UPPSALA DISSERTATIONS IN MATHEMATICS

45

On the pricing equations ofsome path-dependent options

Jonatan Eriksson

Department of MathematicsUppsala University

UPPSALA 2006

List of Papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Eriksson, J. (2005) Monotonicity in the volatility of single-barrieroption prices, to appear in the Int. J. Theor. Appl. Finance.

II Eriksson, J. (2005) When American options are European, sub-mitted to Decis. Econ. Finance.

III Arnarson, T., Eriksson, J. (2005) On the size of thenon-coincidence set of parabolic obstacle problems withapplications to American option pricing, submitted to Math.Scand.

IV Eriksson, J. (2005) Explicit pricing formulas for turbo warrants,submitted to Risk magazine.

Reprints were made with permission from the publishers.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Path-dependent European options . . . . . . . . . . . . . . . . . . . . . . 21.3 Parabolic obstacle problems and free boundary problems . . . . . 31.4 American options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Included papers and description of the results . . . . . . . . . . . . . . . . . 72.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Knock-out options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Knock-in options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Sammanfattning på svenska (Summary in Swedish) . . . . . . . . . . . . . . . 13Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1. Introduction

In this thesis we study financial mathematics in continuous time. This disci-pline of science is typically concerned with the problem of pricing and hedg-ing financial instruments defined in terms of some underlying asset. Exam-ples of such instruments are stock-options and warrants. The value of a stock-option or a warrant is given as the discounted expected future pay-off, butsince the value of the underlying asset typically is unknown at future times,the pay-off is too. The price-evolution has to be modeled with stochastic pro-cesses. However, for the purpose of option pricing not any stochastic processwill do, but there are strict theoretical rules the asset process has to obey toavoid arbitrage in the market. The discounted asset price has to be a martin-gale when pricing financial instruments. The common way of modeling assetprices is to use Brownian motion and more generally solutions to stochas-tic differential equations. In this context the problem of option valuation issimilar to problems of heat conduction and particle motion in physics sinceexpected values of functions of solutions to stochastic differential equationssolve parabolic partial differential equations similar to the heat equation. Prob-lems in finance are however very different in their nature from their physicalcounterparts. In finance, the volatility of the stock, which roughly is the sameas the diffusion coefficient in physics, is most certainly unknown and one hasto rely on historical data and educated guesses.

The implications of model misspecification and misspecification of thevolatility are important and questions about robustness which arise due to theuncertainty of the volatility are connected to properties of the Black-Scholespartial differential equation and preservation of convexity of solution to thatequation, compare [14] and [15].

1.1 Option pricingIn the two seminal papers [3] and [19] the authors describes how to price op-tions on a market with one risky asset and one risk-free asset, such that theno-arbitrage principle holds. An arbitrage opportunity is a risk-free invest-ment strategy, with zero initial endowment, in financial instruments such asstocks, bonds and options, such that the final wealth is non-negative almostsurely and positive with positive probability. The no-arbitrage principle saysthat a financial market should contain no arbitrage opportunities. The resultin the above mentioned papers can be formulated as follows: Assume that

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the market consists of two trades assets B(t) and S(t) evolving according todB(t) = rB(t)dt and dS(t) = µS(t)dt + σS(t)dW (t) where the constants arer ≥ 0, the interest rate, µ , the appreciation rate and σ , the volatility, and whereW is a standard Brownian motion. Then the only price of an option at time t,paying φ(S(T )) at some future time T , and which does not introduce arbi-trage in the market is the discounted expected value of φ(S(T )). However, theexpectation is not calculated under the real-world measure P which is usedto describe the dynamics of S but under the so-called risk-neutral measure Q.This measure is defined as the unique measure making the discounted stock-price a martingale, and the appreciation rate of S(t) under this measure is r.By using the explicit expression for the stock-price process under the measureQ

S(T ) = S(t)e(r− 12 σ2)(T−t)+σ(W Q

T −W Qt )

and by using that the increment of a Brownian motion is normally distributedwith variance equal to the length of the increment, the expected value for theprice

V (s, t) = Es,te−r(T−t)φ(S(T ))

can be calculated explicit in terms of the normal distribution function. Theresults is the famous Black-Scholes formula which was derived for the call-option, φ(s) = (s−K)+, and for the put-option, φ(s) = (K− s)+, in the abovementioned papers.

