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

105

## Homework 1 Bruno Dupire

### 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 xfollows 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 such that yt = f (xt ) is a martingale under

### 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

and

dxt = s ×(x0 + K *-xt ) dWt

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