On the pricing equations of some path-dependent options

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(1)UPPSALA DISSERTATIONS IN MATHEMATICS 45. On the pricing equations of some path-dependent options Jonatan Eriksson. Department of Mathematics Uppsala University UPPSALA 2006.

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(166) List of Papers. This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I II III. IV. Eriksson, J. (2005) Monotonicity in the volatility of single-barrier option prices, to appear in the Int. J. Theor. Appl. Finance. Eriksson, J. (2005) When American options are European, submitted to Decis. Econ. Finance. Arnarson, T., Eriksson, J. (2005) On the size of the non-coincidence set of parabolic obstacle problems with applications to American option pricing, submitted to Math. Scand. Eriksson, J. (2005) Explicit pricing formulas for turbo warrants, submitted to Risk magazine.. Reprints were made with permission from the publishers..

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(168) Contents. 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Path-dependent European options . . . . . . . . . . . . . . . . . . . . . . 1.3 Parabolic obstacle problems and free boundary problems . . . . . 1.4 American options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Included papers and description of the results . . . . . . . . . . . . . . . . . 2.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Knock-out options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Knock-in options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sammanfattning på svenska (Summary in Swedish) . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 1 2 3 5 7 7 8 8 9 10 12 13 15 17.

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(170) 1. Introduction. In this thesis we study financial mathematics in continuous time. This discipline of science is typically concerned with the problem of pricing and hedging financial instruments defined in terms of some underlying asset. Examples of such instruments are stock-options and warrants. The value of a stockoption or a warrant is given as the discounted expected future pay-off, but since 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 processes. However, for the purpose of option pricing not any stochastic process will do, but there are strict theoretical rules the asset process has to obey to avoid arbitrage in the market. The discounted asset price has to be a martingale when pricing financial instruments. The common way of modeling asset prices is to use Brownian motion and more generally solutions to stochastic differential equations. In this context the problem of option valuation is similar to problems of heat conduction and particle motion in physics since expected values of functions of solutions to stochastic differential equations solve parabolic partial differential equations similar to the heat equation. Problems in finance are however very different in their nature from their physical counterparts. In finance, the volatility of the stock, which roughly is the same as the diffusion coefficient in physics, is most certainly unknown and one has to rely on historical data and educated guesses. The implications of model misspecification and misspecification of the volatility are important and questions about robustness which arise due to the uncertainty of the volatility are connected to properties of the Black-Scholes partial differential equation and preservation of convexity of solution to that equation, compare [14] and [15].. 1.1. Option pricing. In the two seminal papers [3] and [19] the authors describes how to price options on a market with one risky asset and one risk-free asset, such that the no-arbitrage principle holds. An arbitrage opportunity is a risk-free investment strategy, with zero initial endowment, in financial instruments such as stocks, bonds and options, such that the final wealth is non-negative almost surely and positive with positive probability. The no-arbitrage principle says that a financial market should contain no arbitrage opportunities. The result in the above mentioned papers can be formulated as follows: Assume that 1.

(171) the market consists of two trades assets B(t) and S(t) evolving according to dB(t) = rB(t)dt and dS(t) = µS(t)dt + σ S(t)dW (t) where the constants are r ≥ 0, the interest rate, µ , the appreciation rate and σ , the volatility, and where W 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 arbitrage in the market is the discounted expected value of φ (S(T )). However, the expectation is not calculated under the real-world measure P which is used to 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 stockprice 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 measure Q Q Q 1 2 S(T ) = S(t)e(r− 2 σ )(T −t)+σ (WT −Wt ) and by using that the increment of a Brownian motion is normally distributed with variance equal to the length of the increment, the expected value for the price V (s,t) = Es,t e−r(T −t) φ (S(T )) can be calculated explicit in terms of the normal distribution function. The results is the famous Black-Scholes formula which was derived for the calloption, φ (s) = (s − K)+ , and for the put-option, φ (s) = (K − s)+ , in the above mentioned papers. By the Feynman-Kac representation formula for solutions to parabolic differential equations, the price V (s,t) can also be computed as the solution to the Black-Scholes equation ∂V ∂V 1 2 2 ∂ 2V σ s − rV + =0 + rs 2 2 ∂s ∂s ∂t. together with the final value V (s, T ) = φ (s), compare [13]. We note that the term ∂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 initial condition as in physics.. 1.2. Path-dependent European options. A European vanilla option is an option with a pay-off that only depends on the stock-price at maturity T . The pay-off can be described as φ (S(T )) for some contract function φ . A European path-dependent option is an option whose pay-off at maturity T depends on the whole path of the stock-price S(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 lookback options, which depend on the maximum and/or the minimum of the stockprice. 2.

