Stochastic Calculus Notes 3/5

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MSc Financial Mathematics - SMM302

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Brownian Motions

Timeline: 1828

1900 1905 1923

the Brownian Motion is introduced by the Scottish royal botanist Robert Brown in an attempt to describe the irregular motion of pollen grains suspended in liquid Louis Bachelier considers Brownian motion as a possible model for stock market values Albert Einstein considers Brownian motion as a model of the motion of a particle in suspension and uses it to estimate Avogadro’s number Norbert Wiener defines and constructs Brownian motion rigorously for the first time. The resulting stochastic process is often called the Wiener process in his honour.

Definition 1 (Brownian motion) The process W := {Wt : t ≥ 0} is called standard Brownian motion if: 1. W0 = 0 2. for s ≤ t, Wt − Ws is independent of the past history of W until time s, i.e. the Brownian motion has increments which are independent of the natural filtration Fs = σ(Wτ : 0 ≤ τ ≤ s) 3. for 0 ≤ s ≤ t, Wt − Ws and Wt−s have the same distribution, which is Gaussian D with mean zero and variance (t − s), i.e. Wt − Ws = Wt−s ∼ N(0, t − s) 4. W has continuous sample paths 1 and 3 imply that Wt ∼ N(0, t). Proposition 2 Cov(Wt , Ws ) = min(s, t) = s ∧ t Proof. Assume that s ≤ t. Then Cov(Ws , Wt ) = = = =

E [(Wt − EWt ) (Ws − EWs )] E [Wt Ws ]
E [Ws (Wt − Ws )] + E Ws2 s.

√ Exercise 1 a) Define a process by Xt := tξ where ξ is a standard Gaussian random variable. Explain why X is not a Wiener process. b) Define a process by Xt := 10Wt +100t where Wt is a standard one-dimensional Wiener process. Compute the probability of the event that X1 < −900. 0

c Laura Ballotta - Do not reproduce without permission.

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Figure 1: Sample trajectories of the Wiener process, the arithmetic Brownian motion and the geometric Brownian motion. Parameter set: T = 1 year; µ = 0.1 p.a.; σ = 0.2 p.a. √ c) Define a process by Xt := t − sξ where ξ is a standard Gaussian random variable. Explain why X is not a Wiener process. The process defined by Bt = µt + σWt is a Brownian motion with drift, i.e. Bt ∼ N(µt, σ 2 t). An important process based on the Brownian motion, which is the basis of modern mathematical finance, is the so called geometric Brownian motion (or Doleans-Dade exponential): Xt = X0 eµt+σWt ; X0 > 0. A sample trajectory of the processes Wt , Bt and Xt are shown in Figure 1. Exercise 2 Let W be a one-dimensional standard Brownian motion. Show that the following processes are Brownian motions. • Bt = −Wt , for t ≥ 0.

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3.1 The Martingale Property 1 • Bt = √ Wct , for t ≥ 0 and c > 0. c  tW1/t for t > 0 • Bt = . 0 for t = 0 Exercise 3 Let X, Y, Z be stochastic processes defined by Xt = eσWt , Yt = eσWt −

σ2 t 2 , Zt

= eµt+σWt

where W is a standard one-dimensional Wiener process and µ, σ are constants. Let Ft denote the history of the Wiener process until time t. i) Calculate E [XT ], E [YT ] and E [ZT ] for a given T ≥ 0. ii) Calculate E [XT | Ft ], E [YT | Ft ] and E [ZT | Ft ] for 0 ≤ t ≤ T. Are X, Y and Z martingales? Justify your answer. Exercise 4 Consider the processes X, Y and Z given in Exercise 2.3. Let Y =

ˆ dP ; dP

ˆ [Xt | Fs ]; E ˆ [Yt | Fs ] and E ˆ [Zt | Fs ] for 0 ≤ s ≤ t. then compute E

3.1

The Martingale Property

Proposition 3 A standard Brownian motion is a martingale. p √ Proof. E|Wt | = tE|Z|, where Z ∼ N(0, 1); this is finite (in fact, it is 2t/π). For s ≤ t, E[Wt |Fs ] = E[Wt − Ws + Ws |Fs ] = E[Wt − Ws ] + Ws = Ws

Exercise 5 Let W be a one-dimensional standard Brownian motion. Show that the process Wt2 − t is a martingale with respect to the natural filtration Ft = σ (Ws : 0 ≤ s ≤ t). Exercise 6 An analyst wishes to use a model which is based on Brownian motion, but which does not become too large and positive for large t. The model proposed is Xt = Wt e−cWt , where Wt is a standard one-dimensional Brownian motion and c is a positive constant. Verify that there is an upper bound which X never exceeds.

