the SV has two noise processes (t and εt), while the GARCH(1,1) model
only has one.
yt= htt (20) ln ht = µ + φ ln ht−1+ εt, |φ| < 1 (21) t∼ N (0, 1) & εt∼ N (0, σ2) Where yt| ht∼ N (0, eht) (22) ht | ht−1, µ, φ, ση ∼ N (µ + φ(ht−1− µ), ση2) (23) h0 | µ, φ, ση ∼ N (µ, ση2/(1 − φ2)) (24)
denote θ = (µ, φ, ση) as a vector of parameters where µ is the level of log-variance, φ
as the log-variance of the persistence and ση the log-variance of the volatility. We have
h = (h0, h1, ..., hn) as unobserved and can be seen as the latent time-varying volatility
process. (Kastner 2016).
2.6
The Bayesian theory
To estimate the volatility is not only limited to the frequentist approach. One can also implement a prior and with that hopefully get better precision using the Bayesian theory. The father of Bayesian statistics was Thomas Bayes (1701-1761). He developed a theorem called Bayes’ Theorem
P (A|B) = P (B|A)P (A)
P (B) (25)
where A and B are events such that the P (A) 6= 0 and P (B) 6= 0.
For a model with the data y and the parameter θ, Bayes’ theorem can be implemented for the posterior of θ given y. Since the data y can be seen as constant, Bayes’ theorem can be written as
P (θ | y) = f (y | θ)f (θ)
For example, using a Bayesian method (Jacquier & Polson 2011) to control for the variating σ and reducing its uncertainty has been done by using the history for different underlying stock returns Ri(i indicates the stock i). σi was assumed to have a conjugate
prior with τ as a location parameter and v as dispersion.
p(σi | Ri) ∝ 1 σ2 i vi+v+22 +1 exp −vis 2 i + vτ 2σ2 i (27) with vi = Ti − 1, Ti indicates the sample size for the returns of asset i. vis2i is seen
as the sum of squared deviations of the Ri for the expected value of their samples.
Changing σ in the Black & Scholes model directly to σi would be inappropriate, as
Jacquier and Polso points out, due to the correlation between the different stock re-turns. They solved it by using the averages for the draws from the Monte Carlo method.
3
Theory & Models
As Abken and Nandi mentions (Abken & Nandi 1996) there is no clear answer to which method is the most optimal one. The rest of this section will present the models that will be used in this thesis.
3.1
Alternative B&S approach: Time determining volatility
As mentioned before, the Black & Scholes model is designed with a constant volatil-ity in mind. That makes it not possible to estimate a time-dependent volatilvolatil-ity and ”plug it in” into the formula. The B&S volatility is assumed to be constant and the GARCH is assumed to be dynamic, the GARCH volatility will most likely converge to an unconditional volatility due to it being a mean-reverting process.
There are several ways to solve it, one of them is to take the volatility as an average value over the time of maturity with a GARCH(1,1) background such as
σGARCH2 = 1 T Z T 0 V (t)dt = T X t=1 σ2t,GARCH (28) where σ2
t,GARCHis the volatility following a GARCH process at time t, V (t) the estimate
of the variance rate and T the time to maturity expressed in days. 3.1.1 GARCH with a Bayesian approach
To be able to estimate σ2
t,GARCH with a Bayesian method, recall the GARCH(1,1)
model being σt2 = α0+ α12t−1+ β1σt−12 (29) = α0+ α1ε2t−1σ 2 t−1+ β1σt−12 (30) with α1 ≥ 0, β1 ≥ 0 and t = εt p σ2
t, εtbeing i.i.d. and follows a Student-t disturbance
(Ardia & Hoogerheide 2010).