(172) The most common types of barrier options are knock-in options and knockout options. As their names suggest a knock-in option is activated when a prescribed barrier is hit by the stock-price, and a knock-out option is extinguished when the barrier is hit. Depending on the relation between the initial stock-price and the barrier these options are usually given names like up-andout 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-andout option is priced under geometrical Brownian motion. A more complete pricing scheme in this case can be found in [21] where the authors make clever combinations of the distribution functions of absorbed geometrical Brownian motions to generate prices for a large numbers of single-barrier options. Compare also [22]. In the paper [22] a sensitivity analysis is performed for barrier options of call and put types (see also [24]), i.e. a calculation of the options ∆ (sensitivity to movements in the underlying stock) and Γ (sensitivity of ∆ to movements in the underlying stock), which is of great interest when performing dynamic hedging. Especially the sign of Γ is important when it comes to the effect on the hedging portfolio of a misspecified volatility. For European vanilla options and for American options it is well-known that convexity of the contract function is enough to ensure a positive Γ which in turn ensures a super-hedging portfolio if the volatility is overestimated, compare [11], [14], [13], [9] and [10]. However, when it comes to barrier options the situation can be very different. Here convexity depends not only on the convexity of the contract function but also on the underlying process and the barrier. This comes as no surprise since an (almost trivial) example of a barrier option with convex contract function and non-convex price is an up-and-out put option with the barrier 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 one of 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-money and a non-constant rebate which is activated if the barrier is hit prior to maturity. This means that if the option is knocked out a small sum is paid to the warrant holder. In the call-case the rebate is given by the difference of the lowest recorded stock-price during a three-hour period after the knock-out event and the strike price. In the put-case it is the highest recorded stock-price which determines the rebate.. 1.3 Parabolic obstacle problems and free boundary problems In a free boundary problem one seeks the solution to a differential equation M f = 0, in some domain Ω where only a part S of the boundary ∂ Ω is given, 3.

(173) whereas the remaining portion Γ is not a priori prescribed. On the boundary of Ω some boundary conditions f = φ are given. To get a well-posed problem an additional condition is given on Γ. This additional boundary condition depends on the particular problem in question, and in financial applications the condition is the so-called high-contact principle , ∇x f = ∇x φ . A solution to a free 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 written as obstacle problems, for instance as the minimization of a functional over a closed convex set K = { f ∈ H : f ≥ φ } in a suitable function space H . Here the a priori given function φ is referred to as the obstacle. The main difference between a general free boundary problem and an obstacle problem is that in an obstacle problem one can analyze the solution without having to analyze the free boundary at the same time. When studying obstacle problems it is often useful to write the problem as a so-called complementary problem.    M f ≤ 0, ( f − φ )M f = 0,   f ≥ φ,. where M is a differential operator corresponding to the above minimization problem. The obstacle problem now becomes to find a function in a suitable function space, e.g. in the Sobolev space Wq2,1 , satisfying the above inequalities in a weak sense such as almost surely, as distributions or in a viscosity 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 solution f satisfies the so-called principle of smooth fit on Γ, that is, the mapping x → ∇x ( f − φ ) is continuous across Γ, at least at points where φ is continuously differentiable. Intuitively the principle of smooth fit is clear, since it in financial applications means that the number of shares of the underlying stock to be held in the hedging portfolio does not jump when the stock passes into the stopping region. Apart from financial applications, obstacle problems arise naturally in physics 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 reduced to an obstacle problem. 4.