By the Feynman-Kac representation formula for solutions to parabolic dif-ferential equations, the price V (s, t) can also be computed as the solution tothe Black-Scholes equation

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σ2s2 ∂ 2V∂ s2 + rs

∂V∂ s

− rV +∂V∂ t

= 0

together with the final value V (s,T ) = φ(s), compare [13]. We note that theterm ∂V

∂ t has the opposite sign of the corresponding term in the heat equation.This is consistent with our specifying a final condition rather than an initialcondition as in physics.

1.2 Path-dependent European optionsA European vanilla option is an option with a pay-off that only depends onthe stock-price at maturity T . The pay-off can be described as φ(S(T )) forsome contract function φ . A European path-dependent option is an optionwhose pay-off at maturity T depends on the whole path of the stock-priceS(t) between t = 0 and t = T . Examples of such options are Asian options,which depend on the average of the stock-price, Barrier options and lookbackoptions, which depend on the maximum and/or the minimum of the stock-price.

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The most common types of barrier options are knock-in options and knock-out options. As their names suggest a knock-in option is activated when aprescribed barrier is hit by the stock-price, and a knock-out option is extin-guished when the barrier is hit. Depending on the relation between the initialstock-price and the barrier these options are usually given names like up-and-out option, up-and-in option, down-and-out option and down-and-in option.

The theory of pricing barrier options goes back to [19] where a down-and-out option is priced under geometrical Brownian motion. A more completepricing scheme in this case can be found in [21] where the authors make clevercombinations of the distribution functions of absorbed geometrical Brownianmotions to generate prices for a large numbers of single-barrier options. Com-pare also [22].

In the paper [22] a sensitivity analysis is performed for barrier options ofcall and put types (see also [24]), i.e. a calculation of the options ∆ (sensitivityto movements in the underlying stock) and Γ (sensitivity of ∆ to movementsin the underlying stock), which is of great interest when performing dynamichedging. Especially the sign of Γ is important when it comes to the effect onthe hedging portfolio of a misspecified volatility. For European vanilla optionsand for American options it is well-known that convexity of the contract func-tion is enough to ensure a positive Γ which in turn ensures a super-hedgingportfolio if the volatility is overestimated, compare [11], [14], [13], [9] and[10]. However, when it comes to barrier options the situation can be verydifferent. Here convexity depends not only on the convexity of the contractfunction but also on the underlying process and the barrier. This comes asno surprise since an (almost trivial) example of a barrier option with convexcontract function and non-convex price is an up-and-out put option with thebarrier in-the-money, i.e. the option has non-zero intrinsic value at the barrier.For certain other types of barrier options convexity is preserved and this is oneof the themes in the thesis and constitutes the content of Paper I.

A new kind of barrier option called turbo warrant is studied in Paper IV.This instrument is essentially a call or a put option with a barrier in-the-moneyand a non-constant rebate which is activated if the barrier is hit prior to ma-turity. This means that if the option is knocked out a small sum is paid to thewarrant holder. In the call-case the rebate is given by the difference of the low-est recorded stock-price during a three-hour period after the knock-out eventand the strike price. In the put-case it is the highest recorded stock-price whichdetermines the rebate.

