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.

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