Reference no: EM132242740

Introduction to Financial Mathematics Assignment -

Let W = (W_t)_t≥0 be a standard Brownian motion.

1. In this exercise we derive the Black-Scholes PDE in another, equivalent manner. We are over the time horizon [0, T] with a stock whose price S_t is given by a geometric Brownian motion,

dS_t = μS_tdt + σ S_tW_t, S₀ > 0,  (1)

where μ ∈ R and σ > 0 are given constants, and a riskless bond (or money market) whose value (or price) B_t is given by

dB_t = rB_t dt, B₀ > 0. (2)

The goal is to find the value C_t of the European call option on S with strike K whose terminal payoff is given by C_T = (S_T - K)⁺. We make various assumptions as listed in the notes. Among these are that there is no arbitrage and that the call price can be written as a function V of time and the stock price, that is, C_t = V(t, S_t) for all t ∈ [0, T]. Thus finding C_t is equivalent to finding the function V.

Consider a portfolio of a_t shares of stock and b_t units of the bond. The value Π_t at time t of this portfolio is

Π_t = a_tS_t + b_tB_t. (3)

The goal is to trade continuously between the stock and the bond, i.e. vary a_t and b_t over time, in such a way that we guarantee that the value of the portfolio at T is exactly equal to the payoff of the call option. That is,

Π_T = (S_T - K)⁺. (4)

Such a portfolio is called a replicating portfolio. The option has no cash flow until terminal time T, and so the portfolio must be continuously rebalanced without cash coming into or flowing out of it until T. This means we require the portfolio to be self-financing in the sense that any change in the value of the portfolio must equal the profit or loss due to changes in the price of the stock or the price of the bond. This requirement translates to

dΠ_t = a_t dS_t + b_t dB_t (5)

Because the self-financed replicating portfolio is equal to the value of the option at T, the assumption of no arbitrage implies that its value is equal to the value of the call option at all previous times t ∈ [0, T). In particular, the initial amount of cash required to finance the portfolio at time zero is the option's price or premium, C₀.

(a) Replace dS_t and dB_t in (5) with their formulas to arrive at the Ito process form for dΠ_t.

(b) Apply Ito's formula to C_t = V(t, S_t) to write dC_t as an Ito process.

(c) The idea of the replicating portfolio Π_t is that it will equal the value of the call at all times, i.e. Π_t = C_t for all t ∈ [0, T]. This implies that all changes must be the same, or dΠ_t = dC_t, which in turn implies that the respective terms in front of dt and dW_t in dΠ_t and dC_t must be the same. Equate the dW_t terms in dΠ_t and dC_t to arrive at an expression for the number of shares of stock a_t.

(d) Now equate the dt terms in dΠ_t and dC_t to obtain an expression for the number of bonds held b_t. (Use your expression for a_t from the last step.)

(e) Because C_t = Π_t for all t, we have that V(t, S_t) = a_t dS_t + b_t dB_t. Insert your expressions for a_t and b_t and obtain an expression for V(t, S_t) in terms of its partial derivatives (evaluated at (t, S_t)), S_t and σ. Note that the B_t term will cancel. Also note that this is an equality of random variables or stochastic processes.

(f) The equality you obtained in the previous step must hold for all t and S_t. The range of S_t is all positive reals, and thus it must hold for arbitrary x ∈ R⁺. Replace S_t by an arbitrary x and obtain the Black-Scholes PDE. Note that this is an equality involving functions.

2. Let's assume a different model for the stock price, the so-called Bachelier model or Brownian motion with drift:

S_t = S₀ + μ d_t + σ dW_t. (6)

Note the difference between this and the GBM. Retrace the steps in exercise 1 to find the "Black-Scholes PDE" for this alternative model.

3. In this exercise we will uncover the celebrated Put-Call Parity formula for European options. We assume the Black-Scholes setting: a horizon [0, T] and a stock and a bond with prices given by (1) and (2). We have a European call option with price C_t and a European put option with price P_t. Both options have strike K and expire at time T. We will consider two portfolios at time t ∈ [0, T]: Portfolio A and Portfolio B.

(a) Portfolio A consists of (long positions in) the European call option and the present value of K dollars to be received at time T. Find an expression for the time T value of this portfolio.

(b) Portfolio B consists of (long positions in) the European put option and the stock. Show that the time T value of this portfolio is equal to the time T value of Portfolio A.

(c) Under the assumption of no arbitrage, argue that the values of Portfolio A and Portfolio B must be equal at all times t ∈ [0, T]. Obtain an expression that relates C_t, P_t, S_t and the present value (at time t) of K dollars to be received at time T.

(d) What is the significance of this formula? (Hint: Suppose you know the price C_t of the European call. What do you know about the price P_t of the European put?)

Attachment:- Assignment Files.rar