1.3 Parabolic obstacle problems and free boundaryproblemsIn a free boundary problem one seeks the solution to a differential equationM f = 0, in some domain Ω where only a part S of the boundary ∂Ω is given,

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whereas the remaining portion Γ is not a priori prescribed. On the boundaryof Ω some boundary conditions f = φ are given. To get a well-posed probleman additional condition is given on Γ. This additional boundary condition de-pends on the particular problem in question, and in financial applications thecondition is the so-called high-contact principle , ∇x f = ∇xφ . A solution to afree boundary problem consists of the function f and of the free boundary Γ.A special class of free boundary problems are the ones which can be writtenas obstacle problems, for instance as the minimization of a functional over aclosed convex set K = f ∈ H : f ≥ φ in a suitable function space H. Herethe a priori given function φ is referred to as the obstacle. The main differencebetween a general free boundary problem and an obstacle problem is that inan obstacle problem one can analyze the solution without having to analyzethe free boundary at the same time.

When studying obstacle problems it is often useful to write the problem asa so-called complementary problem

M f ≤ 0,

( f −φ)M f = 0,

f ≥ φ ,

where M is a differential operator corresponding to the above minimizationproblem. The obstacle problem now becomes to find a function in a suitablefunction space, e.g. in the Sobolev space W 2,1

q , satisfying the above inequal-ities in a weak sense such as almost surely, as distributions or in a viscos-ity sense. The free boundary in the obstacle problem is the set Γ = ∂(s, t) :f (s, t) > φ(s, t), and under some appropriate regularity assumptions the solu-tion f satisfies the so-called principle of smooth fit on Γ, that is, the mappingx → ∇x( f − φ) is continuous across Γ, at least at points where φ is continu-ously differentiable. Intuitively the principle of smooth fit is clear, since it infinancial applications means that the number of shares of the underlying stockto be held in the hedging portfolio does not jump when the stock passes intothe stopping region.

Apart from financial applications, obstacle problems arise naturally inphysics and mechanics. Examples of such situations are:1. The description of a weightless elastic membrane subject to shape change

under constraint.2. Lubrication with oil between two surfaces.3. Melting of ice in water. This problem is however not directly given as an

obstacle problem, rather as a one-phase Stefan problem, but it can be re-duced to an obstacle problem.

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1.4 American optionsAn American option, unlike a European option, can be exercised at any timeprior to the maturity time T . This means that at any instant in time the holderof the option needs to decide if to exercise or to hold the option. This extracomplexity in the problem results in the fact that explicit pricing formulasrarely exists. Only in special cases such as if the American option has thesame value as the corresponding European option or if the maturity time isinfinite is the price explicitly known, compare [11], [25], [20], [7] and PaperII.

Suppose that the contract function is φ , then if the holder decides to exerciseat time τ the received amount is φ(S(τ)). In the papers [1] and [16] it is shownthat the unique arbitrage free price VA at time t of an American option withcontract function φ is given by the optimal stopping problem

VA(s, t) = supτ∈F [t,T ]

Es,te−r(τ−t)φ(S(τ)),

where the supremum is taken over all stopping times with respect to the fil-tration generated by the driving Brownian motion of S(t). Since the decisionon exercising or holding the option only can be based on past information itis intuitively clear that the supremum only can be taken over the above men-tioned stopping times. Notice also that VA always satisfies VA(s, t) ≥ φ(s) andVA(s, t)≥V (s, t) since it is allowed to exercise the option either at t or at T . Be-sides calculating the price of the option it is also interesting to decide a goodstrategy for the option holder. That is to find a stopping time τ∗ realizing thesupremum above. Intuitively it is clear that if VA(S(t), t) > φ(S(t)) it is not op-timal to exercise the option since VA(S(t), t) corresponds to holding the optionand φ(S(t)) corresponds to exercising. In Appendix D in [17] it shown thatan optimal stopping time is given by τ∗ = infu ≥ t : VA(S(u),u) = φ(S(u)),thus the optimal time to exercise the option is when the process (S(t), t) leavesthe continuation region

C = (s, t) : VA(s, t) > φ(s).There is another way of characterizing the American option price. In the

book [2] the connection between optimal stopping problems and free bound-ary problems is established and for the American option pricing problem itcan be shown that VA satisfies Black-Scholes equation at every point in the in-terior of the continuation region and that it is continuous on the whole domain.Moreover, since VA ≥ φ and since VA−φ = 0 on the free boundary the solutionalso satisfies the principle of smooth fit there if it is regular enough. Thus theAmerican option pricing problem can be viewed as a free boundary problemwhere the free boundary is the boundary of C . This free boundary problemcan be rewritten as an obstacle problem where the contract function φ is theobstacle and where the inequality MVA ≤ 0 holds on the whole domain andwhere MVA = 0 holds on the continuation region C .

