PRICING OUTSIDE BARRIER OPTIONS WHEN THE MONITORING OF THE BARRIER STARTS AT A HITTING TIME JACOB MOLETSANE MOFOKENG

Transcription

1 PRICIG OSIE BARRIER OPIOS WHE HE MOIORIG OF HE BARRIER SARS A A HIIG IME by JACOB MOLESAE MOFOKEG submitted in accordance with the requirements for the degree of MASER OF SCIECE in the subject APPLIE MAHEMAICS at the IVERSIY OF SOH AFRICA SPERVISOR: R E M RAPOO FEBRARY 5

2 Abstract his dissertation studies the pricing of Outside barrier call options, when their activation starts at a hitting time. he pricing of Outside barrier options when their activation starts at time zero, and the pricing of standard barrier options when their activation starts at a hitting time of a pre specified barrier level, have been studied previously see [], [4]. he new work that this dissertation will do is to price Outside barrier call options, where they will be activated when the triggering asset crosses or hits a pre specified barrier level, and also the pricing of Outside barrier call options where they will be activated when the triggering asset crosses or hits a sequence of two pre specifed barrier levels. Closed form solutions are derived using Girsanov s theorem and the reflection principle. Existing results are derived from the new results, and properties of the new results are illustrated numerically and discussed. Key terms: Outside barrier option, Reflection principle, Girsanov s theorem, Hitting time. i

3 Acknowledgement Firstly I would like to thank GO for everything, and I would like to express my deepest gratitude to my supervisor r E. Rapoo for her excellent guidance, patience, encouragement and support. Secondly, I would never have been able to finish this dissertation without the support from my wife Manase, she always stood by me through good times and bad times. I would like to also thank ISA for the financial support. Last but not least, I would like to dedicate this dissertation to my daughters thabiseng and Lebohang for their love and understanding, and to my high school maths teacher Mr Mabitsela Koma who introduced me to the beauty of mathematics. ii

4 eclaration I declare that pricing of Outside barrier options when the monitoring of the barrier starts at a hitting time is my own work and that all sources I have used or quoted have been indicated and acknowledged by means of complete references. SIGARE Mr AE iii

5 Contents List of figures Mathematical otation vii viii Introduction. escription of financial derivatives Standard barrier options and Outside barrier options An illustration of an application of Outside barrier options Literature review Outline of the issertation Stochastic Processes 9. Probability spaces and continuous time stochastic processes Girsanov s heorem Reflection principle Ito integral Pricing standard options 3. Pricing standard European options Pricing standard barrier options Standard down barrier options Standard up barrier options Pricing Outside knock-in Barrier options 4 4. Outside barrier call option pricing Changing variables to create Brownian motion with a drift under P Changing variables to create Brownian motion with a drift under P iv

6 4. Outside up-and-in barrier call option Crossing a single barrier Outside up-and-in barrier call option Crossing a sequence of two barriers Outside down-and-in barrier call option Crossing a single barrier Outside down-and-in barrier call option Crossing a sequence of two barriers Pricing Outside knock-out Barrier options Outside up-and-out barrier call option Crossing a single barrier Outside down-and-out barrier call option Crossing a single barrier Outside up-and-out barrier call option Crossing a sequence of two barriers Outside down-and-out barrier call option Crossing a sequence of two barriers Comparison of results with existing closed form solutions 9 6. eriving existing closed form solutions from the dissertation results Outside up-and-in barrier call option from an Outside up-and-in barrier call option where the barrier will start to be monitored at a hitting time crossing a single barrier Standard European call option from an Outside up-and-in barrier call option where the activation of the option will start at a hitting time crossing a single barrier Standard European call option from an Outside up-and-in barrier call option where the activation of the option will start at a hitting time crossing a sequence of two barriers Outside down-and-in barrier call option from an Outside down-and-in barrier call option where the activation of the option will start at a hitting time crossing a single barrier Standard European call option from an Outside down-and-in barrier call option where the barrier will start to be monitored at a hitting time crossing a single barrier Standard European call option from an Outside down-and-in barrier call option where the barrier will start to be monitored at a hitting time crossing a sequence of two barriers Verification of the method used in dissertation, for the case where the triggering price process is identical to the payoff price process Outside up-and-in barrier call option crossing a single barrier Outside up-and-in barrier call option crossing a sequence of two barriers v

7 6..3 Outside down-and-in barrier call option crossing a single barrier Outside down-and-in barrier call option crossing a sequence of two barriers Graphs illustrating properties of pricing formulas 8 Conclusion 7 A Matlab code for graphs 3 A. matlab code for graph A. matlab code for graph A.3 matlab code for graph A.4 matlab code for graph vi

8 List of Figures 7. A plot of IC against expiry time: S =, S = 4, K = 5, =, = 9, r =.5, ρ =.5, =., =., and ranges from.5 to A plot of OC against strike price: S =, S =, =, =, = 9, r =.5, ρ =., =., ranges from. to.6, and K ranges from to A plot of IC against the volatility of asset and volatility of asset : S =, S =, K = 5, =, = 99, =.5, r =.5, ρ =.5, ranges from. to.5 and also ranges. to A plot of OC against the volatility of asset and volatility of asset : S =, S =, K =, = 5, = 95, = 4, r =.5, ρ =.5, ranges from. to.5 and also ranges. to vii

