Máster en Ingeniería Computacional y Matemática
Trabajo Final de Máster
Malliavin calculus applied to option pricing theory
Marc Lagunas Merino
Director del Trabajo Final de Máster: Dr. Josep Vives Santa-Eulalia
Profesor responsable de la Asignatura: Dr. Juan Alberto Rodríguez Velázquez
19 junio 2016
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FICHA DEL TRABAJO FINAL
Malliavin Calculus Applied to Option Pricing Título del trabajo: Theory
Nombre del autor: Marc Lagunas Merino
Nombre del consultor/a: Dr. Josep Vives Santa-Eulalia
Nombre del PRA: Juan Alberto Rodríguez Velázquez
Fecha de entrega (mm/aaaa): 06/2016
Máster en Ingeniería Computacional y Titulación:: Matemática
Área del Trabajo Final: Procesos Estocásticos
Idioma del trabajo: Inglés
Stochastic Volatility, Malliavin Calculus, Palabras clave Stochastic Differential Equations.
Resumen del Trabajo (máximo 250 palabras): Con la finalidad, contexto de aplicación, metodología, resultados i conclusiones del trabajo.
El trabajo que se presenta está enmarcado dentro de la teoría de procesos estocásticos aplicados a la valoración de opciones europeas. El objetivo del documento es presentar una nueva fórmula analítica para escribir la función de densidad del activo subyacente del derivado en términos de la ya conocida función de Black-Scholes más un término de ajuste que viene determinado por la correlación y la volatilidad que dependen del modelo de volatilidad estocástica escogido.
Para lograr llegar a esta nueva expresión analítica para un modelo de volatilidad estocástica generalizado se empieza desde un marco de valoración Black-Scholes con volatilidad constante y se evoluciona hacia un modelo de volatilidad estocástica introduciendo la teoría del cálculo de Malliavin que permite diferenciar funcionales en términos de un proceso de Wiener. Acto seguido, una vez presentado el nuevo marco de valoración con volatilidad estocástica y el propio proceso de volatilidad como un promedio de la volatilidad futura del subyacente, se demuestra el teorema que caracteriza un proceso anticipativo en términos de las derivadas de Malliavin del mismo proceso. Finalmente una vez introducida esta teoría se presentan, por orden, la fórmula de Hull-White con correlación, que da una expresión del valor del derivado en términos de la función de Black-Scholes y una corrección por la correlación entre el proceso del subyacente y el proceso de volatilidad estocástica. A continuación, en el capítulo 3 se presenta el teorema de Breeden y
i
Litzenberger que relaciona el precio de una opción europea y la función de densidad del activo subyacente. Con esto, se acaba deduciendo una nueva expresión analítica para esa función de densidad en términos de la función de Black-Scholes y la corrección por correlación donde intervienen las derivadas de Malliavin y el proceso de volatilidad estocástica.
Abstract (in English, 250 words or less):
The following document belongs to the theory of stochastic processes applied to european option pricing. The goal of the whole project is deducing a new analytical formula that characterizes the underlying asset’s probability density function in terms of the well known Black-Scholes formula plus a correction term that is determined by the correlation between the underlying’s asset model and the generalized volatility model.
In an attempt to deduce the characterization, we start from a constant volatility Black-Scholes model and progressively modify it by replacing the constant volatility by a stochastic volatility model introducing the Malliavin Calculus theory, that will allow to differentiate smooth functionals with respect to the Wiener process. Once this theory is built and the volatility process is settled as an average of the underlying asset’s future volatility, we will give a proof for the Theorem that gives a characterization of an anticipating process in terms of its Malliavin derivatives.
Now the following issues are introduced in order: the correlated version of Hull- White formula that characterizes a derivative’s value in terms of the Black- Scholes formula plus a correction given by the correlation term. Finally in Chapter 3 the Breeden and Litzenberger Theorem is proved in order to relate the option’s price and the underlying’s asset probability density function. With all these tools we end up deducing the new analyitical expression that allow us to write the probability density function in terms of the Black-Scholes formula, the correlation term and the volatility’s Malliavin derivative.
