代写代考 MATH3090/7039: Financial mathematics Lecture 9

MATH3090/7039: Financial mathematics Lecture 9

Arbitrage in continuous time
Black-Scholes: PDE and formula

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Fundamental theorem in continuous time
Solution to the BS PDE

Arbitrage in continuous time
Black-Scholes: PDE and formula
Fundamental theorem in continuous time
Solution to the BS PDE

Consider an Ft-adapted vector price Itˆo process Xt = (Xt1, . . . , XtN ). A portfolio/trading strategy is an Ft-adapted vector process
Θt = (θt1,…,θtN) of quantities held in each asset 1,…,N.
Say that the trading strategy is self-financing if its value Vt = Θt · Xt
satisfies (with probability 1) for all t
dVt = Θt · dXt, equivalently Vt = V0 +
Θu · dXu. Arbitrage is a self-financing trading strategy Θt such that
V0 = 0, and both:
P(VT ≥0)=1 P(VT >0)>0
V0 =0 and P(VT ≥0)=1

Replication
• Definition: a trading strategy Θ replicates a time-T payoff YT if it is self-financing, and its value VT = YT (with probability 1).
• Law of one price: At any time t < T , the no-arbitrage price of an asset paying YT must be the value of the replicating portfolio. • To hedge could mean to [try to] replicate a payoff (or the portion of a payoff attributable to some particular source of risk), but usually it means to [try to] replicate the negative of the payoff. For example, to hedge a position that is short one option usually means to [try to] replicate a position that is long the option. I say “try to” because “hedge” can mean an approximation to replication such as super-replication, or broadly speaking, any strategy to reduce some notion of risk. Arbitrage in continuous time Black-Scholes: PDE and formula Fundamental theorem in continuous time Solution to the BS PDE Motivation for GBM to model a stock price BM is a natural starting point for model-building. But some problems withWt orαt+βWt asamodelforastockprice • BM can go negative, and so can scaled BM with drift αt + βWt. • If dSt = αdt + βdWt then each St+1 − St is independent of Ft. A 10+ dollar move is equally likely, whether St is at 20 or 200. For a GBM S, the drift and diffusion are proportional to S • S stays positive • Each log return log(St+1/St) is independent of Ft. A 10+ percent move is equally likely, whether St is at 20 or 200. Black-Scholes model In continuous time, consider two basic assets: • Money-market or bank account: each unit has price Bt = ert. Equivalently, it has dynamics dBt = rBtdt, B0 = 1 • Non-dividend-paying stock: share price S has GBM dynamics dSt = μStdt + σStdWt, S0 > 0
where volatility σ > 0 and W is BM, under physical probabilities. Find: price Ct of call which pays CT = (ST −K)+ at time T (K > 0).
We first do an intuitive, but flawed, derivation, then do a careful proof.

Plan of intuitive derivation: Replicate B using C and S • We will price options using replication.
The other approach is to use the martingale/risk-neutral pricing approach: Apply Fundamental Thm, and take E of discounted payoff. We do not cover this in this course.
• Construct risk-free portfolio of (C, S). Risk-free means zero dW term
• If self-financing, then it must grow at the risk-free rate r, else there is arbitrage of portfolio vs B.
• On the other hand, if Ct = C(St, t) for some smooth function C, then Itˆo rule says that the portfolio value’s drift can be expressed in terms of C’s partials.
• Therefore C (S, t) satisfies a PDE.
• Solve this PDE to obtain formula for C.

Construct a risk-free portfolio
Use (1 option, -at share), choosing at to cancel the option risk Portfolio value is
Vt = Ct − atSt. So some authors assert that
dVt = dCt − atdSt.
But the Itˆo’s rule says
d(atSt) = atdSt + Stdat + (dat)(dSt),
so it’s not true that d(atSt) = atdSt. Ignoring this point for now . . .

