Modeling the Stock Price Process as a Continuous Time Discrete Jump Process Rituparna Sen Monday, Nov 8
1. Background & Existing Literature • Geometric Brownian motion Black and Scholes(1973), Merton(1973) • Stochastic Volatility Hull and White(1987), Heston(1993),Duffie et. al.(2000) • Jump diffusion Merton(1976),Bakshi and Chen(1997),Ait-Sahalia(2002) • Pure jump processes Eberlein and Jacod(1997),Madan(1999) • Discreteness Harris(1991), Brown et. al. (1991), Gottleib and Kalay(1985), Ball(1988) – Discrete time or Rounding of Continuous path
2. Overview • General discrete jump model • Introducing rare big jumps • Bounds on option price • Real Data Applications • Birth and Death Process • Hedging • Stochastic Intensity • Updating parameters by Filtering • Data Applications for Birth-Death model • Conclusions
3. The General Discrete Jump Model 1 N t = S t Stock Price S t . Tick-size c = 16 c We suppose that there is a risk free interest rate ρ t . N t is modeled as a non-homogeneous ( rate of events per individual depends on t ), linear( rate is proportional to number of individuals present ) jump process. Jumps( Y t ) of size i with probability p i . Rate of events per individual is λ t . Wait for an exponential time with parameter N t λ t and then jump Y t . How to get the correct rate and jump probabilities? Equate the drift and volatility of this model to those of the Black-Scholes model: d < S > t = EY t N t λ t = S t ( � ip i ) λ t � i 2 p i d < S, S > t = EY 2 t N t λ t = S 2 t ( N t ) λ t � p i = 1 , � ip i = r t λ t , � i 2 p i = N t − → Gives B-S with λ t = σ 2 t � p i = 1 , � ip i = r t λ t , � i 2 p i = 1 c − → Gives Cox-Ingersoll-Ross model Rate is uniquely determined. Jump distribution is not unique − → incomplete market.
4. Estimation Have to estimate 2 parameters: intensity rate λ and expected jump size ν . Suppose we observe the process from time 0 to T and the jumps occur at times τ 1 , τ 2 . . . τ k � − 1 �� k By IV.4.1.5 of Andersen et al (1993) ˆ S i − 1 c ( τ i − τ i − 1 ) + S k λ NPMLE = k c ( T − τ k ) i =1 Quasi Likelihood estimators for ν = r λ are obtained as solutions of estimating equations of the form n � w ν ( X i − 1 )( X i − m ν ( X i − 1 )) = 0 i =1 Wefelmeyer Annals of Stat(1996) In our case, the estimator with minimum variance becomes � n i =1 ( X i X i − 1 − 1) ν = ˆ � n 1 i =1 ( X i − 1 )
5. Introducing rare big jumps The jumps come from two competing processes: • Small jumps with rate= N t λ t , E ( Y t ) = r t λ t , E ( Y 2 t ) = N t • Big jumps with rate= µ t , E ( Y t ) = aN t , E ( Y 2 t ) = bN 2 t In the limit this is analogous to the Merton Jump-diffusion model. Estimation can be done by E-M algorithm regarding the data augmented with knowledge of which process it came from as the complete data.
6. Upper and Lower bounds on Option price Eberlein and Jacod Finance and Stochastics(1997) Let f be the payoff of the option. γ ( Q ) = E Q [ e − rT f ( S T )] Assume f is convex, 0 ≤ f ( x ) ≤ x for all x > 0 M T is the class of measures which converge to B-S. Under each Q ∈ M T , e − rt S t is a martingale. Since f in convex, the process A t = f ( e r ( T − t ) S t ) is a Q -submartingale. So γ ( Q ) = e − rT E Q [ A T ] ≥ e − rT f ( e rT S 0 ) We have e − rT f ( S T ) < e − rT S T . So γ ( Q ) < E Q [ e − rT S T ] = S 0
7. An Example S 0 466.77 K 450.00 r 5% T 19 days B-S Price 18.93 Upper bounds E-J 466.77 Cont time 20.38 Discrete time 21.06
8. The algorithm to compute bounds M t =Number of jumps in time 0 to t For each value m of M T For each i ≤ m For each value x of N T i − 1 ( x − K � � c ) + for i = m f i ( x ) = max E ( f i +1 ( x + Y i +1 ) | M T = m, N T i = x ) for 0 ≤ i ≤ m − 1 The maximum value is f 0 ( N 0 ) The problem reduces to: • Get the distribution of M T • Maximize E ( f i +1 ( x + Y i +1 ) | M T = m, N T i = x ) over the distribution on Y i +1
9. Distribution of jump size Let p y = P ( Y t = y | N T i − 1 ) We have the constraints: � p y = 1 � yp y = r � y 2 p y = N T i − 1 λ � y f i ( l + Y t ) p y P ( M T = m | N Ti − 1 = l,Y t = y ) E ( f i ( l + Y t ) | M T = m, N T i − 1 = l ) = � y p y P ( M T = m | N Ti − 1 = l,Y t = y ) � T P ( M T = m | l, y ) = 0 q i,l,y ( t ) Q T − t,l + y ( m − i ) dt where Q t,l ( j ) = P ( M t = k | N 0 = l ) and q i ( t ) is the conditional density of T i given N T i − 1 = l, Y t = y . We have to maximize the ratio of 2 linear functions of p y under three linear constraints. So we have a four point distribution where the maximum is attained. Can argue further that this is indeed a three-point distribution.
Figure 1: Estimation: IBM data Date:June1,2001. Tick= 1 100 . a=Distance from bid-ask midpoint to closest point of prediction interval. b=Length of interval. n=92. Units of tick. Range of option prices $0-$37. Avg bid ask spread=10.
