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Option local and volatility 2 25 2014

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I present a little more details to previous short Cambridge presentation.

I present a little more details to previous short Cambridge presentation.

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  • 1. FAILINGS OF THE OPTION PRICING. Ilya I. Gikhman 6077 Ivy Woods Court Mason, OH 45040, USA Ph. 513-573-9348 Email: ilyagikhman@mail.ru JEL : G12, G13 Key words. Option, derivatives, pricing, hedging, risk-neutral probability, local volatility. Abstract. In this paper we present drawbacks of the Black-Scholes option pricing theory. A shorter version of the this paper were annonceed in [8]. Black-Scholes option theory represents an unified construction of the no-arbitrage pricing. It is common to think that in the theory Black-Scholes price represents the perfect hedging of the stock-bond portfolio. A primary goal of this paper is to highlight drawbacks of this benchmark interpretation of the option pricing concept. In addition we present our critical point on the local volatility adjustment of the Black-Scholes’ pricing theory. In (8) we presented an interpretation of the local volatility concept. Here we pay attention to the mathematical side of the construction. The essence of the option pricing failings is mathematical errors admitted by the authors. Drawbacks Black-Scholes derivations of the option price we announced first in [5]. More details on derivatives pricing were presented in [6,7]. Options is an important class of the derivative instruments traded on the modern markets. Generally speaking a derivative is defined as a financial instrument whose price is changed over the time as the underlying security price changes its value. Derivative contracts are usually settled in a future date called maturity. The modern ‘no-arbitrage’ options pricing concept was first developed in [1]. I. Let { Ω , F , P } denote a complete probability space and let S ( t ) denote a stock price at date t ≥ 0. Suppose that S ( t ) can be interpreted as a random process S ( t , ω ) associated with market scenario ω = ω ( t ). A European call option written on stock S is a financial contract that guarantees a buyer of 1
  • 2. the option the right to buy a security for a known price K at maturity T of the contract. The price K is known as the strike price of the option. Formally payoff of the option at T can be written by the formula C ( T , S ( T ) ) = max { S ( T ) – K , 0 } Therefore buyer of the option can buy stock for K if the S ( T ) > K at maturity receiving a profit S ( T ) – K > 0 and use the right does not buy a stock if S ( T ) < K. In latter case buyer of the call option will lost the premium C ( t , S ( t ) ) which is paid at initial date t to seller of the option. The pricing problem is to define the option price C ( t , S ( t ) ). Let us briefly remind solution of the problem following one of the primary handbook [9]. Black and Scholes introduced a portfolio Π which consists of one short call option − C ( t , S ( t )) and a portion of long underlying stocks  C ( t ,S( t ) ) S J( t , S ( t ) ) = The value of the portfolio at the time t is equal to Π(t,S(t)) = − C(t,S(t)) + J(t,S(t))S(t) (1) They then assumed that dΠ(t,S(t)) = − dC(t,S(t)) + J(t,S(t))dS(t) (2) Then the value of the change in value of the portfolio can be represented by the formula dΠ(t,S(t)) = [  C ( t ,S( t ) ) S 2 ( t ) σ 2  2 C ( t ,S( t ) )  ] dt t 2 S 2 This formula does not contain a term proportional to d w ( t ). In order to exclude arbitrage one needs to assume that dP(t,S(t)) = rP(t,S(t)) dt (3) Here r denotes the risk free interest rate defined by the government bond. In other words rates of return on riskless portfolio and bond should be equal. Otherwise there exists the arbitrage opportunity. Assume that the price of the underlying security is govern by stochastic Ito equation dS(t) = μS(t)dt + σS(t)dw(t) (4) Bearing in mind formulas (1-3) we arrive at the Black-Scholes equation C( t , x ) x 2 σ 2  2 C( t , x )  t 2 x 2 + r C(t, x) x – rC(t,x) = 0 defined in the area t  [ 0, T ), x > 0 with the backward boundary condition C ( T , x ) = max { x – K , 0 } 2 (BSE)
  • 3. The solution of the (BSE) equation can be represented in closed form C ( t , x ) = E exp – r ( T – t ) max ( S r ( T ; t , x ) – K , 0 ) (5) where d Sr ( l ) = r Sr ( l ) d l + σ Sr ( l ) d w ( l ) (6) l ≥ t and S r ( t ) = x . Remark. Pricing formula (5) shows that actual underlying of the call option is the random process S r ( t ) on { Ω , F , P } while according to the general definition of the derivative underlying should be the random process (4). The risk-neutral world was presented as a solution of the confusion. It does not eliminate the fact that underlying of the Black-Scholes pricing formula is the random process (6). First note that the random process S ( t ) is always defined on original probability space { Ω , F , P } regardless whether an option on this stock exists or not. Risk neutral valuation consider equation (4) on the risk-neutral probability space { Ω , F , Q } where the risk-neutral probability measure Q is defined by the formula T Q(A) =  { exp  A 0 μ  r 1 dwQ(t) – σ 2 T  0 ( μ  r 2 ) dt }P(dω) σ for arbitrary F-measurable set A. Here w Q ( t ) denote a Wiener process on { Ω , F , Q }. Then the random process S ( t ) = S ( t , ω ) is a solution of the equation (6) on probability space { Ω , F , P }. Here t w(t) = wQ(t) +  0 μ  r dl σ is the Wiener process on { Ω , F , P }. Thus the essence of the risk neutral valuations is to consider real stock diffusion equation on { Ω , F , Q }. One can completely omit risk neutral valuations starting with the solution of the equation (6). Using explicit formula for the solution of the equation (6) t S r ( t ; 0 , x ) = x exp  (r – 0 1 2  )dl + 2 t   dw(l) 0 the formula (5) can be rewritten in well-known form C ( t , x ) = x N ( d 1 ) – K exp – r ( T – t ) N ( d 2 ) (5) where d1 = 1 σ 2 (T - t ) x [ ln + ] , d2 = d1 –  2 K exp - r ( T - t ) σ Tt 3 T-t