9 Mathematical otation P risk neutral measure. P risk neutral measure. r risk free interest rate. S i intial price of asset i, i =,. i volatility of asset i, i =,. ρ correlation coefficient between asset and asset. µ i drift of the lognormal of asset i under probability measure P, i =,. µ i drift of the lognormal of asset i under probability measure P, i =,. {Wit, t } standard Brownian motion driving asset i price process on a probability space Ω, F, P, i =,. {Wit, t } standard Brownian motion driving asset i price process on a probability space Ω, F, P, i =,. x standard cumulative normal distribution function. x, y, ρ standard cumulative bivariate normal distribution function. C value of a standard European call option. IC value of an up-and-in standard barrier call option. OC value of an up-and-out standard barrier call option. IC value of a down-and-in standard barrier call option. OC value of a down-and-out standard barrier call option. IC value of an up-and-in barrier call option where the option will be activated once the triggering viii

10 asset hits the down barrier. OC value of an up-and-out barrier call option where the option will be activated once the triggering asset hits the down barrier. IC value of a down-and-in barrier call option where the option will be activated once the triggering asset hits the up barrier. OC value of a down-and-out barrier call option where the option will be activated once the triggering asset hits the up barrier. IC value of an up-and-in barrier call option where the option will be activated once the triggering asset hits the down barrier after hitting the up barrier first. OC value of an up-and-out barrier call option where the option will be activated once the triggering asset hits the down barrier after hitting the up barrier first. IC value of a down-and-in barrier call option where the option will be activated once the triggering asset hits an up barrier after hitting the down barrier first. OC value of a down-and-out barrier call option where the option will be activated once the triggering asset hits an up barrier after hitting the down barrier first. ix

11 Chapter Introduction his chapter discusses and describes derivatives, it gives the description of standard European and American options, the definitions of Outside barrier options which are the main concern of this dissertation are given. Also included, is the literature review of option pricing and an illustration of an application where Outside Barrier options which become activated at a hitting time can be applied. he chapter ends with the outline of this dissertation.. escription of financial derivatives erivatives A derivative is a financial contract whose value depends on one or more underlying assets. Common underlying assets include stocks, bonds, interest rates, commodities and market indexes. he price of this contract has to be fair to both the contract writer and the contract buyer. erivative pricing is all about calculating this fair price. Options An option is a derivative contract that offers the buyer the right, but not the obligation to buy call option or sell put option an asset at an agreed price strike price, on a specific date expiry date/exercise date. A stock option is a derivative contract whose value is dependent upon the price of a stock. his dissertation will be pricing stock options. European and American options Standard European options can only be exercised at the expiry date whereas an American option can be exercised at any time up to the expiry date.

12 Exotic options Exotic options, are options whose payoff may depend on the whole underlying asset price process or path, whereas in the case of vanilla options European and American the payoff depends only on the terminal value of the underlying asset price.. Standard barrier options and Outside barrier options Barrier options he type of exotic option this dissertation will deal with is the barrier option. here are two kinds of barrier options, knock-out barrier options which start active and become deactivated or worthless once the underlying asset hits a barrier and knock-in barrier options which start inactive and become active once the underlying asset hits a barrier. Standard knock-out barrier options knock-out options can be classified further into up-and-out barrier options the barrier level is above the initial asset price which start active and become inactive once the underlying asset price hits an up barrier, and down-and-out barrier options the barrier level is below the initial asset price which start active and become worthless once the underlying asset price hits a down barrier. Standard knock-in barrier options knock-in options start worthless and become active once the underlying asset price hits an up barrier or a down barrier. Outside barrier options Inside barrier options are barrier options that knock-in or out when the underlying asset price process crosses a barrier. Sometimes the barrier can be triggered by another asset price process, not the underlying asset price process of the option, and in this case we have an Outside barrier option, see []. In this case one asset triggers the barrier, and the other asset determines the payoff of the option, that is, it determines how much the option is in or out of the money. Outside barrier options are the subject of this dissertation.

13 .3 An illustration of an application of Outside barrier options For a useful application of these options, and using an analogy from Carr [8], suppose a South African gold mining company wants to extend gold mining operations into amibia. his will require the South African company to invest in amibian dollars, so the company will be exposed to the risk of a rise in the amibian dollar against the South African rand, so the company can purchase a call option on amibian ollars, this gives it the possibility to buy a fixed amount of amibian dollars at a fixed exchange rate with respect to the South African rand. Since the company hasn t yet decided about expanding its operations into amibia, it might alternatively decide to buy an Outside barrier option which knocks out if the gold price rises because this will be more than the lost option to fix amibian dollar costs, that is going to mine gold in amibia when the gold price is high will more than make for the money lost when we bought an Outside barrier option. ow we have the gold price process and the South African rand vs amibian dollar exchange rate process. he gold price process is our triggering price process and the exchange rate process is our payoff price process. he mining company can decide to start observing whether the gold price process will rise beyond a pre specifed price level up barrier. his will be the Outside up-and-out barrier call option, where the barrier will start to be monitored at a hitting time. Another example of an application of this type of an option can be the case where a European company buys gold from a South African mine. his company has to worry about the Euro rand exchange rate and the price of gold, so the company can purchase an Outside barrier option which knocks out when the price of gold drops to such an extent that it is cheaper to buy gold, rather than to fix the South African rand costs by buying a call option. he triggering price process will still be the gold price process, and the payoff price process will be the Euro rand exchange rate process. he European company can decide to start observing whether the gold price process will drop beyond a pre specifed price level down barrier. his will be the Outside down-and-out barrier call option, where the barrier will start to be monitored at a hitting time. his dissertation will consider two cases, Outside knock-out barrier options crossing of a single barrier and crossing of two barriers and Outside knock-in barrier options crossing of a single barrier and crossing of two barriers. 3