ii Contents
1 Introduction 8
2 Preliminaries on Malliavin Calculus 12 2.1 Malliavin Derivatives and the Skorohod Integral ...... 12 2.2 Itˆoformula for anticipating processes ...... 14
3 The Hull and White formula. 20
4 Underlying asset pdf formula 24
A 29 A.1 Other definitions and concepts ...... 29
7 Chapter 1
Introduction
The following work is laid on the theory given by Fouque, Papanicolaou and Sircar in Derivatives in Financial Markets with Stochastic Volatility, see [6], and assuming the reader is familiar with stochastic processes, martin- gale theory, financial derivatives and Black-Scholes pricing theory. The later consists in a mathematical model of financial markets that gives investors a theoretical price of European-style options. The model was first published in the Journal of Political Economy in a paper, see [4], written by Fischer Black and Myron Scholes in 1973, named ”The Pricing of Options and Corporate Liabilities”. Robert Merton, see [10], would be the first to publish a paper on the mathematical understanding of this pricing model. This mathematical theory granted them a Nobel Memo- rial Prize in Economic Sciences for their work in 1997, which was only given to Merton and Scholes since Fisher Black had already passed away at that time.
The model assumes that stock prices St satisfy the following stochastic differential equation (SDE).
dSt = µStdt + σStdWt, (1.1)
where µ and σ are constants and Wt is a standard Brownian motion. Now under a certain arbitrage-free market hypothesis, one could solve the SDE and write the price of the call option BS in an analytical formula as
8 CHAPTER 1. INTRODUCTION 9
x −r(T −t) BS(t, x; σ) := e Φ(d+) − Ke Φ(d−), (1.2) where T − t is the option’s time to maturity, x is the logarithm of the stock price St, K is the strike price, µ should be the risk-free rate r that ensures the discounted process is a martingale and Φ is the standard normal distribution of the quantities d+ and d− which are given by
1 2 x − ln(K) + (r ± 2 σ )(T − t) d± = √ . σ T − t
Nowadays, it is widely known that if we take a collection of call options whose time to maturity is T and all are tied to the same underlying asset whose price is St, or equivalently Xt = log(St), and different strike prices, we find out that despite having different market prices, they should all have the same constant volatility under the assumptions of Black-Scholes model, but in fact these volatilities that equate the respective market prices for each call option are not the same. This is widely known as the volatility smile effect, which breaks the Black-Scholes assumption of constant volatility. A first approach to take into account the volatility smile when pricing under the Black-Scholes model is setting the volatility as a deterministic function of time, σ = σt. Under this new approach we can rewrite the under- lying SDE equation (1.1) as
dSt = µStdt + σtStdWt, (1.3)
Now for sake of simplicity in the next sections we will do an algebraic manip- ulation and write the previous expression in terms of the logarithmic price Xt = log(St) under the risk-free measure, this is µ = r, as
dSt = rdt + σtdWt, St
1 Now applying the ItˆoLemma , using Xt = f(t, St) = log(St) as an auxiliary function, we have that
1See Appendix (A.1.1) for more detail on the ItˆoLemma. CHAPTER 1. INTRODUCTION 10
1 dX = ∂ fdt + ∂ fdS + ∂ f(dS )2 (1.4) t t x t 2 xx t 1 dX = rdt + σ dW − σ2dt (1.5) t t t 2 t and integrating both sides of the previous equation we have that we can write XT as
Z T 2 Z T σs 2 XT − Xt = (r − )ds + σs dWs, (1.6) t 2 t
Now it is obvious that the following holds
1 X − X ∼ N ((r − σ¯2)(T − t), σ¯2(T − t)), (1.7) T t 2 t t where
s Z T 1 2 σ¯t := σs ds, T − t t is commonly known as the future average volatility. Now we just have to replace the previous volatility constant in the Black-Scholes equation by the future average volatilityσ ¯t. Hence the derivative’s value Vt, at time t, will be written as
Vt = BS(t, Xt, σ¯t), (1.8) since the processσ ¯t is anticipative because depends on future values of volatil- ity. This is the main reason why we need to use the Malliavin Calculus tool in order to manage anticipative processes like Yt defined by:
Z T 2 2 Yt := (T − t)¯σt = σr dr, (1.9) t CHAPTER 1. INTRODUCTION 11
This kind of analytically tractable stochastic models are covered by A. Gulisas- hvili in [8], see also [7] for an introductory lecture on stochastic volatility models. Chapter 2
Preliminaries on Malliavin Calculus
The following chapter is a brief introduction to the Malliavin Calculus. It covers basic definitions and propositions needed to manipulate stochastic anticipative proceses like volatility. We also build and prove the Hull-White formula that relates the price of an option in terms of the Black-Scholes function, plus a term that depends on the correlation and volatilities between the underlying asset and the volatility process. A more detailed lecture on topics related to Malliavin Calculus that are used in this chapter is [11].
