Options. I. Christopher G. Lamoureux January 9, 2013 Options. I. - PowerPoint PPT Presentation

Options. I. Introduction What? Stochastic Integration Hedge Portfolio Options. I. Christopher G. Lamoureux January 9, 2013

Options. I. Organizing Themes Introduction What? Stochastic The Black-Scholes-Merton option pricing theory is perhaps Integration the most successful model in finance. Hedge Portfolio This may well be because it takes two fundamental market prices as given: the stock price and the risk-free rate. This approach of asking what are the restrictions of the absence of arbitrage on relative prices is even more popular on Wall Street than in academia. We now understand that any financial asset’s price is an expectation in the equivalent risk-neutral measure. This implies that we need two sets of skills. ◮ Changing Measures. ◮ Evaluating stochastic Integrals.

E [max( S − X , 0)] Options. I. Introduction What? Stochastic Integration Where: Hedge Portfolio dS = µ Sdt + σ Sdz It follows that f ( S t | S s ) is lognormal. There are many ways to solve for the value of an option: ◮ Form a risk-neutral portfolio, identify its sde and solve (What Black and Scholes did). ◮ Do a change of measure and take expectations. ◮ Apply Feynman-Kac Theorem. In any case, we have to understand how to take this slide’s eponymous expectation.

Options. I. Itˆ o’s Lemma Introduction What? Stochastic Let G be a function of the random variable x . Integration Hedge Portfolio ∆ G ≈ dG dx ∆ x Taylor Series: d 2 G d 3 G ∆ G = dG dx ∆ x + 1 dx 2 ∆ x 2 + 1 dx 3 ∆ x 3 + . . . 2 6 Now let G be a function of the random variable x and y : ∂ 2 G ∂ y ∆ y + 1 ∆ G = ∂ G ∂ x ∆ x + ∂ G ∂ x 2 ∆ x 2 + . . . 2

Options. I. Itˆ o Introduction What? Stochastic Integration So now, consider: Hedge Portfolio dx = a ( x , t ) dt + b ( x , t ) dz Then: ∂ 2 G ∂ x 2 ∆ x 2 + ∂ 2 G ∂ t ∆ t + 1 ∆ G = ∂ G ∂ x ∆ x + ∂ G ∂ x ∂ t ∆ x ∆ t . . . 2 ∂ 2 G . . . + 1 ∂ t 2 ∆ t 2 + . . . 2

Options. I. Itˆ o Introduction What? Stochastic Integration Hedge Portfolio As ∆ x and ∆ y approach 0: dG = ∂ G ∂ x dx + ∂ G ∂ y dy We can discretize the Itˆ o process: √ ∆ x = a ∆ t + b ǫ δ t

Options. I. Itˆ o Introduction What? The main intuition from Itˆ o’s lemma is that the variance of a Stochastic Wiener process increases at the rate t . So, this means that Integration Hedge Portfolio the expansion: ∆ x 2 = b 2 ǫ 2 ∆ t + . . . We know E ( ǫ 2 ) = 1 (Why?) Itˆ o’s Lemma: ∂ 2 G dG = ∂ G ∂ x dx + ∂ G ∂ t dt + 1 ∂ x 2 dx 2 2 So for dx an Itˆ o process: ∂ 2 G � ∂ G ∂ x + ∂ G ∂ t + 1 � dt + ∂ G ∂ x 2 b 2 dG = ∂ x bdz 2

Options. I. Stock Process Consider that the stock price, S , follows a geometric Introduction What? Brownian motion: Stochastic Integration dS = S µ dt + S σ dz Hedge Portfolio Let G = ln S , then: S , ∂ 2 G since: ∂ G ∂ S = 1 ∂ S 2 = − 1 S 2 , ∂ G ∂ t = 0 . S ( µ Sdt + σ Sdz ) − σ 2 S 2 dG = 1 2 S 2 dt So: µ − σ 2 �� � � T , σ 2 T ln S T − ln S 0 ∼ φ 2 and: µ − σ 2 � � � � T , σ 2 T ln S T ∼ φ ln S 0 + 2

Back to: E [max( S − X , 0)] Options. I. Introduction What? Stochastic Integration � ∞ Hedge Portfolio E [max ( V − K , 0)] = ( V − K ) g ( V ) dV (1) K We know that V is lognormally distributed, so that the mean of ln V is m : m = ln [ E ( V )] − σ 2 2 Now let z = ln V − m , so: σ 1 e − z 2 f ( z ) = √ 2 2 π

Back to: E [max( S − X , 0)] Options. I. So do a change of variable within the integral: Introduction What? � ∞ Stochastic e σ z + m − X Integration � � E [max ( V − K , 0)] = f ( z ) dz ln X − m Hedge Portfolio σ � ∞ � ∞ e σ z + m f ( z ) dz − X = f ( z ) dz ln X − m ln X − m σ σ And: 1 e ( − z 2 +2 σ z +2 m ) / 2 e σ z + m f ( z ) = √ 2 π 1 e ( ( − z − σ ) 2 +2 m + σ 2 ) / 2 = √ 2 π = e m + σ 2 − ( z − σ )2 2 √ e 2 2 π

E [max( S − X , 0)] (Continued) Options. I. Introduction E [max ( V − K , 0)] = e m + σ 2 2 · f ( z − σ ) What? Stochastic � ∞ � ∞ Integration = e m + σ 2 f ( z − σ ) dz − X f ( z ) dz 2 Hedge Portfolio ln X − m ln X − m σ σ And: � ∞ � ln X − m � f ( z − σ ) dz = 1 − N − σ σ ln X − m σ (Why?) Alternately: � ∞ � m − ln X � f ( z − σ ) dz = N + σ σ ln X − m σ or � �  E ( V ) + σ 2  ln � ∞ X 2   f ( z − σ ) dz = N σ ln X − m   σ

E [max( S − X , 0)] (Continued) Options. I. Introduction What? Stochastic Integration Hedge Portfolio Now similarly the second integral: � �  E ( V ) − σ 2  ln � ∞ X 2   f ( z − σ ) dz = N σ ln X − m   σ

Options. I. Black and Scholes Introduction Solving for the expectation is not the contribution of Black What? and Scholes and Merton. The key to option pricing (and Stochastic Integration finance) is ascertaining E ( V ) and σ in the two normal Hedge Portfolio distributions obtained in the preceding slides. Black and Scholes set up a hedge portfolio that entails continual rebalancing the stock and option to replicate a riskless asset. Consider a position that is long ∆ shares of the underlying stock and short one call option. The law of motion for this position is: ∂ 2 C ∂ t dt − 1 − ∂ C ∂ S 2 σ 2 S 2 dt 2 (Since the stock and otion share the same Brownian motion and the position in the stock cancels this term out of the hedge portfolio.)

Options. I. Black and Scholes (Continued) Introduction What? Stochastic Because this position entails investing C − ∂ C ∂ S S , it must be Integration Hedge Portfolio that: ∂ 2 C � � − ∂ C ∂ t dt − 1 C − ∂ C ∂ S 2 σ 2 S 2 dt = ∂ S S dt · r 2 where r is the instantaneous risk-free rate. This sets the stage for the famous Black and Scholes stochastic differential equation: 2 σ 2 S 2 ∂ 2 C ∂ C ∂ t + rS ∂ C ∂ S + 1 ∂ S 2 = rC