14 .4 Literature review he first attempt to price options was done by the French mathematician Louis Bachelier in his Phd thesis in the year 9, see []. He derived closed form formulae for standard European options. he shortcomings of his model were that it ignored discounting and it assumed that stock prices can be negative. he work of Bachelier was adapted to non-negative stock prices by Sprenkle [4], but this model still didn t have the discounting factor. he Sprenkle model was adjusted further by Boness [4], but the formula still had a lot of parameters that had to be estimated. Fisher Black working on a valuation model of stock warrants was joined by Myron Scholes and their work resulted in an improved version of the model developed by Boness, see [3]. heir work was a far much better option pricing model, and is the commonly used model in option pricing. Moving from the pricing of European options to the pricing of barrier options, the first most common way of pricing barrier options is the partial differential equations method see [4]. he basic idea of the partial differential equations method is that all barrier option prices satisfy the Black Scholes partial differetial equation, but with different domains and boundary conditions. he constructed partial differential equation can be solved using analytical or numerical methods. he second most common way of pricing barrier options is the martingale method. In the case of the martingale probabilistic method, the value of the option is calculated as the expected value of the discounted payoff under a risk neutral measure, and this expected value is calculated using probabilistic methods. For a good demonstration of how the martingale method is used to price barrier options see Musiela and Rutkowski [3], and for partial differential equations methods see Zvan, Vetzal and Forsyth, [4]. his dissertation will use the martingale method. Pricing of a single constant barrier option, was first done on a down-and-out European call barrier option in the Black Scholes environment by solving a corresponding partial differential equation with some boundary conditions, where the single constant barrier was continuously monitored for the entire life of the option, by Merton [3]. he closed form formulae of all types of standard barrier options were developed by Rubinstein and Reiner [37]. his was extended by considering the case where the continuous monitoring of the single constant barrier commenced at time zero of the option s life and terminated strictly before the expiry time of the option s life, the other case involved the situation where the continuous monitoring of the single constant barrier started strictly after time zero and terminated at the expiry time, and these options are called partial barrier options, and they were first priced by Heynen and Kat []. heir closed form fromulae were expressed in terms of bivariate normal 4

15 distribution functions. he pricing of window barrier options was done by Armstrong [], and this was an extension of the work of Heynen and Kat [], these barrier options are a variation of partial barrier options, in this case the continuous monitoring of the barrier starts strictly after time zero of the option s life and terminates strictly before the expiry time of the option. he assumptions in this pricing were of constant interest rate, constant volatility and that the asset price follows geometric Brownian motion, and the valuated formulae were expressed in terms of trivariate normal distributions. Further extensions of pricing conitnuously monitored single barrier options were also done where the barrier was not constant, but was a step function, and these were developed by Guillaume [6]. hese are called step up or step down barrier options, which are serial combinations of single constant continuous of barriers. A natural extension of a single continuously monitored barrier option was to evaluate a continuously monitored double barrier option. A double barrier option has a barrier above upper barrier and a barrier below lower barrier the underlying initial asset price, and these were first priced by Kunimoto and Ikeda [7]. heir pricing of this type of a barrier involved evaluating barrier options with two knockout boundaries, where these boundaries were exponential functions of time. he method they employed was the martingale probabilistic method, and the pricing formula was provided as a sum of an infinite series of normal distribution functions. hey established through numerical procedures that this infinite series converges rapidly. Geman and Yor [4], derived closed form solutions of double barrier options using risk neutral valuation techniques, the Markov property and the Cameron-Martin theorem to obtain an expression of the Laplace transform of the option price, but they had to use numerical inversion of the Laplace transform to obtain option prices. Pelsser [34] derived the closed form solutions by determining the inverse Laplace transform analytically using contour integration method, thereby eliminating the need for numerical inversions routine, apart from this he also extended the previous work by valuing double barrier knock-in options and double barrier knock-out options that pay rebates. Li [9], extended the work of Kunimoto and Ikeda to also include rebates for double barrier options that knock-out. Chen, Wang and Shyu [9] determined the closed form solutions of double barrier options by using the martingale method and applying the reflection principle twice, in their valuation they assumed constant continuously monitored barriers. he reflection principle will be repeatedly used in deriving closed form solutions in this dissertation. he literature discussed so far prices barrier options where the barrier will start to be monitored at a predetermined time, and this dissertation will do the pricing when the 5