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When it comes to the free boundary itself, not very much is known in thegeneral case. Questions of regularity are in general difficult to answer. Qual-itative results on the behavior of the boundary in the multi-dimensional caseclose to maturity are given in [23] and on the shape in [25], [20]. However, inthe special case of a one-dimensional geometric Brownian motion there hasbeen a lot of work done. Recently it was shown that the boundary is a convexC∞ function of time, compare [8], [5] and [6].

Due to the difficulty of finding the continuation region one is often inter-ested in either approximations or of finding non-trivial subsets in which oneknows not to exercise the option, compare Paper III.

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2. Included papers and description ofthe results

2.1 Paper IIn this paper we study single-barrier options. The main question is: Underwhich conditions can we guarantee that a convex contract function givesrise to a convex price, i.e. that the options Γ is non-negative? Otherquestions answered in this paper are: When is super-hedging with volatilityover-estimation possible? What can be said about the options sensitivity tomovements in the underlying asset, i.e. the options ∆?

A barrier option is an option which gives a certain pay-off at maturity Tconditioned on some afore-hand determined behavior of the underlying asset.Given a contract function φ and a barrier b > 0, a typical barrier option is anyof the following:• A down-and-out option. It pays φ(S(T )) at maturity T if the underlying

asset S(t) stays above b for all times t ≤ T .• A down-and-in option. It pays φ(S(T )) at T if the barrier is hit from above

at some time τ ∈ [0,T ].• An up-and-out option. It pays φ(S(T )) at T if the underlying asset stays

below b for all times t ≤ T .• An up-and-in option. It pays φ(S(T )) at T if the barrier is hit from below

at some time τ ∈ [0,T ].If the conditions above are not satisfied, nothing is paid to the holder. Thetechniques used in this paper are borrowed from the paper [15] and the mainidea is to use the maximum principle to show that the convexity of the finalvalue is preserved by the option price. In the paper the underlying asset isassumed to solve an SDE of the form

dS(t) = (r−δ )S(t)dt +σ(S(t), t)S(t)dW (t)

where the diffusion coefficient α(s, t) := sσ(s, t) is Hölder(1/2) in space andcontinuous in time and the risk-free rate of return and the dividend rate areconstants.

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2.1.1 Knock-out optionsConsider the case of knock-out options. By introducing the stopping time τb =inft ≥ 0 : S(t) = b we may describe the pay-off at maturity as

φ(S(T ))1τb>T.

The pay-off clearly depends on the whole trajectory of S(t). By risk-neutralvaluation the unique arbitrage-free value of the up-and-out option is given by

U(s, t) = Es,te−r(T−t)φ(S(T ))1τb>T.

Knowing that the option is not yet knocked out, the value function U(s, t) sat-isfies Black-Scholes equation, but with the extra boundary condition U(b, t) =0 for t ∈ [0,T ]. Depending on if it is a down-and-out option or a up-and-outoption the option is alive on s > b or s < b respectively. One of the results inPaper I answers the first of the above questions for knock-out options.

Theorem 1 (Convexity of knock-out options) Assume that the contractfunction is convex on [0,∞) and zero at the barrier. Then a down-and-outoption has a positive Γ if the interest rate is dominated by the dividend rateand an up-and-out option has a positive Γ if the dividend rate is dominated bythe interest rate.

It is worth to notice that if the relation between r and δ goes in the “wrong”direction, a convex contract function which is zero at the barrier need not giverise to a convex price. Thus the theorem may fail if r < δ . The other twoquestions posed in the beginning are answered in the following corollaries.