Construct a risk-free portfolio (cont)
Assume Ct = C(St, t) where C is some smooth function. By Itˆo, dVt = ∂C dt+∂C dSt + 1 ∂2C (dSt)2 − atdSt,
∂t ∂S 2 ∂S2 where C and its partials are evaluated at (St,t).
Now make these cancel by choosing at = ∂C (St, t). Then ∂S
∂C 1∂2C 2 􏰆∂C 1∂2C 2 2􏰇 dVt = ∂tdt+2∂S2(dSt) = ∂t +2∂S2σ St dt
On the other hand, Vt is the value of a risk-free portfolio, so 􏰆 ∂C􏰇
dVt =rVtdt=r C−St∂S dt. Comparing right-hand sides,
∂C + rS ∂C + 1 ∂2C σ2S2 = rC. ∂t t ∂S 2 ∂S2 t

The Black-Scholes PDE and formula
So C solves a PDE
∂C+rS∂C+1∂2Cσ2S2=rC, (S,t)∈[0,∞)×[0,T)
where the terminal condition is given by the payoff function. Solution: the Black-Scholes formula. For t < T , CBS(S,t) = SN(d1)−Ke−r(T−t)N(d2), whereN(x)=√1 􏰞x e−x2/2dx,theCDFofthestandardnormal, ∂t ∂S 2 ∂S2 C (S, T ) = (S − K )+ log(Ser(T −t)/K) σ√T − t d1,2 ≡ d+,− = σ√T − t ± 2 andCBS(S,T)=(S−K)+ =limt→T CBS(S,t). Can directly verify that CBS solves PDE. (How’d we get CBS? Later.) How not to do stochastic calculus? What about the claim that d(Ct − atSt) = dCt − atdSt? Bogus justifications: • The share holdings at are “instantaneously constant”. Nonsense. In fact at is changing (and, moreover, changing so fast that we needed to introduce Itˆo calculus). • Portfolio of (1 option, -at shares) is “self-financing” It’s not. In fact there’s no way to vary this portfolio’s share holdings without outside funding. (The option position does not provide any funding, because it is fixed at 1 unit). The intuitive derivation is useful (and can be improved), but is not a proof. Let’s actually give a proof now. Black-Scholes formula: careful proof Plan: replicate 1 option using a portfolio of (S, B). Let CBS(S, t) be the B-S formula. We are not assuming that CBS(St,t) is the option price; that will be the conclusion. bank acct units . Let’s hold ∂CBS CBS(St,t)−atSt at = ∂S (St,t) shares ,bt = B Portfolio value is then Vt = atSt + btBt = atSt + (CBS(St, t) − atSt) = CBS(St, t). Black-Scholes formula: careful proof (cont) In particular, its terminal value is CBS(ST,T)=(ST −K)+ And it self-finances, because 􏰆∂CBS 1 ∂2CBS 􏰇 ∂CBS dVt=dCBS(St,t)= ∂t+2∂S2σ2St2dt+∂SdSt 􏰆 ∂CBS 􏰇 ∂CBS =r CBS −St ∂S dt+ ∂S dSt = atdSt + btdBt because CBS satisfies the B-S PDE by direct computation. So the portfolio replicates the option Conclusion: at any time t < T , the unique no-arb price of the option equals the portfolio value, which is CBS(St,t). Call price vs S Let K = 100, T − t = 1, σ = 0.2, r = 0.05. Call price CBS(St) and intrinsic value = (St − K)+ and lower bound (St − Ke−r(T −t))+, plotted against St. 60 50 40 30 20 10 50 60 70 80 90 100 110 120 130 140 150 Arbitrage in continuous time Black-Scholes: PDE and formula Fundamental theorem in continuous time Solution to the BS PDE Fundamental theorem No arb ⇔ ∃ equivalent martingale measure P Still holds in continuous time. Proof skipped. Arbitrage in continuous time Black-Scholes: PDE and formula Fundamental theorem in continuous time Solution to the BS PDE The BS PDE Consider the BS PDE ∂C+rS∂C+1∂2Cσ2S2=rC, (S,t)∈[0,∞)×[0,T) ∂t ∂S 2 ∂S2 C (S, T ) = (S − K )+ where the terminal