In the attachment.
h re Question 1: You were given the value of u and d in the binomial tree. This exercise will walk you through two methods for determining u and d. As you know, the Black-Scholes model assumes that the price of a risky asset (the stock SBS , wit cur nt price S0) follows a lognormal distribution. Specifically, in the Black-Scholes model, for any t > 0, ln ( SBSt ) is normally distributed under the risk-neutral measure Q, with mean EQ [ ln ( SBSt )] = ln (S0) + ( r − 12σ 2 ) t and variance VarQ ( ln ( SBSt )) = σ2t: ln ( SBSt S0 ) Q∼ N (( r − 1 2 σ2 ) t, σ2t ) , (1) or, equivalently, SBSt S0 Q∼ LogNormal ( µ̃, σ̃2 ) , (2) where µ̃ := ( r − 12σ 2 ) t, σ̃2 := σ2t, r > 0 is the risk-free rate, and σ > 0 is assumed to be known (stock volatility). You are given the following formulas for the lognormal distribution: EQ [ SBSt S0 ] = eµ̃+ σ̃2 2 = ert and VarQ ( SBSt S0 ) = e2µ̃+σ̃ 2 (eσ̃ 2 − 1) = e2rt(eσ2t − 1). The idea behind the determination of the tree parameters u and d is to choose them so that the mean and variance (under Q) of the stock price on the tree approximately match the mean and variance of the lognormal distribution above. There is some flexibility in this procedure. The two classical methods that we will explore below are easy to implement: the first one assumes ud = 1 (Method 1) and the second one assumes qu = 12 (Method 2). (a) General setup: Consider a multi-period binomial model, with each period of length h (in fractions of a year), and assume that the model is arbitrage-free. The known time-0 price of the risky asset (stock) is S0. At time h (end of the first period), the stock price (Sh) is a random variable that can take the values uS0 or dS0, for some u and d such that u > d > 0. The risk-free rate of interest is r > 0 per year, compounded continuously. More generally, for each n ≥ 1, the stock price at the end of the nth period is given by Snh = ZS(n−1)h, where Z is a random variable that can take the values u or d ∈ (0, u). (i) Compute the risk-neutral mean and variance (i.e., under Q) of the random variable Y1 := Sh S0 in terms of u, d and rh. 2 (ii) (3 points) Let T be a positive integer and STh be the stock price at the end of the T th period in the binomial model. Show that EQ[STh] = EQ[SBSTh ] always holds. In other words, the means of the binomial model and the Black Scholes model always match. Hint: The following fact is useful: SThS0 = ∏T n=1 Snh S(n−1)h and the jump random variables Snh/S(n−1)h, n = 1, . . . , T , are iid (equal to Z). (iii) (3 points) Suppose that we keep t = Th fixed, and let h ↓ 0 (at the same time T → ∞). Intuitively explain why STh is approximately lognormally distributed. Hint: You may use the Central Limit Theorem, which says that the sum of many iid random variables is approximately normally distributed. (b) Method 1: (12 points) In the first method, we assume ud = 1. The choice of u and d are given by u = eσ √ h and d = e−σ √ h. Our goal is to show that this formulation approximately gives us the Black-Scholes model. (i) (2 points) Derive an expression for qu := Q [Sh = uS0] = Q [Y1 = u], as a function of the time step h. (ii) (3 points) Show that qu → 1/2 as h ↓ 0. (iii) (3 points) Show that lim h↓0 VarQ(ln(Sh/S0)) VarQ(ln(SBSh /S0)) = 1. (iv) (2 points) Based on (iii), show that ln(STh) has asymptotically the same variance as ln(SBSTh ), as h ↓ 0. Remark: As a consequence, STh has asymptotically the same distribution as SBSTh , since they are both lognormally distributed with the same mean and the same σ-parameter. (v) (2 points) Given the assumption of no-arbitrage, what condition(s) must the model parameter σ > 0 satisfy, for a given h? What if h ↓ 0? (c) Method 2: (10 points) Consider the same one-period binomial model as in (a). Instead of assuming that u and d are chosen such that ud = 1 as in Method 1, assume now that u and d are chosen such that qu = Q [Sh = uS0] = 12 , that is, the probability of an up-jump is 12 under the risk-neutral measure. (i) (2 points) Given the assumption that qu = 12 , show that there exists δ > 0 such that u = e rh + δ and d = erh − δ. 3 (ii) (3 points) Determine δ, u and d so that the risk-neutral variance (i.e., under Q) of the random variable Y1 matches the risk-neutral variance of SBSh /S0. (ii) (3 points) Show that EQ[(STh)2] = EQ[(SBSTh )2] for any positive integer T . Remark: Similarly to Method 1, because of the Central Limit Theorem and the matching moments, STh has asymptotically the same distribution as SBSTh . (iii) (2 points) In this model, does the