Corollary 1 Assume that the contract function is convex on [0,∞) and zero atthe barrier. Then a down-and-out option has a positive ∆ if the interest rate isdominated by the dividend rate and an up-and-out option has a negative ∆ ifthe dividend rate is dominated by the interest rate.

Corollary 2 Super-hedging of up-and-out options and down-and-out optionsby overestimating the volatility is possible in the cases described in Theorem 1which give rise to convex prices.

2.1.2 Knock-in optionsThe pay-off of a knock-in option can be described by

φ(S(T ))1τb≤T,

and by risk-neutral valuation the value is given by

V (s, t) = Es,te−r(T−t)φ(S(T ))1τb≤T.

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To price these options by the means of differential equations we view theknock-in option as a knock-out contract paying zero at maturity if the barrieris not hit and paying a rebate if the barrier b is hit. The rebate equals thevalue of the corresponding vanilla option at b. More precisely, let f (s, t) =Es,te−r(T−t)φ(S(T )). Then the rebate is given by f (b,τb). Moreover, assumingthat the barrier is not yet hit at present time t, the function V (s, t) satisfiesBlack-Scholes equation with boundary condition V (b, t) = f (b, t) and finalvalue zero. By using the differential equation for V the following answers tothe questions posed in the beginning of this section can be given.

Theorem 2 (Convexity of knock-in options) Assume that the contract func-tion is convex on [0,∞). Then a down-and-in option has a positive Γ if thedividend rate is dominated by the interest rate and an up-and-in option has apositive Γ if the interest rate is dominated by the dividend rate.

Also in this case it can happen that if the relation between r and δ goes inthe wrong direction, a convex contract function need not give rise to a convexprice.

Corollary 3 Assume that the contract function is convex on [0,∞). Then adown-and-in option has a negative ∆ as long as the barrier has not been hitif the dividend rate is dominated by the interest rate and an up-and-in optionhas a positive ∆ as long as the barrier has not been hit if the interest rate isdominated by the dividend rate.

Corollary 4 Super-hedging of up-and-in options and down-and-in options byoverestimating the volatility is possible in the cases described in Theorem 1which give rise to convex prices.

2.2 Paper IIThe main questions in Paper II are the following ones: For which class ofcontract functions is it so that there exist a diffusion model in which the priceof an American option coincides with the price of the corresponding Europeanoption? If we know that there is some diffusion model in which the pricescoincide, what can be said about the contract function?

It is well-known from the paper [11] that if the contract function is convexand zero at the origin and if the dividend rate is zero, then the prices alwayscoincide. However, by relaxing the condition that the equality should holdfor all diffusion models, i.e. for all volatilities, but only demanding that theequality holds for some non-zero volatility satisfying some suitable regular-ity conditions, the class of contract functions becomes strictly larger than theclass of convex ones being zero at the origin. Moreover, if we know that thereis equality for some model, the contract function must belong to this class.This means in particular that outside of this class there is no diffusion model

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in which the prices can coincide, hence that the possibility to exercise earlyalways has a positive value.

The contract functions which qualifies are the ones satisfying the followingconditions:

Condition 1 1. φ is piecewise C2 on [0,∞),2. φ ′

+(a)−φ ′−(a) ≥ 0 for all a ∈ [0,∞)

Condition 2 1. sφ ′−(s)−φ(s) ≥ 0 for all s ≥ 0, and2. if s0φ ′−(s0)− φ(s0) > 0 for some s0 > 0, then sφ ′−(s)− φ(s) > 0 for all

s ≥ s0.

In the paper we assume that the stock-price solves an SDE of the formdS(t) = rS(t)dt +S(t)σ(S(t), t)dW (t) for some Brownian motion W and thatthe volatility satisfy the following conditions.

Assumption 1 1. The diffusion coefficient sσ(s, t) is strictly positive for alls > 0 and all t ∈ [0,T ].

2. There is a constant K > 0 such that |sσ(s, t)| ≤ K(1+ s) for all s ≥ 0 andall t ∈ [0,T ].

3. The function s2σ2(s, t) has a Hölder continuous partial derivative withrespect to s for every t ∈ [0,T ] and is continuous in t.

