FNCE 435 – Empirical Finance Individual Assignment (Section I) 实证金融作业代写 (Important: This assignment is to be implemented on an individual basis. No sharing of material— including da...View details
Dynamic Asset Pricing
Homework 1 Bruno Dupire
动态资产定价代写 1) Trinomial pricing In this economy, a stock and a bond can be freely traded (bought or sold short). Initial dollar price of the stock is 100 and
1) Trinomial pricing 动态资产定价代写
In this economy, a stock and a bond can be freely traded (bought or sold short). Initial dollar price of the stock is 100 and only 3 prices are possible one year from now: 90,100 and 110, with probabilities (under the physical measure P) .20, .40 and .40 respectively. Initial and final dollar price of the bond is 1 (no interest rate).
1) Compute the expectation under P of the pay-off of a Call option with maturity one year and strike price 95
2) Is there a unique possible arbitrage-free price for this Call? Why?
3) Identify the set of probability measures equivalent to P under which the bond and stock dollar prices are martingales
4) What is the range of possible arbitrage-free prices for the Call? Provide a graphical
explanation as well.
5) If the Call trades at the price given by 1), what should you do?
6) Find a numeraire N, portfolio of bonds and stocks such that the prices of the bond and of the stock expressed in terms of N are martingales under P.
2）Move based historical volatility
(hint: delta hedge a Parabola maturity T at the crossing times)
b) How can a) be used to define an estimate of historical volatility? How can you handle price discontinuities?
3) Change of measure x follows the SDE
dxt = a(xt ) dt + b(xt ) dWt
where W is a standard Brownian motion under the probability measure P and a and b are well behaved (say, there are Lipschitz and bounded and b is bounded away from 0 from below).
a) Find a measure Q equivalent to P such that xt is a martingale under Q
b) Find a non constant function f such that yt = f (xt ) is a martingale under P
4) Symmetry in Local Volatility Model 动态资产定价代写
Assume dxt = b(x,t) dWt and define C by C(x,t,K,T) º E[(xT – K)+ xt = x]
1) For K,T fixed, write the (Backward) PDE that C satisfies in (x,t) with final boundary conditions
2) For x0 ,t0 fixed, write the (Forward) PDE that C satisfies in (K,T) with initial boundary conditions
3) For K*,T * fixed, we define f (x,t) º C(x0 ,t0 , x0 + K *-x,t0 + T *-t)
Find the PDE and boundary conditions that f satisfies
4) Compare with 1) and conclude that C(x0 ,t0 ,K*,T*) is the same under
dxt = sxtdWt
dxt = s ×(x0 + K *-xt ) dWt