The Mathematics of Arbitrage

3.3 The Binomial and the Trinomial Model 47 X̂ 1 (x)=−V′ ( ŷ(x) dQ dP ) =−V′(U′(x))c α^1 − 1 U ( dQ dP )α−^11 , =xc−V^1 ( dQ dP ...

48 3 Utility Maximisation on Finite Probability Spaces Coming back to the case of generalα∈]−∞,1[{ 0 },weobtainfrom (3.37) that ...

3.3 The Binomial and the Trinomial Model 49 In the special caseα=^12 ,β=−1, this yields ̂h=^2 xν σ^2 +O ( ∆t 12 ) . We observe f ...

50 3 Utility Maximisation on Finite Probability Spaces For the optimal investment, we obtain X̂ 1 (x)= { x+̂hu, ̃forω=g, x+̂hd, ...

3.3 The Binomial and the Trinomial Model 51 in the case of logarithmic or exponential utility) for the binomial model is also op ...

52 3 Utility Maximisation on Finite Probability Spaces Applying formula (3.29) from Theorem 3.2.1 we find, forω=n,that dQ̂(y) dP ...

3.3 The Binomial and the Trinomial Model 53 the increments are assumed to be independent. To be formal, let (εt)Nt=1be i.i.d. Be ...

54 3 Utility Maximisation on Finite Probability Spaces investment as theFN− 1 -measurable function̂hNdefined bŷhN= ̂h SN− 1 .Ec ...

3.3 The Binomial and the Trinomial Model 55 Let us have a closer look at the constant̂k: the leading term (1−σ 2 β)ν is proporti ...

56 3 Utility Maximisation on Finite Probability Spaces so wUN=exp ( − βν^2 T 2 ασ^2 ) +o(1) =exp ( ν^2 T (1−α)2σ^2 ) +o(1). In t ...

4 Bachelier and Black-Scholes 4.1 Introduction to Continuous Time Models In this chapter we illustrate the theory developed in t ...

58 4 Bachelier and Black-Scholes elementary considerations on the binomial model above (compare Corollary 2.2.12). Theorem 4.2.1 ...

4.3 Bachelier’s Model 59 It is straightforward to derive from (4.4) an “option pricing formula” by calculating the integral in ( ...

60 4 Bachelier and Black-Scholes of the value of the option. By Itˆo’s formula (see, e.g., [RY 91]) dC(St,T−t)= ∂C ∂S dSt+ ( ∂C ...

4.4 The Black-Scholes Model 61 The parameterrmodels the “riskless rate of interest”, while the parameter μmodels the average inc ...

62 4 Bachelier and Black-Scholes After an elementary calculation (see, e.g., [LL 96]) this yields the famous Black-Scholes formu ...

4.4 The Black-Scholes Model 63 always remains positive, while Brownian motion may also attain negative values. This fact has str ...

64 4 Bachelier and Black-Scholes Expandingφ(x) into a Taylor series around zero and usingφ′′(0) =−√^12 π we get the asymptotic e ...

4.4 The Black-Scholes Model 65 The bottom line of these considerations on the role ofris: when we as- sumed thatr= 0 in the abov ...

66 4 Bachelier and Black-Scholes or the “Gamma” of the option price (as a function ofS)attimet, normalised by σ 2 2 S (^2) (in t ...