2.1 Malliavin Derivatives and the Skorohod Integral
Let W be the canonical Wiener process, defined on the canonical space W W W 1 W (Ω , F , P ) , where Ω := C0([0,T ]) is the space of continuous functions on [0,T ], null at the origin. Denote by EW the expectation with respect to PW . Let H = L2([0,T ] , B([0,T ]), l), be a Hilbert space where l denotes the Lebesgue measure on [0,T ] and denote by
1From now on, we will use indistinctly the following notation (Ω, F, P) to refer to the canonical space adapted to the Wiener process.
12 CHAPTER 2. PRELIMINARIES ON MALLIAVIN CALCULUS 13
Z T W (h) := h(s)dWs, 0 the Wiener integral of a deterministic function h ∈ H. Now consider the family of smooth functionals of type
F = f(W (h1),...,W (hn)),
∞ n for any n ≥ 1 and f ∈ Cb (R ). Recall that a smooth functional is such that f and all its derivatives are bounded.
2.1.1 Definition. Given a smooth functional F, we define its Malliavin derivative, DF, as the element of L2(Ω × [0,T ]n) given by
n X ∂f D F = (W (h ),...,W (h ))h (t), t ∈ [0,T ] . (2.1) t ∂x 1 n i i=1 i
The operator D is a closable unbounded operator defined as
2 2 n D· : L (Ω) −→ L ([0,T ] × Ω) n X ∂f F = f(W (h ),...,W (h )) 7→ D F = (W (h ),...,W (h ))h (·) 1 n · ∂x 1 n i i=1 i
Now we can define the iterated derivative operator as
n Dt1,...,tn = Dt1 ··· Dtn F.
This iterated derivative operator Dn is a closable unbounded operator from L2(Ω) → L2([0,T ] × Ω). Now we can define the adjoint of the derivative operator, D, as δ, the Skorohod Integral. Let Domδ be a set of elements u ∈ L2([0,T ] × Ω), such that there exists a constant c satisfying
Z T | E DtF utdt |≤ c||F ||L2(Ω). 0 CHAPTER 2. PRELIMINARIES ON MALLIAVIN CALCULUS 14
Now if u ∈ Domδ, then δ(u) is an element in L2(Ω) characterized by
Z T E (δ(u)F ) = E DtF utdt. (2.2) 0
R T We will make use of the notation δ(u) = 0 utdWt to refer to the Skorohod integral. Finally the following proposition will be used in the next section when treating anticipating processes. A full detailed proof can be found in [3].
2.1.2 Proposition. Let u ∈ Domδ and consider a random variable F ∈ D1,2, where D1,2 denotes the domain of the derivative operator and is a dense 2 2 R T 2 subset of L (Ω), such that E(F 0 ut dt) < ∞. Then
Z T Z T Z T F utdWt = F utdWt − DtF utdt, (2.3) 0 0 0 where F u ∈ Domδ if and only if the right-hand side of (2.3) is square inte- grable.
2 2.1.3 Proposition. Let u ∈ La([0,T ]×Ω), this is the set of square integrable adapted processes (with respect to the filtration generated by W ). Then, for all 0 ≤ t < s ≤ T Dsut = 0.
A detailed proof of this Proposition can be found in [11].
2.2 Itˆoformula for anticipating processes
This section is devoted to prove the following theorem that gives an extension of the Itˆoformula for anticipating stochastic processes, to do so we will follow the proof in [3]. CHAPTER 2. PRELIMINARIES ON MALLIAVIN CALCULUS 15
2.2.1 Theorem. Consider a stochastic process of the form
Z t Z t Xt = X0 + usdWs + vsds, 0 0
2 where X0 is a F0−mesurable random variable and both u, v ∈ La([0,T ] × Ω). R T 1,2 1,2 Consider also a process Yt = t θsds, for some θ ∈ L , where L is the 2 1,2 space of processes u ∈ L ([0,T ] × Ω), such that ut ∈ D . Let F : R3 → R be a twice continuously differentiable function such that there exists a positive constant C such that, for all t ∈ [0,T ] ,F and its derivatives evaluated in (t, Xt,Yt) are bounded by C. Then it follows that
Z t ∂F F (t, Xt,Yt) = F (0,X0,Y0) + (s, Xs,Ys)ds 0 ∂s Z t ∂F Z t ∂F + (s, Xs,Ys)dXs + (s, Xs,Ys)dYs 0 ∂x 0 ∂y Z t 2 ∂ F − + (s, Xs,Ys)(D Y )susds 0 ∂x∂y Z t 2 1 ∂ F 2 + 2 (s, Xs,Ys)usds, 2 0 ∂x
− R T where (D Y )s := s DsYrdr.
