Treball final de grau GRAU DE MATEMÀTIQUES

Facultat de Matemàtiques i Informàtica Universitat de Barcelona

AN INTRODUCTION TO STOCHASTIC MODELS

Autor: Matias Puig

Director: Dr. Jose Manuel Corcuera Realitzat a: Departament de Matemàtiques i Informàtica

Barcelona, 29 de juny de 2017

Contents

Introduction i

1 Historical Background on Stock Price Models 3

2 Stochastic Integration 7 2.1 Martingales and Brownian motion ...... 7 2.2 Integral of mean square integrable processes ...... 9 2.3 Extension of the integral ...... 12

3 Fundamental Theorems 17 3.1 Itô’s formula ...... 17 3.2 Stochastic differential equations ...... 18 3.3 Girsanov’s theorem ...... 19 3.4 Martingale representation theorem ...... 21

4 The Black-Scholes Model 25 4.1 Risk-neutral measure ...... 26 4.2 Arbitrage and admissibility ...... 28 4.3 Completeness ...... 30 4.4 Pricing and hedging ...... 32 4.4.1 Pricing a put ...... 33 4.4.2 Pricing a ...... 34 4.4.3 Hedging ...... 35

5 Stochastic Volatility 39 5.1 Empirical motivations ...... 39 5.2 A general approach for pricing ...... 39 5.3 Heston’s model ...... 42

6 Conclusions 47

7 Bibliography 49 Bibliography: Articles ...... 49 Bibliography: Books ...... 49 Bibliography: Online ...... 50

i

Introduction 1

The main goal of this work is to introduce the stochastic volatility models in mathematical finance and to develop a closed-form solution to option pricing in Heston’s stochastic volatiltiy model, following the arguments in Heston 1993.

No background in mathematical finance will be assumed, so another main goal of this work is to develop the theory of stochastic integration and to introduce the Black-Scholes market model, the benchmark model in mathe- matical finance. Standard topics in the framework of market models, such as trading strategies, completeness and replication, and the notion of arbitrage, will also be reviewed. 2 Introduction Chapter 1

Historical Background on Stock Price Models

Louis Bachelier, in his thesis "Théorie de la Spéculation", made the first contri- bution of advanced mathematics to the study of finance in 1900. This thesis was well received by academics, including his supervisor Henry Poincaré, and was published in the prestigious journal Annales Scientifiques de l’École Normale Supérieure. In this pioneering work, the Brownian motion is used for the modelling of movements in stock prices. In the words of Louis Bachelier, in Bachelier 1900:

La détermination des mouvements de la Bourse se subordonne à un nombre infini de facteurs: il est dès lors impossible d’en espérer la prévision mathématique, [...] et la dynamique de la Bourse ne sera jamais une science exacte. Mais il est possible d’étudier mathématiquement l’état statique du marché à un instant donné, c’est-à-dire d’établir la loi de proba- bilité des variations de cours qu’admet à cet instant le marché. Si le marché, en effet, ne prévoit pas les mouvements, il les consid- ère comme étant plus ou moins probables, et cette probabilité peut s’évaluer mathématiquement.

Bachelier argued that, over a short time period, fluctuations in price are inde- pendent of the current price and the past values of the price, and that these fluctuations follow a zero mean normal distribution with variance propor- tional to the time difference. He also assumed that the prices are continuous, therefore modelled as a Brownian motion (see Bachelier 2011).

Many years later, in the famous article by Black and Scholes, Black and Sc- holes 1973, prices are modeled as a geometric Brownian motion, whose fluctua- tions have a lognormal distribution. This model is based on the assumption that the log returns of a stock price are independent and normally distributed, with variance proportional to the time difference. The log returns are defined as log(p ) log(p ), where p and p denote the prices at times i and j, respec- i j i j tively, with i > j.

The returns of a stock price are defined as the increments

pi pj ri = pj Hence, the approxamation

3 4 Historical Background on Stock Price Models

Figure 1.1: Sample path of a standard Brownian motion, used by Bachelier to model fluctuations in stock prices

log(1 + r) r, when r 1 (1.1) ⇡ ⌧ Gives the approximation

pi pj = ri log(1 + ri)=log(pi/pj)=log(pi) log(pj) (1.2) pj ⇡ So that the lognormal distribution for the stock price increments proposed by Black and Scholes obeys to the intuitive idea that the price returns are inde- pendent and normaly distributed.

Explicitely, as stated in Black and Scholes 1973, the model proposed by Black and Scholes relies on the following assumptions of an "ideal" market:

(a) The interest rate is known and constant through time

(b) The distribution of stock prices at the end of any finite interval is lognor- mal

(c) The stock pays no dividends

(d) The variance rate of the return on the stock is constant

(e) The stock price is continuous over time

Empirical observations of stock price distributions have motivated numerous extensions of the Black-Scholes model in which one or more of the previ- ous assumptions are relaxed. One of the main criticisms of the Black-Scholes model is that the normal distribution of stock price returns does not explain the significant presence of outliers in the distribution of returns (see, for ex- ample, Mandelbrot 1963). Conveniently, relaxing some of the previous as- sumptions results in distributions with a higher presence of outliers, or fatter 5

Figure 1.2: Sample path of a Geometric Brownian motion (with µ = s = 1), used by Black and Scholes to model stock prices tails, which better adjust to the reality of observed prices.

Examples of such extensions include models with dividend payments, stock prices with jumps (non-continuous over time), and models in which the distri- bution of returns is non-Gaussian, among others. These extensions are widely used by practitioners and are described in most text books on mathematical finance (see, for example, Musiela and Rutkowski 2006).

In this project we will concentrate on the stochastic volatility models, in which the assumption that the variance rate of the return is constant is relaxed. 6 Historical Background on Stock Price Models Chapter 2

Stochastic Integration

Let (W, , P) be a probability space, and T an index set. Consider a func- F tion X(t, w) having two arguments in T and W. By fixing t, X(t, ) is · t T a family of random variables defined in W. On the other hand, by fixing2 w, X( , w) can be seen as a family of ’random maps’. We will define a · w W stochastic process2 following the first point of view and show the equivalence between the two.

Definition 2.1 (). Let E be a metric space with the borel s-field. A stochastic process is a family of E-valued random variables (Xt)t T defined in a 2 probability space (W, , P) F In order to define a random map as a function-valued random variable, we must specify the function space and a s-field in this space. The most natural function space to consider is the space ET of all maps from T to E. The s-field will be the product s-field, that is the one generated by cylinder sets of G 1 1 the form p (B1) ... p (Bn), with B1,...,Bn (E) and pt being the t1 \ \ tn 2 B natural projection defined by the rule f f (t). 7!

Let (Xt)t T be a stochastic process and define Y(w)(t)=Xt(w). Is the 2 map Y : (W, , P) (ET, ) -measurable? Indeed, for any cylinder set 1 F ! 1 G G 1 1 1 C = p (B1) ... p (Bn) , Y (C)=X (B1) ... X (Bn) . t1 \ \ tn 2 G t1 \ \ tn 2 F Since is generated by the cylinder sets, we conclude that Y is -measurable. G G On the other hand, if Y : (W, , P) (ET, ) is -measurable, for any F ! G G t T and w W define Xt(w) := Y(w)(t). Then, for any B (E), 21 2 1 1 2 B X (B)= w X (w) B = w Y(w)(t) B = Y (p (B)) . t { | t 2 } { | 2 } t 2 F Given the equivalence between both definitions, we will consider E-valued stochastic processes on T as any of the definitions above. A sample path of X is a function x(t)=X(t, w). We say that a E-valued stochastic process on T has paths in U ET if its sample paths are included in U. A stochastic process X ⇢ is said to be continuous if its paths are included in (T), the set of continuous C functions. The process X is continuous at t T if its sample paths x(t) are 0 2 continuous at t0 almost surely. Two stochastic processes X and Y are said to be versions of each other if P(X = Y )=1 for all t T. The finite dimensional t t 2 distributions of a stochastic process X are the joint distributions of the random vectors (X ,...,X ), t ,...,t T. t1 tn 1 n 2

2.1 Martingales and Brownian motion

We will now consider stochastic processes with the index set T = R+, and with values on R.

7 8 Stochastic Integration

Definition 2.2 (Filtration, adapted process). A filtration ( t)t 0 is an increasing F family of s-algebras included in the s-algebra . A process is said to be adapted to F the filtration ( t)t 0 if for each t, Xt is t measurable. F F

Given a process (Xt)t 0, we can define the natural filtration t = s( Xs, s F {  t ), for which the process X is adapted to. Moreover, it is frequent to consider } the completion of the filtration ( t)t 0- that is, the filtration in which all the F -negligible sets are -measurable. We will refer to this filtration as the one F F0 generated by the process X, without making explicit that it is the completion.

Definition 2.3 (Martingale). A process (Xt)t 0 adapted to a filtration ( t)t 0 is a F martingale if E[ X ] < • t 0 and E[X ]=X a.s for all s < t. | t| 8 t|Fs s The following result will be useful when defining the stochastic integral in the next section:

Theorem 2.4 (Doob inequality). If (Mt)t 0 is a martingale with continuous sam- ple paths, then:

2 2 E sup Mt 4E MT t T | |  | | ⇥  ⇤ Proof. Refer to Lamberton and Lapeyre 2007 (page 58, 13).

We will need some further concepts to define a Brownian motion. A stochastic process is said to have independent increments if for all 0 t < ...< t , X  1 n t1 Xt0 , Xt2 Xt1 ,...,Xtn Xtn 1 are independent. It is said to have stationary increments if for t1 < t2, Xt Xt Xt t X0. 2 1 ⇠ 2 1 Definition 2.5 (Brownian motion). A Brownian motion (also known as ) is a stochastic process (Wt)t 0 with continuous sample paths and indepen- dent and stationary increments.

The following result shows further properties of the Brownian motion which are not explicit in the definition. For a proof of this result, refer to Corcuera n.d.

Proposition 2.6. Let W be a Brownian motion. Then W W N(rt, s2t), for t t 0 ⇠ some r R. 2

A standard Brownian motion is a Brownian motion such that W0 = 0 a.e and W N(0, t). From now on, we will usually refer to a standard Brownian t ⇠ motion as a Brownian motion, without specifying that it is standard.

Proposition 2.7 (Martingale property of Brownian motion). A standard Brow- nian motion is a martingale.

Proof. Let X be a Brownian motion and s < t.

E[X ] X = E[X X ]=E[X X ]=0 t|Fs s t s|Fs t s The second equality is due to the fact that X X is independent to due t s Fs to the independent increments property.

