We start with the description of asset price models. The asset price dynamics pSτqτě0
are governed by a stochastic differential equation (SDE). In this thesis, we introduce the Black&Scholes model, the CEV model, the Heston model and the Merton model. All of these models are described by a SDE of the form
dSτ “ rSτdτ`σpS, τqSτdWτ`Sτ´dJτ, S0“sě0, (2.1a) Jτ “
Nτ
ÿ
i“0
Yi, (2.1b)
withWτ a standard Wiener process,rthe risk-free interest rate and a volatility function σpS, τq. The jump partpJτqτě0 is a compound Poisson process with intensityλě0and
independent identically distributed jumpsYi,iPN, that are independent of the Poisson process pNτqτě0. The Poisson process and the Wiener process are also independent.
If we let the diffusion coefficientσpS, τq be constant and the jump intensityλ“0, then we are in the classical Black&Scholes model of Black and Scholes (1973) and Merton (1973).
As an example of a local volatility model, we begin by presenting the CEV model, which was introduced by Cox (1975). Here, the local volatility is assumed to be a deterministic function of the asset price for the process in (2.1), σpS, τq “ σSτζ´1, 0 ă ζ ă1, σ ą 0 and λ“0.
As an example of a stochastic volatility model, we use the model proposed by Heston (1993). In contrast to the CEV model, the stochastic volatility is driven by a second Brownian motionĂWτ whose correlation withWτ is described by a correlation parameter ρ P r´1,1s, and the model is based on the dynamics of both the stock price (2.1), with jump intensity λ“0, and the variancevτ (2.2),
dvτ “κpγ´vτqdt`ξ?
vτdWĂτ, (2.2)
withσpS, τq “?
vτ, mean variance γ ą0, rate of mean reversion κą0 and volatility of volatility ξą0. Jumps are not included in either of the CEV or Heston models.
The Merton model includes jumps. The log-asset price process is not exclusively driven by a Brownian motion, but instead follows a jump-diffusion process. Thus, in the model of Merton (1976), the volatility of the asset process is still assumed to be constant, i.e.
for all S ą 0 and for all τ ą 0 it holds σpS, τq ” σ ą 0. But being a jump diffusion model, the jump intensity λ ą0 is positive and Nt „ Poisspλtq. The jumps are taken to be independent normally distributed random variables, Yi „Npα, β2q with expected jump size αPRand standard deviation βą0.
After the description of the asset or underlying as a stochastic process, we now focus on option pricing. An option is a derivative whose payoff depends on the performance of the underlyingS. So-called plain vanilla European call or put options have at maturityT, for a pre-specified strikeK, the payoffmaxtST´K,0u(call) ormaxtK´ST,0u(put). Here, the payoff does only depend on the value of the underlying at maturityT. American call or put options have the same payoff function than their European counterpart, however the option holder has the right to exercise the option at any time up to maturity T. In this case, we refer to path-dependent options.
The option is determined by the risk-neutral valuation theory, see Bingham and Kiesel (2004). Here, the basic assumption is that in the determination of the option price the individual risk preferences of a potential investor, may she either be seeking or risk-avers, are not considered. Already implicitly defined by the terminology risk-neutral, for the option price only the expected payoff of the option is important. Furthermore,
to be consistent with this risk-neutral perspective, the expectation is taken under a measure, under which the underlying process is, in expectation, evolving like the risk-free asset. In other words, the with the risk-risk-free interest rate discounted underlying process is a martingale. Bingham and Kiesel (2004) refer to this measure as strong equivalent martingale measure.
Embedding the risk-neutral valuation theory, in the following the option price at time t, for an underlyingS described by (2.1), with a payoff functiong, on a filtered probability space pΩ,F, P,Fq with filtration F “ pFtq0ďtďT, under a strong equivalent martingale measureQis given by
EQre´rpT´tqgpSTq|Fts. (2.3) For notational ease, we use in the followingEr¨sfor the expectation under the risk-neutral measure, EQr¨s.
Before we present three ways to derive this expectation, we introduce the definition of strong solutions based on the following SDE in the one dimensional case
dXt“bpt, Xtqdt`σpt, XtqdWt, (2.4) wherebpt, xq andσpt, xq are Borel-measurable functions fromr0,8q ˆRÑR.
Definition 2.1.1(Strong solution). (Karatzas and Shreve, 1996, Definition 2.1, p. 285) A strong solution of the stochastic differential equation (2.4) on the given probability space pΩ,F, P,Fq with filtration F“ pFtq0ďtďT and with respect to the fixed Brownian motion W and initial condition ζ, is a process X “ tXt; 0 ď t ă 8u with continuous sample paths and with the following properties:
(i) X is adapted to the filtration F“ pFtq0ďtďT
(ii) PrX0“ζs “1 (iii) Prşt
0t|bps, Xs| `σ2px, Xsqudsă 8s “1,0ďtă 8 (iv) the integral version of (2.4)
Xt“X0` żt
0
bps, Xsqds` żt
0
σps, XsqdWs; 0ďtă 8, holds almost surely.
Definition 2.1.2 (Strong uniqueness). (Karatzas and Shreve, 1996, Definition 5.2.3, p. 286) Let the drift vector bpt, xq and dispersion matrix σpt, xq be given. Suppose that, wheneverW is a 1-dimensional Brownian motion on somepΩ,F, Pq,ζ is an independent, 1-dimensional random vector, tFtu is an augmented filtration, and X,X˜ are two strong solutions of (2.4) relative toW with initial conditionζ, thenPrXt“X˜t; 0ďtă 8s “1.
Under these conditions, we say that strong uniqueness holds for the pair pb, σq.
After the introduction of strong, unique solutions, we present the Proposition of Yamada and Watanabe as illustrated in Karatzas and Shreve (1996):
Proposition 2.1.3. (Karatzas and Shreve, 1996, Proposition 2.13, p. 291) Let us sup-pose that the coefficients of the one-dimensional equation
dXt“bpt, Xtqdt`σpt, XtqdWt, satisfy the conditions
|bpt, xq ´bpt, yq| ďK|x´y|, (2.5)
|σpt, xq ´σpt, yq| ďhp|x´y|q, (2.6) for every 0ďtă 8 and xPR, y PR, where K is a positive constant and h :r0,8q Ñ r0,8q is a strictly increasing function withhp0q “0 and for allą0,
ż
p0,q
h´2puqdu“ 8. (2.7)
Then strong uniqueness holds for the equation (2.4).