APPLICATIONS OF MALLIAVIN CALCULUS AND WHITE NOISE ANALYSIS IN INTEREST RATE MARKETS, AND CONVERTIBLE BONDS WITH AND WITHOUT SYMMETRIC INFORMATION By Wong Man Chui A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE DEC 2007 c Copyright by Wong Man Chui, 2007 NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS The undersigned hereby certify that they have read and recommend to the Faculty of Graduate Studies for acceptance a thesis entitled “Applications of Malliavin Calculus and White Noise Analysis in Interest Rate Markets, and Convertible Bonds with and without Symmetric Information” by Wong Man Chui in partial fulfillment of the requirements for the degree of Doctor of Philosophy Dated: Dec 2007 External Examiner: Research Supervisor: Dr Lou Jiann Hua Examing Committee: ii NATIONAL UNIVERSITY OF SINGAPORE Date: Dec 2007 Author: Wong Man Chui Title: Applications of Malliavin Calculus and White Noise Analysis in Interest Rate Markets, and Convertible Bonds with and without Symmetric Information Department: Mathematics Degree: Ph.D Convocation: February Year: 2009 Permission is herewith granted to National University of Singapore to circulate and to have copied for non-commercial purposes, at its discretion, the above title upon the request of individuals or institutions Signature of Author THE AUTHOR RESERVES OTHER PUBLICATION RIGHTS, AND NEITHER THE THESIS NOR EXTENSIVE EXTRACTS FROM IT MAY BE PRINTED OR OTHERWISE REPRODUCED WITHOUT THE AUTHOR’S WRITTEN PERMISSION THE AUTHOR ATTESTS THAT PERMISSION HAS BEEN OBTAINED FOR THE USE OF ANY COPYRIGHTED MATERIAL APPEARING IN THIS THESIS (OTHER THAN BRIEF EXCERPTS REQUIRING ONLY PROPER ACKNOWLEDGEMENT IN SCHOLARLY WRITING) AND THAT ALL SUCH USE IS CLEARLY ACKNOWLEDGED iii Table of Contents Table of Contents iv Acknowledgements vi Summary vii The interest rate markets model framework 1.1 Introduction 1.2 Martingale Modelling Applications of Malliavin calculus to Monte Carlo methods ing interest rate derivatives 2.1 Introduction 2.2 Preliminary Malliavin Calculus 2.3 Application of Malliavin Calculus to Target Redemption Note 2.4 Application of Malliavin Calculus to Callable Libor Exotics 1 in pric 12 16 Applications of Fractional white noise calculus to illiquid interest rate markets 3.1 Introduction 3.2 Preliminary Fractional White Noise Analysis 3.3 Fractional Interest Rate model 3.4 Managing Market Risk Exposure under Fractional Interest Rate Markets 3.5 Convexity Adjustment under Fractional Interest Rate Markets 20 21 22 25 27 29 Convertible Bonds with Symmetric Information 4.1 Introduction 4.2 Some Properties of Convertible Bonds 4.3 Convertible Bonds with Counterparty Risks 34 34 36 46 iv Convertible Bonds with Asymmetric Information 5.1 Introduction 5.2 No Arbitrage Opportunity Pricing with Asymmetric Information 5.3 Pricing Convertible Bonds with Asymmetric Information 57 57 60 65 Bibliography 73 v Acknowledgements I wish to express my sincere gratitude to my mentor Dr Lou Jiann Hua for his guidance and for his help on this thesis vi Summary The Applications of Malliavin calculus and white noise analysis in stock markets has been well-known in the Mathematical Finance’s literature But its application to interest rate markets has been minimal The aim of