Market models

Market models

Wolfgang Runggaldier University of Padova, Italy

www.math.unipd.it/runggaldier

Market models (discrete time ∆ = 1)

• A locally riskless asset (money market account) B_n+1 = B_n(1 + r_n) ↔ B_n+1 − B_n

B_n = r_n (r_n known at n)

• Risky assets (for the moment just one)

S_n+1 = S_n(1 + a_n+1) ↔ S_n+1 − S_n

S_n = a_n+1 l (a_n+1 unknown at n) S_n+1 = S_nξ_n+1

• Example: ξ_n =  u with probability p

d with probability 1 − p

Market models (transition to continuous time)

• Locally riskless asset

B_t+∆− B_t = B_tr_t∆ −→ dB_t = B_tr_tdt (cont. compounding)

• Risky asset (a_t+∆ = a_t∆ + σ_tξ_t+∆ with ξ_t+∆ ∼ N (0, ∆)) S_t+∆ = S_t (1 + a_t∆ + σ_tξ_t+∆)

Let w_t be a process s.t.

∆w_t := w_t+∆ − w_t ∼ N (0, ∆) (Wiener process) S_t+∆ = S_t (1 + a_t∆ + σ_tξ_t+∆)

↓

dS_t = S_t [a_tdt + σ_tdw_t]

Price processes with continuous trajectories

dS_t = S_t [a_tdt + σ_tdw_t]

(∆w_t := w_t+∆ − w_t ∼ N (0, ∆) −→ dw_t ∼ √ dt

→



















d log S_t = a_t − ¹₂σ_t²
dt + σ_tdw_t S_t+∆ = S_t exp

hR t+∆

t a_s − ¹₂σ_s²
ds + R_t^t+∆ σ_sdw_s i

= S_tξ_t+∆

→ All trajectories of S_t are continuous functions of t

Price processes with discontinuous trajectories

• Let T_n (0 = T₀ < T₁ < · · · ) be a sequence of random times where a certain event happens

• Let N_t = n if t ∈ [T_n, T_n+1) ⇔ N_t = P

n≥1 1_{Tn≤t}

→ N_t is a counting process and dN_t ∈ {0, 1}.

• Let S_t change only at times T_n with return γ_n > −1 at T_n S_Tn − S_T⁻

n

S_T⁻

n

= γ_n ↔ S_Tn = S_T⁻

n (1 + γ_n) ↔ dS_t = S_t⁻γ_tdN_t

⇒ S_t = S₀ QN_t

n=1(1 + γ_Tn) = S₀ exp

hPN_t

n=1 log(1 + γ_Tn) i

= S₀ exp

hR t

0 log(1 + γ_t)dN_t i

Combining the two (jump diffusion models)

dS_t = S_t⁻ [a_tdt + σ_tdw_t + γ_tdN_t] implies

S_t+∆ = S_t exp

"

Z t+∆

t



a_s − 1 2σ_s²



ds +

Z t+∆

t

σ_sdw_s

#

N_t+∆

Y

n=N_t

(1 + γ_n)

= S_t exp

h Z t+∆

t



a_s − 1 2σ_s²



ds +

Z t+∆

t

σ_sdw_s +

Z t+∆

t

log(1 + γ_t) dN_t i

More risky assets

• A certain number K of risky assets

dS_tⁱ = S_tⁱaⁱ_tdt + S_tⁱ

M

X

j=1

σ_t^i,jdw_t^j ; i = 1, · · · , K

(dS_t = (diag S_t) A_tdt + (diag S_t) Σ_tdw_t)

with w_t = [w_t¹, · · · , w_t^M]⁰ an M − Wiener process on a filtered probability space (Ω, F , F_t, P ), (F_t = F_t^w).

→ In the classical Black-Scholes model aⁱ_t and σ_t^i,j are deterministic ⇒ S_t : a lognormal process.

• The coefficients may also be stochastic processes that are either:

i) adapted to F_t^w, or

ii) Markov modulated (regime switching models)

dS_t = (diag S_t) A_t(Z_t)dt + (diag S_t) Σ_t(Z_t)dw_t

→ Z_t an exogenous multivariate Markov factor process (volume of trade, level of interest rates or, generically, the “state of the economy”).

→ Z_t may be directly observable or not.

• Price trajectories may exhibit a jumping behaviour, then

dS_t = (diag S_t) A_tdt + (diag S_t) Σ_tdw_t + (diagS_t−)Ψ_t−dN_t with N_t = (N_t¹, · · · , N_t^H)⁰ a Poisson jump process and Ψ^i,j_t > −1 (Jump-diffusion models).

→ More general driving processes are possible (Levy, fractional BM).

• On small time scales prices do not follow continuous trajectories, but rather piecewise constant ones with jumps at random points in time.

→ May be modeled by continuous trajectories sampled at the jumps of a Poisson process.