Ito's lemma and Black-Scholes model(伊藤定理的简单推导与BS模型的推理)

Ito’s lemma and Black-Scholes modelZhiming Liao (chiminhlyao@)Ito’s lemma and Black-Scholes model are indispensable tools in financial applications. Ito lemma, named after its discoverer, Kishi Ito, is of great importance in finding the differential of a function of a particular type of stochastic process. When Fischer Black and Myron Scholes published their work-the pricing of options and corporate liabilities in the Journal of Political Economy in the 1970s, their prominent work immediately drew the interest of the Chicago option market, in 1997, they were awarded Nobel Prize in economics for this significant contribution to financial theory. Since then, Black-Scholes model has been playing a vital role in calculating options. This short paper will address Ito’s lemma and Black-Scholes model and their proofs.Ito’s lemmaSuppose y(t) follows a diffusive stochastic process, that is to saydy t=u y dt+σy dz tHere, u y is the instantaneous expected rate of change in the y and σy is its instantaneous standard deviation.And f y,t is a function differentiable twice in the first argument and once in the second. Then f also follows a diffusive processd f y,t=ðf+ðfu y+1ð2f2σy2dt+ðfσy z tProof (informal derivation using Taylor series expansion formula) Taylor series expansion in two variablesf(x.y)=f(x0.y0)+x−x01!f x x0.y0+y−y01!f y x0.y0+x−x022!f xx x0.y0+x−x0y−y0f xy x0.y0+y−y02f yy x0.y0+x−x03f xxx x0.y0+x−x022!y−y01!f xxy x0.y0+x−x01!y−y022!f xyy x0.y0+y−y03f yyy x0.y0+⋯Assume that we are initially at some α,t and that a short interval of time ∆t passes. During this ephemeral period there will be some associated∆z. Using the Taylor expansion above,f α+u y∆t+σy∆z,t+∆t=fα,t+ u y∆t+σy∆z f y+∆tf t+1u y∆t+σy∆z 2f yy+ u y∆t2+σy∆z∆t f yt+1∆t2f tt+t ird and ig er order termsTherefore,∆f=f α+u y∆t+σy∆z,t+∆t −fα,t= u y∆t+σy∆z f y+∆tf t+12u y∆t+σy∆z 2f yy+ u y∆t2+σy∆z∆t f yt+12∆t2f tt+t ird and ig er order termsSince z is following a standard Brownian motion, ∆z is normally distributed with expected value 0 and variance ∆t. It is easy to check that, E∆z2=∆t.E∆f=1σy2f yy+u y f y+f t∆t+second and ig er order terms in∆tWhen ∆t tends to 0 or small enough, we can ignore higher order terms, so we can get the expected rate of change in f in accord with Ito’s lemma above.Thus,∆f−E∆f=σy∆zf y+second and ig er order terms in∆t and∆zThis e quation is the last part of the Ito’s lemma expression.A formal correct proof can be fo und in Malliaris and Brock’s paper-Stochastic methods in economics and finance.ExampleIn a world with a constant nominal interest rate r, a bond portfolio with value of $1 at time 0 and continuously reinvested coupon payments will be worth B(t)=e rt at time t. Suppose that the price level evolves randomly according to the stochastic processdP=μPdt+σPdzWhere μ is the expected inflation rate and σ is its proportional standard deviation per unit time. The real value of the bond portfolio at time t will beb t=B tWhat is the expected real return on the bonds?Example 2 in the textbook- copula methods in finance-page 9Notice that, givends t=μS t dt+σS t dz tWe can get f S,t=ln S t to obtaind ln S t= μ−12σ2dt+σdz tIf μ and σ are constant parameters, it is easy to obtainln Sτ∼N ln S t+ μ−1σ2τ−t,σ2τ−tWhere N m.s is the normal distribution with mean m and variance s. Then, Pr Sτℑt is described by the lognormal distribution.Black- Scholes modelIn the simplest form, this model involves two underlying assets, a riskless asset like cash bond and a risky asset such as stock. The riskless asset appreciates at the short rate, or riskless rate of