UPPSALA DISSERTATIONS IN MATHEMATICS 45 On the pricing equations of some path-dependent options Jonatan Eriksson Department of Mathematics Uppsala University UPPSALA 2006 ! ""# !$%!& ' ( ' ' ) * + , - ( * -. /* ""#* 0 ( 1 ' 2 * &* 3!4 * * 567 8!2&"#2!4&2"* + ' ' * + ' ( 1 ' 2 * 9 ' : ' ; ' ( : ( ( ' ' , * ) , '' ' ' (* ' - : ' * < , ' , .2' ' * ) , ( ' : ( * < ( '' ' : , ( - '' * ) , : ( ; ' 2 * + ' ' ' , '' 1 !2 = ; , ' , ' ' 2 * ) 9 , ( ' ' , 6 .25 * ) '' 1 1 : , ( ' !" # $%&'(") $ > / -. ""# 557 !"!2"8 567 8!2&"#2!4&2" % %%% 2#$8 ? %@@ *.*@ A B % %%% 2#$8C List of Papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I Eriksson, J. (2005) Monotonicity in the volatility of single-barrier option 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 the non-coincidence set of parabolic obstacle problems with applications 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 . 1 1.1 Option pricing . 1 1.2 Path-dependent European options . 2 1.3 Parabolic obstacle problems and free boundary problems . 3 1.4 American options . 5 2 Included papers and description of the results . 7 2.1 Paper I . 7 2.1.1 Knock-out options . 8 2.1.2 Knock-in options . 8 2.2 Paper II . 9 2.3 Paper III . 10 2.4 Paper IV . 12 Sammanfattning på svenska (Summary in Swedish) . 13 Acknowledgments . 15 Bibliography . 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, 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 pro- cesses. 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 martin- gale when pricing financial instruments. The common way of modeling asset prices is to use Brownian motion and more generally solutions to stochas- tic 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. Prob- lems 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 op- tions 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 invest- ment 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 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 + σS(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 arbi- trage 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 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 measure Q (r− 1 σ 2)(T−t)+σ(W Q−W Q) S(T)=S(t)e 2 T t 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 −r(T−t) V(s,t)=Es,t e 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 call- option, 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 dif- ferential equations, the price V(s,t) can also be computed as the solution to the Black-Scholes equation 1 ∂ 2V ∂V ∂V σ 2s2 + rs − rV + = 0 2 ∂s2 ∂s ∂t together with the final value V(s,T)=φ(s), compare [13]. We note that the ∂V term ∂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 stock- price. 2 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 a prescribed 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 initial stock-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 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. Com- pare 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 func- tion 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 ma- turity. 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 low- est recorded stock-price during a three-hour period after the knock-out event and the strike price.