金融资产定价--新加坡国立mba随机金融最核心讲义.pdf

Stochastic Calculus and ProcessesFE5204Lecture Notes 7Applications to Mathematical Finance§1. Introduction.Take (Ω, F, IP, {Ft}t≥0) to be a filtered probability space, of which thefiltration {Ft}t≥0satisfies the usual conditions.On this filtered probability space, we will define a mathematical market in this set of notes.To begin with, let us consider the celebrated Black-Scholes model, i.e., a market with two investment possibilities S(·) = (S0(·),S1(·)) such that(a) Bond: dS0(t) = rS0(t)dt with S0(0) = 1, and(b) Stock: dS1(t) = µS1(t)dt + σS1(t)dBt,(where {Bt,t ≥ 0} is standard Brownian motion on (Ω, F, IP).We will also assume that the filtration is generated by this Brownian motion.)(This is known as a Black-Scholes model.)Obviously, S0(t) = ert.It has been shown earlier that it can be solved for S1(t), which is the so-called geometric Brownian motion given byS1(t) = S1(0)e(µ−1 2σ2)t+σ Bt.11. Portfolio.θ(t) = (α(t),β(t)), of which• α(t) denotes the number of bonds held at time t, while• β(t) denotes the number of stocks held at t.Given θ = (α,β), the corresponding wealth Wθ(t) at time t isWθ(t) = θ(t) · S(t) = α(t)S0(t) + β(t)S1(t),(1.1)where “·” in θ(t) · S(t) denotes the usual inner product in Euclidian spaces.For easy notation, we will denote by W(t) the corresponding wealth whenever there is no confusion.The portfolio is said to be self-financing ifdWθ(t) = θ(t) · dS(t) = α(t)dS0(t) + β(t)dS1(t),(1.2)(which means that no money is brought in or taken out from the system).Equivalently,Wθ(t) = Wθ(0) +Zt0θ(s) · dS(s)= Wθ(0) +Zt0α(s)dS0(s) +Zt0β(s)dS1(s).(1.3)Remark.Suppose that θ = (α, β) is also an Itˆ o process. SinceWθ(t) = α(t)S0(t) + β(t)S1(t),according to Itˆ o’s formula, we havedWθ(t) = S0(t)dα(t) + α(t)dS0(t) + dhα,S0it+ S1(t)dβ(t) + β(t)dS1(t) + dhβ,S1it.So, under the assumption that θ = (α, β) is an Itˆ o process, the portfoliois self-financing if and only ifS0(t)dα(t) + dhα,S0it+ S1(t)dβ(t) + dhβ,S1it≡ 0.22. Question.What is the relation between the terminal wealth W(T) and theself-financing portfolio θ = (α,β)?To see this more clearly, note that by (1.1) we haveα(t) =W(t) − β(t)S1(t) S0(t).(1.4)Substituting it into (1.2), one getsdW(t) =W(t) − β(t)S1(t) S0(t)dS0(t) + β(t)dS1(t).(1.5)Since dS0(t)/S0(t) = rdt, (1.5) can be re-written asdW(t) = rW(t)dt − rβ(t)S1(t)dt + β(t)[µS1(t)dt + σS1(t)dBt].Hence,dW(t) = rW(t)dt + σβ(t)S1(t)•µ − r σdt + dBt‚ .(1.6)Moving the term rW(t)dt from the right to the left and then multiplying both sides by e−rt,e−rtdW(t) − re−rtW(t)dt = σ e−rtβ(t)S1(t)•µ − r σdt + dBt‚ ,of which the LHS becomes d(e−rtW(t)), and in turn the above equation can now be expressed ase−rTW(T) = W(0) +ZT0σ e−rtβ(t)S1(t)•µ − r σdt + dBt‚ (1.7)at the terminal time T.3Let eBtdef.=µ − r σt + Bt.(1.8)3. Question.IseBta Brownian motion on (Ω, F, IP, {Ft}t≥0)?The answer is no. It is not even a martingale!But, it becomes one under Girsanov transform (to be introduced in §2), which can be described as a change of probability measure fromIP to Q, which is defined bydQ(ω) = e−aBT(ω)−12a2TdIP(ω)= e−aeBT(ω)+12a2TdIP(ω)(1.9)with a =µ − r σ.Put MT= e−aBT−1 2a2T.This means thatE EQ[f] =ZΩf(ω)dQ(ω)=ZΩf(ω)e−aBT(ω)−12a2TdIP(ω)=ZΩf(ω)MT(ω)dIP(ω)= E EI P[f MT].(1.10)Equivalently,dQ dIP|FT= MT.4Using this dQ, we get from (1.7)e−rTW(T) = W(0) +ZT0σ e−rtβ(t)S1(t)deBt.(1.11)Equivalently, puttingW(t) = e−rtW(t),andS1(t) = e−rtS1(t),(discounted value/price processes),we get dW(t) = σ β(t)S1(t)deBt.(1.11)0Remarks:(a) Refer to (1.1).If we take α(t) ≡ 0 and β(t) ≡ 1, then (1.11) becomese−rTS1(T) = S1(0) +ZT0σ e−rtS1(t)deBt.Put S1(t) = e−rtS1(t), the discounted price of the risky asset.Obviously, S1(t) is a martingale w.r.t. Q, which can be expressed asS1(t) = E EQ£S1(T) | Ft⁄.(1.12)Equivalently, S1(t) = ertE EQ£e−rTS1(T) | Ft⁄= e−r(T−t)E EQ[S1(T) | Ft].(1.12)05(b) Check that under Q,eBtfor each t > 0 is indeed a normally distributed r.v. with mean 0 and variance t.To do it, take a bounded Borel function f and observe thatE EQ[f(eBt)] = E EI P[f(eBt)MT]= E EI Ph E EI P[f(eBt)MT| Ft]i= E EI P[f(eBt)Mt]=ZΩf(µ − r σt + Bt(ω))e−aBt(ω)−1 2a2tdIP(ω)(recall a =µ−r σ)=ZI Rf(at + x)e−ax−1 2a2t1√2tπe−x2 2tdx=1√2tπZI Rf(at + x)e−(x2+2axt+a2t2)/(2t)dx=1√2tπZI Rf(at + x)e−(x+at)2/(2t)dx(putting z = at + x)=1√2tπZI Rf(z)e−z2 2tdz.It shows, indeed, that under Q,eBt∼ N(0,t) for each t > 0.(In fact, it can be shown that, as a stochastic process, {eBt;t ≥ 0} is indeed a Brownian motion without drift under Q.)6(c) Recall that S1(t) = S1(0)e(µ−1 2σ2)t+σ Bt,which can now be re-written asS1(t) = S1(0)e−σ2 2t+σ (Bt+µ σt)= ertS1(0)e−σ2 2t+σeBt.(1.13)Observe that dS1(t) = rS1(t)dt + σ S1(t)deBt.(1.13)0This is no surprise, asdeBt=µ − r σdt + dBtand hence, (1.13)0becomesdS1(t) = rS1(t)dt + σ S1(t)•µ − r σdt + dBt‚= µS1(t)dt + σ S1(t)dBt.Under Q,eB is a Brownian motion without drift, and hence the dis- counted price e−rtS1(t) is a martingale under Q, forS1(0)e−σ2 2t+σeBtis a martingale, which has been mentioned in (a), (see (1.12)).Note.As a matte。