  • 4. and N (  ) is the cumulative distribution function of the standard Gaussian random variable. Given formula (5) one can construct portfolio (1) and easy verify that formula (2) that leads us to the equation (BSE) is incorrect. It follows from formula (1) that Π(t,S(t)) = − C(t,S(t)) + J(t,S(t))S(t) = − C(t,S(t)) +  C ( t ,S( t ) ) S(t) S Differential of the function Π (  ) is equal to dΠ(t,S(t)) = − dC(t,S(t)) + J(t,S(t)) dS(t) + S(t)d  C ( t ,S( t ) ) S Two first terms on the right hand side are exist in Black-Scholes’ formula (2). The third term that does not equal to 0 was lost. This explicit error shows that the statement that the portion J ( t , S ( t ) ) of stocks represents perfect hedging of the call option is incorrect. Unfortunately, this college level error for more than forty years could not be recognized. It was taught students in Universities and sophisticated software are selling to investors. II. Other problem that is aimed on adjustment of the Black Scholes pricing concept is the local volatility. This development was presented in [2-4]. Let us briefly outline the problem from mathematical point of view. Consider SDE d S( u ) =  dw(u) S( u ) (7) u > t with initial condition S ( t ) = x. Define function C ( T , K ) = E max { S ( T ; t , x ) - K , 0 } Here ( t , x ) are fixed parameters while T , K are considered as variables. Latter equality can be rewritten in the form  C(T,K) =  (y - K) p(t,x;T,y)dy (8) K Here p ( t , x ; T , y ) is the density function of the stochastic process S ( T ) = S ( T ; t , x ). Twice differentiation in (8) with respect to K leads us to equality  2 C(T, K ) K 2 = p(t,x;T,K) To present local volatility consider a diffusion process K ( T ) , T ≥ t given by equation 4 (9)
  • 5. T K(T) = k +  b(u,K(u))dW(u) (10) t Diffusion coefficient b ( u , K ) is unknown function which should be specified. Assume that there exists a density f ( t , k ; T , K ) of the solution of the equation (10). Then Kolmogorov first equation for the density can be written as following 1 2  f ( t , k;T , K ) [ b 2 ( T, K ) f ( t , k ;T, K ) ]  2 2 K T Then the function f ( t , k ; T , K ) is replaced by the function (11)  2 C(T, K ) . Then K 2 1 2  2 C(T,K )   2 C(T,K ) [ b2 (T, K) ]  2  K2  K2 T  K2 (12) Changing order of differentiations on the right hand side of the latter equation and then integrating twice with respect to variable K lead us to equation 1 2  2 C(T,K ) b (T, K) 2  K2   C(T,K ) T (13) Solving equation (13) for b we arrive at local volatility surface  C(T, K ) T 2  C(T, K ) K2 2 b (T,K )  (14) In original papers variables ( T , K ) in the equation (13) and in the formula (14) were replaced by the variables ( t , S ). Then from (7) and (14) it follows that x  ( t , x ) = b ( t , x ). This transformations of the constant diffusion coefficient  in the stock dynamics (4) to the ‘local volatility surface’ b ( t , x ) in formula (14) might explain observed dependence of the implied volatility on time to maturity and strike price. We should clarify transformations outlined in equalities (10) - (14). Indeed, bearing in mind equality (9) one can easy verify that the second derivative  2 C(T, K ) as a function of the variables ( T , K ) is a K 2 solution of the same Kolmogorov equation as is governing the density function Indeed 5 p ( t , x ; T , K ).
  • 6. 0  = 1 2  2 p ( t , x ;T, K )  p ( t , x ;T, K ) σ  2 2 K T = 1 2  2 C(T,K )   2 C(T,K ) [ σ2 ]  2  K2  K2 T  K2 Therefore similar to transition from the equation (12) to the equation (13) we can conclude that call option price follows to equation 1 2  2 C(T,K ) σ 2  K2   C(T,K ) T Therefore in theory the local volatility surface that is defined in (14) should be equal to the original constant diffusion, i.e. b 2 ( T , K ) =  2. Bearing in mind this remark one can conclude that ‘smile’ effect does not implied by the theory. It rather stems from the option data calculations which do not coincide with the theoretical value. 6
  • 7. References. 1. Black, F., Scholes, M. The Pricing of Options and Corporate Liabilities. The Journal of Political Economy, May 1973. 2. Derman, E., and Kani,I., Riding on a smile, Risk, 7 (1994), p.32-33. 3. Dupire, B., “Pricing and Hedging with Smiles” , Research paper, 1993, p.9. 4. Dupire, B., “Pricing with a Smile”. Risk Magazine, 7, 1994, p.8-20. 5. Gikhman, Il.I. On Black- Scholes Equation. J. Applied Finance (4), 2004, p.47-74, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=500303. 6. http://papers.ssrn.com/sol3/cf_dev/AbsByAuth.cfm?per_id=365639 7. Gikhman, Il.I. Alternative Derivatives Pricing: Formal Approach, LAP Lambert Academic Publishing 2010, p.164. 8. Gikhman, Il.I. Derivatives Pricing Basics and Randomization. 13th International Research Conference on Finance, Accounting, Risk and Treasury Management, Cambridge UK, November 2013, http://www.slideshare.net/list2do. 9. Hull J., Options, Futures and other Derivatives. Pearson Education International, 7ed. 814 p. 7