这个作业是根据数学财务背景来完成期权投资组合的优化和评估

MSc in Financial Mathematics, FM50

第1部分:数学和财务背景
在论文的第一部分,学生应描述在论文的后半部分使用的相关技术和理论。这部分
还应提供有关文献的讨论,并提供适当的参考资料。
关于冷漠定价的经典参考文献包括连续时间的[3]。
财务模型和精算保费计算的[1]。较新的参考文献可以在集合中找到[2]。进一步参考和扩展
流动性差的市场模型可以在[4]中找到。
学习书籍和发表的期刊文章是该课程的重要组成部分
项目。互联网上也有各种来源(例如arxiv.org,
ssrn.com,repec.org。 。 。 ),但应注意的有效性
在未经同行评审的工作中发现的结果。
1个
第2部分:期权投资组合的优化和评估
这部分计算写在上面的欧式期权的无差异价格
标普500指数。套期工具将包括现金以及看涨期权和
将期权置于相同的指数和到期日。所有期权的到期日都是固定的
截至2017年12月15日。以下计算中使用的报价为
在单独的csv文件中给出。它们是在2017年11月15日下午4点获得的。
买入价和卖出价均为一种选择,并以美元报价。数量
可以用最佳买价和卖价来表示的“合约”数量。
一份合约为100张期权。例如,如果对于给定的选项,
10张合约以2,460USD的最高买入价,投资者卖出1张
合约将获得246,000美元。
假设借贷现金的无风险利率
每年为0.043。
基础模拟
投资者认为指数的对数收益log(ST / S0)遵循
t分布参数分别为位置参数µ = 0,比例参数σ= 0.028和自由度ν= 2.958(比例
参数和自由度已校准到10年的历史
数据)。
•在到期时生成100,000个索引对数返回值的样本
并绘制其直方图。解释仿真中涉及的步骤。
•通过重新校准参数µ,σ和来验证仿真结果
通过在Matlab中使用“ fitdist”功能将ν转换为模拟数据。 (在Matlab中
在文档中,t分布称为“ tLocationScale”。总结
表格中的重新校准值
µσν
Table 1: Format to use to display your results
Options portfolio optimisation and valuation
The first step in the calculation of indifference prices involves portfolio optimization with financial liabilities. In this study, we will consider static hedging
where the hedging portfolio is chosen once at the beginning and held constant
until liquidation at maturity. The corresponding hedging problem can be writ2
ten as
minimize V

c −
X
j∈J
P
j
(x
j
)

 over x ∈ D
subject to X
j∈J
S
j
0
(x
j
) ≤ w,
(P)
where c is a random claim to be hedged, w is a given initial wealth, J is the set
of traded assets, S
j
0
(x
j
) is the cost of buying x
j units of asset j at time t = 0,
D ⊂ R
J
is the set of feasible portfolios, P
j
(x
j
) is the payout at time t = 1
when holding x
j units of asset j and V is a nondecreasing convex function on
the space of random cash-flows at time t = 1.
The function V : L
0 → R describes the investor’s preferences over uncertain
net expenditure at time t = 1. In this study, we take
V(c) := log E exp(λc/w),
where λ > 0 is a given “risk aversion” parameter and E denotes the expectation
with respect to the underlying probability measure to be specified below. The
set of feasible portfolios is given by
D := Y
j∈J
[−q
j
b
, qj
a
]
where q
j
b
and q
j
a are the quantities available at the best bid- and ask-prices.
We will assume that the claim c only depends on the value of the underlying
index at time t = 1. It follows that all the random quantities in the above
problem are uniquely determined by the index at maturity. We will model the
index as a random variable. Its distribution is specified according to the views
of the investor.
Consider an investor with an initial wealth w = 100, 000 dollars, c = 0 and
risk aversion factor λ = 2.
• Optimize the investor’s portfolio numerically by approximating the expectation by a sample average and solving the resulting problem using the
Mosek solver. Instruction on using the Mosek solver can be found in the
Appendix.
• Plot the payoff of the optimized portfolio as a function of the index at
maturity.
Indifference pricing
We will denote the optimum value of (P) by
ϕ(w, c) := inf
x∈D
{V(c −
X
j∈J
P
j
(x
j
))|
X
j∈J
S
j
0
(x
j
) ≤ w}.
3
For an agent with financial position described by ¯w units of cash at time t = 0
and a liability of delivering a random cash-flow ¯c at time t = 1, the indifference
price for selling a claim c is defined by
πs( ¯w, c¯; c) := inf{w ∈ R | ϕ( ¯w + w, c¯+ c) ≤ ϕ( ¯w, c¯)}.
This is the least price at which the agent could sell the claim c without worsening
her financial position as measured by the optimum value of (P). Analogously,
the indifference price for buying c is given by
πb( ¯w, c¯; c) := sup{w | ϕ( ¯w − w, c¯− c) ≤ ϕ( ¯w, c¯)}.
The indifference prices can be approximated numerically by line search with
respect to w and solving problem (P) by approximation of the expectation
and solving the resulting optimization problem by using available optimization
packages.
Consider a digital call option that pays 1,000 if the value of the index at
maturity is higher than a strike K. Otherwise, the option payout is zero.
• Consider an investor with initial position ( ¯w, c¯) = (105
, 0) and compute
indifference prices for buying and selling a digital call option with strike
2300, 2400, 2500, 2600, 2700 and 2800 and plot the price as a function of
the strike K.
Part 3
In this part of the thesis, the student is invited to engage in independent
research by developing the above topics in a direction that could be practically
relevant and useful or otherwise interesting. Possible topics could be (but are
not limited to)
• portfolio constraints: margin requirements, different interest rates for
lending and borrowing, . . .
• nonlinear trading costs
• alternative models for the underlying risk factors
• alternative descriptions of risk preferences
• sensitivities
• dynamic trading strategies
• other market sectors
• variance reduction techniques
• duality and optimality conditions
• credit risk


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