Barrier Options Pricing in Uncertain Financial Market

Barrier Options Pricing in Uncertain Financial Market Jianqiang Xu, Jin Peng Institute of Uncertain Systems, Huanggang Normal University, Hubei 438, China College of Mathematics and Science, Shanghai Normal University, Shanghai 34, China peng@hgnu.edu.cn Abstract: In modern finance market, the option pricing problem is one of the most important contents. A barrier option is a derivative contract that is activated or extinguished when the price of the underlying asset passes a predetermined level. In this paper, pricing formulas for barrier options are defined in uncertain financial market and their pricing formulas concerning uncertain stock model are investigated. eywords: finance, barrier options, uncertain process, canonical process, option pricing formula. Introduction Brownian motion was introduced to finance by Bachelier []. Samuelson [9] [] proposed the argument that geometric Brownian motion is a good model for stock prices. In the early 97s, Black and Scholes [3] and, independently, Metron [5] used the geometric Brownian motion to determine the prices of stock options. Stochastic financial mathematics was founded based on the assumption that stock price follows geometric Brownian motion. Different from randomness, fuzziness is another type of uncertainty in the real world. In order to deal with the change of fuzzy phenomena with time, Liu [] proposed Liu process, Liu formula and Liu integral, just like Brownian motion, Ito formula and Ito integral. As a different doctrine, Liu [] presented an alternative assumption that stock price follows geometric Liu process. Moreover, a basic stock model for fuzzy financial market was also proposed by Liu []. We call it Liu s stock model which is a counterpart of Black-Scholes stock model. Some researches surrounding the subject have been made by Peng [6], Qin and Li [7], You [], etc. However, the real life decisions are usually made in the state of uncertainty other than randomness or fuzziness. In order to cope with this kind of complicated uncertainty, Liu [] founded an uncertainty theory that had become a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. Based on Liu s uncertainty theory, uncertain process and uncertain differential equation have been proposed by Liu []. An uncertain process is essentially a sequence of uncertain variables Proceedings of the Eighth International Conference on Information and Management Sciences, unming, China, July -8, 9, pp. 8-86. indexed by time or space. Uncertain differential equation is a type of differential equation driven by canonical process. The canonical process and geometric canonical process which are different from Brownian motion and geometric Brownian motion are fundamental and important uncertain processes. In finance market, barrier options are the most important weakly path-dependent options which come in many flavours and forms, but they have two key features: A knock-out feature causes the option to immediately terminate if the underlying asset reaches a specified barrier level; A knock-in feature causes the option to become effective only if the underlying asset first reaches a specified barrier level. Merton [5] provided the first analytical formula for a down-and-out call option which was followed by the more detailed paper the formulas for all eight types of barriers. This paper focuses exclusively on one-dimensional, single barrier options in uncertain financial market, which include up down)-and-in out) call put) options. Preliminaries In this section, we recalls the fundamental knowledge of uncertainty theory.. Uncertain Variable Definition. Liu []) An uncertain variable is a measurable function ξ from an uncertainty space Γ, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set ξ B = γ Γ ξγ) B ) is an event. An uncertain variable ξ can be used to characterize the uncertain phenomenon.. First Identification Function Definition. Liu []) An uncertain variable ξ is said to have a first identification function λ if i) λx) is a nonnegative function on R such that supλx) + λy)); ) x y

