Stock Hedging Loss and Risk

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A stock is supposed to be bought at time zero with price S0, and to be sold at time T with uncertain price ST. In order to hedge the market risk of the stock, the company decides to choose one of the available put options written on the same stock with maturity at time τ, where τ is prior and close to T, and the n available put options are specified by their strike prices Ki (i = 1,2,··· ,n). As the prices of different put options are also different, the company needs to determine an optimal hedge ratio h (0 ≤ h ≤ 1) with respect to the chosen strike price. The cost of hedging should be less than or equal to the predetermined hedging budget C. In other words, the company needs to determine the optimal strike price and hedging ratio under the constraint of hedging budget. The chosen put option is supposed to finish in-the-money at maturity, and the constraint of hedging expenditure is supposed to be binding.

Suppose the market price of the stock is S0 at time zero, the hedge ratio is h, the price of the put option is P0, and the riskless interest rate is r. At time T, the time value of the hedging portfolio is

S0erT + hP0erT —– (1)

and the market price of the portfolio is

ST + h(K − Sτ)+ er(T − τ) —— (2)

therefore the loss of the portfolio is

L = S0erT + hP0erT − (ST +h(K − Sτ)+ er(T − τ)—– (3)

where x+ = max(x, 0), which is the payoff function of put option at maturity. For a given threshold v, the probability that the amount of loss exceeds v is denoted as

α = Prob{L ≥ v} —– (4)

in other words, v is the Value-at-Risk (VaR) at α percentage level. There are several alternative measures of risk, such as CVaR (Conditional Value-at-Risk), ESF (Expected Shortfall), CTE (Conditional Tail Expectation), and other coherent risk measures.

The mathematical model of stock price is chosen to be a geometric Brownian motion

dSt/St = μdt + σdBt —– (5)

where St is the stock price at time t (0 < t ≤ T), μ and σ are the drift and the volatility of stock price, and Bt is a standard Brownian motion. The solution of the stochastic differential equation is

St = S0 eσBt + (μ − 1/2σ2)t —– (6)

where B0 = 0, and St is lognormally distributed.

For a given threshold of loss v, the probability that the loss exceeds v is

Prob {L ≥ v} = E [I{X≤c1}FY(g(X) − X)] + E [I{X≥c1}FY (c2 − X)] —– (7)

where E[X] is the expectation of random variable X. I{X<c} is the index function of X such that I{X<c} = 1 when {X < c} is true, otherwise I{X<c} = 0. FY(y) is the cumulative distribution function of random variable Y, and

c1 = 1/σ [ln(k/S0) – (μ – 1/2σ2)τ]

g(X) = 1/σ [ln((S0 + hP0)erT − h(K − f(X))er(T − τ) − v)/S0 – (μ – 1/2σ2)T]

f(X) = S0 eσX + (μ−1σ2

c2 = 1/σ [ln((S0 + hP0)erT − v)/S0 – (μ – 1/2σ2)T]

X and Y are both normally distributed, where X ∼ N(0, √τ), Y ∼ N(0, √(T−τ)).

For a specified hedging strategy, Q(v) = Prob {L ≥ v} is a decreasing function of v. The VaR under α level can be obtained from equation

Q(v) = α —– (8)

The expectations can be calculated with Monte Carlo simulation methods, and the optimal hedging strategy which has the smallest VaR can be obtained from (8) by numerical searching methods.

Infinite Sequences and Halting Problem. Thought of the Day 76.0

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In attempting to extend the notion of depth from finite strings to infinite sequences, one encounters a familiar phenomenon: the definitions become sharper (e.g. recursively invariant), but their intuitive meaning is less clear, because of distinctions (e.g. between infintely-often and almost-everywhere properties) that do not exist in the finite case.

An infinite sequence X is called strongly deep if at every significance level s, and for every recursive function f, all but finitely many initial segments Xn have depth exceeding f(n).

It is necessary to require the initial segments to be deep almost everywhere rather than infinitely often, because even the most trivial sequence has infinitely many deep initial segments Xn (viz. the segments whose lengths n are deep numbers).

It is not difficult to show that the property of strong depth is invariant under truth-table equivalence (this is the same as Turing equivalence in recursively bounded time, or via a total recursive operator), and that the same notion would result if the initial segments were required to be deep in the sense of receiving less than 2−s of their algorithmic probability from f(n)-fast programs. The characteristic sequence of the halting set K is an example of a strongly deep sequence.

A weaker definition of depth, also invariant under truth-table equivalence, is perhaps more analogous to that adopted for finite strings:

An infinite sequence X is weakly deep if it is not computable in recursively bounded time from any algorithmically random infinite sequence.

Computability in recursively bounded time is equivalent to two other properties, viz. truth-table reducibility and reducibility via a total recursive operator.

By contrast to the situation with truth-table reducibility, Péter Gacs has shown that every sequence is computable from (i.e. Turing reducible to) an algorithmically random sequence if no bound is imposed on the time. This is the infinite analog of far more obvious fact that every finite string is computable from an algorithmically random string (e.g. its minimal program).

Every strongly deep sequence is weakly deep, but by intermittently padding K with large blocks of zeros, one can construct a weakly deep sequence with infinitely many shallow initial segments.

Truth table reducibility to an algorithmically random sequence is equivalent to the property studied by Levin et. al. of being random with respect to some recursive measure. Levin calls sequences with this property “proper” or “complete” sequences, and views them as more realistic and interesting than other sequences because they are the typical outcomes of probabilistic or deterministic effective processes operating in recursively bounded time.

Weakly deep sequences arise with finite probability when a universal Turing machine (with one-way input and output tapes, so that it can act as a transducer of infinite sequences) is given an infinite coin toss sequence for input. These sequences are necessarily produced very slowly: the time to output the n’th digit being bounded by no recursive function, and the output sequence contains evidence of this slowness. Because they are produced with finite probability, such sequences can contain only finite information about the halting problem.