The following is another important property of the Brownian motion: 2.2 Integral of mean square integrable processes 9

Lemma 2.8. Let (Wt)t 0 be a Brownian motion under the probability P. For any fixed M > 0,

P(max Wt(w) M)=0 t 0 

P(min Wt(w) M)=0 t 0 Proof. This is a consequence of the distribution of the first passage time of a Brownian motion. We define the first passage time T to a level M R as: M 2 T (w)=inf t 0, W (w)=M M { t } It can be proved that the first passage time distribution has the following property (see Karatzas and Shreve 2012, p.80, equation 6.2):

P(TM < t)=2P(Wt > M) 1 !t • ! Since Wt is normally distributed. This proves the result, as any level M will be reached almost surely.

2.2 Integral of mean square integrable processes

The fact that a Brownian motion is almost surely nowhere differentiable - a proof of this result can be found in Karatzas and Shreve 2012-, means that it is not possible to define the integral of a function f over a Brownian motion as T f (s) W ds, for any fixed T > 0. We will construct the stochastic integral 0 | s0| by defining it among a class of simple processes and then extend it to a larger R class (as in the definition of the Lebesgue Integral). From now on, we will consider processes with an index set [0, T].

Definition 2.9 (Mean Square Integrable Process). An adapted process (Xt)t 0 is E T 2 said to be mean square integrable if ( 0 Xt dt) < •. We will denote this class of processes as . S2 R We aim to define the integral for mean square integrable processes. By iden- tifying processes that are versions of one another and defining the natural 2 E T 2 norm X = ( 0 Xt dt), it can be shown that 2 is a Hilbert space, with k kS2 S E T the scalar product R< X, Y >= ( 0 XtYtdt). R For constructing the integral, it is necessary to identify the so-called elemen- tary processes for which the integral can be defined trivially, and so that it can be extended to the whole space. S2

Definition 2.10 (Simple Process). A process (Xt)t T 2 is a simple process if  2 S there exists a partition 0 = t0 < ... < tn = T of [0, T] for which Xt = Xti for t [t , t [. 2 i i+1

Definition 2.11. The integral of a simple process (Xt)0 t T from 0 to T is defined   as:

T I(X)T = Xt dWt := Â Xti (Wti+1 Wti ) (2.1) Z0 i

The integral process is defined as the process (I(X)t)t T, where I(X)t = I(Xs1 s t ).  {  } Note that I(X)t is given by: 10 Stochastic Integration

n 1 I(X)t = Xt (Wt t Wt t) Â i i+1^ i^ i=0 So the integral process is pathwise continuous.

Let be the set of simple processes. Note that the integral of a simple E ⇢ S2 process is a random variable. In addition, it is in L2, and the integral is an isometry between these two spaces:

Lemma 2.12 (Isometry Property for Simple Processes). The integral defines an isometry between and a subspace of L2(W, , P). E FT Proof. Let I be the integral operator. We have to prove that:

T T 2 2 2 2 I(X) L2 = E Xt dW = E Xt dt = X k k 0 0 k kS2 h Z i h Z i Now,

n 1 ( )2 = 2 ( )2 + 2 ( )( ) I X  Xti Wti+1 Wti  Xti Xtj Wti+1 Wti Wtj+1 Wtj i=0 0 i

E X X (W W )(W W ) = E X X (W W ) E W W = 0 ti tj ti+1 ti tj+1 tj ti tj ti+1 ti tj+1 tj Now, applying a similar reasoning,

2 2 2 2 2 E[X (Wt Wt ) ]=E[X ]E[(Wt Wt ) ]=E[X ](t + t ) ti i+1 i ti i+1 i ti i 1 i Because W W N(0, t t ). We obtain: ti+1 ti ⇠ i+1 i n 1 E[ ( )2]= E[ 2 ]( ) I X Â Xti ti+1 ti i=0

Since X is constant on every interval [ti, ti+1[, the expression above is equal to:

n 1 i+1 T 2 2 Â E Xt dt = E Xt dt i 0 i=0 h Z i h Z i

The following result will allow us to extend the integral to the space: S2

Lemma 2.13. The simple processes are dense in S2 The proof of this result requires several steps and will be omitted - it can be found in Karatzas and Shreve 2012 (page 134, problem 2.5). However, to give an idea on how it is accomplished, note that any continuous process (Xt)0 t T   can be approximated in ([0, T]) by simple processes Xn defined as: S2 t T2n 1 n n Xt = Â X(i2 )1]i/2n,(i+1)/2n[(t) i=0 2.2 Integral of mean square integrable processes 11

However, it is not trivial to obtain a similar result for more general adapted processes.

To extend the integral, we apply the general result that given two metric spaces X and Y and a dense subset A of X, any uniformly continuous map f : A Y can be extended in a unique way to X by defining fˆ(x)=lim f (an) ! n where an is any sequence in A converging to x, and this extension is well defined and continuous. In addition, if Y is a vector space, the extension preserves the supremum norm. In our case, we can extend the integral I : L2(W, , ) to Iˆ : L2(W, , ) and it remains an isome- E ! FT P S2 ! FT P try.

T Given a process (Xt)0 t T, we define the process I(X) by I(X)t = Xs1[0,t](s) dWs.   t 0 This process is a continuous martingale: R

( ) Lemma 2.14. [Martingale Property of the Integral] The process I X t is a mar- tingale. Proof. Consider a first case in which X is a simple process, and let s < t. Then, we can assume t = t and s [t , t [. n 2 k k+1 t n Xs dW = Â Xti (Wti+1 Wti ) Z0 i=0

k 1 n = Â Xti (Wti+1 Wti )+Xtk (Wtk+1 Wtk )+ Â Xti (Wti+1 Wti ) (2.2) i=0 i=k+1 Now, the first part of the last equation is an measurable random variable, Fs so its conditional expectation to is the same variable. Fs For the second part, using the martingale property of the Brownian motion,

E[X (W W ) ]=X E[W W ]=X (W W ) tk tk+1 tk |Fs tk tk+1 tk |Fs tk s tk Finally, for j > k, and by the tower property of the conditional expectation,

E[X (W W ) ]=E[E[X (W W ) ] ]=E[X (W W ) ]=0 tj tj+1 tj |Fs tj tj+1 tj |Ftj |Fs tj tj tj |Fs So the last term in the equation is 0. Adding the results, we obtain:

k 1 E[I(X)t s]=Â Xti (Wti+1 Wti )+Xtk (Ws Wtk )=I(X)s |F i=0 We have proven that the integral of a simple process is a martingale. To extend this result to the space, it is sufficient to show that the martingale S2 property is preserved by limits in L2. This is a consequence of the fact that the conditional expectation is continuous in L2. To prove this, recall that for X, Y in L2:

2 2 E E[X t] E[Y t] = E E[X Y t] " |F |F # " |F # ⇣ ⌘ ⇣ ⌘ E E[(X Y)2 ] = E (X Y)2  |Ft Where the inequality is ah consequence ofi Jensen’sh inequality.i 12 Stochastic Integration

( ) Lemma 2.15 (Continuity of the integral). The process I X t has almost surely continuous sample paths. The proof is based on the proof in Lamberton and Lapeyre 2007 (page 38, Proposition 3.4.4).

Proof. Let Xn be a sequence of simple processes converging to X in . Then, S2 by the Doob inequality,

T 2 n+p n 2 n+p n E sup I(X )t I(X )t 4E Xs Xs dWs t T  " Z0 # h  i ⇣ ⌘ T n+p n 2 = 4E Xs X ds 0 s n • Z0 ! ! n+p hn i 2 Thus, sup I(X )t I(X )t converges to 0 in L (W, , ), so there exists a t T F P  subsequence f(n) such that:

f(n+1) f(n) sup I(X )t I(X )t 0, a.e n • t T ! !  Hence, taking a subsequence if necessary, the simple (and continuous) pro- cesses I(Xf(n)) converge uniformly to I(X) almost surely, so X is almost surely continuous.

2.3 Extension of the integral

We would like to extend the integral to the bigger space:

T 2 = X adapted process, Xs ds < • = 1 S { P Z0 | | } If E T X 2ds < •, then the set w, T X (w) 2ds = • must have mea- 0 | s| { 0 | s | } h i T 2 sure zero,R or otherwise the expectationR of Xs ds would be infinity. So 0 | | there is an inclusion . S2 ✓ S R The extension can be achieved by using a technique called localization. For this, we will define the concept of a stopping time: a map t : W T such that ! t s for all s . Moreover, given a process X , we say that a {  } 2 Fs 2 T 2 S sequence of stopping times t is localising for X in if: n S2

1. The sequence (tn)n N is increasing 2 n 2. n, the stopped process Xt = Xt1[t t ] 2 8  n 2 S

3. lim tn = T, a.e. n • ! Let X , and consider the increasing sequence of stopping times defined 2 S by:

t 2 tn = inf t [0, T] Xs ds n 2 Z0 | | n o Defined with the convention inf ∆ = T. These are indeed stopping times. To show this, we begin by observing that t X 2ds is measurable: 0 | s| R Lemma 2.16. Given a process X , t X2ds is -measurable. 2 S 0 s Ft R 2.3 Extension of the integral 13

Proof. Considering the integral as a Riemann integral, it can be expressed as a t 2 limit of partial sums. Then it is clear that 0 Xs ds is measurable, being a limit of measurable functions almost surely. R

Now, for any t T, 2

s t 2 2 tn > t = w s t, Xu(w) du < n = w Xu(w) du < n { } { |8  Z0 | | } { | Z0 | | } Because s X (w) 2du is an increasing function of s.Sot is a stopping time. 0 | u | n R The sequence t is localising for X in . Indeed, since X , n S2 2 S T 2 a.e w W, Nw N, Xs(w) ds < Nw 2 9 2 Z0 | | n So that lim tn = T, a.e. Clearly, the process Xt = Xt1 t t is mean square n • {  n} integrable,! so we can define its integral as usual. Now, we would like to define the integral process of X as the following limit:

t t n I(X)t = XsdWs := lim X dWs,0 t T (2.3) n • s Z0 ! Z0   In order to prove that the integral is well defined, it is necessary to show that the limit exists, almost surely. We state this result in the following theorem:

Theorem 2.17. For any process X , the integral process of X as defined in 2.3 2 S exists almost surely.

