MSc in Financial Mathematics, FM50

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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
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, . . .
• alternative models for the underlying risk factors
• alternative descriptions of risk preferences
• sensitivities
• other market sectors
• variance reduction techniques
• duality and optimality conditions
• credit risk EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

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