(174) 1.4. American options. An American option, unlike a European option, can be exercised at any time prior to the maturity time T . This means that at any instant in time the holder of the option needs to decide if to exercise or to hold the option. This extra complexity in the problem results in the fact that explicit pricing formulas rarely exists. Only in special cases such as if the American option has the same value as the corresponding European option or if the maturity time is infinite is the price explicitly known, compare [11], [25], [20], [7] and Paper II. Suppose that the contract function is φ , then if the holder decides to exercise at time τ the received amount is φ (S(τ)). In the papers [1] and [16] it is shown that the unique arbitrage free price VA at time t of an American option with contract function φ is given by the optimal stopping problem VA (s,t) =. sup Es,t e−r(τ−t) φ (S(τ)),. τ∈F [t,T ]. where the supremum is taken over all stopping times with respect to the filtration generated by the driving Brownian motion of S(t). Since the decision on exercising or holding the option only can be based on past information it is intuitively clear that the supremum only can be taken over the above mentioned stopping times. Notice also that VA always satisfies VA (s,t) ≥ φ (s) and VA (s,t) ≥ V (s,t) since it is allowed to exercise the option either at t or at T . Besides calculating the price of the option it is also interesting to decide a good strategy for the option holder. That is to find a stopping time τ ∗ realizing the supremum above. Intuitively it is clear that if VA (S(t),t) > φ (S(t)) it is not optimal to exercise the option since VA (S(t),t) corresponds to holding the option and φ (S(t)) corresponds to exercising. In Appendix D in [17] it shown that an optimal stopping time is given by τ ∗ = inf{u ≥ t : VA (S(u), u) = φ (S(u))}, thus the optimal time to exercise the option is when the process (S(t),t) leaves the 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 boundary problems is established and for the American option pricing problem it can be shown that VA satisfies Black-Scholes equation at every point in the interior 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 solution also satisfies the principle of smooth fit there if it is regular enough. Thus the American option pricing problem can be viewed as a free boundary problem where the free boundary is the boundary of C . This free boundary problem can be rewritten as an obstacle problem where the contract function φ is the obstacle and where the inequality M VA ≤ 0 holds on the whole domain and where M VA = 0 holds on the continuation region C . 5.

(175) When it comes to the free boundary itself, not very much is known in the general case. Questions of regularity are in general difficult to answer. Qualitative results on the behavior of the boundary in the multi-dimensional case close to maturity are given in [23] and on the shape in [25], [20]. However, in the special case of a one-dimensional geometric Brownian motion there has been a lot of work done. Recently it was shown that the boundary is a convex C∞ function of time, compare [8], [5] and [6]. Due to the difficulty of finding the continuation region one is often interested in either approximations or of finding non-trivial subsets in which one knows not to exercise the option, compare Paper III.. 6.

(176) 2. Included papers and description of the results. 2.1. Paper I. In this paper we study single-barrier options. The main question is: Under which conditions can we guarantee that a convex contract function gives rise to a convex price, i.e. that the options Γ is non-negative? Other questions answered in this paper are: When is super-hedging with volatility over-estimation possible? What can be said about the options sensitivity to movements in the underlying asset, i.e. the options ∆? A barrier option is an option which gives a certain pay-off at maturity T conditioned 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 any of 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. The techniques used in this paper are borrowed from the paper [15] and the main idea is to use the maximum principle to show that the convexity of the final value is preserved by the option price. In the paper the underlying asset is assumed 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 and continuous in time and the risk-free rate of return and the dividend rate are constants.. 7.

(177) 2.1.1. Knock-out options. Consider the case of knock-out options. By introducing the stopping time τb = inf{t ≥ 0 : S(t) = b} we may describe the pay-off at maturity as φ (S(T ))1{τb >T } .. The pay-off clearly depends on the whole trajectory of S(t). By risk-neutral valuation the unique arbitrage-free value of the up-and-out option is given by U(s,t) = Es,t e−r(T −t) φ (S(T ))1{τb >T } .. Knowing that the option is not yet knocked out, the value function U(s,t) satisfies 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-out option the option is alive on s > b or s < b respectively. One of the results in Paper I answers the first of the above questions for knock-out options. Theorem 1 (Convexity of knock-out options) Assume that the contract function is convex on [0, ∞) and zero at the barrier. Then a down-and-out option has a positive Γ if the interest rate is dominated by the dividend rate and an up-and-out option has a positive Γ if the dividend rate is dominated by the 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 give rise to a convex price. Thus the theorem may fail if r < δ . The other two questions posed in the beginning are answered in the following corollaries. Corollary 1 Assume that the contract function is convex on [0, ∞) and zero at the barrier. Then a down-and-out option has a positive ∆ if the interest rate is dominated by the dividend rate and an up-and-out option has a negative ∆ if the dividend rate is dominated by the interest rate. Corollary 2 Super-hedging of up-and-out options and down-and-out options by overestimating the volatility is possible in the cases described in Theorem 1 which give rise to convex prices.. 2.1.2. Knock-in options. The pay-off of a knock-in option can be described by φ (S(T ))1{τb ≤T } ,. and by risk-neutral valuation the value is given by V (s,t) = Es,t e−r(T −t) φ (S(T ))1{τb ≤T } . 8.