16 monitoring of the barrier will start at a hitting time just as in Jun and Ku [4]. A sequential barrier option is a barrier option where the payoff is another barrier option, that is when a knock-in barrier is breached an underlying barrier option is activated by the breaching of this knock-in barrier, as stated in Jun and Ku [4], this type of an option was priced by Pfeffer [35] where he calculated the option price by using Laplace transforms through conditioning on the hitting time. his work was extended by Jun and Ku [4], where they also considered the case where the barrier can start to be continuously monitored, when the underlying asset price involved two hitting times activation of the option will start once the asset has crossed two barrier levels. hey used the martingale method in their valuation with repeated use of the reflection principle and the Girsanov s theorem. Outside barrier options are sometimes referred to as external barrier options see [8] or rainbow barrier options see [8]. Outside barriers can be defined for double barrier options where there is a double barrier for the triggering asset, and they can also be defined for multi asset barrier options where an outside asset triggers the barrier but the payoff is determined by more than asset. Heynen and Kat [] developed closed form solutions of standard European options of a single outside or external barrier option. Kwok, Wu and Yu [8] went beyond the scope of Heynen and Kat [] closed form formulae, their derivation of closed form solutions allowed for more than one underlying asset, and where the barrier can be an exponential barrier, these are exponentially time varying barriers as opposed to time constant barriers. hey derived the closed form formulae using the method of images to find Green s function of the governing differential equation. A further extension was done by the inclusion of standard step up and step down barriers to the pricing of outside barrier options by Guillaume [6]. Another further extension in the pricing of Outside barrier options involved the derivation of closed form formulae, for partial outside double barrier options where the constant barriers will be continuously monitored for the entire existence of the option, these types of barrier options were developed by Banerjee []. he contribution of this dissertation is that it extends the work of Jun and Ku [4], and Carr [8], by deriving closed form formulae of Outside single barrier options, where the option will be activated once the triggering asset first crosses a single barrier level. his dissertation will also cover the case where two hitting times are tied together to activate an Outside barrier option activation commences once the triggering asset crosses a sequence of two barrier levels. o my knowledge no closed form formulae 6

17 exists for this type of Outside barrier options. We will assume constant continuously monitored barriers with no dividends, no rebates, and all of this will be done in a Black Scholes framework. 7

18 .5 Outline of the issertation he new work this dissertation will be doing, is to develop closed form solutions for Outside barrier options when their activation starts at a hitting time by extending the method of [4]. he motivation is this, barrier options are the most actively traded options in over the counter derivatives market, see [], [6] and [34], and as a result ways of properly pricing all different variations of barrier options are essential. his dissertation will be using the Black Scholes model and the martingale method for pricing. Continuous monitoring of the barriers will be assumed, and we will assume that no rebates in our calculations, and all the derivations will be based on the Black Scholes model. he first chapter introduces and explains standard barrier options and Outside barrier options, chapter is about all the mathematics of stochastic processes that will be used in this dissertation, chapter 3 deals with the pricing of standard Europeans options and standard barrier options using martingale methods. Chapter 4 is about the pricing of Outside barrier knock-in call options when the triggering asset has to cross a single barrier, before the option can be activated, and in the same chapter the pricing of Outside knock-in barrier call options is also done when the triggering asset has to cross a sequence of two barriers, before the option can be activated. he fifth chapter deals with the pricing of Outside barrier knock-out call options when the triggering asset crosses a single barrier and a sequence of two barriers, the pricing is done in an analogous manner to the fourth chapter. he sixth chapter derives existing results from the closed form solutions determined in this dissertation, and this chapter also derives the closed form solutions of the [4] paper when the triggering price process and the payoff price process are identical. he graphs demonstrating the properties of the closed form solutions are in chapter 7. Chapter 8 contains the conclusions. 8

19 Chapter Stochastic Processes his chapter introduces the basic concepts of stochastic processes, and all the mathematical concepts that will be used in the dissertation. Familiarity with measure theory and the theory of stochastic processes is assumed, most of the results in this chapter are taken from [5], [3], [39] and [9].. Probability spaces and continuous time stochastic processes efinition... A probability space is a triple Ω, F, P. Ω is the sample space, it is the set of all possible sample paths or the set of all possible outcomes of a random experiment. F is a sigma algebra, it is a collection of subsets of Ω with the following properties:. F.. If A F then its complement A c F. 3. If A, A,... F then i= A i F. P is the probability measure, it is a set function which associates a number P A to each set A F such that: P A, P Ω =, and any sequence of disjoint sets A, A,... F the following result holds: P i= A i = i= P A i. 9

20 efinition... If Ω, F, P is a probability space, then X : Ω R is a random variable if for every Borel set A R, X A F. efinition..3. If Ω, F, P is a probability space, then the random variable X : Ω R is said to be integrable if: Ω X dp <, then EX = Ω XdP exists and is called the expectation of X. efinition..4. Let Ω, F, P be a probability space. he indicator function A : Ω R is defined by:, ω A A ω =, otherwise for A F. efinition..5. If Ω, F, P is a probability space, then the random variable X : Ω R is called square integrable if: X dp <. Ω hen the variance of X is given by V arx = Ω X EX dp, and the family of square integrable random variables will be denoted by L Ω, and the standard deviation of X is given by V arx. efinition..6. he covariance between two random variables Y and Z with the expected values µ Y and µ Z respectively, is denoted by covy, Z = EY µ Y Z µ Z. efinition..7. he correlation between two random variables Y and Z with standard deviations Y and Z is defined as CorrY, Z = covy, Z Y Z.

