Modeling the Stock Price Process As a Continuous Time Discrete Jump Process

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 St Stock Price St. Tick-size c = 16 Nt = c We suppose that there is a risk free interest rate ρt. Nt 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(Yt) of size i with probability pi. Rate of events per individual is λt.

Wait for an exponential time with parameter Ntλt and then jump Yt. How to get the correct rate and jump probabilities? Equate the drift and volatility of this model to those of the Black-Scholes model:

P d < S >t= EYtNtλt = St( ipi)λt

P 2 d < S, S > = EY 2N λ = S2( i pi )λ t t t t t Nt t

P p = 1, P ip = rt , P i2p = N −→ Gives B-S with λ = σ2 i i λt i t t t

P p = 1, P ip = rt , P i2p = 1 −→ Gives Cox-Ingersoll-Ross model i i λt i c

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 τ1, τ2 . . . τk h i−1 ˆ Pk Si−1 Sk By IV.4.1.5 of Andersen et al (1993) λNP MLE = k i=1 c (τi − τi−1) + c (T − τk) r Quasi Likelihood estimators for ν = λ are obtained as solutions of estimating equations of the form n X wν(Xi−1)(Xi − mν(Xi−1)) = 0 i=1 Wefelmeyer Annals of Stat(1996) In our case, the estimator with minimum variance becomes

Pn Xi i=1( − 1) νˆ = Xi−1 Pn ( 1 ) i=1 Xi−1 5. Introducing rare big jumps

The jumps come from two competing processes:

• Small jumps with rate=Ntλt, E(Y ) = rt ,E(Y 2) = N t λt t t

• Big jumps with rate=µt, 2 2 E(Yt) = aNt,E(Yt ) = bNt

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. −rT γ(Q) = EQ[e f(ST )] Assume f is convex, 0 ≤ f(x) ≤ x for all x > 0 MT is the class of measures which converge to B-S. −rt Under each Q ∈ MT , e St is a martingale. r(T −t) Since f in convex, the process At = f(e St) is a Q-submartingale. So

−rT −rT rT γ(Q) = e EQ[AT ] ≥ e f(e S0)

−rT −rT We have e f(ST ) < e ST . So

−rT γ(Q) < EQ[e ST ] = S0 7. An Example

S0 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

Mt=Number of jumps in time 0 to t For each value m of MT For each i ≤ m

For each value x of NTi−1

K (x − c )+ for i = m fi(x) = maxE(fi+1(x + Yi+1) | MT = m, NTi = x) for 0 ≤ i ≤ m − 1

The maximum value is f0(N0) The problem reduces to:

• Get the distribution of MT

• Maximize E(fi+1(x + Yi+1) | MT = m, NTi = x) over the distribution on Yi+1 9. Distribution of jump size

Let py = P (Yt = y | NTi−1 ) We have the constraints: P P r P 2 py = 1 ypy = λ y py = NTi−1 P y fi(l+Yt)py P (MT =m|NTi−1 =l,Yt=y) E(fi(l + Yt) | MT = m, NTi−1 = l) = P y py P (MT =m|NTi−1 =l,Yt=y) R T P (MT = m | l, y) = 0 qi,l,y (t)QT −t,l+y (m − i)dt where Qt,l(j) = P (Mt = k | N0 = l) and qi(t) is the conditional density of Ti given NTi−1 = l, Yt = y. We have to maximize the ratio of 2 linear functions of py 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

1 Date:June1,2001. Tick= 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 1 Date:Dec 2,2000. Tick= 100 . n=12. Range of option prices $0-$1.60. Avg bid ask spread=50. Figure 3: Estimation: Ford Data 1 Date:June1,2001. Tick= 16 . n=34. Range of option prices $0-$11.25. Avg bid ask spread=2. Figure 4: Prediction: Ford Data

1 Date:Dec,2000. Tick= 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

1 Date:June,2001. Tick= 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. Nt = c × St is a birth and death process. Probability that jump size Yt = 1 is pt. Such a process can be considered as a discretized version of the Black-Scholes model if 2 the intensity of jumps is proportional to Nt . 2 Consider processes with intensity λtNt .

• λt is a constant.

• λt a stochastic process and Nt is a birth and death process conditional on the λt process.

