FINANCIAL PRICING MODELS

Transcription

1 Page 1-22 like equions FINANCIAL PRICING MODELS 20 de Setembro de 2013 PhD Page 1- Student 22

2 Contents Page PhD Page 2- Student 22

3 Page 3-22 In 1973, Fischer Black and Myron Scholes presented a paper "The Pricing of Options and Corpore Liabilities". On the same year Robert Merton presented "Theory of rional option pricing"; two paper th are considered as the beginning of Financial Mhemics history. Merton approach (option price model with securities paying dividends) can be seen as generalizion of approach (non-dividends). PhD Page 3- Student 22

4 Page 4-22 Financial Markets are in continuous movement, Financial Engineers continuously construct new financial instruments; and Mhemicians as to ensure the applicability and logical functionality of those instruments; on the other hand there is no such model th describe the real life 100% correctly, therefore, from 1973 up to now, there are several Financial Mhemics models describing Financial Markets and Financial Instruments. In this talk I will present some Financial Mhemics models and show relions between them - starting from Options Pricing. PhD Page 4- Student 22

5 Definitions Page 5-22 Options - financial contracts between two parts (seller and buyer) th gives to the buyer rights without obligions. Example (European Call Options, European Put Options, American Options, etc). Spot Re - the price quoted for an instantaneous settlement on a security or commodity. Discount Bond - is a financial instrument (contingent claim) th promises to pay a certain amount of money (face) in the future time. [default-free discount bond]. PhD Page 5- Student 22

6 Assumptions Page 6-22 The stock price has a drift, volility and the risk-free interest re to be constant. The short selling of securities with full use of proceeds is permitted. There are no transactions cost or taxes, all securities are perfectly divisible. There are no dividends during the life of the derivive. There are no risk-less arbitrage opportunities. Security trading are continuous. PhD Page 6- Student 22

7 BS - Page 7-22 db t = r t B t dt (1) ds t = S t µdt + S t σdw t (2) The assumptions of the model imply th the portfolio must instantaneously earn the same re of return as other short term risk-free securities. The price of the option under BS-Assumptions is given by discounting time-t expected value of max(s T K, 0) under the equivalent martingale measure Q; this is: PhD Page 7- Student 22

8 BS - Page 8-22 C(S t, K, T ) = e r(t t) E Q [max(s T K, 0) conditional] (3) C(S t, K, T ) = S t Φ(d 1 ) e r(t t) K Φ(d 2 ) (4) d 1 = ( ) ( ln St K + r + σ2 2 σ T t ) (T t) and d 2 = d 1 σ T t. PhD Page 8- Student 22

9 General Description Page 9-22 Lets consider a financial market th works without transaction cost and taxes, and where trading takes place when time is in [0, T ]. The asset prices movement is characterized by an probabilistic space (Ω, F, P) th will model the uncertainty and the effect of external inputs to the financial system. PhD Page 9- Student 22

10 General Description Page Lets call S t the asset price time t, governed under probability P by the following stochastic differential equion ds t = µ(t, S t )dt + Σ(t, S t )dw t, (5) where S t = (S 1 (t),, S m (t)) and W t = (W 1 (t),, W b (t)). µ: [0, T ] R m R m and Σ: [0, T ] R m R m b. PhD Page 10-Student 22

11 General Description Page Then We define the contingent claim value as a continuous function depending on time and the price movement of underlying asset V(t, S t ) where V : [0, T ] R m R; has continuous partial derivives V t ; V s i ; 2 V s i s j ; PhD Page 11-Student 22

12 General Description Page Assuming th the financial market does not allow arbitrage opportunity it is proved on Jhon C. Hull (2009) th the contingent claim price sisfies the partial differential equion 1 2 m m b i=1 j=1 k=1 V t + m i=1 µ i (t, S) V S i + Σ ik (t, S)Σ kj (t, S) 2 V S i S j r(t, S)V = 0 subject to the terminal value condition V (T, S) = g(s). Here r(t, S) is the risk free interest re PhD Page 12-Student 22

13 Beyond Page Lets consider the following system of Stochastic Differential Equions: ds t = r t S t dt + σ p t S tdw s t, dr t = λ(θ t r t )dt + ηdw r t, dσ t = k(σ t σ)dt + γσ 1 p t dw σ t, With p = 1; consider the volility to be constant. We need to study the second equion to be able to solve the Stock Price equion. PhD Page 13-Student 22

14 Beyond Page Study of this case is similar to study how to define the price today for default free discount bond th mure future time T. Lets consider the general case where dr t = α(r, t)dt + β(r, t)dw r t (6) Considering the default-free discount bond as financial instrument written on the spot re (different approach from BS). If F(t) is the time-t price of discount bond, then F(t) = f [r(t), t]; 0 t T. Itô Lemma will give us PhD Page 14-Student 22

15 Beyond Page df F for short = 1 f [( F t + α F r + 1 ) 2 β2 2 F r 2 dt + β F ] r dw df F = α f dt + β f dw. Defining portfolio of two discount bonds th mures on different times, if we construct a scenario of no-arbitrage opportunity, then, the excess re of return per unit risk must be the same, therefore, the market price of the risk is defined by ɛ(t) = α f r β f. PhD Page 15-Student 22

16 Beyond Page If we know ɛ(t), then P P under which {W (t)} defined by dw = dw + ɛ(t)dt (7) is a SBM. From here, denomining by B(t) and using Itô division rules df F = df F db ( ) db 2 B + df db B F B th gives df F = β f dw ; 0 t T. Since F is Martingale under P : F = E t [ ] C(T ) = Et B(T ) [ ] 1 = Et B(T ) T e t PhD Page 16-Student 22 r(s)ds (8)

17 Beyond Page We need to find the Stochastic Differential Equion for the spot price under P in order to calcule the bond price (8). From equions (6) and (7) We will have dr t = [α βɛ(t)]dt + βdw t ; 0 t T. (9) The new drift term th We obtain depends only on the ste r(t) (which is Markov process under P ), then, the discount bond price is given by v(t, T ) = E t T e t r(s)ds r(t) ; 0 t T. PhD Page 17-Student 22

18 Beyond Page The Markovians properties will transform our problem into PDE f t + m(r, t) f r + β2 (r, t) 2 f rf (r, t) = 0 (10) 2 r 2 PhD Page 18-Student 22

19 Affine Page Under the risk-neutral probability measure P, supposing th m(r, t) = m 1 (t) + m 2 (t)r and β(r, t) = β 1 (t) + β 2 (t)r; then de default-free discount bond price v(t, T ) is given by PhD Page 19-Student 22

20 Affine Page where v(t, T ) = e A T (t)+b T (t)r(t) T B T (t) = t e s m 2 (u)du t ds; and A T (t) = T t T m 1 (u)b T (u)du t β 2 (u)b 2 T (u)du. PhD Page 20-Student 22

21 Page Bibliografy John C. Hull s (2009) Options, Futures and Other Derivives Seventh Edition, New Jersey. Masaaki Kijima (2000) Stochastic Processes with Applicions to Finance, Capman and Hall/CRC.. Lectures notes in Stochastic Differential Equions, Diffusion Processes, and the Feynman-Kac formula Zhu, Y-I; Wu, X; Chern, I-L, Derivive Securities and Difference Methods Springer 2004 Black, F and M. Scholes (1973). The Pricing of Options and Corpore Liabilities Journal of Political Economy, Vol 81, No. 3, pp PhD Page 21-Student 22

22 Page Thanks For Your Attention!!! PhD Page 22-Student