Pricing Cancellable Barrier Option And Optimal Cancellable Time

An option is one of the most popular financial derivative products.An option is a contract which gives the buyer the right,but not the obligation,to buy or sell an underlying asset or instrument at a specified strike price on or before a specified date.A barrier option is an exotic derivative typically option. A barrier option is cheaper than a typically option,so it is favored by the market. A barrier option has four types,up-and-out,down-and-out,up-and-in, down-and-in.The difference between barrier option and traditional option is that the asset price of barrier option needs to reach a level.After that,barrier option can knock out(or knock in).The economic setting is the standard financial market with constant coefficients.Our underlying risky asset is geometric Brownian motion process whose price satisfies dSt=rStdt+σStdwt,(2.1) here,r is the risk-free rate of interest assumed to be strictly positive,σ is the volatility of the asset’s return assumed to be strictly positive.Wt is a Brownian motion under the risk-neutral measureQ.The solution of(2.1)is St=S0e(r-σ2/2)t+σWt. The payoff of anup-and-out European call barrier option is Z(T)=(S0eσWT-K)+I{WT≥k,MT≤b}, here Wt=Wt+αt,MT=max0≤t≤TWt,B is the barrier leveI,k=1/σlogK/S0,b=1/σlogB/S0.The value function of an up-and-out European call barrier option at time zero is Z(0)=S0[N(δ+(T,S0/K))-N(δ+(T,S0/B))]-e-eTK[N(δ-(T,S0/K))-N(δ-(T,S0/B))]-B(S0/B)2r/σ2[N(δ+(T,B2/KS0))-N(δ+(T,B/S0))]+e-rTK(S0/B)-2r/σ2+1[N(δ-(T,B2/KS0))-N(δ-(T,B/S0))], Here δ±(τ,φ)=1/σ(?)τ[logφ+(r±1/2σ2)τ].The above results of an up-and-out European call barrier option come from thoughts of maximum value function and indicator function. However, when we study knock-in barrier option, this method cannot be used directly. A common solution is to use the relationship between European options and barrier options.Game options have important applications in the field of financial mathematics. The extraordinary difference between game options and traditional options is that the seller of the option has a right to terminate the contract in advance. So, after signing a contract, not only the buyer needs to maximize his benefit, but also the seller does. On the other hand, the game options can also be regarded as a special American Option which allows seller to redeem the rights. Game options are applied to price cancellable American financial derivatives.This article learns from the thought of game options and combines European barrier options with a condition which allows the seller to redeem the right. We want to price up-and-in cancellable barrier option and find a best time for seller to use his right.Z represents the payoff of holder. If the underlying asset does not exceed barrier level before expiring date, Z equals zero. If the underlying asset exceeds barrier level before expiring date, butuntil the end of expiring date, the seller doesn’t use his right to cancel the contract, Z equals to (ST-K)+. Otherwise,the underlying asset exceeds barrier level before expiring date and the seller use his right to cancel the contract, we can get that the payoff of the holder is ((Stc-K)++δ). Above all, payoff of the holder is Ztc=(ST-K)+1{τm<T<tc}+((Stc-K)++δ)1{τm<tc<T}.The value function of an up-and-out European call barrier option at time zero is Results of this article:Theorem1The value function of up-and-in cancellable barrier call option is K++δ1τb<tc<TeαWtc. Inference1The integral of value function Φtc is Φtc=∫0T∫1/σlnK/S0∞e-(r+1/2α2)T(S0eσω-K)eαω2(2b-ω)/t(?)e-(2b-ω)2/2tdωdt∫0tc∫1/σlnK/S0∞e-(r+1/2α2)tc(S0eσω-K+δ)eαω2(2b-ω)/t(?)e-(2b-ω)2/2tdωdt+e-(r+1/2α2)tc∫0tc∫-∞1/σlnK/S0δeαω2(2b-ω)/t(?)e-(2b-ω)2/2tdωdt.Theorem2For cancellable barrier call option, the best cancel time t△satisfies-2S0/(?)e-(r+1/2α2)t-4b2-(σt+αt+2b)2/2te-[1/σlnK/S0-(σt+αt+2b)]2/2t+2(σ+α)S0e-(r+1/2α2)t-4b2-(σt+αt+2b)2/2tN(1/σlnK/S0-(σt+αt+2b)/(?))+2K/(?)e-(r+1/2α2)t-4b2-(αt+2b)2/2te-[1/σlnK/S0-(αt+2b)]2/2t-2Ke-(r+1/2α2)t-4b2-(αt+2b)2/2tN(1/σlnK/S0-(αt+2b)/(?))=0.