The proof is based on the one by Lamberton and Lapeyre 2007. We will need the following proposition:

Proposition 2.18. Let H and t be an -stopping time. Then, 2 S2 Ft T I(H)t = 1 s t HsdWs, a.s. Z0 {  } Proof. For any H , we define 2 S2 T T t HsdWs := HsdWs HsdWs Zt Z0 Z0 Let A . The following property holds: 2 Ft T T 1A Hs1 s>t dWs = 1A HsdWs Z0 { } Zt This is clearly true for simple processes, and the property can be extended to mean-square integrable processes by a density argument.

n

Now, let t = Â ti1Ai , where all Ai are disjoint and ti -measurable. In this i=1 F case, for each i, the process 1A 1 s>t is adapted because 1A is s measurable i { i} i F if s > ti, and the process is zero otherwise. Hence,

T T n n T 1 d = 1 1 d = 1 1 d s>t Hs Ws  Ai s>ti Hs Ws  Ai s>ti Hs Ws Z0 { } Z0 i=1 { } i=1 Z0 { } 14 Stochastic Integration

n T T = = Â 1Ai HsdWs HsdWs i=1 Zti Zt It follows that:

t T HsdWs = 1 s t HsdWs Z0 Z0 {  } Now, consider an arbitrary stopping time t and define a decreasing sequence tn by:

2n 1 (k + 1)T tn = 1 kT (k+1)T Â n n t n k=0 2 { 2   2 } t Clearly, tn converges to t almost surely. Since the map t 0 HsdWs is t t 7! almost surely continuous, n H dW converges to H dW almost surely. 0 s s 0 s s R tn = T By the previous discussion, we know that 0 HsdWs 0 1 s tn HsdWs for R R {  } all n 1. Now, R R

T T 2 T E E 2 1 s tn HsdWs 1 s t HsdWs = 1 t

t T T n L2 HsdWs = 1 t HsdWs 1 t HsdWs s n n • s Z0 Z0 {  } ! ! Z0 {  } tn T In particular, a subsequence of 0 HsdWs converges almost surely to 0 1 s t HsdWs. tn t {  } This concludes the proof, because we also know that HsdWs HsdWs R 0 n • R 0 ! ! almost surely. R R t n Proof of the Theorem. To see that 0 X dW converges, note that because tn are n n+1 increasing, Xt = 1 t t Xt . The previous proposition implies that {  n} R

t t tn n ^ n+1 Xs dWs = Xs dWs Z0 Z0 Hence, on the set w, T X (w)2ds < n , { 0 s } R I(Xn) = I(Xm) for all m n (2.4) t t T 2 T 2 Since w, 0 Xs(w) ds < • = w, 0 Xs(w) ds < n and the set has { } n N{ } n 2 probabilityR 1, the sequence I(X )t convergesS R almost surely.

By construction, and by 2.4, the extended integral has almost surely continu- ous sample paths. Note that the extended integral does not necessarily have the martingale property. However, we will show that it is a local martingale.

Definition 2.19. An adapted process Xt is a local martingale if there exists a se- quence (tn)n N of increasing stopping times such that P(lim tn = T)=1 and such 2 n t that for each n, the stopped process X n (t) := Xt t is a martingale. ^ n

Proposition 2.20. Let X . Then, the integral process I(X) is a local martingale. 2 S t 2.3 Extension of the integral 15

Proof. The proof is based on the one by Capinski, Kopp, and Traple 2012 (p.140, Proposition 4.25).

Consider again the sequence of stopping times

t 2 tn = inf t [0, T] Xs ds n 2 0 | | n Z o t n Recall that I(X)t = lim Mn(t), where Mn(t) := X dWs is a martingale and n • 0 s n ! Xt = Xt1 t t . Now, {  n} R T tk n X (t)=X(t tk)=lim Mn(t tk)=lim 1[0, t ]X dWs n n t k s ^ ^ Z0 ^ t t = lim 1[0,t ]1[0,t ]XsdWs = 1[0,t ]XsdWs = Mk(t) n k n k Z0 Z0 As for n k, t t . n k The following result will be useful in the next sections:

Proposition 2.21. Let M be a non-negative local martingale. Then, M is a super- martingale. Moreover, if E[Mt]=M0 is constant, M is a martingale.

The proof is based on the one in Musiela and Rutkowski 2006 (p. 591, Propo- sition A.7.1). We will need the following result (see Musiela and Rutkowski 2006, p.580, Lemma A.1.2 for a proof):

Lemma 2.22 (Conditional Form of Fatou’s Lemma). Let (Xn)n N be a sequence 2 of random variables in a probability space (W, , P) and let be a sub-s-field of . F G F Suppose that there exists a random variable Z, such that X Z for all n, and such n that E[Z] > •. Then,

E[lim inf Xn ] lim inf E[Xn ] n |G  n |G Proof. (Proposition) Since M is a local martingale, there exists an increasing sequence of stopping times (tn)n N such that P(lim tn = T)=1 and such that for each n, the 2 n tn stopped process Mt is a martingale. Let 0 s t T. Because M is non-negative, and by the conditional form    of Fatou’s lemma,

tn tn E[Mt s]=E[lim inf M s] lim inf E[M s] |F n t |F  n t |F tn = lim inf M = Ms n s So that M is a supermartingale.

Note that in particular, E[M ] M < • for all t. t  0 Finally, in the case of constant expectation, it is clear that if E[M ] M t|Fs  s a.s and E[E[M ]] = E[M ]=E[M ], then E[M ]=M , a.s. t|Fs t s t|Fs s 16 Stochastic Integration Chapter 3

Fundamental Theorems

The standard results that follow are basic for the development of the mathe- matical finance theory.

Let (W, , F =( ), P) be a filtered probability space, and W an -Brownian F Ft t Ft motion. We begin with the following definition:

Definition 3.1 (Itô Process). An Itô process is a process that satisfies the equation:

t t Xt = X0 + asds + bsdWs Z0 Z0 Where:

1. X is -measurable. 0 F0 2. a and b are -adapted and measurable processes. t t Ft 3. T a ds < •, a.s. 0 | s| 4. R T b 2ds < •, a.s. 0 | s| Equivalently,R we say that the process satisfies the equation (in differential notation):

dXt = atdt + btdWt

Lemma 3.2. The expression of an Itô process is unique.

Proof. Refer to Capinski, Kopp, and Traple 2012 (p. 98, Theorem 3.27).

3.1 Itô’s formula

Theorem 3.3 (Itô’s formula). Let g C1,2([0, t] R, R) and (X ) be an Itô pro- 2 ⇥ t cess, with:

dXt = atdt + btdWt

Then Yt = g(t, Xt) is also an Itô process, and satisfies the formula:

1 dY = g + g a + g b2 dt + g b dW (3.1) t t x t 2 xx t x t t Where all partial derivatives of g are evaluated at(t, Xt). For a proof of Itô’s formula, refer to Capinski, Kopp, and Traple 2012 (p. 136, section 4.7). The following example will be particularly relevant:

17 18 Fundamental Theorems

Example 3.4. Let St be an Itô process such that dSt = µStdt + sStdWt, and let g(t, x)=log(x). Then, if Yt = g(t, St),

1 dY =(µ s2)dt + sdW t 2 t Or, equivalently:

1 S = exp (µ s2)t + sW t 2 t We will also require a two-dimensional version of Itô’s formula, adapted from Musiela and Rutkowski 2006.

1 2 Theorem 3.5 (Itô’s formula, two dimensions). Let (Wt ), (Wt ) be Brownian motions, g C1,2([0, t] R2, R) and (X ), (Y ) be Itô processes, with: 2 ⇥ t t

1 dXt = atdt + btdWt

2 dYt = atdt + btdWt

Then Zt = g(t, Xt, Yt) is also an Itô process, and satisfies the formula:

1 1 dZ = g + g a + g a + g b2 + g b2 + g b b dt (3.2) t t x t y t 2 xx t 2 yy t xy t t 1 2 +gxbtdWt + gybtdWt

Where all partial derivatives of g are evaluated at (t, Xt, Yt).

3.2 Stochastic differential equations

In the next chapter we will frequently consider stochastic differential equa- tions of the form:

dXt = a(t, Xt)dt + b(t, Xt)dWt (3.3)

X0 = x0 In this situation, it is important to know if a solution to the equation exists, and in that case, if it is unique. The following theorem, adapted from Capin- ski, Kopp, and Traple 2012 (p.160, Theorem 5.8), provides sufficient conditions for this to happen.

Theorem 3.6. [Existence and uniqueness of stochastic differential equations] Con- sider the stochastic differential equation 3.3, and assume that the coefficient functions a(t, x), b(t, x) are Lipschitz with respect to x and uniformly continuous with respect to t. Moreover, assume that they have linear growth. This means that there exists a constant C > 0 such that:

a(t, x) + b(t, x) C(1 + x ), x R, t [0, T] (3.4) | | | |  | | 8 2 2 Then, 3.3 has a unique solution with continuous paths and that is mean square inte- grable.

Proof. Refer to Capinski, Kopp, and Traple 2012 (p.160, Theorem 5.8). 3.3 Girsanov’s theorem 19

The natural extension of this result to multiple dimensions is also valid. That is, consider a multi-dimensional version of 3.3,

dXt = a(t, Xt)dt + b(t, Xt)dWt (3.5)

X0 = x0

Where (Xt) is a d-dimensional stochastic process (understood as d compo- nents of one dimensional stochastic processes) and Wt is a d-dimensional Brownian motion (understood as d components of one dimensional Brow- nian motions), and x Rd. Then, if the component functions a(t, x), b(t, x) 0 2 are Lipschitz with respect to x and uniformly continuous with respect to t, a unique regular solution to the stochastic differential equation exists (see Musiela and Rutkowski 2006, p.639, Theorem A.3.1).

An important property that we would like the solutions of stochastic differ- ential equations of the form 3.3 to satisfy is the Markov property.

Definition 3.7 (Markov Property). A stochastic process (Xt) in a filtered probabil- ity space (W, , F, P) satisfies the Markov property if, for any bounded measurable F function f : R R and any s t, ! 

E[ f (X ) ]=E[ f (X ) ] (3.6) t |Fs t |FXs Where ( ) is the filtration generated by (X ). FXt t In our case, ( ) is the filtration generated by the Brownian motion (W ). Ft t The following result guarantees that under certain conditions on the coeffi- cient functions, this property is satisfied by solutions of stochastic differential equations. It has been adapted from Capinski, Kopp, and Traple 2012 (p.174, Theorem 5.14).

Theorem 3.8. [Markov Property] Consider a stochastic differential equation of the form 3.3, with coefficients a(t, x) and b(t, x) that are Lipshitz continuous with respect to x, uniformly continuous with respect to t, and satisfy the linear growth condition 3.4. Then, the solution Xt has the Markov property 3.6. That is,

E[ f (X ) ]=E[ f (X ) ] t |FWs t |FXs Proof. Refer to Capinski, Kopp, and Traple 2012 (p.174, Theorem 5.14).

3.3 Girsanov’s theorem

Theorem 3.9 (Girsanov, simple version). Let (Wt) be a standard Brownian Mo- tion and g R. Then, W˜ := W + gt is a standard Brownian motion under the 2 t t probability P˜ defined by:

dP˜ 1 = exp gW g2T dP T 2 Proof. The proof has been adapted from Capinski and Kopp 2012.

