Garmana-Kohlhagena formula:
(Replace dividend rate \(q\) with \(r_f\) - foreign risk-free rate)
\[Call =Se^{-r_fT} N(d_1) - Ke^{-rT}N(d_2)\] \[Put =Ke^{-rT}N(-d_2) - Se^{-r_fT} N(-d_1)\] \[d_1 = \frac{ln\frac{S}{K} + (r - r_f + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\] \[d_2 = \frac{ln\frac{S}{K} + (r - r_f- \frac{\sigma^2}{2})T}{\sigma\sqrt{T}} = d_1 - \sigma\sqrt{T}\] Where:
Black-Sholes formula for options for futures:
\[Call =Fe^{-rT} N(d_1) - Ke^{-rT}N(d_2)\] \[Put =Ke^{-rT}N(-d_2) - Fe^{-rT} N(-d_1)\] \[d_1 = \frac{ln\frac{F}{K} + \frac{\sigma^2}{2}T}{\sigma\sqrt{T}}\] \[d_2 = \frac{ln\frac{F}{K} - \frac{\sigma^2}{2}T}{\sigma\sqrt{T}} = d_1 - \sigma\sqrt{T}\] Where:
Black-Sholes formula for options for bonds:
(Replace dividend rate \(q\) with \(r_f\) - foreign risk-free rate)
\[Call =BN(d_1) - Ke^{-rT}N(d_2)\] \[Put =Ke^{-rT}N(-d_2) - BN(-d_1)\] \[d_1 = \frac{ln\frac{B}{K} + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\] \[d_2 = \frac{ln\frac{B}{K} + (r - \frac{\sigma^2}{2})T}{\sigma\sqrt{T}} = d_1 - \sigma\sqrt{T}\] Where:
If coupon payments from bonds are to be paid within the option period, then before using the formulas, the current value of these payments should be subtracted from B, and volatility σ should be determined for the bond price minus the current value of coupon payments.
Delta is the ratio of the change in the price of the option to the change in the price of the underlying asset:
\(\Delta = \frac{\partial P}{\partial S}\)
Gamma is the rate of change of the options’s delta with respect to the price of the underlying asset:
\(\Gamma = \frac{\partial \Delta}{\partial S} = \frac{\partial^2 P}{\partial S^2}\)
Theta is the rate of change of the value of the option with respect to the the maturity:
\(\Theta = \frac{\partial P}{\partial T}\)
Vega is the rate of change of the value of the option with respect to the volatility of the underlying asset:
\(\nu = \frac{\partial P}{\partial \sigma}\)
Rho is the rate of change of the value of the option with respect to the risk-free rate:
\(\rho = \frac{\partial P}{\partial r}\)
One possible strategy is that you do not take any steps to secure your position. This is called position without coverage (naked position). If an investor holds a short position in option, then upon expiration, he will be forced to buy or sell the underlying at the current market price. If the option is in-the-money, it will incur a loss equal to the difference between the strike price and the price of the underlying instrument at the time of expiry, with the appropriate sign, depending on whether it is a call or put option.
An alternative to the naked position strategy is covered position. An example of this strategy is based on opening position in call option and buying the appropriate number of underlying instruments to which these options refer. It brings appropriate effects if the option is exercised. The investor then delivers the previously purchased underlying assets, and its profit is equal to the premium paid. Otherwise (for naked position), if the price of the underlying instrument falls, the investor will incur a loss.
Imagine the situation of the institution that has short position in call option. The stop-loss strategy is to keep the position naked when the option is out-of-the-money and to change to covered position when the option is in-the-money. This means buying shares when their price increases to K and sales of these shares when their price falls below this level. The initial cost of constructing the strategy is S, when S> K, or 0 otherwise. Therefore, it seems that if the option is in-the-money, the total cost associated with issuing and securing options would be:
\(cost =max(S-K,\ 0)\)
However, in reality, these cash flows should be discounted, and it is not possible to buy and sell shares at the same price K. Therefore, the cost of this strategy becomes high.
Delta hedging is the process of keeping the delta (\(\Delta\)) of a portfolio as close to zero as possible. The strategy allow to immune the portfolio for small movements of underlying price over a short period. It consists in creating a portfolio composed of options and underlying assets, for which the delta coefficient will be equal to 0 (delta neutral). Then the gain or loss on the options position is compensated by the loss or profit in the underlying assets (note that the delta of the underlying assets is equal to 1).