Proof: Fix n ≥ 1 and let ti = it/n. Under these conditions we apply the Taylor expansion up to the second order and obtain that CHAPTER 2. PRELIMINARIES ON MALLIAVIN CALCULUS 16
n−1 X ∂F F (t, X ,Y ) = F (0,X ,Y ) + (t ,X ,Y )(t − t ) t t 0 0 ∂t i ti ti i+1 i i=0 n−1 X ∂F + (t ,X ,Y )(X − X ) ∂x i ti ti ti+1 ti i=0 n−1 X ∂F + (t ,X ,Y )(Y − Y ) ∂y i ti ti ti+1 ti i=0 n−1 1 X ∂2F + (t¯ , X¯ , Y¯ )(t − t )2 2 ∂t2 i ti ti i+1 i i=0 n−1 X ∂2F + (t¯ , X¯ , Y¯ )(t − t )(X − X ) ∂t∂x i ti ti i+1 i ti+1 ti i=0 n−1 X ∂2F + (t¯ , X¯ , Y¯ )(t − t )(Y − Y ) ∂t∂y i ti ti i+1 i ti+1 ti i=0 n−1 1 X ∂2F + (t¯ , X¯ , Y¯ )(X − X )2 2 ∂x2 i ti ti ti+1 ti i=0 n−1 X ∂2F + (t¯ , X¯ , Y¯ )(X − X )(Y − Y ) ∂x∂y i ti ti ti+1 ti ti+1 ti i=0 n−1 1 X ∂2F + (t¯ , X¯ , Y¯ )(Y − Y )2 2 ∂y2 i ti ti ti+1 ti i=0 = F (0,X0,Y0)
+ T1 + T2 + T3 + T4 + T5 + T6 + T7 + T8 + T9,
¯ ¯ ¯ for some intermediate point (ti, Xti , Yti ), between (ti,Xti ,Yti ) and (ti+1,Xti+1 ,Yti+1 ). Now it is clear, by classical arguments, that
Z t ∂F T1 → (s, Xs,Ys)ds 0 ∂s
and
Z t ∂F T3 → (s, Xs,Ys)θsds 0 ∂y CHAPTER 2. PRELIMINARIES ON MALLIAVIN CALCULUS 17
in L1(Ω).
For the addend T2, we can write the following
n−1 X ∂F T = (t ,X ,Y )(X − X ) 2 ∂x i ti ti ti+1 ti i=0 n−1 X ∂F Z ti+1 Z ti+1 Z ti Z ti = (t ,X ,Y ) (X + u dW + v ds) − (X + u dW + v ds) ∂x i ti ti 0 s s s 0 s s s i=0 0 0 0 0 n−1 n−1 X ∂F Z ti+1 X ∂F Z ti+1 = (t ,X ,Y ) v ds + (t ,X ,Y ) u dW . ∂x i ti ti s ∂x i ti ti s s i=0 ti i=0 ti
Again from classical arguments, is easy to see that
n−1 X ∂F Z ti+1 Z t ∂F (t ,X ,Y ) v ds → 1 (s, X ,Y )v ds. ∂x i ti ti s L (Ω) ∂x s s s i=0 ti 0
Now using Proposition (2.1.2) it follows that
n−1 X ∂F Z ti+1 (t ,X ,Y ) u dW ∂x i ti ti s s i=0 ti n−1 X Z ti+1 ∂F = (t ,X ,Y )u dW ∂x i ti ti s s i=0 ti n−1 X Z ti+1 ∂F + D (t ,X ,Y ) u ds s ∂x i ti ti s i=0 ti
Using the chain rule for the derivative operator, see [11], and Proposition (2.1.3) we can deduce that
Z ti+1 ∂F Z ti+1 ∂2F Ds (ti,Xti ,Yti ) usds = (ti,Xti ,Yti )(DsYti )usds. ti ∂x ti ∂x∂y
Using the same arguments we have that CHAPTER 2. PRELIMINARIES ON MALLIAVIN CALCULUS 18
n−1 Z ti+1 2 Z t 2 X ∂ F ∂ F − (t ,X ,Y )(D Y )u ds → 1 (s, X ,Y )(D Y ) u ds, ∂x∂y i ti ti s ti s L (Ω) ∂x∂y s s s s i=0 ti 0 and
Z t 2 1 ∂ F 2 1 T7 →L (Ω) 2 (s, Xs,Ys)usds. 2 0 ∂x
R T 2 Since F and its derivatives are bounded by hypothesis, 0 Evs ds < +∞ R T 2 and 0 E(θs )ds < ∞ it follows that T4 + T5 + T6 + T8 + T9 tends to zero in L1(Ω). Finally, we will prove that for all t ∈ [0,T ] , the process
