# 金融代写|期权定价理论代写Option Pricing Theory代考|MATH1203

## 金融代写|期权定价理论代写Option Pricing Theory代考|Summary and Conclusions

In this chapter we first established the importance of transaction costs in the pricing of options by showing bid-ask spread data from a long series of S\&P 500 European options. These data imply that the equilibrium prices in the option market are only observable within very wide margins of error, especially for OTM options. Yet it is precisely these options that are almost always used in extracting the forward-looking option-implied volatilities. We have also surveyed the few attempts to deal with these imprecisions in the data by explicitly modeling the determination of the spread through extensions of the no arbitrage methodology, which have produced trivial results for realistic trading conditions even in markets that are otherwise complete.

We also surveyed the portfolio selection literature in the presence of proportional transaction costs and its key finding, the existence of an NT region in which the investor refrains from trading. The most powerful results are in discrete time, since they offer greater flexibility in handling important features of the problem such as the length of the investment horizon and the accuracy of the approximations compared to the more elegant continuous time solutions. Nonetheless, a simultaneous equilibrium in the underlying and the option market in order to derive SD option pricing bounds by applying the LP mcthod of the prcvious chaptcr did not succeed in achieving meaningful results under general trading conditions and proportional transaction costs. In the next two chapters we show that an alternative formulation of the $\mathrm{SD}$ approach can, indeed, produce nontrivial option pricing results, which have been used to demonstrate some highly surprising empirical findings.

## 金融代写|期权定价理论代写Option Pricing Theory代考|European Index Option Bounds

The derivation of the bounds follows an extension of the method applied in the first section of Chap. 2, which in turn requires the value function $V\left(x_t, y_t, t\right)$ of an investor or trader holding only a risky asset (the index or stock) and a riskless bond. We adopt the same notation as in Sect. $3.2$ of the previous chapter, with some modifications in order to recognize the role of dividends that will be important in the case of American options. At date $t$, the cum dividend stock price is $\left(1+\gamma_t\right) S_t$, the cash dividend is $\gamma_t S_t$, and the ex dividend stock price is $S_t$, where the dividend yield parameters $\left{\gamma_t\right}_{t=1, \ldots, T^*}$ are assumed to satisfy the condition $0 \leq \gamma_t<1$ and be deterministic and known to the trader at time zero. 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. The trader enters the market at date $t$ with dollar holdings $x_t$ in the bond account and $y_t / S_t$ ex dividend shares of stock. The endowments are stated net of any dividend payable on the stock at time $t .{ }^1$

As before, the trader 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 $v_t$ is constrained to be measurable with respect to the information up to date $t$. The bond and stock account dynamics are somewhat different from Eqs. (3.8) and (3.9) of the previous chapter: $$\begin{gathered} x_{t+1}=\left{x_t-v_t-\max \left[k_1 v_t,-k_2 v_t\right]\right} R+\left(y_t+v_t\right) \frac{\gamma_{t+1} S_{t+1}}{S_t}, \quad t \leq T^{\prime}-1 \ y_{t+1}=\left(y_t+v_t\right) \frac{S_{t+1}}{S_t}, \quad t \leq T^{\prime}-1 . \end{gathered}$$
We adopt the same assumptions as in Sect. $3.2$ of Chap. 3 about the liquidation of the portfolio at the terminal date and the recursive maximization of the expectation of the increasing and concave terminal utility function $E\left[u\left(x_{T^{\prime}}+y_{T^{\prime}}-\max \left[-k_1 y_{T^{\prime}}, k_2 y_{T^{\prime}}\right]\right) \mid S_t\right]$.

# 期权理论代考

## 金融代写|期权定价理论代写Option Pricing Theory代考|European Index Option Bounds

\begin{gathered} x_{t+1}=\left{x_t-v_t-\max \left[k_1 v_t,-k_2 v_t\right]\right} R+\left(y_t+v_t\right) \frac{\ gamma_{t+1} S_{t+1}}{S_t}, \quad t \leq T^{\prime}-1 \ y_{t+1}=\left(y_t+v_t\right) \frac{ S_{t+1}}{S_t}, \quad t \leq T^{\prime}-1 。\end{聚集}\begin{gathered} x_{t+1}=\left{x_t-v_t-\max \left[k_1 v_t,-k_2 v_t\right]\right} R+\left(y_t+v_t\right) \frac{\ gamma_{t+1} S_{t+1}}{S_t}, \quad t \leq T^{\prime}-1 \ y_{t+1}=\left(y_t+v_t\right) \frac{ S_{t+1}}{S_t}, \quad t \leq T^{\prime}-1 。\end{聚集}

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