A stock is supposed to be bought at time zero with price S_{0}, and to be sold at time T with uncertain price S_{T}. 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 K_{i} (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 S_{0} at time zero, the hedge ratio is h, the price of the put option is P_{0}, and the riskless interest rate is r. At time T, the time value of the hedging portfolio is

S_{0}e^{rT} + hP_{0}e^{rT} —– (1)

and the market price of the portfolio is

S_{T} + h(K − S_{τ})^{+} e^{r(T − τ)} —— (2)

therefore the loss of the portfolio is

L = S_{0}e^{rT} + hP_{0}e^{rT }− (S_{T} +h(K − S_{τ})^{+} e^{r(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

dS_{t}/S_{t} = μdt + σdB_{t} —– (5)

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

S_{t} = S_{0} e^{σBt} + (μ − 1/2σ^{2})^{t} —– (6)

where B_{0} = 0, and S_{t} 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}}F_{Y}(g(X) − X)] + E [I_{{X≥c1}}F_{Y} (c_{2} − 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. F_{Y}(y) is the cumulative distribution function of random variable Y, and

c_{1} = 1/σ [ln(k/S_{0}) – (μ – 1/2σ^{2})τ]

g(X) = 1/σ [ln((S_{0} + hP_{0})e^{rT} − h(K − f(X))e^{r(T − τ)} − v)/S_{0} – (μ – 1/2σ^{2})T]

f(X) = S_{0} e^{σX + (μ−1σ2)τ}

c2 = 1/σ [ln((S_{0} + hP_{0})e^{rT} − v)/S_{0} – (μ – 1/2σ^{2})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.