These notes begin with an elementary approach to continuous time methods of calculating Itˆo due to F¨ollmer. The section on Feynman-Kac connects the martingale approach used explicitly in these notes to the more classical approach based on partial differential equations.
Brief Sketch of Lebesgue’s Integral
For any sequence (Xn) of random variables bounded from below, one has. ii). On L2, the vector space of quadratically integrable random variables, there even exists a scalar product defined by .
Convergence Concepts for Random Variables
We will then write Xn−→X weak or Xn−→D X. Relationships between the different concepts of convergence. a) a.s.-convergence and convergence in probability (i) Xn−→X P-a.s. b) Convergence in probability and L1 convergence AssumeXn−→X in L1. X(t, ω) dP(ω) is differentiable in t and its derivative is. c) Convergence in distribution and convergence in probability.
The Lebesgue-Stieltjes Integral
Let FV(IR+) denote the set of all real-valued right-continuous functions on IR+= [0,∞[ of finite variation. Therefore the Lebesgue-Stieltjes integral for any B-measurable real-valued function on IR+ is well defined as.
Exercises
The first elementary applications to option pricing in this chapter deal with the standard Black-Scholes model (Black-Scholes (1973)), first through the classical PDE approach (Sect. 2.5), then through the use of the Martingale -approach (Sect. .
Stochastic Calculus vs. Classical Calculus
Let F ∈C2(IR) be a doubly continuously differentiable real function on the real line IR. Simply, when forming the differential dF(Xt), the second term of the Taylor formula can no longer be neglected, since the term (∆Xt)2, the quadratic variation of Xt, does not disappear at ∆t→ 0.
Quadratic Variation and 1-dimensional Itˆ o-Formula
The above statement implies that the functions Xt with positive quadratic variation Xt are of unbounded total variation. This is the case for the second integral which is well defined as the Lebesgue-Stieltjes integral, since the quadratic variation Xt is of finite variation (see Section 1.3).
Covariation and Multidimensional Itˆ o-Formula
Note that Xt can be of infinite variation, since the integral on the right is well defined as Stieltjes integral, for Yt ∈F V. By this observation, Wiener first obtained a "stochastic" integral for the integrator Xt = Bt and Yt=h(t) a deterministic function, the so-called Wiener integral.
Examples
It is the standard model used in the financial industry for the return of a security price process St with infinitesimal drift µ(t)dt and stochastic noise σ(t)dBt, where σ(t) is called the volatility of St.
First Application to Financial Markets
Consider the following portfolio strategy: buy one contingent claimH at priceV,. saleφ=Vx shares of security X. The value of this portfolio is Therefore, portfolio Π is risk-free and, according to the no-arbitrage principle, its return must be equal to the risk-free interest rate, i.e. 18).
Stopping Times and Local Martingales
For a stochastic process X on the probability space (Ω,F, P), consider the following filtration:. Ft) is called the natural filtration created by (Xt). The following processes are martingales with respect to filtration (Ft):. i) Procedures are clearly adapted. One of the most useful theorems of probability theory is the following so-called "choice stopping theorem".
According to Lebesgue's theorem, the optional stopping theorem also holds for finite stopping times T < ∞, P-a.s., for uniformly integrable (XT∧n).
Local Martingales and Semimartingales
If the game is "fair", then for any b > 0 the payoffs will exceed b in the finite time with probability one. But even for arbitrarily small b > 0, the average wait for this to happen is infinite. Consider the stopping time Vn=Sn∧Un∧Tn, where Tn is the localization sequence of bounded stopping times for (Xt).
Then Mt =X(1)t − X(2)t as the difference of two Itˆo integrals is a local martingale with paths of finite variation.
Itˆ o’s Representation Theorem
Let B be a Brownian motion on (Ω, (Ft), P) with "natural" filtering (generated by (Bt) and all P-null groups) and M an L2-martingale on (Ft, P). Then there exists an adapted process H with.
Application to Option Pricing
While the original Black-Scholes approach resolves PDEs under boundary constraints, the martingale technique leads to option prices as expectations below the martingale measure. The technique to convert P into an “equivalent martingale size” is developed in chapter. Using only elementary facts from independent normally distributed random variables leads to the exponential Doleans-Dade as new density under a measure change for Brownian motion.
This section, which appears rather abstract at first glance, is fundamental to the funding applications in Chapters 4 and 5, where the general Girsanov transformation is repeatedly used.
Heuristic Introduction
The General Girsanov Transformation
Application to Brownian Motion
Xtd) with X0= 0 the following statements are equivalent:. ii) X is a federal local martingale mitXi, Xjt=δij·t. In this chapter, the methods developed in Chapters 2 and 3 are used to derive the fundamentals of “financial economics” in continuous time, such as the market price of risk, the no-arbitrage principle, the fundamental rule of pricing, and its invariance with respect to numerical value. changes and forward measure, a useful tool for dealing with stochastic interest rates. 3.2, the rather complicated problem of numeraire change can be treated in a rigorous way, and the so-called "two-state" or "Siegel" paradox serves as an illustration.
The section on Feynman-Kac relates the martingale approach used in these notes to the more classical approach based on partial differential equations.
The Market Price of Risk and Risk-neutral Valuation
Such an arbitrage opportunity would exist if, at any time t and in any condition ω, one could construct a risk-free portfolio that has a higher return than the risk-free rate r(t, ω). Note that the equilibrium process has not changed; it is the same under both measures P and P∗. The great advantage of the representation (5) under the (martingale) measure P∗ is that we do not need to know anything about the individual expectationsP and risk attitude of investors.