return r t, which is assumed to be nonrandom, although possibly time-varying. Thus, the price of the riskless at time t is assumed to satisfy the differential equationdB t=r t B t Whose solution for the value B 0=1is B t =exp r s ds t0 ProofdB t dt =r t B t ⇔dB t B t =r t dt ⇒dln B tdt =r t ⇔ln B t = r s tds +c⟹B t =exp r s t0 ds (As B 0=1, c=0)The share price S t of the risky asset stock at time t is assumed to follow a stochastic differential equation of the formdS t =μt S t dt +σS t dW t ,Where W t is a standard Brownian motion, μt is a nonrandom function of t, and σ>0 is a constant called the volatility of the stock. Lemma 1If the drift coefficient function μt is bounded, then the SDE(stochastic differential equation) has unique solution with initial condition S 0; and it is given byS t =S 0exp σW t −σ2t2+ μs ds tMoreover, under the risk-neutral measure, it must be the case thatr t =μt .Proof.dW t =0+1dW t (u w =0,σw =1) Consider function S W t ,t =S 0exp σW t −σ2t 2+ μsds t0 Applying Ito’s Lemma,S W =S t σ S WW =S t σ2S t =S t μt −σ22Then,d S W t ,t = ðS ðt +ðS ðW u w +12ð2S ðW 2σw 2 dt +ðS ðW σWdW t = S t μt −σ22 +12S t σ2 dt +S t σdW t =μt S t dt +σS t dW tSo dS t =μt S t dt +σS t dW t , thus proved the first argument of this lemma.Under risk-neutral assumption, the discounted share price of stock must be a martingale. The discount factor is B t , so the discounted share price of stock isS t ∗=S t t =S 0exp σW t −σ2t2+ μs ds t 0 exp r sds t 0=S 0exp σW t −σ2t + μS −r s ds t 0 Applying Ito’s lemma once more, we can havedS t ∗=σS t ∗dW t +S t ∗ μS −r s dtIn order that S t∗ be a martingale, the dt term must be 0; this implies that r t=μt.Lemma 2Under the risk-neutral measure, the ln of the discounted stock price at time t is normally distributedwith mean lnS0−σ2t2 and variance σ2t.Proof,Consider function f S t∗,t=lnS t∗,From the proof of Lemma 1, we can saydS t∗=σS t∗dW t+S t∗μS−r s dt=σS t∗dW t (Under risk-neutral assumption,μS−r s=0) That is dS t∗=σS t∗dW tApplying Ito’s lemma, dlnS t∗=−σ22dt+σdW tAnalogous with example 2 in Ito’s lemma,E(lnS t∗)=E lnS0−σ22t+σdW t=lnS0−σ22tVar(lnS t∗)=Var lnS0−σ22t+σdW t=VarσdW t=σ2Var dW t=σ2tThe Black-Scholes formula for the price of a European Call optionEuropean Call on the asset Stock with strike K and expiration date T is a contract that allows the owner to purchase one share of stock at price K at time T. Thus, the value of the Call at time T is S T−K+. According to the fundamental theorem of arbitrage pricing, the price of the asset Call at time t=0 must be the discounted expectation, under the risk-neutral measure, of the value at time t=T, which, byLemma 1, isC S0,0;K,T =E S T∗−KB T+Where S T∗ has the distribution specified in Lemma 2. A routine calculation, using integration by parts, shows that C x,0;K,T may be rewritten asC x,0;K,T=xΦz−KB TΦ z−σTWhere z=ln xB TK+σ22TσTand Φis the cumulative normal distribution function.ProofAs proved before,S t∗=S tB t=S0exp σW t−σ2t2+μS−r s dstUnder the risk-neutral measure, we haveS T∗=S0exp σW T−σ2T 2Where W T follows standard Brownian motion, W T~N0,T Let W T=g T, so g~N0,1ThereforeS T∗=S0exp σg T−σ2T2g~N0,1C X,0;K,T=E S T∗−KT+S T∗−KT≥0 ⟺xexp σg T−σ2T≥KT⟹σg T−σ2T≥lnKT⟺g≥σ2T2−lnxB TKσTDenote z=ln xB TK+σ22TσT, then S T∗−KB T≥0 ⟺g≥σT−ZC X,0;K,T=E S T∗−KB T+=E xexp σg T−σ2T2−KB T+= xexp σy T−σ2T2−KB Te−y222π∞σT−Z= xexp σy T−σ2T2e−y222π∞σT−Z −KBe−y222π∞σT−Z=x exp σy T−σ2T2e−y222π−∞σT−Z KB Te−y222π∞σT−Z=x exp −σy T−σ2T2e−y222π−z−σT −∞KB T−y222πz−σT−∞=x exp −σy T−σ2T e−y222π−KTΦ z−σTz−σT −∞While x exp −σy T−σ2T2−y222πz−σT −∞=x2π −σ s−σT T−σ2T2− s−σT22d s−σTz(Substitute y=s−σT)=x12π−s22ds z−∞=xΦz。