Ardia and Hoogerheide show how a GARCH(1,1) model with Student-t innovations for εt can be seen as
εt= t v − 2 v λtσ 2 t 1/2 t = 1, 2, ..., T (31) t ∼ N (0, 1) (iid) (32) λt∼ IG v 2, v 2 (iid) (33) σ2t = α0+ α12t−1+ β1σ2t−1 (34)
α0 > 0, v is the degrees of freedom (v > 2) and IG refers to the Inverse Gamma
Now define the vectors as = (1, ..., T)0, λ = (λ1, ..., λT)0, α = (α0, α1)0 and ψ =
(α, β, v). Consider a diagonal (T × T ) matrix Σ, as
Σ(ψ, λ) = diag λt v − 2 v σ 2 t(α, β) T t=1 ! (35)
where σ2t(α, β) = α0 + 2t−1+ β1σt−12 (α, β). The likelihood for (ψ, λ) can be expressed
as L(ψ, λ | ) ∝ (detΣ)−1/2exp −1 2 0 Σ−1 (36) Using Bayes’ rule to get the posterior density
p(ψ, λ | ) = L(ψ, λ | )p(ψ, λ)
R L(ψ, λ | )p(ψ, λ)dψdλ (37)
For the GARCH parameters α and β the priors follow truncated normal distributions. p(α) ∝ N2(α | µα, Σα) I[α ∈ R2+] (38)
p(β) ∝ N1(β | µβ, Σβ) I[β ∈ R+] (39)
with µα, µβ and σα, σβ as the hyperparameters, I[·] the indicator function and N2 and
N1 as normal densities.
Due to that λtis both identically distributed and independent from the IG distribution,
it will yield that the prior distribution for λ conditional on v is
p(λ | v) =v 2 T v2 h Γv 2 i−T XT t=1 λt !−v2−1 × exp " −1 2 T X t=1 v λt # (40)
Following Philippe Deschamps (Deschamps 2006) we assume the distribution to be translated as exponential with the parameters θ > 0 and δ ≥ 2
p(v) = θ exp[−θ(v − δ)] 1[v > δ] (41) We can form the joint prior distribution by assuming prior independence between the parameters. But because of the GARCH(1,1) variance equation, the joint posterior and the full conditional densities can not be expressed in a closed form. Simulations are needed to be able to estimate the parameters. Due to no conjugate prior, the basic Gibbs sampler is ruled out and instead the Markov Chain Monte Carlo (MCMC) will be used as a simulator to approximate the posterior density.
3.2
Alternative B&S approach: Stochastic volatility
Stochastic Volatility (SV) makes the assumption that one will not be able to perfectly predict the future level of volatility, by using the information available today. The volatility in the model is driven by a random source that is different from the asso-ciation driving the asset returns, even if they might be correlated with each other. The stochastic volatility process is vulnerable to two scenarios: the risk of the asset price and the risk of the volatility. Contrary to the Black & Scholes model, a risk-free portfolio cannot be created. The option price for a SV model can change without any change in the asset price, due to the option price being driven by two random variables: the asset price and its volatility. (Abken & Nandi 1996)
For the frequentist SV in this thesis, the values for the parameters will be simulated using the Kalman Filter and smoothened, afterwards be estimated with the maximum likelihood estimation method (see Appendix 6.2 for a more thorough explanation). Noting the similarities to time-series models, the SV will also be used with the slightly adjusted model σ2SV = 1 T Z T 0 V (t)dt = T X t=1 σ2t,SV (42) where σ2
t,SV is the volatility following a SV process at time t, V (t) the estimate of the
variance rate and T the time to maturity expressed in days. 3.2.1 SV with a Bayesian approach
The prior information will be seen as independent, i.e. p(θ) = p(µ)p(φ)p(ση). µ will be
assumed to have a normal prior, µ ∼ N (bµ, Bµ), it has been shown in previous studies
Kastner (2016) that the values of bµ and Bµ have shown to have a rather influential
impact with empirical data. So
p(µ) = √ 1 2πσ2 exp (µ − bµ)2 2Bµ (43) The parameter φ [φ ∈ (−1, 1)] will have (φ + 1)/2 ∼ Beta(a0, b0), i.e.
p(φ) = 1 2Beta(a0, b0) 1 + φ 2 a0−1 1 − φ 2 b0−1 (44) a0 and b0 can be seen as positive hyperparameters and Beta(·) as a beta function. The
expected value for φ is then E(φ) = 2a0
a0+b0− 1 with the variance V (φ) =
4a0b0
(a0+b0)2(a0+b0+1).