The main result of the paper is the following one:

Theorem 3 Assume that the pay-off function φ is piecewise C2. Then φ satis-fies Conditions 1 and 2 if and only if there is a model such that the prices ofthe European and American options with pay-off φ are the same.

2.3 Paper IIIThis paper deals with the size of the non-coincidence set of certain parabolicobstacle problems. This is the set on which the solution to the obstacle prob-lem is strictly larger than the obstacle. The operators in the paper have theform

L f =n

∑i, j=1

ai j∂ 2 f

∂xi∂x j+

n

∑i=1

bi∂ f∂xi

+ c f − ∂ f∂ t

.

We work under the assumption that the lower order coefficients are Dini-continuous and that the top-order coefficients have Dini-continuous spatialderivatives of first order. Dini-continuity is weaker than the more commonHölder continuity and is defined in the following way.

Definition 1 A modulus of continuity α is called a Dini modulus of continuityif

∫ ε0

α(s)s dt < ∞ for all ε > 0 small enough, and a function h is called Dini-

continuous if h has a Dini modulus of continuity.

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The assumptions on the coefficients are the following.

Assumption 2 The coefficients ai j(x, t) are Dini-continuous in x and t andhave Dini-continuous first-order partial derivatives with respect to x. The co-efficients bi(x, t) and c(x, t) are all Dini-continuous in x and t.

The obstacle problem considered in this paper is the following one:

L f ≤ 0,

( f −φ)L f = 0,

f ≥ φ ,

in some domain Ω ⊂ Rn ×R, together with the initial condition f (x,0) =

φ(x,0), where φ is a C2,1-function with Dini-continuous partial derivatives.The non-coincidence set is the set C = (x, t) ∈ R

n ×R : f (x, t) > φ(x, t)and the positivity set is defined as the set where φ is a strict sub-solution toL f = 0, i.e. the set U = (x, t)∈R

n×R : L φ(x, t) > 0. The time-sections Ut

and Ct are defined as Ut = x ∈ Rn : (x, t) ∈U and similarly for Ct . It is clear

from the equation that the inclusion U ⊂ C holds and the purpose of Paper IIIis to show that if the boundary of U is smooth enough, then the inclusion isstrict. More precisely, the main result of Paper III is the following:

Theorem 4 Assume that the lateral part of the boundary of U can be repre-sented locally by a surface which is C1-Dini is space and Lipschitz in time.Then for each t > 0 there is δ (t) > 0, independent of x, such that the distancebetween the boundaries of the time-sections Ut and Ct is greater than δ (t).

The main tool in showing Theorem 4 is the Hopf boundary point lemma,which states that a positive supersolution, to a parabolic equation, which van-ishes at a point on the boundary of a domain must have a positive inwarddirectional derivative at the same point. However, in the standard literature,e.g. [12] or [18], one assumes that the boundary has a strong interior sphereproperty (at least in the spatial variables) which essentially means that oneshould be able to touch the boundary from the inside of the domain with asphere. But by using the C1-Dini assumption on the top-order coefficient andTheorem 1.5.10 in [4] and a change of variables, a Hopf-type lemma can beobtained for domains satisfying only the interior C1-Dini condition in Theo-rem 4.

The results of Theorem 4 can be used to show that the boundary of the pos-itivity set cannot touch the boundary of the continuation region in Americanoption pricing and that it must lie on a uniform distance in space from the freeboundary.

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2.4 Paper IVThe final paper is a paper in which explicit pricing formulas are derived forturbo warrants of put and call types in the classical Black-Scholes model. Inthis paper the stock-price process is assumed to evolve according to

S(t) = S(0)e(r− 12 σ2)t+σW (t)

for some Brownian motion W . Given a barrier b > 0 and a strike price K < b,a turbo call pays

(S(T )−K)

at maturity T if the underlying stock stays above the barrier b at all timesbefore maturity. If the barrier is hit, say at τb, then a rebate

(m(τb +δ )−K)+

is paid at time τb +δ , i.e. δ time units after the first time S hits b.The pay-off of a turbo put is defined in an analogous way. In this case K > b

and the pay-off at maturity T is

(K −S(T ))

if the stock-price stays below the barrier at all times before maturity. If thebarrier is hit at some time τb prior to maturity a rebate

(K −M(τb +δ ))+

is paid at τb +δ . Here m(t) = min0≤u≤t S(t) and M(t) = max0≤u≤t S(t) denotesthe running minimum and the running maximum respectively.