Let the measure associated with the process Nt 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

(n) (n) Nt Let us deﬁne Xt = ln( n ) Z t ∗(n) (n) (n) 1 Xt = Xt − X0 − [pu,Nu log(1 + ) 0 Nu

1 2 2 +(1 − pu,Nu ) log(1 − )]Nu σudu Nu 1 ρt where pt,Nt = (1 + 2 ). 2 Ntσt Let C be the class of functions g that satisfy the following: R P 2 (i) | gˆ(x) | dx < ∞, uniformly in C, and { u xugˆ(x), g ∈ C} is uniformly integrable (here,g ˆ is the Fourier transform of g, which must exist for each g ∈ C); or (ii) g nd g00 bounded, uniformy in C, and with g00 equicontinuous almost everywhere (under Lebesgue measure). Under assumptions (I1) and (I2), for any g ∈ C, ∗(n) Eg(XT ) = Eg(N(0, λT )) + o(1/n) (I1) There are k, k,¯ k < λT < k¯ so that (n) (n) ¯ (n) ∗(n) ∗(n) n lT − λT I(k ≤ lT ≤ k) is uniformly integrable, where lT = (X ,X )T (I2) For the same k, k¯, ∗(n) ∗(n) ¯ P(k ≤ (X ,X )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 F2(x, t),F3(x, t) be the prices of two options at time t when price of stock is cx. R t Let F1(x, t) = cx be the price of the stock and F0(x, t)B0 exp{− 0 ρsds} be price of the bond. Assume Fi are continuous in both arguments. We shall construct a self ﬁnancing risk-less portfolio

3 X (i) V (t) = φ (t)Fi(x, t) i=0

(i) (i) φ (t)Fi(x,t) P (i) Let u (t) = V (t) be the proportion of wealth invested in asset i. u = 1 Since Vt is self ﬁnancing,

3 dV (t) X dF (x, t) = u(i)(t) V (t) F (x, t) i=0 1 = u(0)(t)ρ dt + u(1)(t) (dN − dN ) t cx 1t 2t 3 X (i) + u (t)(αFi (x, t)dt + βFi (x, t)dN1t + γFi (x, t)dN2t) i=2 Vt is risk-less =⇒ (1) 1 P3 (i) u (t) cx + i=2 u (t)βFi (x, t) = 0, (1) 1 P3 (i) −u (t) cx + i=2 u (t)γFi (x, t) = 0 (0) P3 (i) No arbitrage =⇒ u (t)ρtdt + i=2 u (t)αFi (x, t) = ρt The hedge ratios are:

(2) αF2 γF2 + βF2 −1 u = [(1 − − xβF2 )(1 − )] ρ γF3 + βF3

(3) αF3 γF3 + βF3 −1 u = [(1 − − xβF3 )(1 − )] ρ γF2 + βF2

(0) 1 (2) (3) u = − (u αF2 + u αF3 ) ρt (1) (2) (3) u = −x(u βF2 + u βF3 ) 13. Stochastic Intensity

Now we consider the case where the unobserved intensity λt is a stochastic process. We ﬁrst assume a two state Markov model for λt. Naik(1993) Later we describe how we can have similar results for other models on λt.Hull and White(1987) Suppose there is an unobserved state process θt which takes 2 values , say 0 and 1. The transition matrix is Q. When θt = i, λt = λi. Counting process associated with θt is ζt.

Let us denote by {Gt} the complete ﬁltration σ(Su, λu, 0 ≤ u ≤ t) and by P the probability measure on {Gt} associated with the process (St, λt). 14. Risk-neutral distribution

Let us assume that the risk-neutral measure is the measure associated with a birth-death ∗ 2 ∗ 1 ρt ∗ process with event rate λt Nt and probability of birth pt = (1 + ∗ ) where λt is a 2 λt Nt Markov process with state space {λ1, λ2}. We get two diﬀerent values of the expected price under the two values of θ(0). The θ process is unobserved. We cannot invert an option to get θ(0) because it takes two discrete values. Need to introduce πi(t) = P (θt = i | Ft) where Ft = σ(Su, 0 ≤ u ≤ t) ˆ As shown in Snyder(1973), under any P ∈ P the πit process evolves as: dπ1t = a(t)dt + b(t, 1)dN1t + b(t, 2)dN2t where a(t) and b(t, i) are Ft adapted processes. We still need one market traded option and the stock to hedge an option. But to get the hedge ratios, we need π∗(t), a(t), b(t). • The hedge ratios involve a(t), b(t). So we need them to be predictable. But if we have to invert an option to get them, then we need to observe the price at time t to infer πt and from there to get at and bt. So they are no more predictable.

• Given a value of π(t0) , the whole process π(t), t > t0 is determined by the conditional distribution of the θ process given the observed process St. π(t) is completely determined by historical data. We do not have the freedom of determining π(t) by inverting options. Thus, inferring π(t) at each time point t independently will give incorrect prices and lead to arbitrage. 15. Bayesian Framework

As shown in Yashin(1970) and Elliott et. al.(1995), the posterior of θj(t) is given by:

Z t X πj(t) = πj(0) + qijπi(u)du 0 i Z t ¯ 2 + πj(u)(λ(u) − λj)Nudu 0 X + bj(u) 0

λ p (S →S ) ¯ P j λj u− u where λ(t) = πi(t)λi and bj(u) = πj(u−) P − 1 i i πi(u)λipλi (Su−→Su) P R t ¯ 2 Thus, aj(u) = i qijπi(u) + 0 πj(u)(λ(u) − λj)Nu Now we can hedge as in the constant intensity case with modiﬁed hedge ratios. 16. Pricing