The choice of a0 and b0 need to be carefully considered with small datasets, but the
data used has a large number of dates ( 500) and will therefore assume a default value based on a previous study (Kim et al. 1998) where a0 = 20 and b0 = 1.5.
For ση, the volatility of the logvariance, a prior such as ση ∼ Bση × χ
2
1 = G(12, 1 2Bση)
is motivated by Fr¨uhwirth-Schnatter and Wagner (Fr¨uhwirth-Schnatter & Wagner 2009).This prior is not a conjugate prior, but it does not bound ση away from a zero,
like the more common Inverse-Gamma prior for ση.
The prior distribution for ση will now be
p(ση) = 1 2 1 2Bση Γ(12) σ 1 2−1 η e− 1 2Bσηση (45)
with a hyperparameter Bση that previously has shown evidence of not having a notable
impact on empirical applications as long as it is not too close to zero. The joint posterior distribution will now be
p(µ) × p(φ) × p(ση) = 1 √ 2πσ2 exp (µ − bµ)2 2Bµ × 1 2Beta(a0, b0) 1 + φ 2 a0−1 1 − φ 2 b0−1 × 1 2 1 2Bση Γ(12) σ 1 2−1 η e− 1 2Bσηση (46)
This does not seem to follow any identifiable distribution. Therefore, one needs to utilize the MCMC sampling from the joint posterior densities of the desired random variables, i.e. the latent log-variances h and the parameter vector θ.
4
Data
To estimate the models the statistical software used was RStudio with R v.3.0.3, the R-code can be supplied on request. The data used is the daily buy/sell closing prices for the Swedish OMXS30 from the dates 04/01/2010 to 31/08/2010. The variables included was the date of the event, the date of maturity, an indication if the event was a put or call option, the strike, bid and ask price and finally the close price. The data contained 11,197 observations.
The OMXS30 call option data was then paired with stock data from 2010-01-04 to 2012-01-26.
Table 1. Description of data
Variable Mean Std. dev. Min Max Skewness Kurtosis
Strike price 1,024 79.3 420 1,360 0.496 0.506
Close price 1,007 40.9 923.4 1,075 -0.238 -1.13
Time until maturity 0.492 0.428 0.004 2.66 1.22 1.923
Bid price 45.7 38.1 0.01 525.8 1.19 4.46
Ask price 47.6 38.9 0.03 530.8 1.17 4.29
The evolution of stock prices can be seen in figure 2.
5
Results and analysis
To be able to evaluate and compare the different methods, a measurement is needed. The bid and ask prices will be functioning as a measurement to justify if the call price is acceptable. If the call price lies below the bid price, it will be named ”under bounds” and will be seen as an underevaluation of the call price. If the call price lies above the ask price, it will be ”over bounds” and an overevaluation of the call price. An estimated call price which lies between the bid and ask price is ”in bounds” and will be an acceptable price estimate.
Figure 3: The estimated volatilities for the OMXS30 market 04/01/2010 to 26/01/2012 The estimated volatilities can be seen in figure 3, where the volatilities have been plot-ted against the dates of the gathered stock data. From figure 3, the GARCH(1,1) with a Bayeisan prior and the frequentist GARCH(1,1) estimate follows a close pattern, even if the Bayesian GARCH shows a tendency of overestimating compared to the fre-quentist GARCH. The stochastic volatilities seems to follow a less sensitive patterns, where their changes do not spike as much as the GARCH(1,1) models. The Bayesian stochastic volatility shares a pattern that is similar to the GARCH models, while the frequentist SV have a rougher pattern.
5.1
The GARCH(1,1) model results
The frequentist GARCH(1,1) estimated volatility have less than 1 percentage more in bounds than the Bayesian GARCH(1,1), as can be seen in table 2. With the Bayesian GARCH overestimating the bounds with approximately 67 %, it is higher than the frequentist GARCH which shows an overestimate of 61 %.
Table 2. Bounds for GARCH(1,1) (%)
Approach Under In Over
Frequentist 29.5 9.34 61.1
Bayesian 23.9 8.95 67.1
The frequentist GARCH seems to underestimate a higher percentage than the Bayesian GARCH, with 29.5 % compared to 23.9 %.