this work is to fill in this gap In recent years it has become clear that there are various applications of Malliavin Calculus as far as the integration by parts formula is concerned One of its successful applications is to compute the Greeks (i.e., price sensitivities) of Financial derivatives in stock markets In fact, the exotic products created in interest rate markets are as complicated as in the stock markets Target Redemption Notes are one of the good examples for this application due to their discontinuous payoff In the first of this thesis, we will provide two of its applications to the interest rate derivatives Fractional Brownian motion has been applied to describe the behavior to prices of assets and volatilities in stock markets The long range dependence self similarity properties make this process a suitable model to describe these quantities In interest rate markets, we can also observe the same behavior To model the bond prices driven by Fractional Brownian, we apply the multi-dimensional Wick-Itˆ integral as o it precludes arbitrage opportunities This framework is particular useful if the market is illiquid as the trader cannot really observe the true market price and he is forced to quote the market price when his client is asking for it We will demonstrate how two financial problems can be solved under this model framework in the second of this thesis A convertible bond has many of the same characteristics as an ordinary bond but vii viii with the additional feature that the bond may, at any time of the owner’s choosing, be exchanged for a specified asset Moreover, it is also common that the issuer may have the right to terminate the contract In other words, this contract enables both their buyer and seller to stop it at any time It is in fact a subset of Game Options In Chapter 4, we are going to look at some of the properties of this derivative We will also consider the case when there is default risk involved In the final chapter, we suggest a method how to price this derivative when there is insider information Chapter The interest rate markets model framework In this chapter we review the classical HJM model which captures the full dynamics of the entire forward rate curve, while the short-rate models only capture the dynamics of a point on the curve [10] The key to these techniques is the recognition that the drifts of the no-arbitrage evolution of certain variables can be expressed as functions of their volatilities and the correlations among themselves In other words, no drift estimation is needed The importance of the HJM model lies in the fact that virtually any (exogenous term-structure) interest-rate model can be derived within such a framework 1.1 Introduction Let W be a d -dimensional standard Brownian motion given on a filtered probability space (Ω, F, P) As usual, the filtration F = FW is assumed to be right continuous and P− completed version of the natural filtration of W D Heath, R.A Jarrow and A Morton in their paper [10] assumed that, for a fixed maturity T ∈ [0, ∞), the instantaneous forward rate f (t, T ) evolves, under a given measure, according to the following diffusion process: df (t, T ) = α(t, T )dt + σ(t, T ) · dW (t), ∀t ∈ [0, T ], (1.1.1) for a Borel-measurable function