21 efinition..8. A filtration on a probability space Ω, F, P is an increasing family of sub algebras {F t, t } of F, such that for every t, {F t, t } is included in F and if s t, F s F t F. A filtration represents the increasing information through time. efinition..9. A continuous time stochastic process {X t, t } on a probability space Ω, F, P is adapted to a filtration {F t, t } if for all t, X t is F t -measurable, this means that {Xt B F t } for every Borel set B in R. In this case we say that {X t, t } is F t -adapted. efinition... Let {X t, t } be a continuous time stochastic process on a probability space Ω, F, P, then the filtration generated by this stochastic process is F t = X u, u t, that is the smallest sigma algebra containing all events of the form {Xu B F u, u t} for every Borel set B in R. efinition... A standard Brownian motion is a continuous time process {W t, t } on a probability space Ω, F, P such that:. W =, with probability.. he sample paths t W t are continuous with probability. 3. For any finite sequence of times t t t 3... t n the random variables W t n W t n, W t n W t n,..., W t W t are independent. 4. W t h W t is normally distributed with mean and variance h. efinition... A -dimensional continuous time process {W t, W t, t } on a probability space Ω, F, P is a -dimensional standard Brownian motion, if each W it is a standard Brownian motion and W it are independent of each other, where i =,. efinition..3. Let {W t, t } be a standard Brownian motion on a probability space Ω, F, P, then {X t = expµt W t, t } is called a Geometric Brownian motion on Ω, F, P where µ R, >. his is a non-negative variation of Brownian motion, appropriate for modeling stock prices, and X t has a log normal distribution.

22 efinition..4. Let Ω, F, P be a probability space, and let X be a random variable such that E X <. If H F, then the conditional expectation of X given H is denoted by EX H, and is defined as follows:. EX H is H-measurable.. H EX HdP = H XdP, for all H H, and the conditional expectation has the following properties:. EaX by H = aex H bey H where a, b R and Y is a random variable with E X <.. EEX H = EX. 3. EX H = X if X is H-measurable. 4. EX H = EX if X is independent of H. 5. EY X H = Y EX H if Y is H-measurable. For proofs of the above properties see [3], page 95. efinition..5. A continuous time stochastic process {Y t, t } on a probability space Ω, F, P, is a continuous time martingale with respect to a filtration {F t, t }, if. EY t <, for all t. EY ts F t = Y t, for all t, s. efinition..6. A random variable { : Ω [, } is a stopping time with respect to a filtration {F t, t } if {ω : ω t} F t for all t. hat is, to be able to decide whether is a stopping time or not with respect to {F t, t }, just check whether {ω : ω t} is F t -measurable or not, for all t. efinition..7. Probability measures P and P on a measurable space Ω, F are equivalent if, for any A F, we have P A = if and only if P A =. efinition..8. Let P be a measure on a probability space Ω, F, P, then the probability measure P is said to be risk neutral if

23 . P and P are equivalent.. he discounted asset price process is a martingale under P. efinition..9. An arbitrage is a portfolio value process V t satisfying V = and also satisfying for some time >, P V = and P V > >. efinition... Let ft be a function defined for { t }. he quadratic variation of ft up to is: [f, f] = lim P j= n ft j ft j where P = { = t t t... t n = } is a partition of the interval [, ], and P = max j=,...,n t j t j is the maximum step size of the partition. heorem.. Levy s characterisation of Brownian motion One dimension. Let {M t, t } be a stochastic process on a probability space Ω, F, P and {F t, t } be the filtration generated by this process, then {M t, t } is a Brownian motion process if and only if the following conditions hold:. M = almost surely. he sample paths t M t are continuous with probability 3. he process {M t, t } is a martingale with respect to the filtration {F t, t } 4. [M, M] t = t, for all t. For the proof see [39], page 68. efinition... Let {W t, t } be a standard Brownian motion on a probability space Ω, F, P. A continuous and adapted stochastic process {X t, t } with respect to the filtration {F t, t } which is generated by W t, is an Ito process if it can be expressed in the form: X t = X t v s dw s s u s ds 3

24 in integral form, and in differential form, dx t = u t dt v t dw t, where u t and v t are adapted stochastic processes such that and P P u s ds <, = vsds <, =. heorem.. One dimensional Ito s formula. Suppose that {X t, t } is an Ito Process. Let ft, x be a twice differentiable function with respect to x and once differentiable with respect to t, then the process Y t = ft, X t is an Ito process with the representation: dft, X t = f t t, X tdt f x t, X tdx t f x t, X tdx t, where dx t can be computed using the product rule, For the proof see [3], page 46. dw t dw t = dt, dtdw t =, dw t dt =, dtdt =.. Girsanov s heorem he importance of Girsanov s theorem in derivative pricing is that it gives us the ability to change from one probability measure to another, and in the case of this dissertation our aim is to change a Brownian motion with a drift to a standard Brownian motion under a different probability measure, so that the reflection principle can be applied. heorem.. One dimensional Girsanov s theorem. Let {W t, t } be a standard Brownian motion on a probability space Ω, F, P, and {F t, t } be a filtration generated by W t. Let, be a fixed time and let {θ t, t } be an F t -adapted stochastic process such that it satisfies the following condition, 4

