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