It is clear that W˜ has continuous sample paths. We will prove directly that W˜ t has independent and normally distributed increments, by computing the 20 Fundamental Theorems

P˜ n ˜ ˜ probability (A) where A = i=1 Ai and Ai = Wti Wti 1 ai , for any \ {  } given partition t ,...,t of , and every a ,...,a R. By definition, 0 n T 1 n 2

1 2 P˜ (A)=E 1 = EP exp gW g T 1 P˜ A T 2 A ⇥ ⇤ n h i n Now, WT can be expressed as Âi=1 Wti Wti 1 . Similarly, T = Âi=1(ti ti 1) n and 1A = ’ 1Ai . We obtain: i=1 n 1 P˜ ( )=E ( ) 2( ) A P ’ exp g Wti Wti 1 g ti ti 1 1Ai " i=1 2 # The increments Wti Wti 1 are independent, and the indicators sets Ai can be expressed as:

Ai = Wti Wti 1 + g(ti ti 1) ai {  } So, because of the independence property,

n 1 n P˜ ( )= E ( ) 2( ) = P˜ ( ) A ’ P exp g Wti Wti 1 g ti ti 1 1Ai ’ Ai (3.7) i=1 2 i=1 h i

Now, because the increments Wti Wti 1 are distributed as N(0, ti ti 1), 1 P E 2 (Ai)= P exp g(Wti Wti 1 ) g (ti ti 1) 1Ai 2 h i 2 1 2 1 x = exp gx g (ti ti 1) exp dx x+g(ti ti 1) ai 2 2p(t t ) 2(ti ti 1) Z{  } i i 1 p 2 1 x g(ti ti 1) = exp dx 2( ) Z x+g(ti ti 1) ai 2p(ti ti 1) ti ti 1 {  } ⇣ ⌘ p 1 z2 = exp dz (3.8) 2( ) Z z ai 2p(ti ti 1) ti ti 1 {  } ⇣ ⌘ p ˜ ˜ Equation 3.8 proves that Wti Wti 1 are distributed as N(0, ti ti 1), and equation 3.7 proves that they are independent.

A generalized version of Girsanov’s theorem will be required in the Stochastic Volatility chapter. The theorem has been adapted from Musiela and Rutkowski 2006 (p.648, Theorem A.15.1).

Theorem 3.10 (Girsanov’s theorem, a generalized version). Let (Wt) be a d- dimensional Brownian motion in the filtered probability space (W, , F, P), and g be F an Rd-valued F-adapted stochastic process, such that:

T T 1 2 E exp gsdWs gs ds = 1 (3.9) " 0 2 0 # ⇣ Z Z ⌘ ˜ t P˜ Define Wt := Wt + 0 gsds, and define the probability by:

R ˜ T T dP 1 2 = exp gsdWs gs ds (3.10) dP 0 2 0 ⇣ Z Z ⌘ Then, (W˜ t) is a Brownian motion under the probability P˜ . 3.4 Martingale representation theorem 21

3.4 Martingale representation theorem

In this section, we will prove the martingale representation theorem. These re- sults have been adapted from Corcuera n.d., whose notations and arguments will be followed closely. We will need some preliminary results.

p Theorem 3.11 (Martingale L -convergence). Let (Mn)n N be a martingale in a 2 filtered probability space (W, , F =( n), P). Assume that, for some p > 1,Mn p p F F p 2 L = L (W, , P) n N, and sup Mn Lp < •. Then, for some M• L , F 8 2 n N k k 2 2 Lp Mn M• n • ! ! Proof. Refer to Kallenberg 2006 (p.109, Corollary 6.22).

Lemma 3.12. Let ( ) be a filtration in a probability space (W, , P) and let X Gn F 2 L2 = L2(W, , P). Then, F L2 E[X n] E[X •] n • |G ! ! |G Where = s( , n N). G• Gn 2 Proof. Define X := E[X ]. Clearly, X is a martingale with respect to the n |Gn n filtration ( ), and by the properties of the conditional expectation, Gn

sup Xn L2 X L2 < • n N k k  k k 2 By theorem 3.11,

L2 Xn Y n • ! ! For some Y L2. It remains to show that Y = E[X ]=: X . By the 2 |G• • continuity of the conditional expectation in L2 (this was proved in 2.14),

E[Y n]=E[ lim Xm n]= lim E[Xm n]=Xn m • m • |G ! |G ! |G By the tower property of the conditional expectation, we also have that

E[X ]=E E[X ] = X •|Gn |G• Gn n So that, for all n N, h i 2 E[Y ]=E[X ] |Gn •|Gn Hence, for every G n NGn, 2[ 2

E[Y1G]=E[X•1G] (3.11) Define the collection

= G E[Y1 ]=E[X 1 ] (3.12) C { 2 G• | G • G } Clearly, by 3.11,

n NGn • (3.13) [ 2 ⇢ C ⇢ G We wish to show that = . C G• 22 Fundamental Theorems

Now, if we prove that is a s-algebra, we will have, by equation 3.13 and the C fact that generates , that = . [nGn G• C G•

Condition (1) is clear since X n NGn . For condition (2), if B A , 2[ 2 ⇢ C ⇢ 2 C

E[Y1A B]=E[Y1A] E[Y1B]=E[X•1A] E[Y1B]=E[X•1A B] \ \ So that A B . For condition (3) note that, if (An)n N is a sequence in , \ 2 C 2 C by the dominated convergence theorem,

E[Y1 A ]=E[lim Y1 n A ]=lim E[Y1 n A ] [n n n [m=1 m n [m=1 m

= lim E[X•1 n A ]=E[X•1 A ] n [m=1 m [n n Hence, A , and = . [n n 2 C C G• Now, B := X Y > 1 , so n { • n } 2 G•

E[X•1Bn ]=E[Y1Bn ] This implies that P(B )=0 n N, so that P( B )=0. Hence X Ya.s, n 8 2 [n n •  and a similar argument shows that X Ya.s. •

Lemma 3.13. Let (W ) be a Brownian motion in the filtered probability space (W, , F, P), t F where F is generated by (W ). Consider the set of stepwise functions f : [0, T] t J ! R of the form:

n = l 1 f  i ]ti 1,ti] i=0 With l R and 0 = t < ...< t = T. For each f , define i 2 0 n 2 J T T f 1 2 T = exp f (s)dWs f (s)ds E Z0 2 Z0 n f o Let Y L2( , P), and assume that Y is orthogonal to for all f . Then, 2 FT ET 2 J Y = 0. f Proof. Let f and Y L2( , P) orthogonal to . Define := s(W ,...,W ). 2 J 2 FT ET Gn t0 tn By assumption,

n E exp  li Wti Wti 1 Y = 0 i=1 ! n o Taking the conditional by we obtain: Gn n E E exp  li Wti Wti 1 [Y n] = 0 i=1 |G ! n o Let X : W Rn be defined by: !

X =(Wt1 , Wt2 Wt1 ,...,Wtn Wtn 1 ) Decomposing Y as Y = Y+ Y , we get:

n n E E E E exp  li Wti Wti 1 [Y+ n] = exp  li Wti Wti 1 [Y n] |G ! |G ! n i=1 o n i=1 o (3.14) 3.4 Martingale representation theorem 23

Applying the pushforward measure theorem to 3.14, and because of the fact that E[Y n]=E Y 1 = a1,...,Wn Wn 1 = an : |G W a1=W1,...,an=Wn Wn 1 h n i X exp lixi E[Y+ n](x1,...,xn)dP (x1,...,xn) (3.15) Rn  |G Z n i=1 o n X = exp lixi E[Y n](x1,...,xn)dP (x1,...,xn) (3.16) Rn  |G Z n i=1 o Note that 3.15 and 3.16 are, respectively, the Laplace transforms of E[Y+ n](x1,...,xn) X |G and E[Y n](x1,...,xn) with respect to the measure P . We admit the result |G on the uniqueness of the Laplace transform, which in this case implies that:

X E[Y+ n](x1,...,xn)=E[Y n](x1,...,xn), P a.s (3.17) |G |G n So that E[Y+ n](x1,...,xn)=E[Y n](x1,...,xn) for all (x1,...,xn) R |G n X |G 2 \ A for some set A R with P (A)=0. This means that E[Y+ n](w)= ⇢ 1 |G E[Y+ ](w), w X (A). Finally, |Gn 8 2

E[Y+ ]=E[Y+ ] P a.s. (3.18) |Gn |Gn Since this is true for any as defined above, by 3.12 we have that Gn

E[Y n] E[Y s( n; n N)] = Y ± n • ± ± |G ! ! | G 2 Hence, Y = 0.

Proposition 3.14. Let (W, , F =( ), P) be a filtered probability space with = F Ft F , let (W ) be a Brownian motion and let F L2(W, , P). Then, there exists an FT t 2 FT adapted, mean square integrable process (Yt) such that

T F = E[F]+ YtdWt Z0 Proof. Consider the Hilbert space of centered random variables in L2(W, , P), H FT and its subspace consisting of the random variables of the form T Y dW , I 0 t t for some adapted, mean square integrable process (Y ). Note that proving the t R proposition is equivalent to proving that = . I H

If, on the contrary, ( , then there would exist a centered, non-trivial ran- I H dom variable Z orthogonal to . We will prove that this is not possible. 2 H I f f Indeed, suppose that such Z exists. Take Y := , with as defined in t Et Et Lemma 3.13. Then,

T f E Z t dWt = 0 · 0 E h Z i And also

T f E Z 1 + t dWt = 0 (3.19) · Z0 E h i Now, a similar argument as the one in 3.4 shows that f is the solution to the Et following stochastic differential equation:

f f d = f (t) dW Et Et t 24 Fundamental Theorems

In particular,

T f f T = 1 + f (t) t dWt (3.20) E Z0 E From 3.19 and 3.20 we conclude:

f E[Z ]=0 (3.21) ET Since this is true for any f defined as in 3.13, by 3.13 we conclude that Z = 0.