25 E exp θt dt <. efine a process by L t = exp t θ s dw s t θsds. hen the process W t = W t t θ sds is a Brownian motion with respect to the probability measure Q defined by L = dq dp, and this expression means that for all A F, we have QA = A L tdp. he above statement of the one dimensional Girsanov s theorem is modified from Oksendal [3] page 55, which discusses the n-dimensional Girsanov theorem. heorem.. wo dimensional Girsanov s theorem. Let {W t = W t, W t, t } be a -dimensional standard Brownian motion on a probability space Ω, F, P, and {F t, t } be a filtration generated by W t. Suppose θ t = θ t, θ t is a two dimensional process that is F t -adapted and satisfies the following condition, efine t E exp θ s ds t θ s ds <. M t = exp and the probability measure Q by i= W it = W it t t θ is dw is θ is ds, i =,, t θ is ds, t, M = dq dp. 5

26 hen, the process { W t = W t, W t, t }, is a two dimensional Brownian motion under the probability measure Q. For the proof see Oksendal [3] page Reflection principle he reflection principle shows us that the reflected standard Brownian motion after hitting a barrier has the same probability law as the original standard Brownian motion. he path dependent properties of barrier options can be easily examined using this principle, by finding distributions of the maximum or the minimum of the underlying asset price process. heorem.3. Reflection Principle. Let {W t, t } be a standard Brownian motion, on a probability space Ω, F, P. For each path of this Brownian motion that hits level y before time t >, and ends up below level x y at time t, then there is a Brownian motion path with the same probability law that hits level y before time t, and ends up above y x at time t. hat is: P W t x, M t y = P W t > y x for every x, y R such that y and y x, where M t is the maximum value of the process {W t, t } by the time t. Another formulation of this result is the following, the reflection principle states that for each path of the Brownian motion that hits level y before time t >, and ends up above level x y at time t, then by the reflection principle there is a Brownian motion path with the same probability law that hits level y before time t, and ends up below y x at time t. hat is: P W t x, m t y = P W t y x for every x, y R such that y and y x, where m t is the minimum value of the process {W t, t } by the time t. For proof see [36], page 5. Here we will prove the reflection principle for Brownian motion with a drift X t = µt W t using Girsanov s theorem, since the reflection principle is frequently applied to Brownian motion with a drift in this dissertation. 6

27 heorem.3. Reflection Principle. Let X t = µt W t be a Brownian motion with a drift. hen the joint distribution of X t and M t is given by the formula: P X t x, M t y = expyµ x y µt t for every y and y x, where M t is the maximum value of the process {X t, t } by the time t, and a = a π exp t dt. Proof. We have that, we define an equivalent probability measure by: P X t x, M t y = E {Xt x,m t y}, d ˆP dp = exp µw µ. hen {X t = µt W t } is a standard Brownian motion under ˆP by Girsanov s theorem, see heorem.., and continuing we have, after substituting X. dp = exp µx d ˆP µ, Continuing with the calculation of the joint distribtion of X t and M t : dp E {Xt x,m t y} = Ê = Ê exp d ˆP {X t x,m t y} µx µ {Xt x,m t y}, hen E {Xt x,m t y} = Ê exp µy X µ {y Xt x,m t y} = expyµê exp µx µ {Xt y x}. 7

28 efine another equivalent probability measure by: d P = exp µx d ˆP µ, then { W t = X t µt} is a standard Brownian motion under P by Girsanov s theorem, see heorem.., and continuing we have, E {Xt x,m t y} = expyµê d P d ˆP {X t y x} = expyµẽ {X t y x} = expyµ P X t y x = expyµ P W t µt y x = expyµ P W t y x µt x y µt = expyµ. t his proof is modified from [3], page Ito integral he Ito integral is about cases where the integrands and integrators are stochastic processes, and in this dissertation the integrator will be the standard Brownian motion. he problem is that Brownian motion paths are nowhere differentiable, and Brownian motion has unbounded variation, so integrals with repect to Brownian motion cannot be defined in the Riemann integral sense, that is why we need the Ito integral. Let {W t, t } be a Brownian motion process on a probability space Ω, F, P, and we start with the description of a function for which the Ito integral is defined. his function ft, ω should satisfy the following conditions: ft, ω is F t -adapted where F t is the filtration generated by the Brownian motion process, and E S ft, ω dt <. 8

29 In the construction of the Ito integral we firstly define elementary functions φt, ω, their Ito integrals and establish the Ito isometry property, the elementary functions are also used to approximate ft, ω, and S ft, ωdw t will be defined as the limit of a sequence of elementary functions as their sequence approaches ft, ω. efinition.4.. Let {W t, t } be a standard Brownian motion on a probability space Ω, F, P, then φt, ω is called an elementary function if it can be expressed as φt, ω = j e jω {tj,t j }t. ote that φt, ω must be F t -measurable since each e j ω is F tj -measurable. efinition.4.. For elementary functions the integral is defined as follows φt, ωdw tω = j e jωw tj W tj ω. heorem.4. Ito isometry. If φt, ω is elementary and bounded then E For proof see Oksendal, page 6. φt, ωdw t = E φt, ω dt. Ito isometry implies: lim E φ n t, ω φ m t, ωdw t = lim E φ n t, ω φ m t, ω dt n,m n,m As a result φ nt, ωdw t is a Cauchy sequence in L Ω, and it has a limit denoted by ft, ωdw t, and this is the integral of ft, ω with respect to W t. his can be summarized by the following definition: =. efinition.4.3. he Ito integral of the general integrand ft, ω is defined by ft, ωdw t = lim φ n t, ωdw t n where {φ n t, ω} is a sequence of elementary functions such that as n E ft, ω φ n t, ω dt 9