f (0, ·) : [0, T ] → R, the market instantaneous forward curve at time t = 0, and some functions α : C x Ω → Rd , σ : C x Ω → Rd where C = {(u, t)|0 ≤ u ≤ t ≤ T } Moreover for any maturity T, α(·, T ) and σ(·, T ) follow adapted processes, such that T T |α(u, T )|du + |σ(u, T )|2 du < ∞, P − a.s A zero coupon bond of maturity T is a financial security paying to its holder one unit of cash at a pre-specified date T in the future The price of a zero coupon bond of maturity T at any instant t ≤ T will be denoted by P (t, T ); it is obvious that P (T, T ) = We will usually assume that, for any fixed maturity T , the bond price P (·, T ) follows a strictly positive and adapted process on a filtered probability space (Ω, F, P) Moreover we have a one to one relationship between zero-bond prices and forward rates f (t, T ) = −∂ ln P (t, T ) ∂T − P (t, T ) := e T t f (t,u)du (1.1.2) One can show that in order for the dynamics in (1.1.1) arbitrage free, the function α cannot be arbitrarily chosen and it must have the following form under the risk neutral measure Q T α(t, T ) = σ(t, T ) · σ(t, s)ds t (1.1.3) 62 Hence −1 pA ≥ sup EQFT [B0 (τ (ω))(exp−r(T −τ (ω)) −γSτ (ω) )+ ] τ ≤T where we have used the facts that {τ ≤ t} ∈ Ft ⊆ Gt and θ ∈ Ft ⊆ Gt and the stochastic integrals defined under FT remains unchanged under an initial enlargement In this remaining section, we are going to apply a weaker assumption, namely the regular conditional distribution of G given Ft , t ∈ [0, T ], is absolutely continuous with respect to the law of G for P− almost all ω ∈ Ω One example where the random variable G satisfies this assumption is the sum of the endpoint WT and a random variable ∼ N (0, 1) which is independent of FT Then we have px = t T +1 (x − Wt2 ) x2 exp − + T −t+1 2(T − t + 1) 2(T + 1) (5.2.1) and this implies that the information drift is µx = s x − Ws T −s+1 (5.2.2) where s ∈ [0, T ] As we know that when there are dividend payments, it is the discounted gain process required to be a martingale under QF Moreover if we have a stochastic dividend processes, would theorem 5.2.1 remain valid? The next result that we are going to show tells us that in fact it still remains true For simplicity, we consider only European contingent claims But first let us state the following result from [6] on risk neutral valuation under the assumption that the price dynamics, under the objective probability measure, is given by dSt = St αdt + St σdWt (5.2.3) and the dividend structure is assumed to be of the form dDt = St δdt + St γdWt (5.2.4) Proposition 5.2.2 Under the above assumption of the stock and dividend processes, the fair price V0 of a claim process Φ(ST ) is V0 = exp−r(T −t) EQ [Φ(ST )] (5.2.5) 63 where the Q dynamics of S and D are given by dSt = St αγ + σr − δσ σ+γ ˆ dt + St σdWt (5.2.6) dDt = St δσ + γr − γα σ+γ ˆ dt + St γdWt (5.2.7) Proof See [6] Proposition 5.2.3 Under the assumptions that (1)the regular conditional distribution of G given Ft , t ∈ [0, T ], is absolutely continuous with respect to the law of G for P− almost all ω ∈ Ω (2)the P dynamics of S and D are given by equation 5.2.3 T x and 5.2.4 respectively (3)the information drift satisfies EP [exp (µs ) ds ] < ∞, P almost surely then in the absence of an arbitrage