To price the warrant one prices the two parts separately, i.e. the knock-outpart and the rebate part. The key observation to make here is that the rebatepart can be viewed in the following way. Consider first the call-case:• Let Vc(b) = Ebe−rδ (m(δ )−K)+, i.e. the value of the rebate when the bar-

rier is hit.• The rebate has the same value as an American digital option paying Vc(b)

at the first hitting time τb given that the stock-price is above, and has nothit the barrier yet. The value is given by Vc(b)Es,te−r(τb−t)1τb≤T.

The put-case is similar:• Let Vp(b) = Ebe−rδ (K −M(δ ))+.• The rebate has the same value as an American digital option paying Vp(b)

at the first hitting time τb given that the stock-price is below and has not hitthe barrier yet. The value is given by Vp(b)Es,te−r(τb−t)1τb≤T

Now since the densities for τb, m(t) and M(t) are explicitly known, the prob-lem of finding the price is just a matter of integration. Notice that the de-composition made above strongly depends on the time-homogeneity and theMarkov property of the underlying process S(t). With a time-inhomogeneousprocess the decomposition is no longer possible and we really need the jointdensity of (m(t),τb) and the joint density of (M(t),τb).

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Sammanfattning på svenska (Summary inSwedish)

I denna avhandling bestående av en introduktion och fyra artiklar studerasfinansiell matematik i kontinuerlig tid. Huvudtemat för avhandlingen ärvägberoende optioner och de optioner som studeras är barriäroptioner ochamerikanska optioner. I finansiell matematik är man ofta intresserad av attprissätta finansiella instrument definierade i termer av någon underliggandetillgång såsom en aktie. I och med att den underliggande tillgångens framtidavärde i allmänhet inte är känt så modelleras den med en stokastisk process.I denna avhandling används lösningar till stokastiska differentialekvationersom modell för aktiepriset.

Priset för en option kan i allmänhet beräknas som ett väntevärdeav en kontraktsfunktion av den underliggande stokastiska processen.Kontraktsfunktionen är en på förhand specificerad funktion som angerhur utbetalningsstrukturen för kontraktet ser ut. I och med den välkändakopplingen mellan paraboliska partiella differentialekvationer och ovannämnda väntevärde kan optionspriset även beräknas som lösningen till endifferentialekvation inte helt olik värmeledningsekvationen.

En stor skillnad mellan finansiella och fysikaliska problem är att aktiensframtida volatititet (motsvarar diffusionskoefficienten i fysikaliska problem)i allmänhet är okänd. Man får förlita sig på historiska data eller intelligentagissningar och en viktig fråga i finansiella tillämpningar är därför hur op-tionspriset beror på volatiliteten och hur en felspecificerad volatilitet påverkarpriset. I papper I undersöks denna fråga för barriäroptioner av europeisk typ.En barriäroption är en option med en extra klausul som säger att optionen för-faller värdelös om en på förhand given barriär nås av aktiepriset någon gångföre slutdatumet för optionen. Det visar sig att liksom i fallet med vanliga eu-ropeiska och amerikanska optioner är konvexitet hos optionpriset i aktieprisetavgörande för hur optionspriset beror på volatiliteten. Om optionspriset ärkonvext är det också växande i volatiliteten. Konvexitet hos barriäroptionerstuderas i detta papper med hjälp av paraboliska differentialekvationer ochdet visas att om kontraktsfunktionen är konvex och noll i barriären och omden riskfria räntan och utdelningstakten förhåller sig till varandra på ett visstsätt så är barriäroptionspriset konvext i aktiepriset.