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Exercise 7 The evolution of a stock price S is modelled by St = eµt+σWt where Wt is a standard one-dimensional Brownian motion, µ and σ are fixed parameters and the initial value of the stock is S0 = 1. 1. Derive an expression for P (St ≤ x) . 2. Derive an expression for the median and the expectation of St . 3. Determine an expression for the conditional expectation E (St |Fu ), where u < t and F denotes the filtration associated with the process S. 4. Find conditions on µ and σ under which the process S is a martingale.

3.2

Construction of a Brownian motion

There are many constructions of Brownian motion, none easy. The approach used here refers to the limit of random walks when we take the limit of the step interval finer and finer. Consider a sequence {Xi }i∈N of i.i.d. random variables with EXi = 0, Consider the random walk Sn =

V ar(Xi ) = 1. n X

Xi .

i=1

Now define the process

S[N t] ZN (t) = √ , N where [Nt] denotes the largest integer which is
less than or equal to Nt. Thus, the process √ and jumps at k/N by an amount Xk / N . ZN (t) stays constant over the interval Nk , k+1 N

Theorem 4 As N tends to infinity, the distribution of {ZN (t) : t ≥ 0} converges to that of {Wt : t ≥ 0}.

Proof. Clearly EZN (t) = 0. P s] Xk √ For s ≤ t, we have Cov(ZN (t), ZN (s)) = V ar [N = [NNs] , which converges to s as i=1 N N → ∞. Finally we note that the Central Limit Theorem guarantees that the limiting distribution of ZN (t) for each t is N(0, t). Therefore the limiting process is a Gaussian process with the same expectation and covariance function as Brownian motion, which is enough to prove that it is a Brownian motion.

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3.3 The variation process of a Brownian motion

3.3

The variation process of a Brownian motion

Definition 5 Consider a time interval [0, T ], partitioned into n smaller sub-intervals. For p > 0, let n X (p) Wt − Wt p . V = i i−1 i=1

Then V

(p)

is called the pth variation process of W . In particular:

• for p = 1, V (1) is the total variation process; • for p = 2, V (2) is the quadratic variation process.

V (1) and V (2) represent different measures of how much Wt varies over time. From here we can gather few pieces of crucial information about the inner structure of the Wiener process, which are described in the following. Theorem 6 Let {Wt : t ≥ 0} be a standard Brownian motion. Consider a time interval [0, T ] and its partition 0 = t0 < t1 < t2 < · · · < tn = T in such a way that def τn = sup1≤i≤n (ti − ti−1 ) → 0. Then 2 p P (i) ni=1 Wti − Wti−1 −→ T as n → ∞ a.s. P (ii) ni=1 Wti − Wti−1 −→ ∞ as n → ∞, as long as τn ≤ Cn−α for some C, α > 0. Proof.

(i) Convergence in probability is hard to check. But convergence in mean square works  and implies convergence in probability. Hence we want to show that E |V (2) − T |2 → 0 as n → ∞ as long as (ti − ti−1 ) → 0. Let n X (2) ∆Wti = |Wti − Wti−1 |, so that V = ∆Wt2i 1

∆ti = ti − ti−1 , so that T = Then

n X

∆ti .

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  2  2  n n n X X X E  ∆Wt2i − T  = E  ∆Wt2i − ∆ti  1 1 1  2  n X
2  = E ∆Wti − ∆ti  1  2  n X  = E Zi  1

=

n X 1

E(Zi2 ) +

XX i

j6=i

E(Zi Zj ),

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3 BROWNIAN MOTIONS where Zi = ∆Wt2i − ∆ti .

Note that, by the independent increment property for Brownian motion, E(Zi Zj ) = E(Zi )E(Zj ) = 0 for i 6= j. Now,

h X X 2 i E V 2 − T = E(Zi2 ) = 2 (∆ti )2 i

≤ τn

i

X

∆ti = τn T,

i

which by hypothesis decreases to 0 as n → ∞. (ii) by contradiction: consider the set ( A :=

) n X Wt − Wt < ∞ . ω: i i−1 i=1

Now, consider the quadratic variation process of W , V (2) . By construction, it follows that n n X X Wt − Wt 2 ≤ max Wt − Wt Wt − Wt . V (2) = i

i−1

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i

i

i−1

i

i−1

i=1

(2) Since W is a continuous = 0, which contradicts the previous a.s.V Pn process, then result. Therefore, i=1 Wti − Wti−1 −→ ∞.