∂F (t ,X ,Y )u ∂x i ti ti sI[0,T ] belongs to the domain of δ and that
Z t ∂F (s, Xs,Ys)usdWs 0 ∂x Z t ∂F = F (t, Xt,Yt) − F (0,X0,Y0) − (s, Xs,Ys)ds 0 ∂s Z t ∂F Z t ∂F − (s, Xs,Ys)vsds − (s, Xs,Ys)dYs 0 ∂x 0 ∂y Z t 2 ∂ F − − (s, Xs,Ys)(D Y )susds 0 ∂x∂y Z t 2 1 ∂ F 2 − 2 (s, Xs,Ys)usds. (2.4) 2 0 ∂x
Now applying Lemma 1 in [3] to the sequence of processes
2n−1 X ∂F Φn = u (t ,X ,Y ) (s), s s ∂x i ti ti I[ti,ti+1] i=0 CHAPTER 2. PRELIMINARIES ON MALLIAVIN CALCULUS 19
n 2 0 we have that Φ converges in L ([0,T ] × Ω) to Φ = usF (ti,Xti ,Yti )I[0,t](s). as n tends to infinity. From the previous steps we have that
n X Z ti+1 ∂F δ(Φn) = (t ,X ,Y )u dW ∂x i ti ti s s i=0 ti converges in L1(Ω) to a random variable equal to the right-hand side of equation (2.4). Then in order to end the proof it suffices to prove that the right-hand side of equation (2.4) belongs to L2(Ω). This follows easily from the hypotheses of the theorem. 2 Chapter 3
The Hull and White formula.
It has already been proved that if σ is a stochastic process uncorrelated with price, i.e. ρ = 0, we have that the value of the derivative at time t is written as
Vt = Et(BS(t, Xt, σ¯t)), where BS stands for the Black-Scholes formula in equation (1.8). This is the classical Hull-White formula that covers non correlated stochastic volatility models. Now since the future average volatility is an anticipative process, as proved in the previous section, we will make use of the Malliavin-Skorohod calculus tool to solve the case where volatility and the underlying are cor- related. So following [12], see also [1], we will introduce the following theo- rem that allows us to write an option’s value at time t as the Black-Scholes function plus a correction term by means of the correlation and stochastic volatility of the underlying asset.
3.0.2 Theorem. Assume that
2 1,2 − (A1): σ ∈ LW , 1,2 − (A2): σ ∈ LW
Then we have that,
Z T ρ −r(T −t) 3 2 Vt = Et [BS(t, Xt, σ¯t)] + Et e (∂x − ∂x)BS(s, Xs, σ¯s)Λsds ,(3.1) 2 t
20 CHAPTER 3. THE HULL AND WHITE FORMULA. 21 where Z T 2 Λs := ( Dsσr dr)σs. (3.2) s
Proof: This proof is based on a decomposition method technique. Recall that XT + VT = (e − K) = BS(T,XT , σ¯T ), hence
−r(T −t) Vt = e Et [BS(T,XT , σ¯T )] −rt −rT e Vt = e Et [BS(T,XT , σ¯T )] −rt −rT e Vt = Et e BS(T,XT , σ¯T ) .