5.2
SV results
Table 3 shows that the overall estimation of the stochastic volatility had a higher out of bounds than the GARCH volatility models. The frequentist stochastic volatility has an overall in bounds of 6.89 %, compared to the Bayesian SV with 7.02 - a small difference, but a difference nonetheless. The stochastic volatility shows evidence of an even spread with just a few percentage decimals difference between the under and over bounds estimates.
Table 3. Bounds for SV (%)
Approach Under In Over
Frequentist 42.5 6.89 50.6
Bayesian 41.9 7.02 51.1
The over and under bounds for stochastic volatility differs approximately overall with 8-9 %, while for the GARCH(1,1) model the percentage difference between the under and over bounds for the frequentist and Bayesian methods are 31.6 respectively 43.2 %. i.e. the under and over bounds for the overall stochastic volatility with both Bayesian and frequentist methods show signs of having an even spread between under and over estimation of the bounds.
5.3
Periodical analyses
Due to the results in the previous subsection, a sensitivity analysis was performed by sorting the dates into three time periods, each with a general attribute as can been seen in figure 4. The Blue period covers the options that have a lifetime which ranges from 2010-01-18 to 2010-04-15 and have the attributes of being a short time period
with a rather calm volatility, i.e. the volatility does not suffer from any sudden and significant changes. The Black period ranges from 2010-04-10 to 2010-09-18 and have the attribute of having a notable spike in the middle of the options’ lifetime. The Green period covers 2010-08-01 to 2011-06-01 in which the attributes are a normal volatility pattern but also that it covers a long time period, compared to the other two.
Figure 4: The periods for the estimated volatilities for the OMXS30 market
5.3.1 The GARCH(1,1) models
For the frequentist GARCH(1,1) model in table 4, the Blue period appears to show signs of the volatility underestimating the bounds, although the percentage of in bounds are notably higher with approximately 25 % compared to the overall’s 9 %. On the other hand, the Black period has a similar in bounds percentage as the overall but has a heavy overestimation with 78.8 %.
Table 4. Bounds for freq. GARCH(1,1) (%)
Period Under In Over
Overall 29.5 9.34 61.1
Blue 46.0 24.8 29.2
Black 10.6 10.5 78.8
The Green period also has a low in bounds percentage, however instead of a high overestimation it has a high underestimation.
The Bayesian GARCH(1,1) model follows the opposite pattern of its frequentist coun-terpart for the Blue period. In table 5 it shows that the GARCH methods for the Blue period have about the same in bounds percentage, but the Bayesian GARCH seems to overestimate with approximately 46 %, while the frequentist GARCH underestimates with 46 %.
Table 5. Bounds for Bayesian GARCH(1,1) (%)
Period Under In Over
Overall 23.9 8.95 67.1
Blue 28.2 26.3 45.5
Black 9.42 11.6 77.0
Green 54.3 9.63 36.0
For the Black period the Bayesian and frequentist GARCH follow a similar pattern with approximately 10 % in bounds and a heavy overestimation with at least 75 %. The Green period also has a similarity with the in bounds percentage for the Bayesian and frequentist GARCH model with approximately 10 %, but the Bayesian GARCH does not overestimate the model as heavily as the frequentist GARCH with 54 % compared to 62 %.
5.3.2 The stochastic volatility models
Overall the stochastic volatility models show a rather even distribution between over-and underestimating the bounds for both the frequentist over-and Bayesian estimation meth-ods. This can be seen in table 6 and also table 3 and 7. The frequentist stochastic volatility increases the in bounds percentage during the Blue period from approxi-mately 7 to 12 percentage. However, the overestimation in the Blue period is as high as 83.2 percentage, which is the highest out of bounds estimation in this whole analysis.
Table 6. Bounds for freq. SV (%)
Period Under In Over
Overall 42.5 6.89 50.6
Blue 4.43 12.4 83.2
Black 23.5 12.9 63.7
Green 69.1 10.6 20.3
The Black period for frequentist SV has a slightly higher in bounds percentage than the Blue period with just 0.5 %. The overestimation is smaller with 63.7 %, compared
to the previous 83.2 %. The Green period shows a large underestimation with 69.1 % instead of the Blue and Black’s significant overestimations of bounds. The Green period also has a smaller in bounds percentage than the previous periods but a larger percentage than the overall.