8 JIANQIANG XU, JIN PENG ii) for any set B of real numbers, we have Mξ B = sup λx), x B if sup λx) <.5 x B sup x B c λx), if sup λx).5. x B 3).5 Canonical Process Definition.6 Liu [4]) An uncertain process C t is said to be a canonical process if i) C = and almost all sample paths are Lipschitz continuous, and ii) C t has stationary and independent increments, and iii) every increment C s+t C s is a normal uncertain variable with expected value and variance t, whose uncertainty distribution is Φx) = + exp πx )), x R. 6) 3t Theorem. Liu []) First Measure Inversion Theorem) Let ξ be an uncertain variable with first identification λ. Then for any set B of real numbers, Mξ B can be expressed by 3)..3 Expected Value Definition.3 Liu []) Let ξ be an uncertain variable. Then the expected value of ξ is defined by E[ξ] = Mξ rdr Mξ rdr 4) provided that at least one of the two integrals is finite. Theorem. Liu []) Let ξ be an uncertain variable with uncertainty distribution Φ. If the expected value E[ξ] exists, then E[ξ] =.4 Uncertain Process Φx))dx Φx 5) Definition.4 Liu [4]) Given an index set T and an uncertainty space Γ, L, M). An uncertain process is a measurable function from T Γ, L, M) to the set of real numbers. That is to say an uncertain process Xt, γ) is a function of two variables such that the function Xt, γ) is an uncertain variable for each t. For simplicity, sometimes we simply use the symbol X t instead of longer notation Xt, γ). Definition.5 Liu [4]) An uncertain process X t is said to have independent increments if X t X t, X t X t,, X tk X tk are independent uncertain variables for any times t < t < < t k. An uncertain process X t is said to have stationary increments if, for any given t >, the X s+t X s are identically distributed uncertain variables for all s >. The canonical process plays the role of a counterpart of Brownian motion. Definition.7 Liu [4]) Let C t be a canonical process. Then the uncertain process G t = expet + σc t ) is called a geometric canonical process, where e is called the log-drift and σ is called the log-diffusion. The geometric canonical process is expected to model stock prices in an uncertain environment..6 Uncertain Integral Definition.8 Liu [4]) Let X t be an uncertain process and let C t be a canonical process. Then the uncertain integral of X t with respect to C t is b a X t dc t = lim i= k X ti C ti+ C ti ) 7) provided that the limit exists almost surely and is an uncertain variable..7 Chain Rule Theorem.3 Liu [4]) Let C t be a canonical process, and let ht, c) be a continuously differentiable function. Define X t = h t, C t ). Then it holds the following chain rule dx t = h t t, C t)dt + h c t, C t)dc t. 8) 3 Liu s Stock Model Let X t be the bond price, and Y t the stock price. Assume that stock price follows a geometric canonical process. Then Liu [4] characterizes the uncertain price dynamics as follows, dxt = rx t dt dy t = ey t dt + σy t dc t 9) where r is the riskless interest rate, e is the stock drift, σ is the stock diffusion, and C t is the canonical process.

BARRIER OPTIONS PRICING IN UNCERTAIN FINANCIAL MARET 83 4 nock-out Barrier Options Barrier options are the most popular path-dependent options traded in exchanges worldwide and also in over-the-counter markets. They have two features: A knock-out feature causes the option to immediately terminate if the underlying asset reaches a specified barrier level a before the expiration date T ; A knock-in feature causes the option to become effective only if the underlying asset first reaches a specified barrier level a before the expiration date T. = exp rt ) M max Y t a M et + σc T lnu du = exp rt ) M max Y t a + exp πlnu et ) )du. 4. Up-and-Out Call Option Definition 4. The price of up-and-out call option with the f = E[exp rt )Y T ) + ] M max Y t a ) Theorem 4. Suppose that X t and Y t satisfy the price dynamics described by the Liu s stock model 9), then the upand-out Proof. f = exp rt ) M max Y t a + exp πlnx et ) E[exp rt )Y T ) + ] M max Y t a = exp rt ) M max Y t a E [ exp et + σc T ) ) +] = exp rt ) M max Y t a M exp et + σc T ) x dx = exp rt ) M max Y t a M exp et + σc T ) u du ) Example : Suppose that a stock is presently selling for a price of = 35, the riskless interest rate is r is.8 per annum, the stock drift e is.6 and the stock diffusion σ is.3. The barrier level a is 38. Suppose that max Y t a is equivalent to Y T a. We would like to find up-and-out call barrier option price that expires in half a year and has a strike price of = 4. y= 35 exp-.8.5)./+exppi logx)-.6.5)./sqrt3).3.5)))) -./+exppi log38./35)-.6.5)/sqrt3).3.5))))) ; f = quady,4/35,) The calculation result shows that f =.4694. This means the appropriate call option price in the example is about 47 cents. 4. Up-and-Out Put Option Definition 4. The price of up-and-out put option with the f = E[exp rt ) Y T ) + ] M max Y t a ) Theorem 4. Suppose that X t and Y t satisfy the price dynamics described by the Liu s stock model 9), then the upand-out f = exp rt ) M max Y t a + exp πet lnx) 3)