Now we will apply the previous anticipative Itˆoformula to the process
−rs e BS(s, Xs, σ¯s), between t and T, then take its conditional expectation Et and multiply it rt by e . Then we will obtain an expression for Vt. Now the anticipative Itˆo formula proved in section (2.1) from [3], can be adapted to our case. Define
Z T 2 2 Yt := (T − t)¯σt = σr dr. (3.3) t
Using (A1), we are under the conditions in Theorem (2.2.1) also found 1,2,2 in [1], and so, for any smooth functional F ∈ Cb ([0,T ] × R × [0, ∞)), we have
Z t W,B F (t, Xt,Yt) = F (0,X0,Y0) + ∂sF (s, Xs,Ys)ds + δt (∂xF (·,X·,Y·)σ·) 0 Z t 2 Z t σs 2 + ∂xF (s, Xs,Ys)(r − )ds − ∂yF (s, Xs,Ys)σs ds 0 2 0 Z t Z t 1 2 2 + ρ ∂xyF (s, Xs,Ys)Λsds + ∂xF (s, Xs,Ys)σs ds. 0 2 0 CHAPTER 3. THE HULL AND WHITE FORMULA. 22
Now we will apply this result to the function that concerns us, the Black- Scholes function. This is
r y F (s, x, y) := ersBS(s, x, ). (3.4) T − s
The problem now is that this function doesn’t satisfy the required condi- tions of the previous Itˆoformula since its derivatives are not bounded, so we will apply a small correction in the parameters of the previous function to overcome this problem. This is known as the mollifier argument. For n ≥ 1 and δ > 0, we will consider the following approximation,
r y + δ x F (s, x, y) := e−rsBS(s, x, )φ( ), (3.5) n,δ T − s n
2 where φ ∈ Cb (R), such that
1 if |z| ≤ 1 φ(z) = [0, 1] if |z| ∈ [1, 2] (3.6) 0 if |z| > 2
Lets plot the previous function in order to recall that it is bounded as well as its derivatives.
Figure 3.1: Adjustment function used to ensure all derivatives are bounded. CHAPTER 3. THE HULL AND WHITE FORMULA. 23
Then conditions to apply the anticipative Itˆoformula to Fn,δ(s, Xs,Ys) are met. Taking the conditional expectation Et, using that Skorohod type integrals have zero expectation and multiplying by ert, we obtain
X e−r(T −t)BS(T,X , σ¯δ )φ( T ) Et T T n X = BS(t, X , σ¯δ)φ( t ) Et t t n Z T −r(s−t) + Et e An(s)ds t Z T ρ −r(s−t) 3 2 δ Xs + Et e (∂x − ∂x)BS(s, Xs, σ¯t )φn( )Λsds 2 t n Z T ρ −r(s−t) 3 2 δ 1 0 Xs + Et e (∂x − ∂x)BS(s, Xs, σ¯s ) φ ( )Λsds , 2 t n n
where
rY + δ σ¯δ := s , s T − s and
σ2 X A (s) := s ∂ BS(s, X , σ¯δ)φ0( s ) n n x s s n σ2 1 X X + s BS(s, X , σ¯δ)( φ00( s ) − φ0( s )) 2n s s n n n r X + BS(s, X , σ¯δ)φ0( s ). n s s n
Finally, the result follows from the dominated convergence theorem, tak- ing limits first on n ↑ +∞ and then on δ ↓ 0. The dominated convergence runs thanks to the properties of Black-Scholes function and (A2). For the left hand side and the two first terms on the right hand side we use the fact tat x function BS(t, x, σ) is bounded by e +K and its derivative (∂xBS)(t, x, σ) is bounded by ex. For the last two terms on the right hand side we use Lemma 2 in [2]. 2 Chapter 4
Deriving the analytical form of the underlying probability density function.
The purpose of this section is to derive a new analytical form of writing the underlying probability density function in terms of the correlation between the underlying process and the stochastic volatility process and in terms of its Malliavin derivatives. To do so we need to introduce the so called Breeden and Litzenberger formula, see [5], that will allow us to relate the option’s present value with the underlying asset’s probability density function.
4.0.3 Theorem. Assume that the underlying asset, St, follows a Markov process with a probability density function fST (x) for ST . Then it holds that Z ∞ −r(T −t) Ct(T,K) = e fST (x)(x − K)dx (4.1) K Z ∞ ∂Ct −r(T −t) (T,K) = −e fST (x)dx (4.2) ∂K K ∂C2 t (T, x) = e−r(T −t)f (x). (4.3) ∂K2 ST
Now we derivate the Hull-White formula for correlated stochastic volatil- ity model with respect to K as it follows.