30 Chapter 3 Pricing standard options he aim of this chapter is to illustrate how the techniques of Girsanov s theorem and the reflection principle can be used in option pricing. In this chapter we will show how to price standard European options and standard barrier options using these techniques. See [3] and [3]. All the results in section 3. are given without proof in [8]. Since we will be proving similar but new results in later chapters, we illustrate the techniques that we will use in later chapters by giving detailed proofs here. 3. Pricing standard European options Let {W t, t } be a standard Brownian motion on a probability space Ω, F, P, and we also assume that our Black Scholoes model consists of a single risky asset {S t, t } following the geometric Brownian motion, ds t = µs t dt S t dw t 3. and a risk free asset {B t, t } that evolves acoording to the following ordinary differential equation, db t = rb t dt, B = where > is the volatility constant, µ R is the drift rate and r is a riskless interest rate, other assumptions of the Black Scholes model are that, there are no arbitrage opportunities, trading takes place continuously, the risky asset pays no dividends, there are no transaction costs and no taxes.

31 he Black Scholes model assumes derivatives can be replicated by portfolios of other securities. In order to price the option we need to construct a portfolio that will replicate the option exactly. Let C be an F measurable random variable that represents the derivative payoff at time, and then we construct a replicating portfolio {V t, t } process that replicates the derivative payoff at every time, so that {V t = C t, t }. Otherwise an arbitrage opportunity exists if V t < C t for some t, then one can sell the derivative and buy the replicating strategy thereby making a profit, and if V t > C t then one can sell the derivative and buy the replicating again making a profit. ue to the no arbitrage principle the portfolio will replicate the derivative at every instant. he construction of the portfolio begins with an initial V, and at each time t the portfolio has {θ t, t } shares of stock where θ t is adapted to the filtration generated by W t, and the remainder of the portfolio value {V t θ t S t, t } is invested at the risk free interest rate r. he constructed portfolio will be self financing meaning that, changes in the of value of the portfolio are entirely due to changes in value of the assets and not to the injection or removal of wealth from outside. As a result the evolution of the portfolio value V t is given by: dv t = θ t ds t rv t θ t S t dt. Firstly we find the probability measure under which the discounted stock price process is a martingale and thereafter we show that the discounted portfolio value process is a martingale under the same measure. We start by applying applying Ito s formula to the discounted asset price process exp rts t : dexp rts t = r exp rts t dt exp rtds t = r exp rts t dt exp rts t µdt dw t = r exp rts t dt µ exp rts t dt exp rts t dw t = µ r exp rts t dt exp rts t dw t, and finally µ r dexp rts t = exp rts t dt dw t.

32 We can see that E exp µ r dt <, t, we can put µ r L t = exp W t µ r t for t and define an equivalent probability measure by: dp dp = exp µ r W µ r. 3. hen from Girsanov s theorem heorem.., {W t Brownian motion on a probability space Ω, F, P. = W t µ r t, t } is a standard From. we have dexp rts t = exp rts t dw t exp rts t = S t exp rus u dw u, so we have shown that exp rts t is a martingale under the measure P, this measure is called a risk neutral measure and is equivalent to the original measure P. ext we show that the discounted portfolio value process {exp rtv t, t } is a martingale under P, by applying Ito s product formula: dexp rtv t = dexp rtv t exp rtdv t dexp rtdv t = r exp rtv t dt exp rtθ t ds t rv t θ t S t = r exp rtv t dt exp rtθ t µs t dt S t dw t rv t θ t S t µ r = r exp rtv t dt exp rt rv t dt θ t S t dt dw t µ r = θ t S t dt dw t, after substitution of W t we end up with the following: exp rtv t = V t θ u S u dw u.

33 So exp rtv t is a martingale under the risk neutral measure P, and this implies the following: exp rtv t = E exp r V F t = E exp r C F t, and so from above, the value of an option at time zero, is the expected value of the payoff discounted at the risk free interest rate r under the risk neutral measure P, that is: C = E exp r C. he risky asset s evolution under the probability measure P is: µ r ds t = µs t dt S t dw t = µs t dt S t dw t dt = rs t dt S t dwt. We can find the solution of this stochastic differential equation using Ito s lemma, see heorem..3 to the function ft, x = ln x, and we get: dst = rdt dwt S t d ln S t = S t ds t St dt S t = S t ds t dt. he above equation can be rewritten as: and so the solution is: ln St S t = rt Wt St ln = r t Wt, S S t = S exp r t W t. 3.3 When the asset price S of a standard European call option at the expiry date > is less than the strike price K, that is S < K, then the option is worthless, if S > K the option is exercised and has the value S K. 3