84 JIANQIANG XU, JIN PENG Proof. E[exp rt ) Y T ) + ] M max Y t a = exp rt ) M max Y t a E [ exp et + σc T )) +] = exp rt ) M max Y t a M exp et + σc T ) x dx = exp rt ) M max Y t a M exp et + σc T ) u du = exp rt ) M max Y t a M et + σc T lnu du = exp rt ) M max Y t a + exp πet lnu) )du. Example : Suppose that a stock is presently selling for a price of = 4, the riskless interest rate is r is.8 per annum, the stock drift e is.6 and the stock diffusion σ is.3. The barrier level a is 38. Suppose that max Y t a is equivalent to Y T a. We would like to find up-and-out put barrier option price that expires in half a year and has a strike price of = 35. y= 4 exp-.8.5)./+exppi.6.5- logx))./sqrt6).3.5)))) -./+exppi log38./4)-.6.5)/sqrt3).3.5))))) ; f = quady,,35/4) The calculation result shows that f =.55. This means the appropriate put option price in the example is about 5 cents. 4.3 Down-and-Out Call Option Definition 4.3 The price of down-and-out call option with the f = E[exp rt )Y T ) + ] M min Y t a 4) Theorem 4.3 Suppose that X t and Y t satisfy the price dynamics described by the Liu s stock model 9), then the downand-out f = exp rt ) M min Y t a + exp πlnx et ) 5) Example 3: Suppose that a stock is presently selling for a price of = 35, the riskless interest rate is r is.8 per annum, the stock drift e is.6 and the stock diffusion σ is.3. The barrier level a is 38. Suppose that min Y t a is equivalent to Y T a. We would like to find up-and-out call Option barrier Option price that expires in half a year and has a strike price of = 4. y= 35 exp-.8.5)./+exppi logx)-.6.5)./sqrt3).3.5))))./+exppi log38./35)-.6.5)/sqrt3).3.5)))) ; f = quady,4/35,) The calculation result shows that f =.469. This means the appropriate call option price in the example is about 4 cents. 4.4 Down-and-Out Put Option Definition 4.4 The price of down-and-out put option with the f = E[exp rt ) Y T ) + ] M min Y t a 6) Theorem 4.4 Suppose that X t and Y t satisfy the price dynamics described by the Liu s stock model 9), then the downand-out f = exp rt ) M min Y t a + exp πet lnx) 7)

BARRIER OPTIONS PRICING IN UNCERTAIN FINANCIAL MARET 85 Example 4: Suppose that a stock is presently selling for a price of = 4, the riskless interest rate is r is.8 per annum, the stock drift e is.6 and the stock diffusion σ is.3. The barrier level a is 38. Suppose that min Y t a is equivalent to Y T a. We would like to find up-and-out put barrier option price that expires in half a year and has a strike price of = 35. y= 4 exp-.8.5)./+exppi.6.5- logx))./sqrt3).3.5))))./+exppi log38./4)-.6.5)/sqrt3).3.5)))) ; f = quady,,35/4) The calculation result shows that f =.796. This means the appropriate put option price in the example is about 8 cents. 5 nock-in Barrier Options 5. Up-and-In Call Option Definition 5. The price of up-and-in call option with the f = E[exp rt )Y T ) + ] M max Y t a 8) Theorem 5. Suppose that X t and Y t satisfy the price dynamics described by the Liu s stock model9), then the upand-in f = exp rt ) M max Y t a + exp πlnx et ) 5. Up-and-In Put Option 9) Definition 5. The price of up-and-in put option with the f = E[exp rt ) Y T ) + ] M max Y t a ) Theorem 5. Suppose that X t and Y t satisfy the price dynamics described by the Liu s stock model 9), then the upand-in f = exp rt ) M max Y t a + exp 5.3 Down-and-In Call Option πet lnx) ) Definition 5.3 The price of down-and-in call option with the f = E[exp rt )Y T ) + ] M min Y t a ) Theorem 5.3 Suppose that X t and Y t satisfy the price dynamics described by the Liu s stock model 9), then the downand-in f = exp rt ) M min Y t a + exp πlnx et ) 5.4 Down-and-In Put Option 3) Definition 5.4 The price of down-and-out put option with the f = E[exp rt ) Y T ) + ] M min Y t a 4) Theorem 5.4 Suppose that X t and Y t satisfy the price dynamics described by the Liu s stock model 9), then the downand-out f = exp rt ) M min Y t a 6 Conclusions + exp πet lnx) 5) In this paper, one-dimensional, single barrier options pricing formulas, which include eight possible types, were defined. And we investigated their pricing problems based on the Liu s stock model in uncertain financial market. Ground on the results, some potential applications of uncertain stock models will be an interesting topic of future research.

