**IEOR 4706 : Final **

金融数学代考 Consider a discrete-time binomial model with T = 2 and R = 1 (meaning that the interest rate is constant equal to 0), u = 2 and d = 1/2.

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### 1) Fix *K*2 *> K*1 *> *0. A collar option with maturity *T > *0, strikes *K*1 and *K*2, written on the underlying asset *S*, is an option with payoff at time *T *given by min(max(*S**T **, K*1)*, K*2)*. *

Assume that *S *does not deliver dividends and has no storage costs.

(a) Under no arbitrage condition, denote *CL**t*(*T, K*1*, K*2*, S*) at time *t **∈ *[0*, T*] for the price of the collar option. Find an expression of *CL**t*(*T, K*1*, K*2*, S*) in terms of *K*1*, K*2*, d*(*t, T*),*C**t*(*T, K*1*, S*) and *C**t*(*T, K*2*, S*).

(b) What purpose do you see for holding such an option ?

(c) Consider a discrete-time binomial model with *T *= 2 and *R *= 1 (meaning that the interest rate is constant equal to 0), *u *= 2 and *d *= 1*/*2. The initial risky asset price is *S*0 = 20. 金融数学代考

(c1) Are there arbitrage opportunity in this market ? Why ?

(c2) Give the price at time 0 of a collar option with strikes *K*1 = 8 and *K*2 = 20. Provide as well the replicating strategy.

### 2) Assume that there are *T *periods, and *r**t *= (*r**t*1*, . . . , r**tn*) is the vector of returns of *n *financial assets at time *t*. The Kelly criterion consists of solving the problem

(a) Use Taylor expansion of log for approximation, and derive an explicit formula for the portfolio *x *= (*x*1*, . . . , x**n*). Explain how this is applied for data.

(b) The data consists of the prices of Meta, Disney, Moderna and Netflix from 1/1/2022 to 11/30/2022. Taking *T *= 11 (one period corresponds to one month), compute the Kelly portfolio specified in (a). What is the annualized Sharpe ratio ?

(c) Compute the minimum variance portfolio of (b). What is the (in-sample) annualized Sharpe ratio ? Comment.

### 3) Consider the stochastic differential equation 金融数学代考

*dZ**t *= (2023 *− *log *Z**t*)*Z**t**dt *+ *Z**t**dB**t **, Z*0 = 1*. *

(a) Apply Itô’s formula to log *Z**t *, and get an expression for *Z**t *. Do you recognize log *Z**t *?

(b) Compute E*Z**t *and *V ar*(*Z**t*). What are the limits of E*Z**t *and *V ar*(*Z**t*) as *t **→ ∞*?

(c) What is the quadratic variation of *Z**t *?

4) Given a Black-Scholes financial market. Suppose *f*(*t, S**t*), a “price” at time t for a European type derivative with an expiration date *T*, is given as ,where *σ *is the volatility for the underlying asset, *r *is the risk-free interest rate, and *S**t *is the price of the underlying asset at time *t*.

(a) What is the payoff of this derivative at time *T *?

(b) Is it an arbitrage free price ? Why ?

(c) Suppose *T *= 1 year, and suppose that the underlying price today is 5 dollars and the interest rate is 0 percent. If your portfolio is to long one unit of this derivative and short *N*(*t*) shares of the underlying. Find *N*(*t*) in terms of *t *and *S**t *so that the portfolio is risk-free.

### 5) Given a Black-Scholes financial market. Let *r *be the risk free rate, *σ *be the volatility,*K *be the strike and *T *be the maturity. The payoff function is given by [(*S**T **− **K*)+] 2 . **金融数学代考**

(a) Compute explicitly the price formula *C**t*(*S**t **, r, σ, K, T*) for any *t **≤ **T*. What is the replicating strategy ?

(b) Set *r *= 0*.*02, *σ *= 0*.*15, *K *= 100 and *T *= 1. Write down the Feynman-Kac PDE for the price *v*(*t, x*) at time *t *given the underlying stock price is *x*. Compute numerically *v*(0*, x*) for *x **∈ {*95*, *96*, . . . , *104*, *105*}*, and compare with (a).

(c) As in (b), use the Monte Carlo method to compute *v*(0*, x*) for *x **∈ {*95*, *96*, . . . , *104*, *105*}*,and comment.

(d) Back to the standard Black-Scholes financial market, and the European option. The data consists of option trading for Apple on 10/19/2021. Plot the implied volatility sur-face, and what is the estimated implied volatility with strike 160 and maturity 1/7/2022 ?(This question should be treated separately to (a) – (c)).