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
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