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option volatility and pricing mouna haddadi phd student faculty of science and technology of marrakesh cadi ayyad university morocco youngwomeninprobability2014 abstract part 1 historical volatility part 2 implied volatility this ...

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                                                                                     Option Volatility and Pricing
                                                                                                              Mouna HADDADI (PhD student)
                                                                Faculty of Science and Technology of Marrakesh, Cadi Ayyad university, Morocco.
                                                                                               YOUNGWOMENINPROBABILITY2014
                                                  Abstract                                                                  Part 1. Historical volatility                                                       Part 2 : Implied volatility
                  This poster discusses three types of volatility : the historical, implied           Thehistorical volatility reflects the past price movements of the under-           The implied volatility is often interpreted as estimation of the future
                  and stochastic volatility, and the concept of local volatility. Finally the         lying asset, it is calculated as a standard deviation of a stock’s returns        volatility. It means that this volatility is a volatility anticipated by the
                  pricing formula of Vanilla options with stochastic volatility model such            over a fixed number of days.                                                       market maker. In other words it is the value of σ that equalizes the
                  as the model of Heston is presented.                                                The estimate of historical volatility starting from data :                        price calculated by Black & Scholes model with the observed prices on
                                                                                                      n+1:thenumberof observations;                                                     the market
                                                                                                      S : the price at the time t;                                                                observed                Black−Scholes              impl
                                                                                                       t                                                                                        C         (S ;T;K) = C                   (S ;T;K;σ       )
                                                                                                      u : the return at time t;                                                                   t          t            t                t
                                                Introduction                                           t
                                                                                                      τ : duration of the time intervals in year.
                                                                                                      Then                                                                            Estimation of implied volatility
                                                                                                                                             S                                           Wecan find the value of an European call, by using the Risk-Neutral
                  The study of the volatility of certain financial assets became very im-                                          u =ln        t
                                                                                                                                   t        S
                  portant subject in finance. This importance comes from the possibility                                                      t−1                                        valuation method i.e by considering that price of a call corresponds to
                  to measuretheuncertaintyofevolutionoftheyieldonanasset(shareor                      for t = 1;2;:::;n                                                                 its expected value of future return discounted :
                                                                                                      Theestimate s of the standard deviation of u is given by this formula :
                  index) and the fact that the fluctuations of prices can not be neglected.                                                          t                                                  −rT b                         −rT b             +
                                                                                                                                                                                                C0 = e      E[max(S −K;0)]=e              E (S −K)
                  Nowadays, any investor is conscious of these fluctuations which intro-                                          v                                                                                    T                         T
                                                                                                                                 u          n
                  duce an element of risk into its portfolio. That is why the investors wish                                     u 1 X                 2
                                                                                                                            s = t             (u −u)                                            b
                  to choose the degree ”of exposure” at the risk compatible with their                                              n−1         t                                       where E is the expectation operator under the probability risk–neutral.
                  level of tolerance to this risk. Thus the study of the volatility plays an                                               t=1
                  essential role in evaluation and hedging of risk of an investment.                  where u is the average of u                                                       The value of call and put option a time 0 is given by :
                                                                                                                                  t
                                                                                                      The Black and Scholes model assumes the following dynamic of the                                      C = S Φ(d )−KerTΦ(d )
                                                                                                                                                                                                              0     0    1              2
                                                                                                      stock price :                                                                                         P = KerTΦ(−d )−S Φ(−d )
                                                                                                                              dS =µSdt+σSdz                                                                   0                2     0       1
                                       The definition of the volatility                                                           t       t        t  t
                                                                                                      where z is a Wiener process                                                       where
                                                                                                              t                                                √                                                S           σ2
                                                                                                            dS                                                                                               ln( 0) + (r +    )T               √
                                                                                                      Since   t behaves like a normal distribution N(µdt;σ       dt), the stan-                        d =      K √ 2 ,d =d −σ T
                                                                                                            St                                   √                                                       1                         2     1
                  Thevolatilityisameasureforvariationofpriceofafinancialinstrument                     dard deviation of the return is equal to σ   dt                                                               σ T
                  over time.                                                                          Therefore s is an estimator of σ√τ. Thus we estimate σ by σb where :              and Φ is a the normal probability distribution function.
                  The types of volatility :                                                                                                 s                                           It is not possible to reverse the preceding equation and to express σ
                                                                                                                                      σb = √                                            according to S , K, r, T and C . However, it is possible to determine
                  –Historical volatility                                                                                                     τ                                                          0                 0
                  –Implied volatility                                                                                                                                                   the value of this implied volatility by using methods of interpolation
                  –Stochastic volatility                                                                                                                                                like the method of Newton & Raphson.
                                          Part 3 : Smile volatility                                                          Part 4 : Local volatility                                                           Part 5 : Heston model
                  The implied volatility of an option evolves according to the strike and            Dupire formula :                                                                   Heston model (1993)
                  the maturity of the option, now when we draw the implied volatility                                                                                                   The Heston stochastic volatility model is based on the following stock
                  according to strike for a given maturity, generally we do not obtain a             Fokker-Planck equation :                                                           price and variance dynamics
                  horizontal line, which corresponds to the assumption of consistency of             (∂f + r ∂ (xf) − 1 ∂2 (x2σ2(x;T)f) = 0                                                            dS(t) = µ(t)S(t)dt+pv(t)S(t)dZ
                  implied volatility.                                                                  ∂T      ∂x          2∂x2                                                                                                               1
                                                                                                       f(x;t) = δ(S −x)          sur [0,+∞]x[t,+∞]
                                                                                                                     t                                                                                                              p
                                                                                                     where δ is the Dirac function                                                                        dv(t) = κ(θ −v(t))dt +σ       v(t)dZ
                                                                                                                                                                                                                                               2
                                                                                                                                                                                        where hdZ ;dZ i = ρdt , θ : the long-run average of v(t),
                                                                                                      Theorem : for every (t,s) fixed the function :                                                1    2
                                                                                                                                                                                        κ : controls the speed by which v(t) returns to its long-run mean
                                                                                                                                                                                        and σ : the volatility of volatility.