It is the same equilibrium pricing process as under P, but in a simpler representation under P∗.P∗ is called the “equivalent risk-neutral measure” or the “P-equivalent martingale measure”.
The Fundamental Pricing Rule
Let Z(t) be the price process of a security with continuous dividend payments given by the dividend rate dZ(t) which is proportional to Z(t).1 Assume that the dynamics of the (ex dividend) process is given by.
Connection with the PDE-Approach
In this case, the Feynman-Kac formula also allows backtracking: the conditional expectation (12) can be calculated as the solution of the PDE (9) under the boundary constraint (10). Thus, the solution of (14) reduces to the solution of the partial differential equation (15) under the boundary constraint (16).
Currency Options and Siegel-Paradox
Change of Numeraire
4.1, 'no arbitrage' and Girsanov's theorem applied to the process 'market price of risk' λ(t) under the Novikov condition (3) together with Prop. Let CT ∈ FT be a conditional claim with CT = VT(φ) for some self-financing portfolio strategy φ. Conclusion: P∗ is a martingale measure with respect to (X0) as numerical if, and only if, Q∗i is a martingale measure with respect to (Xi) as numerical.
Solution of the Siegel-Paradox
Admissible Strategies and Arbitrage-free Pricing
However, there may be many different equivalent martingale measures in M(X), as is the case when the market is "incomplete". In incomplete markets, therefore, the price of an arbitrary contingent claim cannot be determined by "no arbitrage" arguments. The price Πt of any attainable conditional demand CT is uniquely determined by no-arbitrage and is given by the relation (19), where the expectation is taken for arbitrary P∗ ∈ M(X).
Whether or not a market's property is "arbitrage-free" should not depend on the choice of the numeraire.
The “Forward Measure”
Option Pricing Under Stochastic Interest Rates
Term structure models are considered one of the most complex and mathematically demanding subjects in finance. One of the first term structure models was the short-rate model by Vasicek (1977). A disadvantage of short-term interest rate models is that they cannot be adjusted to the overall maturity structure (they only capture the first point on the "yield curve").
For a comprehensive treatment of fixed income markets and thermal structure models, we refer to Musiela-Rutkowski (1997), who also provides a detailed overview of the historical development of this complex topic.
Different Descriptions of the Term Structure of Interest
This approach finally led to the so-called “LIBOR” or “market” models of Sandmann-Sondermann-Miltersen (1995) and Miltersen-Sandmann-Sondermann (1997), further developed by Brace-Gatarek-Musiela. (1997). These models have become very popular in the financial industry because they are stable and arbitrage-free, generate non-negative interest rates and, most importantly, reflect the market practice of Black's caplet formula (see Sections 5.5 and 5.6). ). Looking at the continuous-time term structure model in the general form of Heath-Jarrow-Morton (1992) as an infinite collection of assets (zero bonds of various maturities), the methods developed in Chap.
Readers who have reviewed the original HJM articles may appreciate the simplicity of this approach, which leads to the basic HJM results in a straightforward manner.
Stochastics of the Term Structure
4.1, no arbitrage implies that the risk pricing process for all “assets” is the same, i.e. does not depend on the expiration date. 2) Since there is now a continuum of assets, but only d factors, NV imposes restrictions on the driving terms µ(t, T) (term structure models with finitely many factors are “over”-complete).
The HJM-Model
No arbitrage (see Sections 4.1 and 5.2) implies the existence of a risk price process λ(t), which is independent ofT, with. Since we have already done this measure change from P to P∗ for the bond prices, we can directly derive the dynamics of f(t, T) under P∗ using the same technique as above. Hence, forward rates and spot rates under the (spot) martingale measure P∗ are given by.
Note: Note that the volatility τ of forward rates and the volatility σ of zero bonds have opposite signs, as both depend on the same Brownian movement.
Examples
The “LIBOR Market” Model
Caps, Floors and Swaps
So the question of whether we need stochastic calculations in finance boils down to the question: Are graphs of stock prices, exchange rates or interest rates really of limitless variation? Such functions, such as the Weierstraß function or a path of Brownian motion, are pure mathematical constructs. 1 A colleague once made the self-ironic remark: the older you get, the better you get at drawing paths of Brownian motion.
The Buy-Sell-Paradox
Local Times and Generalized Itˆ o Formula
Notation: µ=F−·λ (measure of the second derivative of F) Remark 6.1. Generalized formula Itˆo for convex functions) Let F :IR−→IR be a convex function.
Solution of the Buy-Sell-Paradox
Arrow-Debreu Prices in Finance
Consider the following special conditional statement Ht(x)∈ Ft. Arrow-Debreu security depends on the value of X at time).
The Time Value of an Option as Expected Local Time
We say that x is of quadratic variation along(τn) if this holds for all real functions xi, xi+xj (1≤i, j≤n). Let us also mention the processes of finite energy X=M +A, where M is a local martingale and A is a process with paths of quadratic variation 0 along the dyadic subdivisions. Carr-Jarrow (1990): “The stop-loss start-gain paradox and option valuation: A new decomposition into intrinsic value and time value,” review by Fin.
Delbaen-Schachermayer (1995): “The No-Arbitrage Property under a Change of Numerical,” Stochastics Stochastic Rep. The Banach Space of Workable Contingent Claims in Arbitration Theory,” Ann.
![Fig. 6.3. U t (a, ω) = # up-crossings from X t (ω) of the interval [a, a + ]](https://thumb-ap.123doks.com/thumbv2/azpdfco/10574163.0/125.659.139.509.73.313/fig-6-u-ω-crossings-x-ω-interval.webp)
