Continuous Time Finance
This chapter gives a brief survey of continuous time finance. We categorize diffusion models according to the nature of their volatility coefficient. Models whose volatility coefficient does not exhibit randomness are treated in Sect. 3.1. Models whose vo
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This cha pter gives a bri ef survey of cont inuous time finan ce. We categorize diffusion models according to the nature of their volatility coefficient. Models whose volatility coefficient does not exhibit randomness are t reated in Sect. 3.1. Mod els whose volatility coefficient follows a st ochastic pro cess are discussed in Sect. 3.2. The mat erial pr esent ed in this chapte r is st andard. Uncert ainty volatility mod els, on which the origin al work of this book is grounded , are discussed in Cha pter 4.
3.1 Deterministic Volatility Most of our work is based on equity j FX Black-Scholes models. For t his reason, Black-Scholes ana lysis is reviewed in grea ter detail. Since t he software on t he accompa nying CD can also be used to calibrate int erest rat e mod els, we also describe some interest rat e models. 3.1.1 One-Factor Black-Scholes Analysis
There are severa l ways to derive t he Black-Scholes par ti al different ial equat ion. References [10], [26], [43], [78], for example, all use stochastic calculus, and in parti cular It o's formula (a good source for which is [64]). In [25] an alte rnative derivation is used: t he Black-Scholes formul a can also be int erpret ed as t he cont inuous-t ime limit of a bin omial random walk model. Given is a filtered probability space (n , F , {Fd , P) and a finite time horiz on T . In this probabili ty space, let B = {Bd , B o = 1 be the pric e pro cess of a riskless asset (B for Qond) , and let S = {Sd be t he secur it y pri ce pro cess:
dB = r. B , dS = St(J.Lt dt
+ O"t dW )
(3.1)
W is a Brown ian motion and Tt, J.L t and O"t are sufficient ly well-behave d funct ions. Let X be a nonn egat ive F r-measurable ran dom variable t hat represent s t he payoff st ructure of a cont ingent claim on S . R. Buff, Uncertain Volatility Models - Theory and Application © Springer-Verlag Berlin Heidelberg 2002
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3 Continuous Time Finance
(3.2) and
(t
= exp {
-it
¢u dWu -
~
it¢~
(3.3)
dU}
define a martingale measure Q equivalent to P via Q(A) = E p ((tlA) for all A E F . The arbitrage-free pri ce 7r of the cont ingent claim X is given by 7r
(3.4)
= E Q (f3r X )
where f3 = {f3t} , f3t = l /B t , is the discount pro cess belonging t o B. In order to compute n ; a replicating strat egy for X is const ruc te d explicitely. Let f(St , t) denote the (yet unknown) pri ce of X at time t for security pri ce Sf , with final value f(ST ,T) = X. Let F = {Ft} be the associated price pro cess: F, = f(St ,t). Assume for the mom ent that f is twic e differentiabl e. Define the ]R2- valued pro cess e = {( ey, ef)} of a position ey in the riskless asset and a position ei in the security as follows: 1
The portfolio impli ed by
e replicates F
a
et = as f (St, t)
and
(3.5)
at all t imes and thus X at t ime T :
(3.6) Now notice that , with Ito's formul a ,
(3.7) This impli es together with the definit ion of of t he valu e of the portfolio (eyB t , el St ) is
e that t he inst ant an eous change
+ el dS = (eY rtBt + eill-tSt) dt + eiO"tst dW =
ey dB
2
af 1 2 2a f dF -
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