opportunity an investor with insider information and an ordinary investor have the the same fair price Proof It suffices to show that the process St and Dt follows the same dynamics under probability measure QG as in proposition 5.2.2 Define t x t x dPM |Gt := Mt = exp− (µs )dWs − (µs ) ds dP Then t ¯ (µx )ds Wt = Wt − s is a (P, Gt )− wiener process Hence the process St and Dt can be re-written as ¯ dSt = St (α + σµx )dt + St σdWt s ¯ dDt = St (δ + γµx )dt + St γdWt s The Gain process, i.e., G(t) = St B0 (t) + t dDs B0 (s) can also be written as dSt −rSt dt dDt + + B0 (t) B0 (t) B0 (t) x x ¯ (α + σµs + δ + γµs − r)St dt + (γ + σ)St dWt = B0 (t) dG(t) = Define φs := α+δ−r γ+σ + µx Since EP [exp s stants, it implies that EP [exp T (φs ) ds T x (µs ) ds ] < ∞ and σ are just some con- < ∞]So we can define 64 ˜ dQ |gt := exp− dP t (φs )dWs − t (φs ) ds ˜ ˜ ¯ And under this probability measure Q, Wt = Wt + process ˜ Also under Q, we have t ˜ φs ds is a (Q, Gt ) Wiener ˜ dSt = St (α + σµx )dt + St σ dWt − s α+δ−r + µx dt s γ+σ ˜ dDt = St (δ + γµx )dt + St γ dWt − s α+δ−r + µx dt s γ+σ Hence after simplification, we have dSt = St ( αγ + σr − δσ ˜ )dt + St σdWt σ+γ (5.2.8) δσ + γr − γα ˜ )dt + St γdWt σ+γ (5.2.9) dDt = St ( ˜ Hence Q(f ((St , Dt )) = Q(f ((St , Dt )) for any f : R2 → R (5.2.10) 65 5.3 Pricing Convertible Bonds with Asymmetric Information In this paper, we are particularly interested in the following payoff process: R(σ, τ ) = max(γSσ , F )1σ ∞}, and that the marginal utility tends to zero when wealth tends to infinity, i.e., U (∞) := limx→∞ U (x) = Regarding the behavior of the (marginal) utility at the other end of the wealth scale we shall distinguish two cases Case 1(Negative wealth not allowed): We assume that U satisfies the condition U (x) = −∞, for x < 0, while U (x) > −∞, for x > 0, and the so-called Inada condition U (0) := lim U (x) = ∞ (5.3.1) xα α where α ∈ (−∞, 1) \ {0} and x→0 For example U (x) = ln(x) for x > 0, or U (x) = x > Case 2(Negative wealth allowed): We assume that U (x) ≥ −∞, for all x ∈ R, and that 67 U (−∞) := lim U (x) = ∞ (5.3.2) x→−∞ For example U (x) = − exp−γx , where γ > and x ∈ R Then we find a saddle point (σ , τ ) in the following sense: If for all (σ, τ ) ∈ Π1 x Π2 , where Π1 and Π2 are the GT − and HT − stopping 0T 0T 0T 0T times respectively, we have EP [U1 (− exp−r((σ ∧τ ∧T )) R(σ , τ )|G0 ) ≥ EP [U1 (− exp−r((σ∧τ ∧T )) R(σ, τ )|G0 )] (5.3.3) and EP [U2 (exp−r((σ ∧τ ∧T )) R(σ , τ )|H0 ) ≥ EP [U2 (exp−r((σ ∧τ ∧T )) R(σ , τ )|H0 )] (5.3.4) If such a saddle point exists, then we can apply utility maximization problem to compute the utility indifference premium based on the following: Define p as the initial capital and h as the utility indifference premium which satisfies the following equation: σ ∧τ ∧T θ∈Θ σ ∧τ ∧T θs dSs −R(σ , τ ))|G0 ] = sup EP [U1 (p+ sup EP [U1 (p+h+ θ∈Θ θs dSs |G0 ] (5.3.5) where Θ represents all those elements which are Ht − predictable and S− integrable and satisfy Θ = {θ ∈ L(S)| t θ dSu u is bounded uniformly in t and ω} So it is important to show that such a saddle point exists which satisfies equation 5.3.3 and 5.3.4 And the next result shows that it does exist But first let us assume 68 the interest rate r = and define the followings: For = t0 < t1 < · · · < tk = T recursively: σk = tk and τk = tk , σi−1 := ti−1 : ω ∈ Ai−1 , (5.3.6) otherwise σi−1 := σi if ω not ∈ Ai−1 τi−1 := ti−1 : ω ∈ Bi−1 , (5.3.7) otherwise τi−1 := τi if ω not ∈ Bi−1 where Ai−1 and Bi−1 are Gti−1 − and Hti−1 − measurable respectively and satisfy ˆ Ai−1 = {EP [U1 (R(σi , τi ))|Gi−1 ] ≥ EP [U1 (Uti−1 )|Gi−1 ]}\Bi−1 ˆ Bi−1 = {EP [U2 (R(σi , τi ))|Fi−1 ] ≤ EP [U2 (Lti−1 )|Fi−1 ]} (5.3.8) (5.3.9) ˆ ˆ where Uti−1 = max(γSti−1 , F ) and Lti−1 = γSti−1 Theorem 5.3.1 The (σ , τ ) defined recursively in 5.3.3 and 5.3.4 is the saddle point which satisfies equation 5.3.1 and 5.3.2 Proof For any arbitrary σ and τ ∈ Π1 x Π2 , it is sufficient to prove that for any 0T 0T i = 1, , k we have EP [U1 (R(σi , τi ))|Gti )] ≤ EP [U1 (R(σ, τi ))|Gti ] (5.3.10) 69 and EP [U2 (R(σi , τi ))|Hti ] ≥ EP [U2 (R(σi , τ ))|Hti ] (5.3.11) We are going to prove it by backward induction It is obviously true for i = k Let us assume that it is true for i = k − 1, we are going to show that it is also true for i = k − too For any A ∈ Gtk−2 ˆ EP [U1 (Ltk−2 )|Gtk−2 ]dP EP [U1 (R(σk−2 , τk−2 ))|Gtk−2 ]dP = A∩{τk−2 =tk−2 } A ˆ min{EP [U1 (Utk−2 )|Gtk−2 ] + A∩{τk−2 >tk−2 } , EP [U1 (R(σk−1 , τk−1 ))|Gtk−2 ]}dP ˆ EP [U1 (Ltk−2 )|Gtk−2 ]dP ≤ A∩{τk−2 =tk−2 } ˆ EP [U1 (Utk−2 )|Gtk−2 ]dP + A∩{τk−2 >tk−2 }∩{σ=tk−2 } + EP [U1 (R(σk−1 , τk−1 )) A∩{τk−2 >tk−2 }∩{σ>tk−2 } |Gtk−2 ]}dP ˆ EP [U1 (Ltk−2 )|Gtk−2 ]dP ≤ A∩{τk−2 =tk−2 } ˆ EP [U1 (Utk−2 )|Gtk−2 ]dP + A∩{τk−2 >tk−2 }∩{σ=tk−2 } + EP [U1 (R(σ, τk−2 )) A∩{τk−2 >tk−2 }∩{σ>tk−2 } |Gtk−2 ]}dP The last term is due to the assumption that the inequality 2.3.5 holds when i = k − and on set {τk−2 > tk−2 } we have τk−2 = τk−1 Hence EP [U1 (R(σk−1 , τk−1 ))]dP = EP [EP [U1 (R(σk−1 , τk−1 ))|Gtk−1 ]|Gtk−2 ]dP ≤ EP [U1 (R(σ, τk−2 ))|Gtk−2 ]dP To prove the second part, once again we use backward induction It is obviously true for i = k Let us assume that it is true for i = k − 1, we are going to show that it is also true for i = k − too Let A ∈ Htk−2 70 ˆ EP [U2 (Utk−2 )|Htk−2 ]dP EP [U2 (R(σk−2 , τk−2 ))|Htk−2 ]dP = A∩{σk−2 =tk−2 } A ˆ max{EP [U2 (Ltk−2 )|Htk−2 ] + A∩{σk−2 >tk−2 } , EP [U2 (R(σk−1 , τk−1 ))|Htk−2 }dP ˆ EP [U2 (Utk−2 )|Htk−2 ]dP ≥ A∩{σk−2 =tk−2 } ˆ EP [U2 (Ltk−2 )|Htk−2 ]dP + A∩{σk−2 >tk−2 }∩{τ =tk−2 } + EP [U2 (R(σk−1 , τk−1 )) A∩{σk−2 >tk−2 }∩{τ >tk−2 } |Htk−2 ]}dP ˆ EP [U2 (Utk−2 )|Htk−2 ]dP ≥ A∩{σk−2 =tk−2 } ˆ EP [U2 (Ltk−2 )|Htk−2 ]dP + A∩{σk−2 >tk−2 }∩{τ =tk−2 } + EP [U2 (R(σk−2 , τ )) A∩{σk−2 >tk−2 }∩{τ >tk−2 } |Htk−2 ]}dP And the rest of the arguments same as above For example, if we have a binomial tree model with γ = 1, ST −1 (ω) = 1.5, ST up = 2, ST down = 1, pup = pdown = 0.5, F = 1.5, r = 0, U1 (x) = U2 (x) = − exp−x and GT −1 contains the information that ST = with probability From equations (5.3.8) and (5.3.9), it is easily to see that ω ∈ AT −1 and hence the issuer will terminate the contract at time T − 1, one period before the contract expiration date When the historical probability measure P collides with the martingale probability measure Q it is obvious that equation (5.3.5) can be solved for a strictly increasing and concave function U1 , and h is equal to 71 EQ [exp−r((σ ∧τ ∧T )) R(σ , τ )|G0 ] For general P , the key part is to find a function V (x, y, t) to be defined later which solves the left hand side of (5.3.5) by first representing it as a HJB equation We provide a sketch as below: Due to the Markov property of the stock process and equations (5.3.8) and (5.3.9), we can construct a fixed domain G in {0, , T } × R2 as follows: G := {i − 1, s, R|Sti−1 = s ∈ (Ai−1 ∪ Bi−1 )c } Then we extend G to a domain G in R × R2 such that it is continuous with respect to time t Let ∂G = ∂G1 ∪ ∂G2 where ∂G1 and ∂G2 are the continuous versions constructed from the equalities in (5.3.8) and (5.3.9) respectively Now we define X := (X1 , X2 ), where X1 (s) = Ss and the wealth process dX2 (s) = rX2 (s)ds + (α − r)θs ds + σθs dWs , t ≤ s ≤ T with X2 (t) = x2 By assuming that r = and U1 (x) = − exp−γx with the risk aversion parameter γ > Hence the HJB equation with solution V (x1 , x2 , t) corresponds to the left hand side of (5.3.5) can be written as: 1 Vt + max( σ θ2 Vx2 x2 + σ St θVx1 x2 + αSt Vx1 + θαVx2 + σ St2 Vx1 x1 ) = 0, θ 2 ˆ ˆ and V (x1 , x2 , t) = − exp−γ(x2 −Ut ) if (t, x1 , x2 ) ∈ ∂G1 and V (x1 , x2 , t) = − exp−γ(x2 −Lt ) if (t, x1 , x2 ) ∈ ∂G2 The maximum can be achieved when θ(x1 , x2 , t) = − σ St Vx1 x2 +αVx2 σ Vx x and it yields 72 Vt − (σ St Vx1 x2 + αVx2 )2 2 + σ St Vx1 x1 + αSt Vx1 = σ Vx2 x2 Suppose that V (x1 , x2 , t) has the following separable form: V (x1 , x2 , t) = − exp−γ(x2 ) ψ(x1 , t) Then by substitution, we arrive 2 2 ψx1 α2 ψt + σ St ψx1 x1 − St σ = ψ 2 ψ σ2 ˆ with the boundary conditions: ψ(x1 , t) = expγ Ut if (t, x1 , R) ∈ ∂G1 and ψ(x1 , t) = ˆ expγ Lt if (t, x1 , R) ∈ ∂G2 This quasi-linear equation can be solved numerically Bibliography [1] Kyle A., Continuous auctions and insider trading, Econometrica 53 (1985) [2] Jurgen Amendinger, Initial enlargement of filtrations and additional information in financial markets, Thesis, der Technischen Universitat Berlin (1999) [3] Moshe Zakai A.Suleyman Ustunel, Transformation of measure on wiener space, Springer, 2000 [4] Eric Benhamou, Pricing convexity adjustement with wiener chaos, (1999) [5] Francesca Biagini and Bert Oksendal, Minimal variance hedging for fractional brownian motion, Methods and Applications of Analysis (2003) [6] Thomas Bjork, Arbitrage theory in continuous time, Oxford University Press, 2004 [7] Kabanov Y.-Runggaldier W Bjork T., Di Masi G., Towards a general theory of bond markets, Finance and Stochastics (1997) [8] Gatarek D Brace A and Musiela M., The market model of interest rate dynamics [9] Cajueriro D and B Tabak, Testing for fractional dynamics in the brazilian term structure of interest rates 73 74 [10] R.A Jarrow D Heath and A Morton, Bond pricing and the term structure of interest rates: a new methodology, Working Papaer, Cornell University (1987) [11] Christoph K uhn, Game contingent claims in complete and incomplete markets, ă Journal of Mathematical Economics (2004) [12] J Duan and K.Jacobs, Short and long memory in equilibrium interest rate dynamics, CIRANO (2001) [13] Pierre Collin Dufresne, Pricing and hedging in the presence of extraneous risks, Working Paper (2003) [14] Andreas E.Kyprianou, Some calculation for israeli options, Finance and Stochastic (2004) [15] Robert J Elliot and John van der Hoek, Using the hull and white two factor model in bank tresury