(178) To price these options by the means of differential equations we view the knock-in option as a knock-out contract paying zero at maturity if the barrier is not hit and paying a rebate if the barrier b is hit. The rebate equals the value of the corresponding vanilla option at b. More precisely, let f (s,t) = Es,t e−r(T −t) φ (S(T )). Then the rebate is given by f (b, τb ). Moreover, assuming that the barrier is not yet hit at present time t , the function V (s,t) satisfies Black-Scholes equation with boundary condition V (b,t) = f (b,t) and final value zero. By using the differential equation for V the following answers to the questions posed in the beginning of this section can be given. Theorem 2 (Convexity of knock-in options) Assume that the contract function is convex on [0, ∞). Then a down-and-in option has a positive Γ if the dividend rate is dominated by the interest rate and an up-and-in option has a positive 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 in the wrong direction, a convex contract function need not give rise to a convex price. Corollary 3 Assume that the contract function is convex on [0, ∞). Then a down-and-in option has a negative ∆ as long as the barrier has not been hit if the dividend rate is dominated by the interest rate and an up-and-in option has a positive ∆ as long as the barrier has not been hit if the interest rate is dominated by the dividend rate. Corollary 4 Super-hedging of up-and-in options and down-and-in options by overestimating the volatility is possible in the cases described in Theorem 1 which give rise to convex prices.. 2.2. Paper II. The main questions in Paper II are the following ones: For which class of contract functions is it so that there exist a diffusion model in which the price of an American option coincides with the price of the corresponding European option? If we know that there is some diffusion model in which the prices coincide, what can be said about the contract function? It is well-known from the paper [11] that if the contract function is convex and zero at the origin and if the dividend rate is zero, then the prices always coincide. However, by relaxing the condition that the equality should hold for all diffusion models, i.e. for all volatilities, but only demanding that the equality holds for some non-zero volatility satisfying some suitable regularity conditions, the class of contract functions becomes strictly larger than the class of convex ones being zero at the origin. Moreover, if we know that there is 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 9.

(179) in which the prices can coincide, hence that the possibility to exercise early always has a positive value. The contract functions which qualifies are the ones satisfying the following conditions: 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, and 2. if s0 φ− (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 form dS(t) = rS(t)dt + S(t)σ (S(t),t)dW (t) for some Brownian motion W and that the volatility satisfy the following conditions. Assumption 1 1. The diffusion coefficient sσ (s,t) is strictly positive for all s > 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 and all t ∈ [0, T ]. 3. The function s2 σ 2 (s,t) has a Hölder continuous partial derivative with respect 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 satisfies Conditions 1 and 2 if and only if there is a model such that the prices of the European and American options with pay-off φ are the same.. 2.3. Paper III. This paper deals with the size of the non-coincidence set of certain parabolic obstacle problems. This is the set on which the solution to the obstacle problem is strictly larger than the obstacle. The operators in the paper have the form n n ∂f ∂2 f ∂f L f = ∑ ai j + ∑ bi +cf − . ∂ xi ∂ x j i=1 ∂ xi ∂t i, j=1 We work under the assumption that the lower order coefficients are Dinicontinuous and that the top-order coefficients have Dini-continuous spatial derivatives of first order. Dini-continuity is weaker than the more common Hölder continuity and is defined in the following way. Definition 1 A modulus of continuity α is called a Dini modulus of continuity if 0ε α(s) s dt < ∞ for all ε > 0 small enough, and a function h is called Dinicontinuous if h has a Dini modulus of continuity. 10.