The Bayesian stochastic volatility’s different periods can be seen in table 7. Like its frequentist counterpart, the pattern for the overall bounds are similar. The Bayesian SV heavily underestimates the bounds for the Blue period, while the frequentist SV instead overestimates them notably. For the Black period, the Bayesian and frequentist SV both overestimates the over bounds with 70.3 respectively 63.7 %.
Table 7. Bounds for Bayesian SV (%)
Period Under In Over
Overall 41.9 7.02 51.1
Blue 78.4 14.2 7.34
Black 18.3 11.4 70.3
Green 76.3 7.26 16.5
The same can be said for the Green period, in which the Bayesian underestimates it with 76.3 percentage while the frequentist SV also underestimates it with the slightly smaller 69.1 %. The overall Bayesian stochastic volatility’s in bounds estimation was the smallest with 7.02 %, the Green period had a slightly higher in bounds with 0.26 % while the Blue and Black period had a higher in bounds percentage with 14.2 re-spectively 11.4 %.
Overall the SV shows an even distribution of over and under the bounds estimation compared to the GARCH estimates, which has a tendency to over estimate the call options. However, the two GARCH models perform better during the Blue period and overall regarding the estimated in bounds percentage than the stochastic volatility. The frequentist stochastic volatility has a lower percentage of in bounds overall and especially in the Blue period. Nonetheless, it achieved the highest percentage of in bounds during the Green and Black period.
6
Discussion and conclusions
In this thesis we discuss different methods to estimate the volatility to be utilized in the Black & Scholes model for options pricing for the Swedish OMXS30 market. By taking the average of the volatility estimates we still fulfill the constant volatility assumption of the Black & Scholes model meanwhile controlling for some of the time variation in the volatility of the stock market. Since we do not know the true risk-free rate and dividend yield for each observation, the results could vary from what they originally should have been. The limited time period could also be a possible risk factor for the results.
Sensitivity analyses show that options that have a lifetime during a spike in the volatil-ity have a significantly lower percentage of in bounds, compared to options which have a lifetime during a calm volatility period. This strengthens the theory above and the recommendation of using models that can take spikes or jumps into consideration. Based on the frequentist stochastic volatility’s results compared to the other meth-ods, it might indicate that an estimation method using a smoothing attribute can lead to a better volatility average estimate by smoothing out the spikes and sudden changes. This theory is strengthened by graphically analyzing the estimated volatilities and comparing them to the bounds results - the more sensitive an estimation method is to a sudden change, the worse the in bounds estimate become for periods with a long or spiked attribute. Estimation methods that are not sensitive to sudden changes preform worse overall and during calm, short periods.
A possible problem with taking the average of the volatility might be the Black Scholes model’s constant volatility assumption; if one examine the graph of volatilities it can be seen that it is changing with possible jumps. Averaging over a time period with one possible jump could distort the mean estimate and with that giving a misleading num-ber. Methods that can control for sudden jumps or further smooth out the volatility time series can be recommended as a study in the future.
The purpose of this thesis was to analyze different estimation methods for the volatil-ity to be utilized in the Black & Scholes model. The main goal was to compare and evaluate these methods and with that contribute to existing research in this field, some-thing that has been accomplished. Further studies will be needed to decide whether the average volatility of the estimated variances are worth considering. Examples of other approaches that can be made are for instance; control for each observation’s risk-free rate and dividend yield, using other orders of GARCH and ARCH terms, trying different priors for the Bayesian analysis, extending the methodological approaches to different stock market which has a calmer or rougher change in volatility over time, and on a longer time period.