Theorem 3.15 (Martingale Representation Theorem). Let (Mt) be a square inte- grable martingale with respect to the filtration ( ), and (W ) a Brownian motion. Ft t Then, there exists an adapted, mean square integrable process (Xt) such that, in differential notation:

dMt = XtdWt, a.s

Proof. Applying the previous proposition applied to MT, we obtain the exis- tence of a mean square integrable process (Yt) such that:

T T MT = E[MT]+ YtdWt = M0 + YtdWt Z0 Z0 Since (M ) is a martingale, for any t [0, T] we have: t 2 T t Mt = E[MT t]=E M0 + YtdWt t = M0 + YtdWt |F Z0 F Z0 h t i Where we have used the fact that 0 YtdWt is a martingale (see 2.14). R Chapter 4

The Black-Scholes Model

The Black-Scholes model consists of two stocks S and b, in a time frame [0, T], where b is a bank account such that b(t)=ert, and the risky stock S satisfies the differential equation:

dSt = µStdt + sStdWt (4.1)

Where W is a Brownian motion in the filtered probability space (W, , F, P), t F and P represents the so-called empirical or physical probability. The dis- 1 counted stock price is defined as St⇤ = bt St.

Define F =( ) to be the filtration generated by the Brownian motion (W ). Ft t We assume that = ∆, W and = . The coefficients of the stochastic F0 { } FT F differential equation 4.1, according to the notations in 3.3, are

a(t, x)=µx (4.2)

b(t, x)=sx

These functions clearly satisfy the regularity conditions in 3.6, so there exists a unique mean-square integrable solution St with continuous sample paths. By theorem 3.8, this solution also satisfies the Markov property.

An explicit solution for this stochastic differential equation was found in 3.4:

1 S = S exp (µ s2)t + sW (4.3) t 0 { 2 t} The following are some implicit assumptions of this model, as argued in Cor- cuera n.d.:

it has continuous trajectories •

St Su its returns are independent of s(Ss,0 s u). Indeed, • Su  

St Su St 1 2 = 1 = exp (µ s )(t u)+s(Wt Wu) 1 Su Su { 2 }

Which is independent of s(S ,0 s u). s   its returns are stationary: •

St Su St u S0 Su ⇠ S0

25 26 The Black-Scholes Model

A trading strategy or strategy is a random vector f =(f1, f2) with values in R2, adapted to the filtration ( t)0 t T. The value of the strategy is the process F   V (t) := f (b(t), S ). f t · t A strategy is self-financing if:

1 2 dVf(t)=ft db(t)+ft dSt (4.4) 1 This is a natural extension of the discrete time expression DnVf = fnDnb + f2D S. The discounted value process is defined as V (t)=f (1, S ), where n n f⇤ t · t⇤ St⇤ = St/b(t) is the discounted stock price. For 4.4 to make sense, we need to require that:

1. T f1 dt < •, a.s 0 | t | R T 2 2 2. 0 (ft ) dt < •, a.s R We will denote be the set of self-financing strategies as F. We will discuss further restrictions on the set of strategies considered in the model later on.

The following characterization of self-financed strategies will be useful:

Lemma 4.1. Let f be a strategy satisfying integrability conditions (1) and (2). Then, f F iff 2 ˜ 2 ˜ dVt(f)=ft dSt

Proof. The proof has been adapted from Lamberton and Lapeyre 2007 (p. 65, Proposition 4.1.2).

Suppose that f is self-financing. The Itô formula gives

rt dV˜ (t)= rV˜ (t)dt + e dV (t) f f f Imposing the self-financing condition, we obtain:

rt 1 rt 2 rt 1 2 dV˜ (t)= re (f e + f S )dt + e (f db(t)+f dS ) f t t t t t t

2 rt rt 2 = f ( re S dt + e dS )=f dS˜ t t t t t The converse can be proved similarly.

4.1 Risk-neutral measure

A risk-neutral measure is a measure P˜ equivalent to P such that the discounted stock price is a martingale under P˜ . The following results guarantee the exis- tence and uniqueness of a risk-neutral measure. We begin with the following lemma:

Lemma 4.2. Given a Brownian motion (W ), the process M := S exp 1 s2t + t t 0⇤ { 2 sW is a martingale. t} 4.1 Risk-neutral measure 27

Proof. For any s < t,

1 E M /M = E exp s2(t s)+s(W˜ W˜ ) t s|Fs { 2 t s }|Fs h i h i 1 = E exp s2(t s)+s(W˜ W˜ ) (because W˜ W˜ ) {2 t s } t s?Fs h i Now,

1 E exp s2(t s)+s(W˜ W˜ ) {2 t s } h i 1 1 x2 = exp s2(t s)+sx exp dx R ( ) Z 2 2p(t s) 2 t s n o p 1 (x s(t s))2 = exp dx = 1 R 2p(t s) 2(t s) Z n o p

Proposition 4.3. A risk neutral measure exists, defined by:

dQ µ r 1 (r µ)2 = exp WT⇤ T (4.5) dP t s 2 s2 ⇣ ⌘ ⇣ ⌘ µ r Where the process Wt⇤ := Wt + s t is a Brownian motion under P.

Proof. In view of Girsanov’s theorem, we seek a value g such that by defin- ing W := W + gt, the stock price is a martingale under the probability , t⇤ t Q defined as in Girsanov’s theorem. The stock price evolves according to:

dS⇤ =(µ r + gs)S⇤dt + sS⇤dW⇤ t t t t It is a martingale under Q iff the drift term µ r + gs = 0.

Indeed, let r = µ r + gs. Then,

1 2 1 2 S⇤ = S⇤exp (r s )t + sW⇤ = exp rt S⇤exp s t + sW⇤ t 0 { 2 t } { } 0 {2 t } The previous lemma implies that exp 1 s2t + sW is a martingale under { 2 t⇤} Q, which proves the result.

Remark. The stock price St evolves according to the equation:

dSt = rStdt + sStdWt⇤ (4.6)

Theorem 4.4. A unique neutral measure Q exists.

Proof. Refer to Capinski and Kopp 2012 (p. 51, Theorem 3.12) 28 The Black-Scholes Model

4.2 Arbitrage and admissibility

An arbitrage opportunity is an opportunity to have positive returns in an in- vestment with no risk. More specifically, an arbitrage strategy is defined as following:

Definition 4.5. An arbitrage strategy is a self-financing strategy f =(f1, f2) such that V (0)=0,V (T) 0, a.s and P(V (T) > 0) > 0. f f f The following example shows that the market model described above, with the set of self-financing strategies, has arbitrage opportunities. In order to eliminate these opportunities, it will be necessary to restrict the set of so- called admissible strategies, introduced below.

Theorem 4.6. Arbitrage opportunities exist within the class of self-financing strate- gies.

This example is based on the suicide strategy described in Capinski and Kopp 2012 (p.24-27).

We begin by considering the strategy f given by:

2 1 ft = sS˜ pT t t The risk free component is determined by the self-financing condition and an (arbitrary) initial value. Now, the strategy is not almost surely square integrable, but it will be modified later on. Firstly, we will study its properties.

Lemma 4.7. For each M 0, P(min t : V˜ (t) V˜ (0) M T)=1 { f f } 

P(min t : V˜ (t) V˜ (0) M T)=1 { f f  }  Proof. ˜ 2 ˜ 2 ˜ 1 dVf(t)=ft dSt = ft sStdWt⇤ = dWt⇤ pT t So that:

t ˜ ˜ 1 Vf(t) Vf(0)= dWu⇤ 0 pT u Z Now, let

t 1 g(t)= du 0 pT u Z Then, V˜ (t) V˜ (0) and W (g(t)) have the same distribution. A proof of this f f ⇤ result can be found at Capinski and Kopp 2012 - here we will focus on its consequences. Recall that - see Lemmma 1.8, for any fixed M > 0:

Q(max Wt⇤ M)=0 t 0 

Q(min Wt⇤ M)=0 t 0 And since P and Q are equivalent, 4.2 Arbitrage and admissibility 29

P(max Wt⇤ M)=0 t 0 

P(min Wt⇤ M)=0 t 0 Clearly, g(t) •, so that !t T !

P(max V˜f(t) M)=P(max W⇤(g(t)) M)=P(max W⇤(t) M)=0 t T  t T  t 0   

P(min V˜f(t) M)=P(min W⇤(g(t)) M)=P(min W⇤(t) M)=0 t T t T t 0  

Proof. (Theorem)

Let V˜f(0)=1 and take M = 1, so that:

P(min t : V˜ (t) 0 T)=1 { f  }  Let

t = min t : V˜ (t)=0 T, a.s { f }  Define the self-financing strategy q as:

2 2 qt = ft 1 t t {  } And such that V (0)=1. Since t T almost everywhere, it is clear that q  T 2 2 (qt ) dt < •, a.e Z0 So that q F. This suicide strategy begins with a positive wealth and ends 2 almost surely in bankrupcy. As we will see, this cannot be admissible. In- deed, we can construct a new strategy from q that is an arbitrage opportunity. Define:

c1 = q1 + 1 t t c2 = q2 t t Then V (0)= V (0)+1 = 0 and V (T)= V (T)+erT = erT. Clearly c c q c q 2 F, because, just like q, it is self-financing and satisfies integrability conditions. The previous properties of c show that it is an arbitrage opportunity.

We will now define which strategies are considered admissible in the model. Although several alternatives have been proposed in the literature, here we follow the definition from:

Definition 4.8. A self-financing strategy f is admissible if it is bounded by below. We will denote the set of admissible strategies as F F. 0 ⇢ The next result guarantees that no arbitrage opportunities exist within the admissible strategies: 30 The Black-Scholes Model

Proposition 4.9. No admissible strategy is an arbitrage opportunity.

Proof. Suppose that f F is an admissible strategy and an arbitrage oppor- 2 0 tunity - we will argue by contradiction. In that case, Vf(t) is a local martingale with respect to Q which is bounded by below, so there exists a constant L such that Vf(t)+L is non-negative. Hence, Proposition 1.21 implies that Vf(t) is a super martingale under Q. In particular, E [V (T)] V (0)=0, which Q f  f combined with the fact that P(Vf(T) > 0) > 0 and that P and Q are equiv- alent, implies that P(Vf(T) < 0) > 0, which condradicts the assumption V (T) 0. f Definition 4.10 (European Option). A European option with maturity T is defined by a non-negative, -measurable random variable H = h(S ), that expresses its FT T payoff.

Definition 4.11. A strategy f F replicates the with payoff H if H = 2 0 Vf(T). The market model is complete if every European option can be replicated.

We are now able to state a fundamental result in the Black-Scholes model, relating the price process of a derivative to the value process of the replicat- ing strategy. The proof of the result will be outlined - however, substantial technical details that have been omitted can be found at Capinski and Kopp 2012.

