4. Black-Scholes Models and PDEs. Math6911 S08, HM Zhu

Transcription

1 4. Black-Scholes Models and PDEs Math6911 S08, HM Zhu

2 References 1. Chapter 13, J. Hull. Section.6, P. Brandimarte

3 Outline Derivation of Black-Scholes equation Black-Scholes models for options Implied volatility 3

4 4. Black-Scholes Models and PDEs 4.1 Derivation of the Black-Scholes-Mertion differential equation Math6911 S08, HM Zhu

5 Assumptions for Black-Sholes Equation Asset price follows the lognormal random walk The risk-free interest rate r and the asset volatility σ are known functions of time over the life of an option No transaction cost or taxes No arbitrage possibilities No dividends during the life of an option Trading of the underlying asset is continuous Short selling is permitted and the assets are divisible 5

6 Concepts underlying the Black- Scholes equation The option price & the stock price depend on the same underlying source of uncertainty We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate This leads to the Black-Scholes differential equation 6

7 1 of 3: The Derivation of the Black-Scholes Differential Equation Model of the asset price: ds = µ S dt + σs dz (4.1) Option value V using Ito's ˆ lemma: V V 1 V V dv = µ S + + σ S dt S dz ( 4. ) + σ S t S S We set up a portfolio consisting of short one option V long number of underlying asset S

8 of 3: The Derivation of the Black-Scholes Differential Equation The value of the portfolio Π is given by V Π= V+ S (4.3) S The change in its value in time dt is given by V dπ= dv+ ds (4.4) S Substituting Eq's (4,1) and (4.) to Eq (4.4) gives V t 1 dπ= + σ S d V t (4.5) S

9 3 of 3: The Derivation of the Black-Scholes Differential Equation The return of the riskless portfolio Π would see a growth of dπ= rπdt (4.6) over a time dt, where r is risk-free interest rate. Substituting Eq's (4.6) and (4.1) to Eq (4.5) leads to the Black-Scholes equation: V 1 V V t S S + σ S + rs rv = 0 (4.7)

10 The Black-Scholes Differential Equation Any derivative security whose price is dependent only on the current stock price and t, which is paid for up-front, must satisfies the Black-Scholes differential equation or its variations Other options, for example, American options that depend on both the history and present values of the asset, can also fit into the Black-Scholes framework

11 4. Black-Scholes Models and PDEs 4. More on the Black-Scholes-Mertion differential equation Math6911 S08, HM Zhu

12 The Derivative terms The delta given by is a measure of the correlation/sensitivity between the movements of the option and the underlying asset The linear differential operator 1 + σ S + rs r t S S is a measure of the difference between the return on a hedged option portfolio and the return on a bank deposit = V S return on hedged option portfolio return on a bank deposit 1

13 Risk-Neutral Valuation The variable µ does not appear in the Black-Scholes equation The only parameter from the stochastic differential equation involved is the volatility σ It is independent of risk preference We can assume that all investors are risk-neutral Therefore, the expected return on any securities is the risk-free interest rate r

14 Boundary and final conditions The Black-Scholes equation is a common type of PDEs, called backward parabolic equation For such an equation to have a unique solution, we will need the boundary and final conditions of V(S,t) For example, a typical boundary conditions are V V ( a, t) = Va ( t) ( b, t) = V ( t) A typical final condition is the value of V(S, t) at t = T V b ( S, T ) = V ( S ) T

15 4. Black-Scholes Models and PDEs 4.3 Black-Scholes analysis for European options Math6911 S08, HM Zhu

16 Boundary and final conditions for European call options The value of a European call option satisfies the Black-Scholes equation C 1 C C + σ S + rs rc = t S S s.t. ( S, T )? Final condition : C = 0 Boundary conditions : C C ( S, t) ( 0, t) = =?? as S

17 The Black-Scholes Formulas for C(S,t) (See p 48-49, [4]; proof of solving PDE, p76-79, [4]) ( t) ( ) rt C S,t = S N(d 1) K e N(d ) where 1 x y ( ) N x = e dy π ln( S / K ) + ( r + σ / )( T t) d1 = σ T t ln( S / K ) + ( r σ / )( T t) d = = d1 σ T t σ T t

18 Approximate the cumulative normal distribution function N(x) (See p 97, Hull) N 1 x y ( x ) = e dy π A polynomial approximation that gives 6 decimal place accuracy: 1 N N( x) = where 1 k = γ = 1+ γx a a 4 1 = , = , ( )( x a k + a k + a k + a k + a k ) a a 1 N 3 ( x) = , = , 4 a 3 5 N if x if x = x / ( x) = e π 0 0

19 The Black-Scholes Formulas for C(S,t) (Proof p 95, Hull; using risk-neutral evaluation) Another way to derive the Black-Scholes equation is to use risk-neutral valuation where ( ) ( ) ( ) ( 0) ( t) = ˆ rt C S,t e E max ST K, = rt t rt t e Se N(d 1) K N(d ) N(d ): probability that the option will be exercised in ( t) a risk-neutral world K N(d ): probability that the strike price will be paid rt Se N( d ) : expected value of a variable which equals S if S > K 1 T T and is zero otherwise in a risk-neutral world

20 European call value C(S,t) as a function of S and t (r = 0.1, σ=0.4, K=1)

21 Boundary and final conditions for European put options The value of a European call option satisfies the Black-Scholes equation P 1 P P + σ S + rs rp= t S S s.t. 0 Final condition: P ( S,T ) =? Boundary conditions: ( 0 ) P,t =? P( S,t )? as S

22 The Black-Scholes Formulas for P(S,t) (See pages 48-49, [4]) Put-call parity: ( t) S+ P C = Ke rt Using the put-call parity we can obtain: ( ) rt t P= K e N( d ) S N( d 1)

23 European put value P(S,t) as a function of S and t (r = 0.1, σ=0.4, K=1)

24 4. Black-Scholes Models and PDEs 4.4 Implied Volatility Math6911 S08, HM Zhu

25 Implied Volatility Although volatility can be estimated from a history of the stock price, traders often work with implied volatility The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price There is a one-to-one correspondence between prices and implied volatilities; therefore, an iterative search procedure can be used to find the implied volatility Often used to monitor the market s opinion about the volatility of a particular stock