# 金融代写|期权定价理论代写Option Pricing Theory代考|MATH451

## 金融代写|期权定价理论代写Option Pricing Theory代考|Super Replication and the Failure of No Arbitrage

Super replication refers to an alternative and more general approach to deal with the binomial model in the presence of transaction costs, originally introduced by Bensaid, Lesne, Pagès and Scheinkman (1992) for European options and extended to American options by Perrakis and Lefoll $(2000,2004)$. This approach does not necessarily replicate the option at every node of the binomial tree but derives instead perfect hedging policies for an intermediary that issues the options that are specific to the type of settlement of the option upon maturity. Nonetheless, for physical delivery options such as options on equities it was shown that super replication coincided with replication at every node. ${ }^3$

In super replication models transaction costs are represented by the convex function $\phi(y)=(1+k) y$ if $y \geq 0, \phi(y)=(1-k) y$ if $y \leq 0$. At any period $j \in[0, n]$ of the $n$ time periods to option expiration the hedging portfolio must contain enough cash in the riskless asset to cover the subsequent position, including the cost of rebalancing the portfolio. Let $\omega_j$ denote a particular path from 0 to $j$, that is, a particular sequence of up and down moves, $S_j\left(\omega_j\right)$ the corresponding stock price and $\left(N_j, B_j\right)$ the optimal hedging portfolio; although the latter may depend on $\omega_j$, this dependence can be shown not to exist. For every $\omega_j$ there are two successor paths $\omega_{j+1}$ corresponding to the two values of $S_{j+1}, u S_j$ and $d S_j$. For perfect hedging, therefore, we must have $R B_j \geq B_{j+1}+\phi\left(N_{j+1}-N_j\right) S_{j+1}$, and the optimal path-dependent portfolios are found by evaluating the following function for each path.

The function $Q_j$ represents the minimal cash needed to hedge the option and cover the transactions costs. It can be shown that the program (3.7) yields an efficient solution for all types of settlement for European options. The case of American options, in which early exercise must be included in all nodes, is considerably more complex because of nonconvexities in the objective function. Nonetheless, an efficient algorithm also exists for both American calls with dividends and American physical delivery puts.

## 金融代写|期权定价理论代写Option Pricing Theory代考|Asset Allocation in Discrete Time

Consider a class of traders who invest only in a risky asset and a riskless bond, a condition that will be assumed throughout this section. Each trader makes sequential investment decisions in the primary assets at the discrete trading dates $t=0,1, \ldots, T^{\prime}$, where $T^{\prime}$ is the terminal date and is finite. A bond with price one at the initial date has price $R, R>1$ at the end of the first trading period, where $R$ is a constant; without loss of generality bond trades are assumed to be traded without frictions. On the other hand, the risky asset’s trades incur proportional transaction costs charged to the bond account. At each date $t$, the trader pays $\left(1+k_1\right) S_t$ out of the bond account to purchase one share of stock and is credited $\left(1-k_2\right) S_t$ in the bond account to sell (or sell short) one share of the risky asset. It is assumed that there are no dividends within the option’s time to expiration $T<T^{\prime} .4$

The trader enters the market at date $t$ with dollar holdings $x_t$ in the bond account and $y_t / S_t$ shares of stock and increases (or decreases) the dollar holdings in the stock account from $y_t$ to $y_t{ }^{\prime}=y_t+v_t$ by decreasing (or increasing) the bond account from $x_t$ to $x_t{ }^{\prime}=x_t-v_t-\max \left[k_1 v_t,-k_2 v_t\right]$. The decision variable is constrained to be measurable with respect to the information up to date $t$. The bond account dynamics $v_t$ are $$x_{t+1}=\left{x_t-v_t-\max \left[k_1 v_t,-k_2 v_t\right]\right} R, \quad t \leq T^{\prime}-1,$$
and the stock account dynamics are
$$y_{t+1}=\left(y_t+v_t\right) \frac{S_{t+1}}{S_t}, \quad t \leq T^{\prime}-1 .$$

# 期权理论代考

## 金融代写|期权定价理论代写Option Pricing Theory代考|Asset Allocation in Discrete Time

$$y_{t+1}=\left(y_t+v_t\right) \frac{S_{t+1}}{S_t}, \quad t \leq T^{\prime}-1 .$$

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