Theorem 4.12. Let H be the payoff of a derivative which is replicated by the strategy f. Assuming that the option price is an Itô process, the No Arbitrage Principle implies that the price of the option at time t, V , is equal to V (t) for all t T. t f 2 Proof. This proof is based on the one in Capinski and Kopp 2012 (p.21, Theo- rem 2.16). Assume that this is not the case, and let t0 be any time in which a difference between Vf(t) and Vt appears with positive probability. Consider a strategy y with zero initial value and that buys the cheaper of the two and sells the most expensive short at time t0 and invests the remaining money in the bank account. The value Vy(T) of this strategy is positive with a positive probability. Indeed, assuming without a loss of generality that Vf(t0) > Vt0 ,

r(T t ) V (T)= V (t ) V e 0 V (T)+V (4.7) y f 0 t0 f T r(T t ) = V (t ) V e 0 (because V = H = V (T)) (4.8) f 0 t0 T f Which is greater than zero with positive probability. Now, if we show that y is an admissible strategy, this would violate the No Arbitrage Principle.

The strategy is self-financing by construction, and it remains to show that it is bounded by below. The reader can refer to Capinski and Kopp 2012 (p.30, Theorem 2.16) for a proof of this detail.

4.3 Completeness

We aim to prove that any square integrable European option can be replicated by an admissible strategy. The proof of the following theorem is based on the proof in Capinski and Kopp 2012:

Theorem 4.13 (A Completeness Theorem). For any square integrable European option H with respect to Q, there exists an admissible strategy f that replicates H. 4.3 Completeness 31

Proof. We will see that, in fact, such a f exists that satisfies the additional property that Vf⇤(t) is a martingale under Q, and

V (t)=b(t)V⇤(t)=b(t)EQ H⇤(t) (4.9) f f |Ft h i Where H⇤(t)=H/b(t). Hence we seek an admissible strategy f with value given by the previous expression. The martingale representation theorem guarantees the existence of an adapted, mean square integrable process X(t) such that:

d EQ H⇤(t) = X(t)dW⇤ (4.10) |Ft t ⇣ h i⌘ Returning to 4.9 and differentiating on both sides, we have:

1 2 1 2 dVf(t)=ft db(t)+ft dSt = ft rb(t)dt + ft (rStdt + sStdWt⇤) (Self-financing condition)

d b(t)EQ H⇤(t) = b(t)d EQ H⇤(t) + EQ H⇤(t) db(t) (Itô Formula) |Ft |Ft |Ft ⇣ h i⌘ ⇣ h i⌘ h i

= b(t)X(t)dW⇤ + rb(t)EQ H⇤(t) dt t |Ft h i Equating both sides and rearranging, we obtain:

1 2 2 0 = rb(t)ft + rft St rb(t)EQ H⇤(t) t dt + ft sSt b(t)X(t) dWt⇤ " |F # " # h i By the uniqueness of the expression of an Itô process,

1 2 rb(t)f + rf S rb(t)EQ H⇤(t) = 0 t t t |Ft h i f2sS b(t)X(t)=0 (4.11) t t 2 Isolating ft in the second equality gives:

b(t)X(t) f2 = (4.12) t sS(t) And substituting in the first equality gives:

1 f = EQ H⇤(t) X(t)/s t |Ft h i We have obtained a unique strategy that may attain H - now we need to verify all the conditions in the theorem. Firstly, the following calculation shows that f attains H:

V (t)=f (b(t), S )=b(t))EQ H⇤(t) b(t)X(t)/s + b(t)X(t)/s = b(t)EQ H⇤(t) f t ⇤ t |Ft |Ft h i h i In particular, Vf(T)=H.

The strategy is admissible because V (t)=E H is clearly a martingale. f⇤ Q ⇤|Ft h i For the self-financing condition, 32 The Black-Scholes Model

dVf⇤(t)=X(t)dWt⇤ Note that, from 4.12, we have:

2 X(t)=sft St⇤ So that:

2 2 dVf⇤(t)=sft St⇤dWt⇤ = ft dSt⇤

4.4 Pricing and hedging

We have shown that the Black Scholes model has no arbitrage opportunities and is complete in the sense that every european option that is square in- tegrable with respect to Q is replicable. We also know that the price of a replicable european option at any time is determined by the conditional ex- pectation of the value of the replicating strategy at the given time, under the risk neutral probability. In this section, we will develop pricing formulas for european options and to obtain the replicating strategy.

Let H be the payoff of a European option with maturity T. We will assume that H = h(S ) for some function h. For a call option h(x)=(x K)+ and T for a h(x)=(K x)+. Recall that if f is an admissible strategy that replicates H, then the value of the option at time t < T is equal to the value of the strategy f at time t, Vf(t) (see Theorem 4.12). In particular, if Vt denotes the value of the option:

rt rt rt rT r(T t) V = e V˜ (t)=e EQ[V˜ (T) ]=e EQ[e H ]=EQ[e H ] t f f |Ft |Ft |Ft (4.13) Note that H should be square-integrable with respect to Q, due to the condi- tions in Theorem 4.13. In that case, the theorem guarantees the existence of a replicating strategy f, validating the previous argument. Now, if H = h(ST), with h : R R being a bounded, measurable function, we can express V ! t as a function of St and t - the condition that h is bounded will be needed to apply the Markov property. Indeed, following the arguments in Lamberton and Lapeyre 2007 (p. 69, Remark 4.3.3),

2 r(T t) r(T t) (r s /2)(T t)+s(W⇤ W⇤) Vt = EQ[e H t]=EQ e h Ste T t t |F " F # ⇣ ⌘ Note that S is -measurable, and W W is independent of under Q. t Ft T⇤ t⇤ Ft Hence,

2 r(T t) (r s /2)(T t)+s(W⇤ W⇤) Vt = EQ e h Ste T t t " F # ⇣ ⌘

2 r(T t) (r s /2)(T t)+s(W⇤ W⇤) = EQ e h Ste T t St (because of the Markov property) " # ⇣ ⌘ 4.4 Pricing and hedging 33

2 r(T t) (r s /2)(T t)+s(W⇤ W⇤) = EQ e h xe T t St = x " # ⇣ ⌘ x=St r(T t) (r s2/2)(T t)+s(W W ) = EQ e h xe T⇤ t⇤ " # ⇣ ⌘ x=St Where the last equality is a consequence of the fact that if W W is inde- T⇤ t⇤ pendent of , then it is independent of S . Now, we can express V as: Ft t t

Vt = P(t, St) (4.14)

Where

r(T t) (r s2/2)(T t)+s(W W ) P(t, x)=EQ e h xe T⇤ t⇤ (4.15) " # ⇣ ⌘ And, since W W is distributed as (0, T t) under Q, the previous ex- T⇤ t⇤ N pectation is expressed by the following integral:

• y2/2 r(T t) (r s2/2)(T t)+sypT t e P(t, x)=e h xe dy (4.16) • p2p Z ⇣ ⌘ In the case of calls and puts, P(t, x) can be calculated explicitely, giving rise to the Black-Scholes pricing formulas. Firstly, we will need to prove that call and put options are square integrable with respect to Q. The case of a put option with payoff H = h(S )=(K S )+, this is clear since the payoff is T T bounded by K. We admit the following result, which shows that this is also the case for a call option, with payoff H = h(S )=(S K)+. T T

Lemma 4.14. The payoff H = h(S )=(S K)+ of a call option is square inte- T t grable with respect to Q.

Proof. Refer to Capinski and Kopp 2012 (p.55-56, Call options).

4.4.1 Pricing a put option We can apply the results obtained in 4.15 to the case of a put option with h(x)=(K x)+, since the function h is bounded. In this case, + r(T t) (r s2/2)(T t)+s(W W ) P(t, x)=EQ e K xe T⇤ t⇤ " # ⇣ ⌘ W W Let q = T t and z = T⇤ t⇤ , a standard normal random variable under Q. pq Then, + rq spqz s2q/2 P(t, x)=E Ke xe " # ⇣ ⌘ Define:

log(K/x) (r s2/2)q d = (4.17) spq 2 Note that Ke rq xespqz s q/2 0 z d. We obtain: () 

rq spqz s2q/2 F(t, x)=E Ke xe 1 z d " {  }# ⇣ ⌘ 34 The Black-Scholes Model

d y2/2 rq spqy s2q/2 e = Ke xe dy • p2p Z ⇣ ⌘ d y2/2 d y2/2 rq e spqy s2q/2 e = Ke dy xe dy • p2p • p2p Z Z rq The value of the first integral is Ke times the cumulative distribution func- tion of a standard normal random variable evaluated at d: N(d). For the second integral, the change of variable t = y + spq shows that its value is xN(d spq). Finally,

r(T t) P(t, x)=Ke N(d) xN(d spT t) (4.18)

Figure 4.1: Black-Scholes pricing of a put option at time t = 0, with K = 100, T = 2, r = 0.04 and s = 0.1.

4.4.2 Pricing a call option

A call option has a payoff H = h(S ), where h(x)=(x K)+. Because h is T not bounded, we can’t use the results in 4.15 to obtain a price for the option, as in the case of put options. However, the price will be obtained by means of a relationship between the price of a call option and the price of a put option for a given K, known as the call-put parity. The following theorem has been adapted from Capinski and Kopp 2012 (p.56, Theorem 3.16):

Theorem 4.15 (Call-put parity). Let Ct, Pt be the price of a call option and a put option, respectively, at time t, both with strike price K and maturity T. Then,

r(T t) C P = S Ke (4.19) t t t 4.4 Pricing and hedging 35

Proof. We follow closely the arguments and notations in Capinski and Kopp 2012 (p.56, Theorem 3.16).

Note that

S (S K)+ +(K S )+ = K (4.20) T T T This equation can be deduced by separately considering the cases S K 0 T and S K 0. Hence, T  S C + P = K (4.21) T T T rT Multiplying both sides by e ,

rT rT rT S⇤ e C + e P = Ke (4.22) T T T rt rt Now, (St⇤), (e Ct), and (e Pt) are Q-martingales (this assertion relies on the fact that call and put options are replicable), so

rT rT rT rT rt rt e = E[e ]=E[S⇤ e C + e P ]=S⇤ e C + e P |Ft T T T|Ft t t t (4.23) The result follows by multiplying both sides by ert.

