Asset Pricing in Continuous Time
MATH0085
MSc Examination
2020
1. Let (Ω , F, (Ft ), P) be a filtered probability space where (Ft )t≥0 is generated by the Brow- nian motion (Wt )t≥0 and P is the real-world probability measure.
Consider the pricing kernel model (πt )t≥0 defined by
where α and β are constants.
(a) [U] Assume and show that
[10 marks]
(b) [SS] Derive the interest rate (rt )t≥0 and the market price of risk (λt )t≥0 processes associated with the pricing kernel (2). What must hold for β such that the market price of risk is financially meaningful? [10 marks]
(c) [U] Denote the price process of a discount bond with maturity T by (PtT)0≤t≤T. As- suming the pricing kernel model (2) it follows that
where . What can be concluded for the short-rate of interest that underlies the bond price? Explain your answer. [5 marks]
2. Let (Ω , F, (Ft)t≥0, Q) be a probability space equipped with the filtration generated by (WtQ)t≥0 where the risk-neutral probability measure Q is equivalent to P, and (WtQ) is a Q-Brownian motion.
Let 1{A} denote the indicator function of an event A. Let (rt) and (σt) be (Ft)-adapted and positive processes. Consider the following model for the price process (St ) of an asset:
where S0 is a constant.
(a)[SS] Derive the stochastic differential equation for the process (St ). [5 marks]
(b)[SS] Construct the dynamics of the process (St ) under the “real” probability mea- sure P assuming a market price of risk (λt )t≥0 is given. Apply Girsanov’s Theorem and the Sharpe Ratio at the appropriate steps in your derivation (write down explicitly the theorem). Here, you may assume that Novikov’s Condition is satisfied. [10 marks]
(c)[SS] Let rt and σt be deterministic functions. Calculate the price Ht at time t of an option with payoff function HT = 1{ST < x} at the maturity date T, where x > 0 and 0 ≤ t ≤ T. [10 marks]
3. Consider a call option with strike price K and expiry date T written on an underlying asset with price process (St )0≤t≤T . The call option price at t = 0 is denoted C(0, T, S0 , K), which is twice differentiable in K. Let q(0, T, S0 , y) denote the (transition) density given by
where Q is a risk-neutral measure. Furthermore, we assume a constant interest rate r.
(a)[SS] Show that
[6 marks]
Consider dSt = rSt dt + σ(t, St )StdWt , where (Wt )t≥0 is a Q-Brownian motion. The associated Kolmogorov forward equation is given by
where q(t,T, x, y) is given by Eq. (4) and limt→T q(t,T, x, y) = δ(y).
(b)[SS] Show that
[7 marks]
(c)[SS] By use of the results in (a) and (b),show that
What integration assumptions need to hold at infinity? [12 marks]
4. Let (Ω , F, (Ft), P) be a filtered probability space where (Ft) is generated by a Brownian motion (Wt)0≤t. Let the money market process (Bt)0≤t satisfy dBt = rBtdt where r ≠ 0 is a non-zero constant, and let the asset price process (St)0≤t be given by
St = σBtWt , (8)
where σ is the constant volatility parameter. Furthermore, we introduce a derivative with maturity T > 0 and price Ct at time t given by Ct = C(t, St ) for 0 ≤ t < T, and CT = C(ST ) for t = T. We assume that the function C(t, x) is once continuously differentiable in t and twice continuously differentiable in x, fort ∈ [0, T) and x ∈ R.
(a)[SS] Show that
dSt = rSt dt + σBtdWt. (9)
[5 marks]
(b)[SS] Show that the SDE satisfied by (Ct )0≤t
[5 marks]
(c)[SS] Let (φt )0≤t≤t
Vt = φtSt + ψtBt ,
where VT = C(T, ST ). Show that (φt ) and (ψt ) must, respectively, satisfy
and derive the PDE satisfied by the derivative price. Give the terminal boundary condition of the PDE.
[15 marks]
5. Let (Ω , F, (Ft), P) be a filtered probability space where (Ft)t≥0 is generated by the Brow- nian motion (Wt )t≥0 and P is the real-world probability measure. Denote by (PtT)0≤t≤T the price process of a discount bond. Consider
where (ΣtT )0≤t≤T is the volatility of the discount bond price process and (λt)t≥0 is the market price of risk. Moreover, the pricing kernel process (πt )t≥0 satisfies
where (rt )t≥0 is the short-rate of interest process.
(a) [SS] By use of Equations (10) and (11) show that
[10 marks]
(b)[S] By closely inspecting the SDE (12), explain how (ΣtT), (λt) and (rt) can be respec- tively interpreted as the bond price volatility, the market price of risk and the short-rate of interest. [6 marks]
(c) [SS] Let the (Ft )-adapted process (At )t≥0 bedefined by
dAt = vt dWt , (13)
where (vt )t≥0 is a stochastic process. Consider a discount bond price process of the form
where P0t, for 0 ≤ t ≤ T, and b(t) are deterministic functions, and where T is the bond maturity. Derive the expression for the bond volatility process (ΣtT)0≤t≤T . [9 marks]