Theorem 6.(i) simply tells us that the Brownian motion accumulates quadratic variation at rate 1 per unit of time, i.e. Wt − Wt 2 i i−1 ≈ 1; ti − ti−1

informally, we can then write (dWt ) (dWt ) = 1 × dt. Note that most functions which are continuous (and have continuous derivatives) have zero quadratic variation. The paths of the Brownian motion are unusual in that their quadratic variation is not zero. This is one of the reasons why ordinary calculus does not work for Brownian motions. The other important reason is in the property of the total variation. Consider some ordinary function, continuous and differentiable, then the Mean Value Theorem says that in each subinterval [ti−1 , ti ] of our partition, there exists a t∗i such that f (ti ) − f (ti−1 ) = f ′ (t∗i ) . ti − ti−1

This implies that the total variation of the function f can be written as V (f ) =

n X i=1 T

=

Z

0

|f (ti ) − f (ti−1 )| = |f ′ (t)| dt,

n X i=1

|f ′ (t∗i ) (ti − ti−1 )|

3.4 The Reflection Principle and Functionals of a Brownian Motion

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where the last equality follows by recognizing the Riemann sum of the function f . However, for the case of the Brownian motion, we have just shown that V −→ ∞, which essentially implies that there are some problems with the quantity “|Wt′ |”. In fact, the Brownian motion is not differentiable, as we will show in more details in Unit 4. This is consistent with intuition. In fact, the differential of a function tells you the angle of the tangent to the curve in any specific point. But if you are able to calculate such a quantity, you know where the function is going to be in the next dt period of time. However, you should not forget that the function in our case is represented by a random process. Since it is random, you cannot predict the outcome in the next period of time. Other variation processes that are of interest for us are the following:


P • Cross Variation: limn→∞ ni=1 Wti − Wti−1 (ti − ti−1 ) = 0, due to the fact that the Brownian motion is a continuous process. Informally, we can write (dWt ) (dt) = 0;

• limn→∞

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i=1

|ti − ti−1 |2 = 0, which is trivial, and implies (dt) (dt) = 0.

The Reflection Principle and Functionals of a Brownian Motion

Definition 7 (First passage time) For a Brownian motion we define the first passage time to a ∈ R to be the stopping time Ta given by Ta = inf{t ≥ 0 : Wt = a}, with the condition Ta = ∞ if Wt never hits a.

Proposition 8 (The Reflection Principle) Let W be a standard Brownian motion and let Ta be as above. Then, for any x ≤ a, P[Ta < t ∩ Wt < x] = P[Ta < t ∩ Wt > 2a − x]

Proof. By conditioning on the value of Ta . Let fa denote the probability density

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= P[Ta < t ∩ Wt > 2a − x].

Proposition 9 Let {Wt : t ≥ 0} be a standard Brownian motion and Ta the first passage time to a. Then P[Ta < t] = 2P[Wt > a] and Ta is almost surely finite. Proof. For a > 0, P[Ta < t] = P[Ta < t ∩ Wt ≥ a] + P[Ta < t ∩ Wt < a].

3.4 The Reflection Principle and Functionals of a Brownian Motion

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Since {Wt ≥ a} implies that {Ta < t}, it follows that P[Ta < t ∩ Wt ≥ a] = P[Wt ≥ a]. In addition, the Reflection Principle tells us that P[Ta < t ∩ Wt < a] = P[Ta < t ∩ 2a − Wt < a] = P[Ta < t ∩ Wt > a], which is also equal to P[Wt ≥ a]. Therefore P[Ta < t] = 2P[Wt ≥ a]. Now Wt ∼ N(0, t), from which it follows that Z ∞ Z ∞ 1 1 −y2 /2 2 √ e−y /2 dy = 1 P[Ta > t] = 2 √ √ e dy → 2 2π 2π a/ t 0 as t → ∞. Lemma 10 Let {Wt : t ≥ 0} be a standard Brownian motion and define the running maximum of W to be M0t = sup Ws . 0≤s≤t

Then

 a ≤ a] = 2N √ − 1, t where N denotes the distribution function of the standard normal distribution. P[M0t