The price of a call option follows from 4.18 and the call-put parity. Indeed,

r(T t) C = P + S Ke (by 4.19) t t t

r(T t) r(T t) = Ke N(d) S N(d spT t)+S Ke (by 4.18) t t

r(T t) = S (1 N(d spT t)) Ke (1 N(d)) t

r(T t) = S N( d + spT t) Ke N( d) (4.24) t In particular, the price of a call option is a function of t and St:

Ct = C(t, St) (4.25) Where

r(T t) C(t, x)=xN( d + spT t) Ke N( d) (4.26)

4.4.3 Hedging Now that the pricing formulas have been justified, we aim to obtain explicit formulas for replicating (or hedging) strategies in the same setting of a Eu- ropean option with an -measurable payoff H = h(S ) which is square FT T integrable with respect to Q. We follow the arguments in Fouque, Papanico- laou, and Sircar 2000, which begin by recalling that the value of the option is equal to:

Vt = P(t, St) (4.27)

The fact that Vt is a function of t and St was proven in 4.15 assuming that h was a bounded function. However, we saw in 4.25 that this is also true for 36 The Black-Scholes Model

Figure 4.2: Black-Scholes pricing of a call option at time t = 0, with K = 100, T = 2, r = 0.04 and s = 0.1. call options. Hence, for the following, we assume that either h is bounded or h(x)=(x K)+. Now, let f be a replicating strategy for H. Such a strategy exists by theorem 4.13. By Theorem 4.12, P(t, St) is equal to the value of the strategy Vf(t):

rt 1 2 e ft + ft St = P(t, St) (4.28) Differentiating on both sides, and applying the self-financing condition on the left side, we obtain:

1 (rertf1 + f2µS )dt + f2sS dW =(P + µS P + s2S2P )dt + sS P dW t t t t t t t t x 2 t xx t x t (4.29) Where all the partial derivatives of P are evaluated at (t, St) - we will use this abbreviation later on without further mention. By the uniqueness of the expression of an Itô process, we conclude that:

2 ft = Px(t, St) (4.30) And from 4.28 we deduce that:

1 rt f = e (P(t, S ) P (t, S )S ) (4.31) t t x t t The previous argument also shows the relationship between the pricing func- tion P(t, x) and a certain PDE (the Black-Scholes PDE). Indeed, substituting the value of the hedging strategy in 4.29, we obtain the formula:

r(P P S + P µS )dt + P sS dW (4.32) x t x t x t t 4.4 Pricing and hedging 37

1 = P + P µS + s2S2P dt + sS P dW t x t 2 t xx t x t Equating the drift terms in both expressions, we obtain the equation:

1 P rP + rP S + s2S2P = 0 t x t 2 t xx In particular, the drift terms in equation 4.32 will be equal if P satisfies the following PDE:

1 2 2 P rP + rxP + s x P = 0, t, x R+ (4.33) t x 2 xx 8 2

P(t, x)=h(x), t, x R+ 8 2 Where all the partial derivatives of P are evaluated at (t, x). 38 The Black-Scholes Model Chapter 5

Stochastic Volatility

5.1 Empirical motivations

In the Black-Scholes model,

St 1 2 = exp (µ s )(t u)+s(Wt Wu) Su { 2 } So that

St Su St 1 2 log( )=(µ s )(t u)+s(Wt Wu) Su ⇡ Su 2 Which is distributed as N µ 1 s2)(t u), s(t u) . Hence, the variance 2 rate of the returns is approximately⇣ given by the volatility⌘ . In a Stochastic Volatility framework, the stock price is modelled as:

dSt = µStdt + stStdWt (5.1)

Where st is now a process instead of a constant. Since the variance rate of re- turns in the Black-Scholes model is approximately the volatility s, the stochas- tic volatility model is indeed a generalization of the Black-Scholes model in which the variance rate of the returns is no longer assumed to be constant, as argued in Chapter 1.

Moreover, as argued in Gatheral 2011, mixing distributions with different variances produces distributions with higher peaks and fatter tails, which means that the modeling of random volatility produces distributions in re- turns that are better adjusted to the observed stock price distributions (Man- delbrot 1963). The randomness of the volatility parameter also allows us to incorporate more information into the model, such as correlation between the volatility and the stock price, or diffusion properties of the volatility such as mean reversion, which will be introduced in the . Further argu- ments and empirical motivations for the Stochastic Volatility models, such as the observation of implied volatilities, are out of the scope of this project, but can be found in detail in Gatheral 2011 and Fouque, Papanicolaou, and Sircar 2000.

5.2 A general approach for pricing

In this section, we follow closely the arguments and notations in Fouque, Pa- panicolaou, and Sircar 2000 (p. 46, section 2.5). Consider a stochastic volatility model of the form:

39 40 Stochastic Volatility

dSt = µStdt + stStdWt (5.2)

Where (St) represents the stock price, st = f (Yt) for some positive function f , and Yt is an Ornstein-Uhlenbeck process, defined by:

dY = a(m Y )dt + b dZˆ (5.3) t t t t Where a and m are constants with am > 0, and (bt) is an adapted process. Intuitively, this process pulls toward or reverts to m, known as the long-run mean level of (Yt), with a velocity a, known as the rate of mean reversion. We assume that the Brownian motion (Zˆ t) satisfies:

Zˆ = rW + 1 r2Z (5.4) t t t q Where (Z ) is a standard Brownian motion independent of (W ), and r t t 2 [ 1, 1]. This implies that the correlation between (Zˆ ) and (W ) is r. Indeed, t t

E ˆ E ˆ E Zt [Zt] Wt [Wt] E[Zˆ W ] rt ( ˆ )= = t t = = cor Zt, Wt h i r (5.5) sd(Zˆ t)sd(Wt) ptpt t

Thus, this model allows us not only to incorporate randomness into the volatility, but also to specify the correlation between the volatility and the stock price, or skewness, as argued in the previous section.

As usual, the filtered probability space is (W, , F, P), in this case being F F =( ) the filtration generated by the two Brownian motions. More ex- Ft plicitely, is generated by the sets w W W < a, F < b for s t and the Ft { 2 | s s }  P-null sets. In particular, (bt) is required to be F-adapted.

In the Black-Scholes model, we were able to obtain a pricing formula for European options with a payoff H = h(ST) which is square integrable with respect to Q after proving the existence of the unique risk neutral measure Q. In this case, we will first assume the existence of a certain risk neutral measure Q, under which the discounted stock price St⇤ is a martingale. As in the Black-Scholes model, the No Arbitrage Principle implies that the option price Vt is given by:

r(T t) V = EQ[e H ] t |Ft So, for each risk neutral measure Q, we are able to find a reasonable option price Vt. We will now find a family of equivalent risk neutral measures, by means of the multidimensional Girsanov theorem.

µ r Indeed, let q := , and define: t st

t Wt⇤ = Wt + qsds Z0 And for an arbitrary adapted, square integrable process (gt), define:

t Zt⇤ = Zt + gsds Z0 Let Qg be defined by: 5.2 A general approach for pricing 41

T T T dQg 1 = exp q dW g dZ (q2 + g2)ds (5.6) P s s s s s s d Z0 Z0 2 Z0 ! Assuming certain regularity conditions for f - for example, that it is bounded away from 0 - and according to Girsanov’s theorem in two dimensions, each measure Qg is a risk neutral probability, under which Wt⇤ and Zt⇤ are inde- pendent Brownian motions. The process (gt) is called the risk premium factor or the market price of volatility risk, and parametrizes the space of risk neutral measures. The Itô formula shows that the stochastic processes (St), (Yt), and ˆ (Zt⇤) are driven by the following dynamics under the risk neutral measure Qg:

dSt = rStdt + f (Yt)StdWt⇤, (5.7)

2 dY = a(m Y ) b (rq + g 1 r dt + b dZˆ ⇤, (5.8) t t t t t t t h q i 2 Zˆ ⇤ = rW⇤ + 1 r Z⇤ (5.9) t t t ⇣q ⌘ Now, assume that gt = g(t, St, Yt), bt = b(t, St, Yt). Then, the option price V = E [e r(T t)H ] is also a function of t, S and Y , because of the t Qg |Ft t t Markov property. Indeed,

r(T t) Vt = EQg e h ST t " F # ⇣ ⌘ r(T t) = EQg e h ST St,Yt (Because of the Markov property) " F # ⇣ ⌘ r(T t) = EQg e h ST t = s, Yt = y " S # ⇣ ⌘ s=St,y=Yt Under these assumptions, we can proceed as in the previous section to show that the function P defined by Vt = P(t, St, Yt) follows a certain PDE. Indeed,

rt rt rt d(e P(t, S , Y )) = re P(t, S , Y )) + e dP(t, S , Y ) t t t t t t And the two dimensional Itô formula gives:

rt d e P(t, St, Yt) = (5.10) rt 2 = e Pt rP + rStPs + a(m Yt) bt rqt + gt 1 r Py " h ⇣ q ⌘i

1 2 2 1 2 + f (Yt) St Pss + bt Pyy + r f (Yt)StbtPsy dt 2 2 #

rt rt ˆ +e f (Yt)StPs dWt⇤ + e nybtPy dZt⇤ ⇣ ⌘ ⇣ ⌘ Note: we frequently abbreviate P(t, s, y) as P. We will not abbreviate in cases where confusion may arise or to clarify the dependence on t, s, y.

rt In particular, e P(t, St, Yt) will be a local Qg-martingale if P(t, s, y) is a solu- tion of the following PDE: 42 Stochastic Volatility

µ r P rP + rsP + a(m y) b(t, s, y) r + g(t, s, y) 1 r2 P (5.11) t s f (y) y h ⇣ q ⌘i 1 1 + f (y)2s2P + b(t, s, y)2P + rb(t, s, y)sf(y)P = 0 2 ss 2 yy sy

5.3 Heston’s model

Heston’s model, as described in Heston 1993, is a particular case of the general model described above, in which f (y)=py and b(t, s, y)=spy. Hence, the dynamics of the processes (St) and (Yt) are give by:

dSt = µStdt + YtStdWt (5.12) p dY = a(m Y )dt + s Y dZˆ (5.13) t t t t Where, as before, Zˆ t is defined as: p

Zˆ = rW + 1 r2Z (5.14) t t t Note that 5.12 and 5.13 can be expressedq in the form 3.5, with:

a(t, s, y)=(µs, a(m y)) (5.15) b(t, s, y)=(pys, spy) In this case, the coefficients are not Lipschitz. However, 5.13 can be rewritten in the form (see Heston 1993):

d Y = b Y dt + ddZˆ (5.16) t t t Indeed, applying Itô’s formulap one canp obtain an expression or Yt similar to 5.13. In this form, the coefficients of the stochastic differential equation satisfy the regularity conditions in 3.5, so that a unique solution exists and satisfies the Markov property.

In this section, we wish to obtain a pricing function P(t, St, Yt) for a European call option with maturity T and payoff H = h(S )=(S K)+, where S is t t t modelled according to Heston’s model. Steven L. Heston, in his article Heston 1993, derives a closed-form solution for the call option price C(t, s, y) in this model. We will follow closely its arguments to achieve the desired formula.