24 CHAPTER 4. UNDERLYING ASSET PDF FORMULA 25
2 2 Z T ∂ Vt ∂ ρ −r(s−t) 3 2 2 = 2 Et [BS(t, Xt, σ¯t)] + Et e (∂x − ∂x)BS(s, Xs, σ¯s)Λsds ∂K ∂K 2 t
∂2V ∂2 t = (BS(t, X , σ¯ )) (4.4) ∂K2 Et ∂K2 t t Z T 2 ρ −r(s−t) 3 2 ∂ + Et e (∂x − ∂x) 2 (BS(s, Xs, σ¯s))Λsds , (4.5) 2 t ∂K where
∂ ∂ (BS(t, X , σ¯ )) = (eXt Φ(d ) − e−r(T −t)KΦ(d )) ∂K t t ∂K + − ∂d ∂d = eXt φ(d ) + − e−r(T −t)Kφ(d ) − − e−r(T −t)Φ(d ), + ∂K − ∂K − since
∂d −1 ∂d + = √ = − , ∂K Kσ¯t T − t ∂K we can rewrite the previous equation as it follows
∂ ∂d (BS(t, X , σ¯ )) = + (eXt φ(d ) − e−r(T −t)Kφ(d )) − e−r(T −t)Φ(d ) ∂K t t ∂K + − − −r(T −t) = −e Φ(d−),
Xt −r(T −t) where e φ(d+) − e Kφ(d−) = 0, (see A.1.2). Now for instance we will compute the second order derivative of BS(t, Xt, σ¯t) with respect to K.
∂2 ∂ (BS(t, X , σ¯ )) = −e−r(T −t)Φ(d ) ∂K2 t t ∂K − ∂d = −e−r(T −t)φ(d ) − − ∂K e−r(T −t)φ(d ) = √ − Kσ¯t T − t CHAPTER 4. UNDERLYING ASSET PDF FORMULA 26
∂2 e−r(T −t)φ(d ) √ − 2 (BS(t, Xt, σ¯t)) = . (4.6) ∂K Kσ¯t T − t
Finally replacing the previous equation in equation (3.4) we have that.
∂2V e−r(T −t)φ(d ) t √ − 2 = Et (4.7) ∂K Kσ¯t T − t Z T −r(T −s) ρ −r(T −s) 3 2 e φ(d−) + Et e (∂x − ∂x) √ Λsds (4.8) 2 t Kσ¯s T − s
Now if the derivative we are pricing is a call option we have that Vt = Ct and replacing the previous expression in equation (3.3) we can deduce by means of the Breeden-Litzenberger formula a new expression for the proba- bility density function of fST (x)
∂2V f (x) = er(T −t) t ST ∂K2 −r(T −t) r(T −t) e φ(d−) = e Et √ Kσ¯t T − t Z T −r(T −s) r(T −t) ρ −r(T −s) 3 2 e φ(d−) + e Et e (∂x − ∂x) √ Λsds 2 t Kσ¯s T − s φ(d−) = Et √ Kσ¯t T − t Z T ρ −r(T −s) 3 2 φ(d−) + Et e (∂x − ∂x) √ Λsds 2 t Kσ¯s T − s
Finally this can be rewritten as
φ(d−) fST (x) = Et √ Kσ¯t T − t Z T −r(T −s) ρ e 3 2 + Et √ (∂x − ∂x)(φ(d−)) Λsds (4.9) 2 t Kσ¯s T − s CHAPTER 4. UNDERLYING ASSET PDF FORMULA 27
4.0.4 Remark. Note that if σ is deterministic, the previous formula says that the law of ST is lognormal, as expected. To prove so we will start by the uncorrelated case were volatility is constant and will try to derive the Black-Scholes underlying distribution. Now if ρ = 0 andσ ¯t = σ is constant we have that (4.9) is written as
φ(d ) f (x) = √− , ST xσ τ where τ = T − t. Lets start by proving that this follows a lognormal distri- bution by checking that
φ(d ) f (x) = √− ∼ LN(µ, σ2) ST xσ τ
In order to see that fST (x) follows a log-normal distribution as one would expect since we have come back to the basic model of Black-Scholes, we have to ensure that fK fulfills the following
φ(d ) 1 (ln(x)−µ)2 − − 2 f ∼ LN ⇔ fS (x) = √ ∼ LN ⇔ fS (x) = √ e 2σ , T xσ τ T xσ 2π
In order to simplify the calculus we will suppose, without loss of gener- ality, that T = 1 and t = 0, hence τ = 1. On the other hand recall that φ is the standard normal probability density function. Now all is left to do is checking the following equality
φ(d ) 1 (ln(x)−µ)2 − − 2 fS (x) = = √ e 2σ (4.10) T xσ xσ 2π
1 (ln(x)−µ)2 − 2 ⇔ fS (x) = √ e 2σ T xσ 2π 1 −1 ln(S) − ln(K) + (r − σ2/2)τ 2 ⇔ fS (x) = √ exp{ √ } T xσ 2π 2 2σ τ 1 ln(K) − ln(S) − (r − σ2/2)2 ⇔ fS (x) = √ exp{ } T xσ 2π 2σ2 1 (ln(x) − µ)2) ⇔ fS (x) = √ exp{ , } T xσ 2π 2σ2 CHAPTER 4. UNDERLYING ASSET PDF FORMULA 28 where µ = (ln(S)+(r−σ2/2). This ends the proof that the probability density function of the non correlated case follows a log-normal density function. Now we will do the analogous task by proving that the underlying asset’s probability density function is log-normal when considering the uncorrelated case and deterministic volatility instead of constant volatility as in the pre- vious case. We know from equation (1.6) that
Z T 2 Z T σs XT = Xt + (r − )ds + σsdWs, t 2 t
Let us consider t=0 without loss of generality and the following stands.