assumption of no arbitrage imply any restriction on the choice of the model parameter σ > 0, for a given h? Question 2: [30 points] Consider two standard one-dimensional Brownian motions {W Pt }t≥0 and {V Pt }t≥0 on a filtered space space (Ω, {Ft}t≥0,F ,P). Suppose that the filtration {Ft}t≥0 is a filtration for the two Brownian motions and that the two Brownian motions {W Pt }t≥0 and {V Pt }t≥0 have a correlation of ρ > 0. Consider a market model consisting of a risky asset whose value at time t is St and a bank account worth ert at time t. The risky asset does not pay dividends. Assume that the stochastic process {St}t on the filtered space (Ω, {Ft}t≥0,F ,P) satisfies S0 > 0 and has the following stochastic differential equation: dSt = αStdt+ σStdW P t , where α, σ ∈ R+. A newly designed financial contract (derivative) pays off at maturity an amount that depends on the value of the risky asset, as well as the level of inflation in the market. Assume that inflation is modeled by a stochastic process {It}t on the same space that satisfies I0 > 0 and has the following stochastic differential equation: dIt = βItdt+ ηItdV P t , where β, η ∈ R+. At time T > 0 (the maturity of the contract), the derivative’s payoff is given by ΦT = max ( ST IT e−aT , S0 I0 egT ) , where a ≥ 0 and g > 0 are given constants. This contract guarantees a minimum payoff of S0I0 e gT at maturity. Define the stochastic process {Zt}t by Zt = St It , for all t ≥ 0. (a) (5 points) Derive an expression for the dynamics of the stochastic process {Zt}t under the probability measure P (the stochastic differential equation that {Zt}t satisfies under P). (b) (6 points) For a given T > 0, derive an expression for the expected value and the variance of the random variable ZT under the probability measure P, assuming that the Brownian motions {W Pt }t≥0 and {V Pt }t≥0 are independent. 4 (c) (7 points) For a given T > 0, derive an expression for the expected payoff of this derivative at time T under the probability measure P, assuming that the Brownian motions {W Pt }t≥0 and {V Pt }t≥0 are independent. (d) (7 points) Now, instead of having a stochastic inflation rate, the inflation rate is assumed constant over the lifetime of this derivative contract, so that It = I0 > 0 for t ∈ [0, T ]. Consider a contract Ψ that has the following payoff at maturity: ΨT = max (√ ST I0 e−aT , S0 I0 egT ) . Suppose also that this market model is a Black-Scholes model. Under the risk-neutral probability measure Q, the risky asset’s price follows the stochastic differential equation dSt = rStdt+ σStdW Q t , where {WQt }t≥0 is a standard Brownian motion under Q, and r, σ > 0. Derive an expression for Ψ0, the time-0 price of this financial derivative, as a function of S0, I0, a, g, r, σ, and T . Hint: Show that Ψ0 can be written in the form A×N (d1) +B ×N (d2) , for some functions A, B, d1, and d2 of the parameters S0, I0, a, g, r, σ, and T , where N denotes the standard normal CDF. (e) (5 points) Define the Vega of a security’s price, and explain what it measures. Compute the Vega of Ψ0, the time-0 price of the derivative above in (d). 5 Formula Sheet Black-Scholes Partial Differential Equation for a Derivative The market consists of a dividend-paying stock with price process {St}t and continuous dividend yield δ ≥ 0, and a risk-free bank account with price process {Bt}t, where Bt = ert for each t ≥ 0, and r > 0 is the continuously compounded interest rate. The price V (t, St) at time t of a European derivative on the stock, with payoff XT at time T , is said to satisfy the Black-Scholes PDE if{ Vt + (r − δ)StVS + 12σ 2S2t VSS = rV (t, St), with boundary condition V (T, ST ) = XT , where VS = ∂V∂St (t, St), Vt = ∂V ∂t (t, St), and VSS = ∂2V ∂S2t (t, St). Call and Put Prices in the Black-Scholes framework The Black-Scholes formula for the price at time zero of a European call option on a non-dividend-paying stock (i.e., δ = 0) is given by c(0, S0,K, T ) = S0N(d1)−Ke−rTN(d2), where N is the standard normal cumulative distribution function, d1 = ln(S0/K) + (r + σ 2/2)T σ √ T and d2 = d1 − σ √ T . The Black-Scholes formula for the price at time zero of a European put option on a non-dividend-paying stock is given by p(0, S0,K, T ) = Ke −rTN(−d2)− S0N(−d1). The following equation might be useful: S0N ′(d1) = Ke −rTN ′(d2), where N ′(x) = ddxN(x) is the standard normal density function. Some Results on Normal and Lognormal Distributions Let X be a Gaussian (normal) random variable with mean m and variance σ2 (this distribution is denoted by N