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MATH 375 Stochastic modelling in insurance and finance

创建日期：2024-04-24 17:34:32

所属分类：数学mathematics

标签： MATH 375

Stochastic modelling in insurance and finance

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MATH 375
Stochastic modelling in insurance and finance
Time allowed: 1 hour and 30 minutes
INSTRUCTIONS TO CANDIDATES: Full marks will be given for complete
answers to all questions.
Paper Code MATH 375 Page 1 of 2 CONTINUED
1. Let α, β, γ, λ, x0, be given positive constants, and (W˜ (t), t ≥ 0) a standard
Brownian motion under the risk-neutral probability measure P˜. Let (x(t), t ≥ 0)
be the solution to the following equation: dx(t) = αdt+ βdW˜ (t), t ≥ 0,
x(0) = x0.
Consider the following interest rate model:
r(t) := γt+ λx(t), t ≥ 0.
i) Can the process (r(t), t ≥ 0) take a negative value for some t ≥ 0? Justify your
answer. [6 marks]
ii) Derive the price p(t, T ) of the zero-coupon bond at time t ∈ [0, T ]. [9 marks]
iii) Let α = 1, β =
√
3, γ = 1, λ = 1, x0 = 0.03. Consider a forward contract
on the zero-coupon bond of part (ii) with maturity T = 1, the delivery date of
which is T1 = 0.5 and the delivery price is K = 0.5. Find the value of this forward
contract at time t = 0 for the holder with a short position. [10 marks]
2. State the three types of recovery rules for the intensity based credit risk
models. If the intensity is constant, i.e. γ(t) = λ > 0 for all t ≥ 0, then what
is the expected value and the variance of default time τ under the risk-neutral
probability measure? [8 marks]
3. Consider a market of a bank account B(t) and a stock S(t), that satisfy
the equations: 
dB(t) = rB(t)dt, t ∈ [0, T ],
dS(t) = S(t)[µdt+ σdW (t)], t ∈ [0, T ],
B(0) = 1, S(0) = S0.
Here r, µ, σ, S0 are known positive constants, and (W (t), t ∈ [0, T ]) is a standard
Brownian motion. Let 0 < K1 < K2 be two given constants. Consider a contract
with the following terminal payoff:
X :=
S(T )
K1
I(K1Find the price X(t) at time t ∈ [0, T ] of this contract. [17 marks]
Paper Code MATH 375 Page 2 of 2 END

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