I papper IV studeras en annan typ av barriäroption, en så kallad turbowar-rant. Detta är ett relativt nytt instrument och skillnaden mot vanliga barriärop-tioner är att optionsinnehavaren har möjlighet att få en liten summa pengar

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även om den underliggande tillgången slår i barriären innan mognadsdatumet.Denna summa beror på det lägsta eller det högsta aktiepriset under en kortperiod efter det att barriären slagits i (typiskt 3 timmar) beroende på om detär en warrant av köp- eller säljtyp. Syftet med papperet är att härleda explicitaprisformler för turbowarranter när den underliggande aktien antas följa en ge-ometrisk Brownsk rörelse som är fallet i Black och Scholes klassiska modell.

En annan typ av vägberoende option är den så kallade amerikanska optio-nen. En amerikansk option kan till skillnad mot en europeisk option lösas innär som helst innan mognadsdatumet. Förutom att bestämma priset av optio-nen är det här också intressant att bestämma en optimal strategi, dvs bestämmanär optionen skall lösas in så att detta görs på ett optimalt sätt. På grund avatt optionen kan lösas in när som helst kommer värdet alltid att ligga ovanförkontraktsfunktionen. Intuitivt är det klart att om optionsvärdet är strikt störreän kontraktsfunktionen så skall optionen behållas. Klassiska resultat säger atten optimal tidpunkt att lösa in optionen är första gången optionsvärdet är likamed kontraktsfunktionens värde. Området där optionen skall behållas kallasför fortsättningsområdet och spelar en avgörande roll i bestämmandet av op-timala strategier. I allmänhet är detta området mycket svårt eller omöjligt attbestämma explicit. Det är därför intressant att få kvalitativ information om hurdet ser ut och hur stort det är.

I papper II studeras under vilka förutsättningar som det amerikanska option-spriset sammanfaller med motsvarande europeiska pris. Kända resultat sentidigare ger att om kontraktsfunktionen är konvex och noll i origo så sam-manfaller priserna i alla diffusionsmodeller. I papper II studeras vilka kon-traktsfunktioner som har den egenskapen att det finns någon modell i vilkenpriserna sammanfaller. Det visar sig att denna klass av funktioner innehållerfunktioner som ej är konvexa. Det visar sig också att om det är så att prisernasammanfaller i någon modell så måste kontraktsfuntionen komma från dennaklass. Speciellt betyder detta att utanför denna klass av funktioner är alltid enamerikansk option värd mer än en europeisk.

I papper III studeras storleken på fortsättningsområdet för amerikanska op-tioner i fallet med många underliggande tillgångar. Precis som för europeiskaoptioner kan priset av en amerikansk option beskrivas med hjälp av lösnin-gen till en parabolisk differentialekvation. Möjligheten att lösa in optionennär som helst gör dock att randen till området där ekvationen skall lösas ärokänd och måste bestämmas som en del av lösningen. Priset uppfyller ett såkallat fritt randproblem. Detta kan skrivas om till en variationsolikhet som isin tur är ett hinderproblem. I papper III visas att det områden som utgörs avde punkter i vilka kontraktsfunktionen (dvs hindret) är en strikt sublösningtill Black-Scholes ekvation, kallat positivitetsområdet, är en äkta delmängdav fortsättningsområdet och att för en given tidpunkt finns ett minsta avståndmellan dessa två mängder som kan väljas lika för alla värden på aktiepriset.

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Acknowledgments

I am most grateful and indebted to my adviser Johan Tysk for his guidance,his support and for sharing his deep mathematical knowledge and insight withme. His pushing me forward has been invaluable to me and to the process ofwriting this thesis. His comments on the manuscripts have greatly improvedthe work.

I would also like to thank FMB (the Graduate School in Mathematics andComputing) for financial support, my colleagues at the Department of Mathe-matics, my friends and family, and last but not least Salla for always believingin me and bearing with me during these five years.

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