Z T 2 Z T σs XT = X0 + (r − )ds + σsdWs 0 2 0 Z T Z T 1 1 2 1 2 XT − X0 = rT − ( σs ds)T + ( σs ds)T 2 T 0 T 0 1 X − X ∼ N(rT − σ¯2T, σ¯2T ). T 0 2 0 0
Recall that the future average volatility has already been defined as
s Z T 1 2 σ¯t := σs ds. T − t t
Now since XT is the logarithm of the price process, ST , and follows a Normal distribution, is obvious that ST follows a Log-normal distribution. Appendix A
A.1 Other definitions and concepts
A.1.1 Lemma. The following Lemma is known as the ItˆoLemma, see [9] for more detail. Let St be an Itˆoprocess given by an SDE of the following form
dSt = a(t, St)dt + b(t, St)dWt, and let Xt be another stochastic process written as a function of the previous process f(t, St). Then Xt is also an Itˆoprocess given by
∂f ∂f 1 ∂2f dX = (t, S )dt + (t, S )dS + (t, S )(dS )2. t ∂t t ∂x t t 2 ∂x2 t t A.1.2 Lemma.
−r(T −t) Stφ(d+) = e Kφ(d−). (A.1)
Proof:
−r(T −t) Stφ(d+) = e Kφ(d−) S φ(d ) ⇔ t er(T −t) = − K φ(d+) S d2 − d2 ⇔ log( t ) + r(T − t) = + − , K 2
29 APPENDIX A. 30
Now in addition we have that the right hand side of the last equation is
d2 − d2 1 + − = (d + d )(d − d ) 2 2 + − + − 1 √ √ = (2d − σ T − t)σ T − t 2 + S 1 1 = log( t ) + (r + σ2)(T − t) − σ2(T − t) K 2 2 S = log( t ) + r(T − t). K 2 Bibliography
[1] E. Alos` , A generalization of Hull and White formula and applications to option pricing approximation. Finance and Stochastics 10 (3): 353-365, 2006.
[2] E. Alos,` J. A. Leon,´ J. Vives, On the short-time behavior of the implied volatility for jump diffusion models with stochastic volatility. Fi- nance and Stochastics 11 (4): 571-589, 2007.
[3] E. Alos,` D. Nualart, An extension of Itˆoformula for anticipating processes. Journal of Theoretical Probability. 11 (2): 492-514, 1998.
[4] F. Black, M. Scholes, The pricing of options and corporate liabilities. Journal of Political Economy, 637-654, 1973.
[5] D.T. Breeden, R.H. Litzenberger, Prices of state-contingent claims implicit in option prices. Journal of business, 51:621-651, 1978.
[6] J.P. Fouque, G. Papanicolaou, K.R. Sircar, Derivatives in fi- nancial markets with stochastic volatility. Cambridge University Press, 2000.
[7] J. Gatheral, The Volatility Surface: A Practitioner’s Guide, Wiley, 2006.
[8] A. Gulisashvili, Analytically Tractable Stochastic Stock Price Models, Springer, 2012.
[9] K. Itoˆ, Stochastic Integral, Proceedings of the Imperial Academy, 1944.
[10] R. Merton Theory of Rational Option Pricing, The Bell Journal of Economics and Management Science, 1973.
[11] D. Nualart The Malliavin Calculus and Related Topics, Springer, 2005.
31 BIBLIOGRAPHY 32
[12] J. Vives, Decomposition of the pricing formula for stochastic volatility models based on Malliavin-Skorohod type calculus., 2015.