7
Appendix
7.1
Proof of the Black Scholes formula
When S is lognormally distributed with the standard deviation of ln(S) = σ then E[max(S − K, 0)] = E(S)N (d1) − KN (d2) where d1 = ln(E(S)K )+ σ2 2 σ and d2 = ln(E(S)K )−σ2 2 σ
Set g(S) as a probability density function of S, then it follows
E[max(S − K, 0)] = Z ∞
K
(S − K)g(S)d(S) (47)
The mean of ln(S) is m = ln[E(S)] − σ22
Define the variable Q = ln(S)−mσ which follows a standard normal distribution. Then the density function h(Q) = √1
2πe −Q2/2
Using the above results with substitution will give E[max(S − K, 0)] = Z ∞ (ln(K)−m)/σ (eQσ+m− K)h(Q)d(Q) (48) = Z ∞ (ln(K)−m)/σ (eQσ+m)h(Q)d(Q) − Z ∞ (ln(K)−m)/σ (K)h(Q)d(Q) (49) Note that eQσ+mh(Q) = √1 2πe (−Q2+2Qσ+2m)/2 (50) = √1 2πe (−(Q−σ)2+2m+σ2)/2 = e m+σ2/2 √ 2π e (−(Q−σ)2)/2 = em+σ2/2h(Q − σ) (51)
Then we can see that
E[max(S − K, 0)] = eQσ+m Z ∞ (ln(K)−m)/σ h(Q − σ)d(Q) − K Z ∞ (ln(K)−m)/σ h(Q)d(Q) (52) If we set N (x) as P (X < x) where the variable X follows a standard normal distribu-tion, then the first of the above integrals can be rewritten as
1 − N (ln K − m) σ − σ) = N −(ln K + m) σ + σ)
Substituting m will now give N " −(ln K + ln[E(S)] − σ2 2 ) σ + σ) # = N ln[E(S)/K] + σ 2/2 σ = N (d1)
Similarly, the second integral can be shown to be the same as N (d2). Therefore we will
now have that
E[max(S − K, 0)] = em+σ2/2N (d1) − KN (d2) = eln[E(S)]N (d1) − KN (d2)
E[max(S − K, 0)] = E(S)N (d1) − KN (d2) (53)
7.2
The Kalman Filter
It is a method to determine the most optimal value estimations for the state vector θt given the information at the current time t, the information at the time is denoted
as It. The Kalman Filter is designed on two equations: The prediction equation and
updating equation. Assume
mt = E[θt|It], the optimal estimator of θt based on the information It at time t and
Ct= E[(θt− mt)(θt− mt)0|It], which is the mean square error matrix of mt.
For the prediction equation, assume that the mt−1 and Ct−1 at the time t − 1 are
given, then the optimal predictor of θt and the neighboring mean square error matrix
are
mt|t−1 = E[θt|It−1] = Gtmt−1 (54)
Ct|t−1 = E[(θt− mt−1)(θt− mt−10 )|It−1] = GtCt − 1G0t+ Wt (55)
Then the optimal vector for yt, given the information at t − 1 can be written as
yt|t−1 = E[yt|It−1] = Ftmt|t−1 (56)
with the prediction error et= yt− yt|t−1 = yt− Ftmt|t−1 = Ft(θt− mt|t−1) + vt
and its MSE matrix E[ete0t] = Qt= FtCt|t−1Ft0+ Vt.
The updating equations are implemented when new observations (yt) become
avail-able. mt|t−1 and its mean square error are updated by
mt= mt|t−1+ Ct|t−1Ft0Q −1 t (yt− Ftmt|t−1) (57) = mt|t−1+ Ct|t−1Ft0Q −1 t vt (58)
and
Ct= Ct|t−1− Ct|t−1Ft0Q −1
t FtCt|t−1 (59)
When all data (IT) is observable, the Kalman smoothing recursions will be
imple-mented to compute the most optimal estimates for E[θt|IT] with
E[θt|IT] = mt|T = mt+ Ct∗(mt+1|T − Gt+1mt) (60)
E[(θt− mt|T)(θt− mt|T)0|IT] = Ct|T = Ct+ Ct∗(Ct+1|T − Ct+1|t)Ct∗0 (61)
Ct∗ = CtG0t+1C −1
t+1|t (62)
By setting mT |T = mT and CT |T = CT the algorithm will start and then proceed
backwards for t = T −1, T −2, ..., 1. The final step is to simply estimate the parameters by the maximum likelihood estimation. (Zivot & Yollin 2012)
8
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