(180) The assumptions on the coefficients are the following. Assumption 2 The coefficients ai j (x,t) are Dini-continuous in x and t and have Dini-continuous first-order partial derivatives with respect to x. The coefficients 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) ∈ Rn × R : f (x,t) > φ (x,t)} and the positivity set is defined as the set where φ is a strict sub-solution to L f = 0, i.e. the set U = {(x,t) ∈ Rn ×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 III is to show that if the boundary of U is smooth enough, then the inclusion is strict. 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 represented 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 distance between 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 vanishes at a point on the boundary of a domain must have a positive inward directional derivative at the same point. However, in the standard literature, e.g. [12] or [18], one assumes that the boundary has a strong interior sphere property (at least in the spatial variables) which essentially means that one should be able to touch the boundary from the inside of the domain with a sphere. But by using the C1 -Dini assumption on the top-order coefficient and Theorem 1.5.10 in [4] and a change of variables, a Hopf-type lemma can be obtained for domains satisfying only the interior C1 -Dini condition in Theorem 4. The results of Theorem 4 can be used to show that the boundary of the positivity set cannot touch the boundary of the continuation region in American option pricing and that it must lie on a uniform distance in space from the free boundary. 11.

(181) 2.4. Paper IV. The final paper is a paper in which explicit pricing formulas are derived for turbo warrants of put and call types in the classical Black-Scholes model. In this paper the stock-price process is assumed to evolve according to S(t) = S(0)e(r− 2 σ 1. 2 )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 times before 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 the barrier 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) denotes the running minimum and the running maximum respectively. To price the warrant one prices the two parts separately, i.e. the knock-out part and the rebate part. The key observation to make here is that the rebate part can be viewed in the following way. Consider first the call-case: • Let Vc (b) = Eb e−rδ (m(δ ) − K)+ , i.e. the value of the rebate when the barrier 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 not hit the barrier yet. The value is given by Vc (b)Es,t e−r(τb −t) 1{τb ≤T } . The put-case is similar: • Let Vp (b) = Eb e−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 hit the barrier yet. The value is given by Vp (b)Es,t e−r(τb −t) 1{τb ≤T } Now since the densities for τb , m(t) and M(t) are explicitly known, the problem of finding the price is just a matter of integration. Notice that the decomposition made above strongly depends on the time-homogeneity and the Markov property of the underlying process S(t). With a time-inhomogeneous process the decomposition is no longer possible and we really need the joint density of (m(t), τb ) and the joint density of (M(t), τb ). 12.

(182) Sammanfattning på svenska (Summary in Swedish). I denna avhandling bestående av en introduktion och fyra artiklar studeras finansiell matematik i kontinuerlig tid. Huvudtemat för avhandlingen är vägberoende optioner och de optioner som studeras är barriäroptioner och amerikanska optioner. I finansiell matematik är man ofta intresserad av att prissätta finansiella instrument definierade i termer av någon underliggande tillgång såsom en aktie. I och med att den underliggande tillgångens framtida värde i allmänhet inte är känt så modelleras den med en stokastisk process. I denna avhandling används lösningar till stokastiska differentialekvationer som modell för aktiepriset. Priset för en option kan i allmänhet beräknas som ett väntevärde av en kontraktsfunktion av den underliggande stokastiska processen. Kontraktsfunktionen är en på förhand specificerad funktion som anger hur utbetalningsstrukturen för kontraktet ser ut. I och med den välkända kopplingen mellan paraboliska partiella differentialekvationer och ovan nämnda väntevärde kan optionspriset även beräknas som lösningen till en differentialekvation inte helt olik värmeledningsekvationen. En stor skillnad mellan finansiella och fysikaliska problem är att aktiens framtida volatititet (motsvarar diffusionskoefficienten i fysikaliska problem) i allmänhet är okänd. Man får förlita sig på historiska data eller intelligenta gissningar och en viktig fråga i finansiella tillämpningar är därför hur optionspriset beror på volatiliteten och hur en felspecificerad volatilitet påverkar priset. I papper I undersöks denna fråga för barriäroptioner av europeisk typ. En barriäroption är en option med en extra klausul som säger att optionen förfaller värdelös om en på förhand given barriär nås av aktiepriset någon gång före slutdatumet för optionen. Det visar sig att liksom i fallet med vanliga europeiska och amerikanska optioner är konvexitet hos optionpriset i aktiepriset avgörande för hur optionspriset beror på volatiliteten. Om optionspriset är konvext är det också växande i volatiliteten. Konvexitet hos barriäroptioner studeras i detta papper med hjälp av paraboliska differentialekvationer och det visas att om kontraktsfunktionen är konvex och noll i barriären och om den riskfria räntan och utdelningstakten förhåller sig till varandra på ett visst sätt så är barriäroptionspriset konvext i aktiepriset. I papper IV studeras en annan typ av barriäroption, en så kallad turbowarrant. Detta är ett relativt nytt instrument och skillnaden mot vanliga barriäroptioner är att optionsinnehavaren har möjlighet att få en liten summa pengar 13.

