Reset Call Option's Price Properties And Warrant's Dilution

In this paper we consider two part of contents, the first part is mainly consider reset option and the second part we mainly consider one of our country's new financial product-Warrant.In the first part we consider the reset call option which is a financial contract that when the asset's price fall to a certain price, according to the contract the option holder can reset the strike price so that the option holder can get more profit. According to the reseting time,the reset option can be divided into three part:The first one is reset options with predetermined dates: The process of the reset price is predetermined by fixed dates.Taking call options for example, We set 0 < t₁ <…< t_n < T,the initial time t=0 and the strike price is K,according to the contract if the underlying assets price S(t₁) is below K at time t₁,then the new strike price is S(t₁),if not the strike price still is K. In the next time,we consider the new strike price analogy to the first time.Generally speaking,when it is time t = t_m,the reseted strike price is K_m,andThe second one is reset options with predetermined levels:the process of the reset price is predetermined by fixed levels.Taking call options for example, we set K > K₁ > K₂>…> K_n.the initial time t=0 and the strike price is K,according to the contractif the underlying assets price fall to a predetermined level K₁ ,the the new strike price is K₁. The next time is analogy to the first one. Generally speaking,if the underlying assets' price fall from K_m-1to K_m,then the new strike price is K_m(1≤m≤n) The third one is reset options in a determined time: Before it's matury time, the holder of the options can reset option strike price in order to maxmine his profile,and the strike price and reset time are both not predetermined. And we mainly consider in type of option in the first part of this paper. Without lossing of generelity,we set the initial strike price K = 1,T is the matury time, t is the time we consider option's price V(S, (?))is option price. Where S is the underlying assets price, and (?) is time t to matury time.We also set our problem in Black-Scholes framwork,it is saying that the risk-neutral price process follow Geometric Brown motionWhereZ_tis standard Winer process,σ> 0 is fixed volatility,r > 0is the risk-neutral interest and q≥0 is the divident.When we reset the strike price, the option become a at-the-money call op-tion,and the price of the option isS(?)(r).WhereBecause the holders want to maximin their profile,so we have the following model:WhereiE|^is the risk-neutral probability,t^*the optimal stop time between t and T,and Options price also equal to the following modelFrom above we can see that the differnce between standard option and reset option is that they have different obtacle function.In our paper,we first consider following varialtional inequality with zero boundary valueWhereWe finally prove the existence and uniqueness of the solution and get the boundary estimation of the solution.From above we can see that the problem(4.3) has the following obtacle function.Which equivalent to the obtacle problem(2.8).So we can know that the solution of problem (4.3)is the solution to problem (2.8).From the transition (2.6) We can know that the solution U(S, (?)) to the problem (2.8) is exist and uniqe,also from the inequalityWe can get thatis also existe and unique, which can deduce that the price function of reset call option is exsited and uniqe. From the boundary of U(S, (?)), we can get the result that V(S, (?)) is also boundary.In the second part of the first topic,we disccus properties of the price function.First we give some properties of the function d/d_?((e^q(?)(?)((?))).(1)If r≥q,thene^9(?)(?)(?) is increasing for (?)∈(0, +∞] ;(2)If r≤q,then there existe a unique.point (?)^*∈(0, +∞),so that when(?) > (?)*, e^9(?)C((?))is decreasing for(?),when(?) < (?)*,e^q(?)(?)((?))is increasing for (?).(3)There exist (?)^** > 0 so thatand when|r -q|≤σ²/2,(?)^**= +∞,(4)If r-q > 0 then there exist a constant(?)* > 0 so that (5)If r - q = 0,then(?)'((?)) > 0, 0 < (?) < +∞. (112)(6)If r - q < 0 then(?)'((?)) >δ₀e^-r-q(?), 0 < (?) < +∞. (113)Whereδ₀ is a positive constant.In the second we give the relation that the price function is decreasing for time and increasing for stock price, we also give the properties theorem of the free boundary:Theorem 1 If 0 < r -q≤σ²/2 ,then the free boundary x = h((?)) is decreasing in 0 < (?)≤(?)^*..And the following theromTheorem 2 If 02/2, then(1) h((?)) is smooth in(0,(?)^*);(2) x = h((?)) is monotonic increasing;(3) h((?))∈C[0,(?)^*)∩C^∞(0,(?)^*).Theorem 3 If -σ²/2≤r-q≤0, then for any T > 0, h(≤) is monotonic decreasing,and h(0) = 0, h(≤)∈C[0, T]∩C^∞(0, T).When r - q < -σ²/2, the free boundary has following properties. Theorem 4 The free boundary x = h((?)) is bounded, and there exist a constant R₀ > 0 which is independent of T,such that00, 0<(?)≤T. (114) Theorem 5 If r - q < -σ²/2 ,then h((?)) is monotonic decreasing in [0, (?)^**],and for any T, h((?))∈C[0, T]∩C^∞(0, T]In the second part of the thesis we consider the influence of warrant issuing to other financial products especially to the stock price,stock index,and the pricing of stock index option. When the company issue warrant,it got primium from investor, so the financial framwork of the company is changed. If not change the pricing formula which set before warrant issuing, there would exist a arbitray opportunity.In our paper we find that after warrant issuing, the company has a implied delt which can not be seen in the market.In ordinary text book we alway assume that the stock process is geometric Brown motion,but the issue of warrant influence the company's financial framwork, so the former assumption of stock process is not exist. In order to give a much accurater pricing formula, we assume the underlying asset is the company's asset,finally we give the pricing formula which is the combination of standard option price and compound option price. For the simplity,we assume just one stock in our stock index and the stock index option is European.Assume warrant issue at time t₀, after that time the financial framwork of the company changes as following From the change of stock price we get the change of stock index priceThen using no-arbitray principle we get warrant price function after it issueWe also prove the existance,uniqueness and bounded estemision of the warrant premiumTheorem 6 The price function of warrantis exist and unique,morever the function has following propertyFinally we use compound option method,we get the polished European stock index call option price formula as followWhere C(Y_t Y_T0^*, T₀) is standard BS formula and CC(Y_t, K^cc, T₀, K, T) is Geske's compound option pricing formula Where r is bank interest,σis stock volatility and N(a, b,p) is multiple Normal distributionFor the European stock index put option, we also get the polished pricing formula after warrant issueWhere...