Figure 2: Estimation: ABMD Data Date:Dec 2,2000. Tick= 1 100 . n=12. Range of option prices $0-$1.60. Avg bid ask spread=50.
Figure 3: Estimation: Ford Data Date:June1,2001. Tick= 1 16 . n=34. Range of option prices $0-$11.25. Avg bid ask spread=2.
Figure 4: Prediction: Ford Data Date:Dec,2000. Tick= 1 16 . n=578. Range of option prices $0-$16. Avg bid ask spread=2. a=4.33, b=17.14 In learning sample, a=3.27, b=19.21
Figure 5: Prediction: IBM Data Date:June,2001. Tick= 1 100 . n=1823. Range of option prices $0-$80. Avg bid ask spread=10. a=55.68, b=273.17 In Learning Sample a=21.23 b=345.76
10. Birth & Death Model Perrakis(1988), Korn et. al.(1998) Jump size ± c . N t = c × S t is a birth and death process. Probability that jump size Y t = 1 is p t . Such a process can be considered as a discretized version of the Black-Scholes model if the intensity of jumps is proportional to N 2 t . Consider processes with intensity λ t N 2 t . • λ t is a constant. • λ t a stochastic process and N t is a birth and death process conditional on the λ t process. Let the measure associated with the process N t be P . Let ξ ( t ) be the underlying process of event times. So dξ ( t ) = 1 if there is a jump at time t. dS ( t ) = cY ( t ) dξ ( t )
11. Edgeworth expansion for Option Prices = ln( N ( n ) Let us define X ( n ) n ) t t � t [ p u,N u log(1 + 1 X ∗ ( n ) = X ( n ) − X ( n ) − ) t t 0 N u 0 +(1 − p u,N u ) log(1 − 1 )] N 2 u σ 2 u du N u ρ t where p t,N t = 1 2 (1 + t ). N t σ 2 Let C be the class of functions g that satisfy the following: u x 2 � g ( x ) | dx < ∞ , uniformly in C , and { � (i) | ˆ u ˆ g ( x ) , g ∈ C} is uniformly integrable (here, ˆ g is the Fourier transform of g , which must exist for each g ∈ C ); or (ii) g nd g ′′ bounded, uniformy in C , and with g ′′ equicontinuous almost everywhere (under Lebesgue measure). Under assumptions (I1) and (I2), for any g ∈ C , E g ( X ∗ ( n ) ) = E g ( N (0 , λT )) + o (1 /n ) T (I1) There are k, ¯ k < λT < ¯ k, k so that � � l ( n ) I ( k ≤ l ( n ) ≤ ¯ k ) is uniformly integrable, where l ( n ) = ( X ∗ ( n ) , X ∗ ( n ) ) T n − λT T T T (I2) For the same k, ¯ k , P( k ≤ ( X ∗ ( n ) , X ∗ ( n ) ) T ≤ ¯ k ) = 1 − o (1 /n )
12. Hedging The market is complete when we add a market traded derivative security. We can hedge an option by trading the stock, the bond and another option. Let F 2 ( x, t ) , F 3 ( x, t ) be the prices of two options at time t when price of stock is cx . � t Let F 1 ( x, t ) = cx be the price of the stock and F 0 ( x, t ) B 0 exp {− 0 ρ s ds } be price of the bond. Assume F i are continuous in both arguments. We shall construct a self financing risk-less portfolio 3 � φ ( i ) ( t ) F i ( x, t ) V ( t ) = i =0 be the proportion of wealth invested in asset i . � u ( i ) = 1 Let u ( i ) ( t ) = φ ( i ) ( t ) F i ( x,t ) V ( t ) Since V t is self financing, 3 dV ( t ) u ( i ) ( t ) dF ( x, t ) � = V ( t ) F ( x, t ) i =0 u (0) ( t ) ρ t dt + u (1) ( t ) 1 = cx ( dN 1 t − dN 2 t ) 3 � u ( i ) ( t )( α F i ( x, t ) dt + β F i ( x, t ) dN 1 t + γ F i ( x, t ) dN 2 t ) + i =2
V t is risk-less = ⇒ cx + � 3 u (1) ( t ) 1 i =2 u ( i ) ( t ) β F i ( x, t ) = 0 , cx + � 3 − u (1) ( t ) 1 i =2 u ( i ) ( t ) γ F i ( x, t ) = 0 ⇒ u (0) ( t ) ρ t dt + � 3 i =2 u ( i ) ( t ) α F i ( x, t ) = ρ t No arbitrage = The hedge ratios are: u (2) = [(1 − α F 2 ρ − xβ F 2 )(1 − γ F 2 + β F 2 )] − 1 γ F 3 + β F 3 u (3) = [(1 − α F 3 ρ − xβ F 3 )(1 − γ F 3 + β F 3 )] − 1 γ F 2 + β F 2 u (0) = − 1 ( u (2) α F 2 + u (3) α F 3 ) ρ t u (1) = − x ( u (2) β F 2 + u (3) β F 3 )
13. Stochastic Intensity Now we consider the case where the unobserved intensity λ t is a stochastic process. We first assume a two state Markov model for λ t . Naik(1993) Later we describe how we can have similar results for other models on λ t . Hull and White(1987) Suppose there is an unobserved state process θ t which takes 2 values , say 0 and 1. The transition matrix is Q . When θ t = i , λ t = λ i . Counting process associated with θ t is ζ t . Let us denote by {G t } the complete filtration σ ( S u , λ u , 0 ≤ u ≤ t ) and by P the proba- bility measure on {G t } associated with the process ( S t , λ t ).
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