risk management, Mathematical Finance Bachelier Congress 2000 (2000) [16] Jerome Lebuchoux-Pierre-Louis Lions Eric Fournie, Jean-Michel Lasry, Applications of malliavin calculus to monte-carlo methods in finance 2, Finance and Stochastic (2001) [17] Damir Filipovic, Consistency problems for heath-jarrow-morton interest rate models, Springer, 2001 [18] Bert Oksendal Francesca Biagini, A general stochastic calculus approach to insider trading, University of OSLO (2005) [19] Yaozhong Hu and Bert Oksendal, Fractional white noise calculus and applications to finance, (2000) [20] Peter Imkeller, Malliavin calculus in insider models: additional utility and free lunches, Humboldt-Universitat zu Berlin (2002) 75 [21] Steven E.Shreve Ioannis Karatzas, Methods of mathematical finance, Springer [22] Christoph K uhn Jan Kallsen, Convertible bonds: Financial derivatives of game ă type, Exotic Option Pricing and Advanced Levy Models (2005) [23] Arturo Kohatsu-Higa-David Nualart Jose M Corcuers, Peter Imkeller, Additional utility of insiders with imperfect dynamics information, (2002) [24] Yuri Kifer, Game options, Finance and Stochastic (2000) [25] Arturo Kohatsu-Higa and Miquel Montero, Malliavin calculus in finance, Universiat Pompeu Fabra (2004) [26] Gregory F Lawler, Introduction to stochastic processes, Chapman and Hall/CRC Probability Series, 1995 [27] F A Longstaff and R.S Schwartz, Valuing american options by simulation:a simple least-square approach, Review of Financial Studies (2001) [28] H.P Jr McKean, A free-boundary problem for the heat equation arising from a problem in mathematical economics, Industr Manag Rev., 6,32-39 [29] David Nualart, The malliavin calculus and related topics, Springer-Verlag, 1995 [30] Bernt Oksendal, Fractional brownian motion in finance, (2004) [31] V Piternarg, The best of wilmott 2, tarns: Models, valuation, risk sensitivities, Wiley, 2006 [32] Vladimir V Piternarg, A practitioner’s guide to pricing and hedging callable libor exotics in forward libor models, Available at SSRN (2003) [33] John Schoenmakers, Robust libor modelling and pricing of derivative products, Chapman and Hall/CRC Probability Series, 2005 76 [34] Henrik Hult Tomas Bjoke, A note on wick products and the fractional blackscholes model, (2004) [35] A.Suleyman Ustunel, An introduction to analysis on wiener space, Springer, 1995 [36] Christian Menn Xin Guo, Robert A.Jarrow, A generalized lando’s formula: A filtration expansion perpective, Working Paper, Cornell University (2006) [37] Xun Yu Zhou, Stochastic controls: Hamiltonian systems and hjb equations (stochastic modelling and applied probability), Springer, 1999 ... Author: Wong Man Chui Title: Applications of Malliavin Calculus and White Noise Analysis in Interest Rate Markets, and Convertible Bonds with and without Symmetric Information Department: Mathematics... ? ?Applications of Malliavin Calculus and White Noise Analysis in Interest Rate Markets, and Convertible Bonds with and without Symmetric Information” by Wong Man Chui in partial fulfillment of the requirements... exactly with a piecewise constant function Chapter Applications of Malliavin calculus to Monte Carlo methods in pricing interest rate derivatives In this chapter we will apply Malliavin calculus in