Adiabaticity Conditions for Volatility Smile in Black-Scholes Pricing Model

Adiabaticity Conditions for Volatility Smile in Black-Scholes Pricing Model L. Spadafora,1 G. P. Berman,2 and F. Borgonovi1, 3 1Dipartimento di Matematica e Fisica, Universit`a Cattolica, via Musei 41, 25121 Brescia, Italy 2Theoretical Division, MS-B213, Los Alamos National Laboratory, Los Alamos, NM, 87545 3I.N.F.N. Sezione di Pavia, Pavia, Italy (Dated: July 8, 2021) Our derivation of the distribution function for future returns is based on the risk neutral approach which gives a functional dependence for the European call (put) option price, C(K), given the strike price, K, and the distribution function of the returns. We derive this distribution function using for C(K) a Black-Scholes (BS) expression with volatility, σ, in the form of a volatility smile. We show that this approach based on a volatility smile leads to relative minima for the distribution function (“bad” probabilities) never observed in real data and, in the worst cases, negative probabilities. We show that these undesirable effects can be eliminated by requiring “adiabatic” conditions on the volatility smile. PACS numbers: 05.10.Gg, 05.40.Jc, 02.50.Le, 89.65.Gh Keywords: Volatility smile, Black-Sholes model, no-arbitrage conditions I. INTRODUCTION One of the simplest “products” on the derivative fi- nancial market is the European call (put) option [1, 2]. Considering the risk neutral approach, the price of the European call option, C ≡C(ST , K, T, r), is defined by C = e−rT Z ∞ K (ST −K)P(ST )dST , (1) where ST is the stock price at time t = T, K is the strike price of the option, T is the expiration time (time to ma- turity) of the option, r is the interest rate and P(ST ) ≥0 is the distribution function of the stock prices in a “risk- neutral world” ( R ∞ 0 P(ST )dST = 1). Eq. (1) is too general since it does not place any re- strictions on the underlying stock price distribution func- tion, P(ST ). To calculate explicitly the option price, C, using Eq. (1), one must know the distribution function, P(ST ). Consequently, one must make some assumptions about the stock prices. An important achievement in the theory of option pricing is the Black-Scholes (BS) theory which gives analytic solutions for the European call and put options [3]. In particular, for the European call option, a solution of the BS equation is given by Eq. (1), if one assumes for the distribution function, P(ST ), a log-normal distribu- tion, P(x) = 1 p 2πσ2(T −t) exp  −(x + σ2(T −t)/2)2 2σ2(T −t)  , (2) where x = ln(ST /S(t))−r(T −t) is the logarithmic return deprived of the risk-free component, S(t) is the stock price at time t and σ is the stock price volatility. For seek of simplicity, in the following we consider t = 0 and we define S0 ≡S(t = 0). Substituting Eq. (2) in (1) an explicit expression for the price of the European call option which satisfies the BS equation [3] is obtained, CBS = S0N(d1) −Ke−rT N(d2), (3) where d1 = ln(S0/K) + (r + σ2/2)T σ √ T , d2 = d1 −σ √ T, N(x) = 1 √ 2π R x −∞dz e−z2/2. (4) The distribution function (2) follows from a stochastic model for stock prices, dS = rSdt + σSdz, (5) where dz is a Wiener increment [4]. It can be shown it is never optimal to exercise an American call option on a non-dividend-paying stock early [1], [5]; therefore Eq. (3) can also be used to estimate the fair value for this kind of options. There are some problems with the expressions for C given by Eqs. (1)-(3). Indeed, one can derive any option price from Eq. (1), using different assumptions about the distribution function, P(ST ). To derive from Eq. (1) a result for C which will even approximately coincide with the real market price, CM, one must specify a distri- bution function for future stock prices, P(ST ). On the other hand, the expression given by (3) is (a) too specific, and (b) derived using rather strong restrictions. Namely, the stochastic process Eq. (5) does not account for cor- relations of returns, x and, moreover, the volatility, σ, and the interest rate, r, are not well-defined parame- ters (given the actual data). As a result, the expression, CBS, often does not coincide (even approximately) with the corresponding market option price, CM. Useful ap- proaches have been developed which partially solve the problems mentioned above. arXiv:1003.3316v2 [q-fin.PR] 24 Mar 2010 2 We shall mention here one analysis which is related to that presented below. This analysis deals with building “implied trees” [6]. There are many variations of this approach, but the main idea is based on the solution of the inverse problem: a search for a stock price model that corresponds to the real market prices of options, CM. A more restricted problem is to search for a stock price model that effectively deals with the volatility smile. In this case, one starts with the BS formula (3), (even for American options) but instead of choosing a fixed volatility, σ = constant, one uses the dependence, σ = σ(K) (volatility smile). To some extent, this dependence corresponds to

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