Proof. {M0t ≥ a} ⇐⇒ {Ta ≤ t}. Therefore      a a t P[M0 ≤ a] = 1 − P[Ta ≤ t] = 1 − 2 1 − N √ = 2N √ − 1 t t Exercise 8 Let Wt be a standard one-dimensional Brownian motion and define Bt = tW 1 t for t > 0, with B0 = 0. 1. Calculate EBt , V arBt , Cov (Bs , Bt ) for s < t. 2. Show that P (Wt < ct for all t ≥ 1) = P (Wt < c for all 0 ≤ t ≤ 1) , where c is a constant. 3. Find an expression for the value of these probabilities by stating the probability density function of M1 = sup0≤t≤1 Wt . Exercise 9 Determine the distribution function of the running minimum of a standard one-dimensional Brownian motion. The results concerning the distribution of the hitting time and the maximum/minimum are relevant for financial application as there are quite few exotic options traded on the OTC markets, which depend on these particular functionals of the Brownian motion. Examples you can think of: barrier options and lookback options.

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Correlated Brownian motions

Proposition 11 In order to construct two Brownian motions, Wt and Xt , such that Corr(Wt , Xt ) = ρ, i.e. Cov(Wt , Xt ) = ρt, we may take Wt = ρXt +

p

1 − ρ2 Zt ,

where Zt is another Brownian motion independent of Xt . Proof. Any process of the form Wt = aXt + bZt possesses the stationary, independent increment property, has continuous sample paths and starts from 0pat time t. The distribution of Wt = aXt + bZt is N (0, (a2 + b2 )t). Choosing a = ρ, b = 1 − ρ2 implies that V ar(Wt ) = t, so that Wt is a standard Brownian motion, and that Cov(Wt , Xt ) = Cov(ρXt +

p

1 − ρ2 Zt , Xt ) = ρV ar(Xt ) = ρt.

Exercise 10 Let Xt be a one-dimensional standard Brownian motion. a) Consider the process (1)

(2)

Wt = Xρt + X(1−ρ)t , where X (1) and X (2) are independent copies of the given process X. i) Show that the process Wt can be represented as Wt =

√

ρBt +

p

1 − ρZt ,

where Bt and Zt are independent standard Brownian motions. ii) Show that the process Wt is a Brownian motion. iii) Calculate Cov (Wt , Bt ) and Cov (Wt , Bs ) (1)

(2)

b) Consider two Brownian motions Wt and Wt with representation as in part (b.i), i.e. p √ (i) (i) Wt = ρBt + 1 − ρZt , i = 1, 2.   (1) (2) Calculate Cov Wt , Wt .

3.6 Simulating trajectories of the Brownian motion - part 1

3.6

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Simulating trajectories of the Brownian motion - part 1

At this point you know few many properties of the Wiener process; but what does it look like? A sample path was presented in Figure 1; how can you obtain such a trajectory? A simple way of generating numerical samples of the Brownian motion is to use the property that the increments of the Brownian motion are independent and follow a normal distribution, with mean 0 and variance equal to the length of the time period over which you observe those increments. So, even if we have shown that this is not entirely correct, we can approximate the increments of the Wiener process between time s and time t by generating a sample variate from the standardized normal distribution, and then multiply √ it by t − s. This approach is known as the sequential algorithm. This is the basis of Monte Carlo simulation; a very simple example is shown in the Excel file available on the module page in CitySpace. A more serious attempt (which in any case is based on the same principle) to generating paths of the Wiener process can be done in Matlab or in C++ and you will see how to implement this in your Numerical Methods modules. The drawback of the proposed simulation procedure is that you need to generate fairly many paths, if you want to evenly cover the full probability space and, consequently reduce the variance of your Monte Carlo estimate. In Figure 3 you can find 10 samples paths for the Wiener process, and the corresponding sample paths of the arithmetic Brownian motion and the geometric Brownian motion. If you observe carefully, you can see that these 10 trajectories are not evenly spread, but concentrate around the mean. However, depending on the final task of your simulation code (for example, approximating the price of some derivative security), the generation of a big number of paths might prove time inefficient. An alternative approach, which solves the problem noted above, makes use of stratification. The idea here is to subdivide the probability space into K strata, and then “force” your sample deviate to be in a specific stratum. The advantage of stratified sampling is shown in Figure 4 for the case of the normal distribution. Using stratification efficiently for Monte Carlo purposes, however, is not so straightforward, as it requires the knowledge of the Brownian bridge, which we will meet in the next unit. For this reason, our discussion will have to wait till then.

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Figure 3: 10 sample trajectories of the Wiener process, the arithmetic Brownian motion and the geometric Brownian motion. Parameter set: T = 1 year; µ = 0.1 p.a.; σ = 0.2 p.a.

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3.6 Simulating trajectories of the Brownian motion - part 1

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Figure 4: Alternative generation methods of random numbers.