Inspired by the results obtained in the previous section, we require that C(t, s, y) satisfies the following PDE:

C rC + rsC + a(m y) l(t, s, y) C (5.17) t s y 1 1 + ys2C + s2yC + rsysC = 0 2 ss 2 yy sy Where

l(t, s, y)=s r(µ r)+pyg(t, s, y) 1 r2 (5.18) ⇣ q ⌘ And with boundary condition C(T, s, y)=(s K)+. 5.3 Heston’s model 43

The price C(t, St, Yt) can also be expressed as follows:

r(T t) C(t, S , Y )=EQ e S K (5.19) t t g T + Ft h i r(T t) r(T t) = EQg e ST1 S K t e KEQg 1 S K t (5.20) { T } F { T } F h i h i e r(T t)S = E T r(T t) E St Qg 1 ST K t e K Qg 1 ST K t (5.21) St { } F { } F h i h i = S P KP(t, T)P (5.22) t 1 2 r(T t) ( )= r(T t) = E e ST Where P t, T e , P1 Qg S 1 ST K t , t { } F h i and P2 = EQg 1 S K t . { T } F h i Now, P(t, T)P2 is the price at time t of an option with payoff H = 1 S K . { T } Indeed,

r(T t) P(t, T)P2 = EQg e 1 S K t (5.23) { T } F h i Thus, the equation for P(t, T)P2 is exactly the same as 5.17. Consider the change of variables x = ln(s), so that:

∂ ∂ 1 = , ∂s ∂x s

∂2 ∂2 1 1 ∂ = ∂s2 ∂x2 s2 s2 ∂x Which gives the following PDE for P2:

∂P 1 ∂P ∂P 1 ∂2P 2 + r y 2 + a(m y) ly 2 + y 2 (5.24) ∂t 2 ∂x ∂y 2 ∂x2 1 ∂2P ∂2P + s2y 2 + rsy 2 = 0 2 ∂y2 ∂x∂y

With boundary condition P2(T, x, y)=1 x log(k) . { } Now, substituting C(t, s, y) in 5.17 and using the previous result, we obtain a PDE for P1:

∂P ∂P ∂P s 1 rsP + rs 1 + P + s a(m y) ly 1 ∂t 1 ∂x 1 ∂y ⇣ ⌘ ⇣ ⌘ 1 ∂2P ∂P 1 ∂P ys 1 + 2 1 + P ys 1 + P 2 ∂x2 ∂x 1 2 ∂x 1 ⇣ ⌘ ⇣ ⌘ 1 ∂2P ∂2P ∂P s2ys 1 + rsys 1 + 1 = 0 2 ∂y2 ∂x∂y ∂y ⇣ ⌘ Reordering this expression, we obtain the following PDE for P1:

∂P 1 ∂P ∂P 1 + r + y 1 + a(m y) ly + rsy 1 (5.25) ∂t 2 ∂x ∂y 1 ∂2P 1 ∂2P ∂2P + y 1 + s2y 1 + rsy 1 = 0 2 ∂x2 2 ∂y2 ∂x∂y With the boundary condition: 44 Stochastic Volatility

P1(T, x, y)=1 x log(K) { } We can summarize 5.24 and 5.25 in the expression:

2 2 2 ∂Pj ∂Pj ∂Pj 1 ∂ Pj 1 ∂ Pj ∂ Pj + r + u y + a b y + y + s2y + rsy = 0, ∂t j ∂x j j ∂y 2 ∂x2 2 ∂y2 ∂x∂y (5.26)

Pj(T, x, y)=1 x log(K) { } Where

1 u = 1 2 1 u = 2 2 aj = am, j = 1, 2 b = a + l rs 1 b2 = a + l

Now, continuing with the same notations, assume that xj, j = 1, 2, is a stochastic process that follows the stochastic differential equation:

dxj(t)= r + ujyj(t) dt + yj(t)dWt (5.27) q dy (t)= a b y (t) dt + s y (t)dZˆ j j j j j t q As argued in Heston 1993, Pj is the conditional probability that the option expires in-the-money:

P (t, x, y)=P x (T) log(K) x (t)=x, y (t)=y (5.28) j j j j ⇣ ⌘ To prove this, define

fj(t, x, y)=E 1 x (T) log(K) xj(t)=x, yj(t)=y { j } h i Assume that fj(t)= fj(t, xj(t), yj(t)) is sufficiently regular to apply the Itô formula. In this case,

∂ fj ∂ fj ∂ fj d f (t)= + r + u y (t) + a b y (t) (5.29) j ∂t j j ∂x j j j ∂y ⇣ 2 2 2 1 ∂ fj 1 ∂ fj ∂ fj + y (t) + s2y (t) + rsy (t) dt 2 j ∂x2 2 j ∂y2 j ∂x∂y ⌘ ∂ fj ∂ fj + r + u y (t) dW + a b y (t) dZˆ j j ∂x t j j j ∂y t ⇣ ⌘ ⇣ ⌘ Where all the partial derivatives of f are evaluated at (t, xj(t), yj(t)).

Now, fj is a martingale, because due to the tower property of conditional expectations and the Markov property,

E fj(t) s]= E E 1 x (T) log(K) xj(t)=x, yj(t)=y s F { j } x=xj(t),y=yj(t) F ⇥ h ⇥ ⇤ i 5.3 Heston’s model 45

E E 1 x (T) log(K) t s = fj(s) { j } F F h ⇥ ⇤ i Thus, the drift term in 5.29 must vanish. We obtain the PDE:

∂ fj ∂ fj ∂ fj + r + u y (t) + a b y (t) (5.30) ∂t j j ∂x j j j ∂y 2 2 2 1 ∂ fj 1 ∂ fj ∂ fj + y (t) + s2y (t) + rsy (t) = 0 2 j ∂x2 2 j ∂y2 j ∂x∂y

Clearly, fj must satisfy the boundary condition

fj(T, x, y)=1 x log(K) { } Hence, assuming regularity conditions on fj,

Pj(t, x, y)=E 1 x (T) log(K) xj(t)=x, yj(t)=y (5.31) { j } h i = P x (T) log(K) x (t)=x, y (t)=y j j j ⇣ ⌘ Now, applying the Itô formula to Pj(t, x, y) we obtain that it is a martingale because the drift term vanishes. Since Pj and fj satisfy the same boundary conditions and are both martingales, Pj = fj and the regularity assumption on fj is validated.

The probabilities 5.31 are not easily obtained in a closed form. However, consider the characteristic function gj defined as:

ifxj(T) gj(t, x, y; f)=E e xj(t)=x, yj(t)=y h i We admit (see Heston 1993, Appendix p.341) that gj can be expressed as:

C(T t:f)+D(T t:f)y+ifx gj(t, x, y; f)=e (5.32) Where

a 1 c edt j Cj(t; f)=rfit + 2 (bj rsfi + dj)t 2log (5.33) s 1 cj ⇣ ⌘ dt bj rsfi + dj 1 e j Dj(t; f)= (5.34) s2 1 cedjt and

bj rsfi + d c = (5.35) j b rsfi d j d = (rsfi b )2 s2(2u fi f2) (5.36) j j j To obtain the desired result,q the characteristic functions can be inverted in the following way:

• iflog(K) 1 1 e gj(t, x, y; f) Pj(t, x, y)= + Re df (5.37) 2 p 0 if Z h i The following are illustrative plots of the call option price in Heston’s model, according to the obtained closed-form solution. The underlying code has been obtained from Roberts n.d. 46 Stochastic Volatility

Figure 5.1: Heston pricing of a call option according to the closed-form solu- tion (the dashed line) at time t = 0, with K = 100, T = 2, r = 0.04, a = 0.5, r = 0.5, Y0 = 0.1, m = 0.1, s = 0.1 compared to the Black-Scholes call price (the regular line) at time t = 0, with K = 100, T = 2, r = 0.04 and s = 0.1, and the payoff function (x K)+ (the thick line).

Figure 5.2: Heston pricing of a call option according to the closed-form solu- tion (the dashed line) at time t = 0, with X0 = 80, K = 100, T = 2, r = 0.04, a = 1, r = 0.5, m = 0.1, s = 0.1 compared to the Black-Scholes call price (the regular line) at time t = 0, with K = 100, T = 2, r = 0.04 and S0 = 80. Chapter 6

Conclusions

The Black-Scholes model, which is the benchmark model in continuous-time market models, relies on a series of assumptions that were introduced in Chapter 1: the interest rate is known and constant through time, the dis- tribution of stock prices at the end of any finite interval is lognormal, the stock pays no dividends, the variance rate of the return on the stock is con- stant and the stock price is continuous over time. In this project we have mathematically formulated the Black-Scholes model, introducing the theory of stochastic integration and the fundamental results on stochastic processes as a necessary background, and taking particular care in examining the un- derlying assumptions of the model, as well as covering the topics of trading strategies, completeness, and the notion of arbitrage.

The model has been extended to incorporate stochastic volatility, particularly in the setting of Heston’s stochastic volatility model. Stochastic volatility models are one of the many extensions of the Black-Scholes model that are frequently employed by practitioners in financial markets. Other models that incorporate jumps, or non-gaussian distributions, among others, are not in the scope of this project but are equally relevant, and can be combined with the stochastic volatility framework adding more precision. We have shown that the stochastic volatility framework not only reduces the assumptions and limitations of the Black-Scholes model, but also allows to capture additional information in the model, such as correlation between the stock price and the volatility, and a mean-reverting structure for the volatility. Moreover, this so- lution captures some aspects of the empirical distribution of stock prices, such as fat tails and higher peaks, more accurately than the Black-Scholes model. This precision of the stochastic volatility models in capturing stock price dis- tributions results in option valuations that are more reliable than the ones obtained from the Black-Scholes model, as argued in Gatheral 2011. Finally, despite the fact that the incorporation of stochastic volatility adds complex- ity to the model, we obtained a closed-form solution for pricing options in Heston’s model. This closed-form solution reduces the complexity of pricing options within Heston’s model and is the cause of the high popularity of this model.

47 48 Conclusions Chapter 7

Bibliography

Bibliography: Articles

Mandelbrot, Benoit (1963). ?The variation of certain speculative prices? In: The journal of business 36.4, pp. 394–419. Black, Fischer and Myron Scholes (1973). ?The pricing of options and corpo- rate liabilities? In: Journal of political economy 81.3, pp. 637–654. Heston, Steven L (1993). ?A closed-form solution for options with stochastic volatility with applications to bond and currency options? In: Review of financial studies 6.2, pp. 327–343. Courtault, Jean-Michel et al. (2000). ?Louis Bachelier on the centenary of The- orie de la speculation? In: 10.3, pp. 339–353.

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Peter Morters, Yuval Peres (2008). Brownian Motion. url: https://www.stat. berkeley.edu/~peres/bmbook.pdf. Corcuera, Jose Manuel. Introduction to Quantitative Finance. url: http://www. ub.edu/plie/personal_PLiE/corcuera_HTML/ifqfe3.pdf. Roberts, Dale. Implementations of the Heston stochastic volatility model. url: https: //github.com/daleroberts/heston.