Risk management is important in the practices of financial institutions and other corporations. Derivatives are popular instruments to hedge exposures due to currency, interest rate and other market risks. An important step of risk management is to use these derivatives in an optimal way. The most popular derivatives are forwards, options and swaps. They are basic blocks for all sorts of other more complicated derivatives, and should be used prudently. Several parameters need to be determined in the processes of risk management, and it is necessary to investigate the influence of these parameters on the aims of the hedging policies and the possibility of achieving these goals.

The problem of determining the optimal strike price and optimal hedging ratio is considered, where a put option is used to hedge market risk under a constraint of budget. The chosen option is supposed to finish in-the-money at maturity in the, such that the predicted loss of the hedged portfolio is different from the realized loss. The aim of hedging is to minimize the potential loss of investment under a specified level of confidence. In other words, the optimal hedging strategy is to minimize the Value-at-Risk (VaR) under a specified level of risk.

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.

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 criterion of optimality is to minimize the VaR of the hedging strategy.

The mathematical model of stock price is chosen to be a geometric Brownian motion, i.e.

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−1/2σ2)t} —– (6)

where B_{0} = 0, and S_{t} is lognormally distributed.

Proposition:

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−1/2σ^{2})τ,

c_{2} = 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 in Proposition can be calculated with * Monte Carlo simulation methods*, and the optimal hedging strategy which has the smallest VaR can be obtained from equation (8) by numerical searching methods….