动态资产定价代写 Dynamic Asset Pricing代写 三项式定价代写

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

dx_t = a(x_t ) dt + b(x_t ) dW_t 

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 x_t is a martingale under Q 

b) Find a non constant function f such that y_t = f (x_t ) is a martingale under P 

4) Symmetry in Local Volatility Model 动态资产定价代写

Assume dx_t = b(x,t) dW_t and define C by C(x,t,K,T) º E[(x_T – K)⁺ x_t = 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

dx_t = sx_tdW_t 

and

dx_t = s ×(x₀ + K *-x_t ) dW_t

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