                                                                                                                                                                                        Thefundamental partial differential equation (PDE) verified by option
                                                                                                                                −r(T−t) Q              +
                                                                                                                  C(T;K)=e              E [(S −K) =S =s]                                price is :
                                                                                                                                               T           t
                                                                                                     is a solution of Dupire equation :                                                       ∂C    1      ∂2C           ∂2C      1    ∂2C        ∂C
                                                                                                                   ∂C              ∂C     1 2          2∂2C                                       + CS2         +ρσvS          + σ2v        +rS       −rC
                                                                                                                   ∂T −r(C −K∂K)−2σ (T;K)K ∂K2 =0                                             ∂t    2      ∂S2          ∂S∂v      2    ∂v2        ∂S
                                                                                                     In particular :                   s                                                                                 ∂C
                                                                                                                                           ∂C+r(C−K∂C)                                                                 −∂v[κ(θ−v)−λv]=0
                                                                                                                           σ(T;K) =      2∂T          ∂K
                                                                                                                                               K2∂2C                                    We seek to solve the preceding PDE in the case of a European call
                                                                                                                                                 ∂K2                                    option of strike K and maturity T, by analogy with the Black & Scholes
                                                                                                                                                                                        formula, the solution of this option is of the form :
                                                                                                                                                                                                           C(S;v;t) = SP −Ke−r(T−t)P
                                                                                                                                                                                                                            1                2
                                                                                                              Part 4 : The link between local volatility and implied
                                                                                                                                     volatility                                         By injecting it in the PDF, and From the Fourier inversion theorem,
                                                                                                                                                                                        we have :                   Z          "             #
                                                                                                                                                                                                       P =1+1 +∞Re e−iφln(K)fj dφ
                                        Implied volatility smile                                                                                                                                         j    2   π                  iφ
                                                                                                     Let us consider a model with non-deterministic volatility. That is the                                           0
                                                                                                     risk-neutral dynamics of S is written as :                                         For j = 1;2 , P are probabilities , and f are characteristics functions
                                                                                                                                                                                                        j                          j
                  The Smile of volatility is a phenomenon observed on the markets of                                                                                                    such as
                                                                                                                               dS
                  options vanillas which contradicts the assumption of Black and Scholes                                          t = µdt +σ dW
                                                                                                                                S              t   t                                                f (x;v;T;φ) = exp(C(τ;φ)+D(τ;φ)v +iφx)
                  according to which the volatility of an option is constant and is not                                           t                                                                  j
                  influenced by the value of other parameters. From a statistical point               Instantaneous volatility σ is a process such as :                                                            h                               dri
                                                                                                                                                                                        and C(τ;φ) = rφiτ + a (b −ρσφi+d)τ −2ln 1−ge
                  of view such a form of curve of volatility according to the strike price                                   Z                                                                                  σ2    j                         1−g
                  corresponds to a value of Kurtosis higher than 3, therefore risk-neutral                                     T
                                                                                                                                 σ2ds < ∞;∀ T > 0                                                                                   "          #
                  dynamics of Black & Scholes and Merton are not compatible with the                                               s                                                                                bj −ρσφi+d 1−edr
                                                                                                                              0                                                                          D(τ;φ) =
                  phenomenon of smile which exists on all markets options.                                                                                                                                                 σ2         1−gedr
                                                                                                     Thus we can define the local variance as conditional expectation of the
                                                                                                     future instantaneous variance                                                           bj −ρσφi+d                      q                                
                                                                                                                                           h            i                              g =                      and     d =     ρσφi−b 2−σ2 2u φi−φ2
                                                                                                                           2            Q     2                                             b −ρσφi−d                                     j            j
                                                                                                                         σ (T;K) = E        σ S =K                                           j
                                                                                                                           L                 T    T
                                                                                                                                                                                              u = 1 , u = −1 , a = κθ , b = κ+λ−ρσ , b = κ+λ
                                                                                                     This definition and that based on Dupire’s formula are equivalent.                         1    2    2     2              1                  2
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...Option volatility and pricing mouna haddadi phd student faculty of science technology marrakesh cadi ayyad university morocco youngwomeninprobability abstract part historical implied this poster discusses three types the thehistorical reects past price movements under is often interpreted as estimation future stochastic concept local finally lying asset it calculated a standard deviation stock s returns means that anticipated by formula vanilla options with model such over xed number days market maker in other words value equalizes heston presented estimate starting from data black scholes observed prices on n thenumberof observations at time t impl c k u return introduction duration intervals year then wecan nd an european call using risk neutral study certain nancial assets became very im ln portant subject nance importance comes possibility valuation method i e considering corresponds to measuretheuncertaintyofevolutionoftheyieldonanasset shareor for its expected discounted theestim...

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