86 JIANQIANG XU, JIN PENG Acknowledgements This work is supported by the National Natural Science Foundation Grant No.7675), the Major Research Program Grant No.Z87) of Hubei Provincial Department of Education, China. References [] Bachelier L, Théorie de la spéculation, Ann. Sci. École Norm. Sup., Vol.7, -86, 9. [] Baxter M and Rennie A, Financial Calculus: An Introduction to Derivatives Pricing, Cambridge University Press, 996. [3] Black F and Scholes M, The pricing of option and corporate liabilities, Journal of Political Economy, Vol.8, 637-654, 973. [4] Etheridge A, A Course in Financial Calculus, Cambridge University Press,. [5] Gao J and Gao X, A new stock model for credibilistic option pricing, Journal of Uncertain Systems, Vol., No.4, 43-47, 8. [6] Gao Xin and Qin Zhongfeng, On Pricing of Barrier Options in Fuzzy Financial Market, www.orsc.edu.cn/process/843.pdf [7] Hull J, Options, Futures and Other Derivative Securities, 5th ed., Prentice-Hall, 6. [8] Li X and Qin ZF, Expected value and variance of geometric Liu process, Far East Journal of Experimental and Theoretical Artificial Intelligence, Vol., No., 7-35, 8. [9] Li X and Liu B, A sufficient and necessary condition for credibility measures, International Journal of Uncertainty, Fuzziness & nowledge-based Systems, Vol.4, No.5, 57-535, 6. [] Liu B, Uncertainty Theory, Springer-Verlag, Berlin, 4. [] Liu B, Uncertainty Theory, nd ed., Springer-Verlag, Berlin, 7. [] Liu B, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, Vol., No., 3-6, 8. [3] Liu B and Liu Y, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems, Vol., No.4, 445-45,. [4] Liu B, Some research problems in uncertainty theory, Journal of Uncertain Systems, Vol.3, No., 3-, 9. [5] Merton R, Theory of rational option pricing, Bell Journal Economics & Management Science, Vol.4, No. 4-83, 973. [6] Peng Jin, A General Stock Model for Fuzzy Markets, Journal of Uncertain Systems, Vol., No.4, 48-54, 8. [7] Qin Z and Li X, Option Pricing Formula for Fuzzy Financial Market, Journal of Uncertain Systems, Vol., No., 7-, 8. [8] Reiner E, and Rubinstein M, Breaking down the Barriers, Risk, Vol.4, No.8, 8-35, 99. [9] Samuelson P A, Proof that properly anticipated prices fluctuate randomly, Industrial Manage Review., Vol.6, 4-5, 965. [] Samuelson P A, Mathematics of speculative prices, SIAM Rev., Vol.5, -4, 973. [] You C, Multidimensional Liu process, differential and integral, Proceedings of the First Intelligent Computing Conference, Lushan, October -3, 7, 53-58. [] Zadeh LA, Fuzzy sets, Information and Control, Vol.8, 338-353, 965.