condition is given by the payoff function. • The BS-PDE is a backward PDE: the terminal condition is given at time T , and we want to obtain the option price at time 0. • We solve the Black-Scholes PDE by reducing to the heat equation which has a fundamental solution. • The reduction of BS to the heat equation can be achieved via a change of variables. • We then need to reverse the change of variable. Solution to the diffusion or heat equation The PDE ∂U − 1σ2 ∂2U = 0 ∂τ 2 ∂z2 f(ζ) dζ. 􏰘􏰗􏰖􏰙 with initial condition (where f satisfies a growth condition) has solution 1􏰑∞ 􏰊(z−ζ)2􏰋 U(z,τ) = √ exp − 2σ2τ U(z,0) = f(z) (The initial condition holds in the following sense: at all z where f is continuous, we have U(y,τ) → f(z) as y → z, τ ↓ 0.) Proof: Direct calculation. Intuitive interpretation The PDE describes the concentration U, at position z and time τ, of a substance (e.g. a dye) propagating in an infinite one-dimensional medium. The PDE comes from physical laws (rate of movement is proportional to the concentration gradient). Solution: a 1-unit point mass of the substance, initially located at ζ diffuses to have at time τ and position z the concentration, 1 􏰊 (z−ζ)2􏰋 √2πσ2τ exp − 2σ2τ So if initial concentration is f(ζ) = U(ζ,0) at each ζ, then the concentration at position z and time τ is the total of the contributions from every ζ: 1􏰑∞ 􏰊(z−ζ)2􏰋 √ exp − 2σ2τ f(ζ) dζ. 􏰘􏰗􏰖􏰙 Intuition of the diffusion kernel 1 􏰊 (z−ζ)2􏰋 √2πσ2τ exp − 2σ2τ . is sometimes called the “diffusion kernel” or “fundamental solution” or “Greens function” for the diffusion PDE. Intuition: • This is the Normal(ζ,σ2τ) density. Regard the initial unit mass at ζ as many many particles each following a Brownian motion. • Alternative derivation: let ζ = 0. Note that if U(z,τ) solves the PDE, then so does U(cz,c2τ). So look for a solution U(z,τ) = 1 u( z2 ). σ√τ σ2τ Plug into PDE to produce an ODE for u, and solve. Application of L9.21 to the Black-Scholes PDE Consider the BS PDE ∂C+rS∂C+1∂2Cσ2S2=rC, (S,t)∈[0,∞)×[0,T) ∂t ∂S 2 ∂S2 C (S, T ) = (S − K )+ Let τ = T − t, z = log(S) + (r − σ2/2)τ, U(z, τ) = erτ C. In other words,andU(z,τ)=erτC(ez−(r−σ2/2)τ,T−τ). Itcanbeshowthat U(z,τ) satisfies the PDE ∂U−1σ2∂2U=0, (z,τ)∈R×(0,T], ∂τ 2 ∂z2 U(z,0)=f(z)=(ez −K)+ . Solve for U using L9.21. Then transform back to obtain the BS formula CBS(·) on L9.12. B-S PDE: comments • To price instead an option paying f (ST ), we use the same PDE, changing only the terminal condition to C(S,T) = f(S). This follows from both the replication argument and the risk-neutral pricing argument • The replication argument proved that: if C(S,t) is a function that satisfies the B-S PDE with terminal data C(S,T) = f(S), then (under technical conditions) a trading strategy of ∂C/∂S shares and (C − St · ∂C/∂S)/Bt units of the bank account replicates a claim paying f (ST ), and self-finances. So to hedge a claim on f (ST ), one can use PDE or risk-neutral pricing to find the option pricing function C which satisfies the B-S PDE. We can then calculate ∂C/∂S to find the delta hedge. A broad picture 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com