(183) även om den underliggande tillgången slår i barriären innan mognadsdatumet. Denna summa beror på det lägsta eller det högsta aktiepriset under en kort period efter det att barriären slagits i (typiskt 3 timmar) beroende på om det är en warrant av köp- eller säljtyp. Syftet med papperet är att härleda explicita prisformler för turbowarranter när den underliggande aktien antas följa en geometrisk Brownsk rörelse som är fallet i Black och Scholes klassiska modell. En annan typ av vägberoende option är den så kallade amerikanska optionen. En amerikansk option kan till skillnad mot en europeisk option lösas in när som helst innan mognadsdatumet. Förutom att bestämma priset av optionen är det här också intressant att bestämma en optimal strategi, dvs bestämma när optionen skall lösas in så att detta görs på ett optimalt sätt. På grund av att optionen kan lösas in när som helst kommer värdet alltid att ligga ovanför kontraktsfunktionen. Intuitivt är det klart att om optionsvärdet är strikt större än kontraktsfunktionen så skall optionen behållas. Klassiska resultat säger att en optimal tidpunkt att lösa in optionen är första gången optionsvärdet är lika med kontraktsfunktionens värde. Området där optionen skall behållas kallas för fortsättningsområdet och spelar en avgörande roll i bestämmandet av optimala strategier. I allmänhet är detta området mycket svårt eller omöjligt att bestämma explicit. Det är därför intressant att få kvalitativ information om hur det ser ut och hur stort det är. I papper II studeras under vilka förutsättningar som det amerikanska optionspriset sammanfaller med motsvarande europeiska pris. Kända resultat sen tidigare ger att om kontraktsfunktionen är konvex och noll i origo så sammanfaller priserna i alla diffusionsmodeller. I papper II studeras vilka kontraktsfunktioner som har den egenskapen att det finns någon modell i vilken priserna sammanfaller. Det visar sig att denna klass av funktioner innehåller funktioner som ej är konvexa. Det visar sig också att om det är så att priserna sammanfaller i någon modell så måste kontraktsfuntionen komma från denna klass. Speciellt betyder detta att utanför denna klass av funktioner är alltid en amerikansk option värd mer än en europeisk. I papper III studeras storleken på fortsättningsområdet för amerikanska optioner i fallet med många underliggande tillgångar. Precis som för europeiska optioner kan priset av en amerikansk option beskrivas med hjälp av lösningen till en parabolisk differentialekvation. Möjligheten att lösa in optionen när som helst gör dock att randen till området där ekvationen skall lösas är okänd och måste bestämmas som en del av lösningen. Priset uppfyller ett så kallat fritt randproblem. Detta kan skrivas om till en variationsolikhet som i sin tur är ett hinderproblem. I papper III visas att det områden som utgörs av de punkter i vilka kontraktsfunktionen (dvs hindret) är en strikt sublösning till Black-Scholes ekvation, kallat positivitetsområdet, är en äkta delmängd av fortsättningsområdet och att för en given tidpunkt finns ett minsta avstånd mellan dessa två mängder som kan väljas lika för alla värden på aktiepriset.. 14.

(184) 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 with me. His pushing me forward has been invaluable to me and to the process of writing this thesis. His comments on the manuscripts have greatly improved the work. I would also like to thank FMB (the Graduate School in Mathematics and Computing) for financial support, my colleagues at the Department of Mathematics, my friends and family, and last but not least Salla for always believing in me and bearing with me during these five years.. 15.

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