R T − ρsds How to get E(e 0 X | F0) under ﬁxed values of Q, λ0, λ1, π0(0)? Fix θ0 = i Generate the θ process. Generate the ξ waiting times as non-homogeneous Poisson process. At each event time St jumps by ±c with probability pt and 1 − pt. Get the expectation under θ0 = i. Now take the average of these with respect to π0. 2 Generating ξ: Generate T0 from Exp(λθ0 N0 ) Let τ0 = inf{t : θt 6= θ0} If T0 < τ0, jump at time T0. Otherwise, generate T from Exp(λ N 2 ) 1 θτ0 τ0

τ1 = inf{t : θt 6= θτ0 } If T1 < τ1, jump at time τ0 + T1 Continue. This is justiﬁed by memorylessness.

We need to infer 5 parameters q01, q10, λ0, λ1, π0(0) ⇒Invert the prices of 5 options with differ- ent maturities at time 0. Using the filtering equations it is possible to infer the required quantities by using one option at 5 points of time close to 0. π0(tj) − π0(t0) = [q00π0(t0) + q10π1(t0) + π0(t0)](tj − t0) + [π0(t0) − π1(t0)]λ2 ... (∗) Another possibility of reducing the number of parameters to be estimated is to assume that the θ process is in equilibrium when we start observing. Then we need to estimate (or invert for) 4 parameters(q01, q10, λ0, λ1). The chain is irreducible and Tii ∼ Exp(q10 + q01) that is non-lattice and finite mean. q10 π0(0) = (e.g. Ross Pg214) q10+q01 17. Generalizations

According to Snyder(1973), the πi(t) process evolves as: dπ(t) = a(t)dt + b(t, 1)dN1t + b(t, 2)dN2t where a(t) and b(t, i) are Ft adapted processes. Even if we do not want to specify the model for the λ process, it is still of that form. Let us see the form of the posterior in the general case:

where λNt,ζt = g(Nt)λtpλt,Nt if ζt = Nt + 1

g(Nt)λt(1 − pλt,Nt ) if ζt = Nt − 1 0 o. w. λ∗ = E(λ (t, N , λ ) | N ) Nt,ζt Nt,ζt t0,t t t0,t

Ψt(v | Nt0,t, λt) = E(exp(iv∆λt) | Nt0,t, λt) For example, let dλt = ft(λt)dt + Gt(λt)dWt, where Wt is Brownian motion. In this case 1 2 2 Ψt(v | Nt0,t, λt) = ivft(λt) − 2 v Gt (λt). As long as the observed process is Markov jump process, the second and third terms are same as in the two state Markov case and hence yield the same function on taking inverse Fourier transform. The ﬁrst term, on taking inverse Fourier transform yields L, the Kolmogorov-Fokker-Plank diﬀerential operator associated with λ. λ p (S −→S ) R t ¯ P j λj u u dπt = L + πj(u)(λ − λj)g(S(u))du + 0 1. But then we no longer have the distribution of jump size from simple martingale considerations. We have to either assume the jump distribution, or estimate it, or impose some optimization criterion to get a unique price. Also, if the jump magnitude can take k values, we need k − 1 market traded options and the stock to hedge an option. Figure 6: Estimation: ABMD Data, general model

Figure 7: Estimation: ABMD Data, Birth Death model with Stochastic volatility

Figure 8: Error in CALL price for training sample of IBM data Figure 9: Error in CALL price for test sample of IBM data 18. Stochastic Intensity rate

• To estimate 5 parameters: λ0, λ1, q01, q10, and π. • The objective is to ﬁnd the parameter set that minimizes the root mean square error between the bid-ask-midpoint and the daily average of the predicted option price, for all options in the training sample.

• We followed a diagonally scaled steepest descent algorithm with central diﬀerence approximation to the diﬀerential. ˆ • The starting values of λ0, λ1 are taken to be equal to the value of the estimator λ obtained in the constant intensity model.

• The starting values of q01, q10 are obtained by a hidden Markov model approach using an iterative method (Ref Elliott 1995).

• We do a ﬁnite search on the parameter π.

• For the ABMD and Ford datasets, the RMSE of prediction obtained from the constant intensity method is less than the bid-ask spread.

• For IBM data, q01 , q10 and π are 8.64e-02, 1.2126 and λ0 and λ1 are 1.042039e-06 and 8.326884e-08. • The RMSE is 29.6799. Compare this to the birth death model with constant intensity (RMSE=52.7729) or Black-Scholes model with constant volatility (RMSE=46.7688).

• This is an ill-posed problem. 19. Conclusions

• Quadratic Variation is not observable

• General model is Nonparametric

• Bounds are small enough to be of practical use

• New Statistical problem of predicting an interval

• General jump models are not amenable to hedging, Birth-Death process is

• Stochastic intensity

• Combining risk neutral estimation with updating by historical data

• Need extra derivative to hedge an option