34 he value of a standard European call option at time zero, is the expected value of the payoff discounted at the risk free interest rate r under the risk neutral measure P and the payoff is: S ω K, S ω > K C ω =, S ω. hroughout this dissertation and,, will represent the standard univariate normal distribution function and the standard bivariate normal distribution function respectively. heorem 3... he value of a standard European call option at time zero is: C = S where x = x π exp t dt. ln S K r exp r K ln S K r, Proof. From the statement just before the theorem we have, C = E exp r S K {S >K} = E exp r S {S >K} E exp r K {S >K} = E exp r S exp r W {S >K} E exp r K {S >K} = S E exp W {S >K} exp r KE {S >K}. We will apply Girsanov s theorem with θ =, to evaluate the first expectation. We can see that E exp dt <, t, so if we define an equivalent probability measure by: then by Girsanov s see heorem.. {W t under P. dp dp = exp W, = W t t, t } is a standard Brownian motion 4

35 Continuing with the calculation of C, after substitution with dp we get: dp dp C = S E dp {S >K} exp r KE {S >K} = S E {S >K} exp r KE {S >K} = S P S > K exp r KP S > K. Substituting the Brownian motion process {W t, t }, the asset price process {S t, t } becomes: S t = S exp r t W t t Starting with the calculation of P S > K: = S exp r t W t. 3.4 W > K = P r K W > ln S K = P W > ln r S = P W < ln S K r P S > K = P S exp r = ln S K r. Similarly, P A = ln S K r. We have proved that the value of a standard European call option at time zero is: C = S ln S K r exp r K ln S K r. 5

36 his result is proved in [3]. 3. Pricing standard barrier options his section starts by defining Brownian motions with a drift under probability measures P and P, these will be frequently used in the pricing of standard barrier options. After this, we show how to price standard barrier options using Girsanov s theorem, the reflection principle and the parity relationship. o get a Brownian motion with a drift under P, we change variables of a geometric Brownian motion in equation 3., by letting {X t = ln St S, t }, and replacing r with µ, ending up with: µ X t = t Wt. 3.5 From equation 3.3 in the proof of heorem 3.., we can, in an analogous way, get a Brownian motion with a drift under P, by replacing r with µ, ending up with: µ X t = t Wt. 3.6 hroughout this chapter and will represent the up and down barriers of the price process {S t, t } respectively, u = ln S will be the up barrier as seen by the process {X t, t }, and d = ln S will be the down barrier as seen by the process {X t, t } and the strike price K will be k = ln K S as seen by the process {X t, t }, and finally we will always assume the intial asset price S, to be greater than and less than throughout the whole chapter. For the rest of this chapter τ d will be the first time the process {X t, t } hits the down barrier d, and τ u will be the first time this process hits the up barrier u τ d = and τ u = will be the case when the barrier is not hit. 6

37 3.. Standard down barrier options he value of a down-and-in barrier call option IC at time zero, is the expected value of the payoff discounted at the risk free interest rate r under the risk neutral measure P, where the payoff is: S ω K, if inf t S t ω, and S ω > K ICω =, otherwise, and the payoff a down-and-out standard barrier call option OC is given by: S ω K, if inf t S t ω >, and S ω > K OCω =, otherwise. heorem 3... he value of a down-and-in barrier call option at time zero, where the strike price K is greater than the down barrier is: µ IC = S ln S K µ µ S K exp r ln S K µ S. Proof. he risk neutral value of IC, can be expressed as follows, IC = E exp r S K {S >K,m d}. As in the proof of heorem 3.., IC = S E {S >K,m d} K exp r E {S >K,m d} = S P S > K, m d K exp r P S > K, m d. It is easy to see that, P S > K, m d = P X > k, m d = E {X >k,m d}. 7

38 Applying the conditions of Girsanov s theorem heorem.. to equation 3.4, we can see that E exp µ dt <, t, where θ = µ, and we can put, µ L t = exp Wt µ t for t and define an equivalent probability measure by: dp dp = exp µ W hen from Girsanov s theorem heorem.., {X t = W t motion on a probability space Ω, F, P, and µ. 3.7 µ t, t } is a standard Brownian P X > k, m d = E dp dp {X >k,m d} µ = E exp X µ {X >k,m d}. efine a new process {X t, t } as, X t X t = d X t t τ d t > τ d. 3.8 By the reflection principle heorem.3., {X t, t } is a standard Brownian motion process under the probability measure P, and substituting with this process we get the following, P X > k, m d = E exp = exp d µ µ d X µ {X d k} E exp µ X µ {X d k}. Clearly, Girsanov s theorem can be applied here, so we define another probability measure by: d ˆP µ dp = exp X µ, 8

39 then {Ŵt = X t µ see heorem.. and t, t } is a standard Brownian motion under ˆP by Girsanov s theorem, P X > k, m d = exp d µ E µ exp X µ d = exp d µ E ˆP dp {X d k} dµ = exp ˆP X d k dµ µ = exp ˆP Ŵ d k dµ µ = exp ˆP Ŵ d k dµ Ŵ = exp ˆP ln S K µ µ = ln S K µ S. {X d k} µ Similarly P X > k, m d = ln S S K µ. hus, µ IC = S ln S K µ µ S K exp r ln S K µ S. ote that this result agrees with the one in [8] heorem 3... he value of a down-and-out barrier call option at time zero, where the strike price K is greater than the down barrier is: ln S K OC = S µ µ S S ln S K exp r K µ ln S K µ µ K exp r S ln S K µ. 9