Abitrage Approach to Pricing Derivatives
Jin-Chuan Duan
Hong Kong University of Science and Technology
Correspondence to:
Prof. Jin-Chuan Duan Department of Finance
Hong Kong University of Science & Technology Clear Water Bay, Kowloon, Hong Kong
Tel: (852) 2358 7671; Fax: (852) 2358 1749 E-mail: [email protected]
Web: http://www.bm.ust.hk/~jcduan
Static vs. dynamic spanning
• Assume that there are N possible states at time 1.
Every security entitles its holder an N-dimensional payoff vector. There are K securities with the N×K payoff matrix A and current K×1 price vetor P.
• If A has a rank N, then the market is complete in the sense that any possbile payoff structure can be
spanned (static) by some portfolio of K securites.
• If A’s rank is less than N, the market is incomplete. A payoff structure can still be priced by arbitrage as long as it falls inside the static spanning.
• Arrow-Debreau equilibrium refers to a complete market competitive equilibrium in which allocations are efficient. Note that no arbitrage is a necessary condition of market equilibrium.
• The price vector P cannot be arbitrary. To say the least, it cannot permit arbitrage in the sense that any two portfolios with an identical payoff vector must has the same current value.
• If one is allowed to trade between time 0 and 1, the spanning set can be enlarged even though the number of securities remains fixed. In other words, one is more likely to be able to price a payoff structure by arbitrage.
Black-Schole dynamic spanning approach to option valuation
Asset price dynamic
t t
t dt dW
S
dS = µ +σ
Its derivative security with payoff function at time T equal to f (ST;θ) has a time-t value expressed as
) , , , ,
; ,
(S t σ µ T r θ
C t or Ct for short.
Consider a dynamically rebalanced portfolio shorting
∆t− units of the underlying asset to hedge the derivative security. The hedged porfolio’s value at time t is
t t t
t C S
V = −∆ − . Applying Ito’s lemma gives rise to
2 . 1
2 1
2 2 2
2
2 2 2
2
t t
t t t
t t t
t t t
t t t
t t t
t t t
t
S dS dt C
S S C t
C
dS dt
S S C S dS
dt C t C
dS dC
dV
− ∆
∂ + ∂
∂ + ∂
∂
= ∂
∆
∂ − + ∂
∂ + ∂
∂
= ∂
∆
−
=
−
−
−
σ
σ
Setting ∆ = ∂Ct yields a locally risk-free hedged
S dt S C C
r dt rV dt
S S C t
C
t t t t t t
t t t
∂
− ∂
=
=
∂ + ∂
∂
∂
2 2 2
2
2 1σ
or (the Black-Scholes PDE)
2 0 1
2 2 2
2 − =
∂ + ∂
∂ + ∂
∂
∂
t t
t t t
t t
t rC
S rS C S
S C t
C σ
The solution to this PDE depends on the terminal
condition: f (ST;θ) . It can be solved using separation of variables, Green’s function or Fourier/Laplace
transformation technique.
A probabilistic way of solving the generalized Black-Scholes PDE
When both µt and σt are functions of S , the Black-t Scholes PDE applies.
2 0 1
2 2 2
2 − =
∂ + ∂
∂ + ∂
∂
∂
t t
t t t
t t
t t rC
S rS C S
S C t
C σ
The solution to the generalized PDE can be obtained by directly applying the backward equation for the Kac functional; that is the following conditional expectation satisfying the Black-Scholes PDE:
{
r T t T t}
t E e f S S
C = − ( − ) ( ;θ)|
with respect to the following artificial diffusion system:
* t t t
t rdt dW
S
dS = +σ
This probablistic solution suggests a new perspective of risk-neutral valuation
Martingale pricing theory
The Kac functional result suggests that e−rtSt is a martingale with respect to the law Q which W is at* standard Brownian motion. Note that CT = f (ST;θ). The same martingale result is thus true for derivatives as well.
Alternatively, one can show this by the Kunita-Watanabe martingale representation theorem (see Harrison and
Kreps (1979), Journal of Economic Theory).