The investor’s position remains delta neutral for only a short period, which results from continuous changes in the delta value. In practice, the use of a delta hedging strategy is associated with the need to periodically adjust the position (rebalancing). Such strategy is called dynamic delta hedging strategy. The delta coefficient can be deduced from the Black-Sholes formulas. In case of the european call and put options for stocks paying dividend at rate \(q\) the delta is:
\[\Delta_{call} = e^{-qT}N(d_1)\]
\[\Delta_{put} = e^{-qT}[N(d_1) -1]\]
In case of the european call and put options for futures for exchange index the delta is (\(r\) is risk-free rate): \[\Delta_{call} = e^{-rT}N(d_1)\]
\[\Delta_{put} = e^{-rT}[N(d_1) -1]\]
Note that delta of the futures is (first derivative of futures price \(F=Se^{(r-q)T}\)) for S:
\(\Delta_{futures} = e^{(r-q)T}\)
The construction of a hedging strategy with respect to the short (long) position in the european call option consists of maintaining a long (short) position in \(N(d_1)\) underlying assets. In the case of the european put option, the delta assumes a negative value, which means that the long (short) position in the put options should be secured with a long (short) position in \((N(d_1)-1)\) shares.
The delta hedging strategy, however, provides hedging only with small changes in the price of the underlying instrument between subsequent position adjustments. A more effective hedging strategy, eliminating part of the delta hedging deficiencies, is a gamma hedging strategy
It involves creating a portfolio with zero gamma and delta coefficients at the same time. The gamma factor of the portfolio consisting of options determines the relative change in the value of the delta coefficient relative to the change in the price of underlying assets. A small value of \(\Gamma\) means that the delta changes very slowly, and rebalancing to the hedged position can be made rarely. When the gamma value is high, the delta shows a high sensitivity to changes in the underlying asset price. In this case, leaving the zero delta portfolio is very risky and leads to potential losses. Due to the fact that the gamma of the positions in the underlying assets and futures contracts for these assets is equal to zero, the only way to change the gamma coefficient is to take the appropriate position in the options.
Let’s assume that the delta neutral portfolio has a gamma equal to \(\Gamma\), and a traded option has a gamma equal to \(\Gamma_T\) . If the number of traded options added to the portfolio is \(w_T\), the gamma of the portfolio is
\(w_T \Gamma_T + \Gamma\).
Hence, the position in the option necessary to make the portfolio gamma neutral is:
\[w_T = -\frac{\Gamma}{\Gamma_T}\]
Entering such a position will naturally change the value of the portfolio delta. It is necessary to adjust the position in the underlying assets in order to restore the portfolio delta equal to zero. In this way, the investor can create a gamma hedging strategy that will hedge the portfolio of options against large and small changes in the value of underlying assets. However, we must remember that the zero value of the gamma can only be maintained for a short period of time and must be corrected over time. The gamma for the european call and put options for stocks can be calculated on the basis of the Black-Scholes formula:
\[\Gamma = e^{-qT}\frac{N^{'}(d_1)}{S\sigma \sqrt{T}}\]
Where
\(d_1 = \frac{ln\frac{S}{K} + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\)
\(N^{'}(y) = \frac{1}{\sqrt{2\pi}}e^{\frac{-y^2}{2}}\)
\(q\) - dividend rae
The gamma for the european call and put options for futures can be calculated on the basis of the Black-Scholes formula:
\[\Gamma = e^{-rT}\frac{N^{'}(d_1)}{S\sigma \sqrt{T}}\]
\(r\) - risk-free rate
The investor has a portfolio with a zero delta (delta neutral) and a gamma value equal to \(-3000\). Delta and gamma for the options included in this portfolio are 0.62 and 1.5 respectively. The investor wants to bring the value of the gamma and delta ratios of the above portfolio to zero. Suggest a strategy.
Calculate the value of the european call option for EURPLN (treat EUR as foreign currency), with the following characteristics: \(S = 4.05\), \(K = 4.0000\), \(r = 7\%\), \(r_f = 3\%\), \(T = 0.5\), \(\sigma = 32\%\). Check how the price of this option will change if its volatility changes to:
a) 22%
b) 41%
You have a portfolio of shares that reflects the composition of S&P500 index. The value of the portfolio is 100 mln USD. S&P500 is at the level of 2900. You would like to insure your portfolio against S&P500 decrease higher than 5% over the next 3 months. Risk-free rate is 2%, dividend ratio is 1% (for S&P500 and your portfolio) and the volatility of the S&P500 is 24%.
a) How much the insurance will cost if you decide to use put option?
b) How the strategy will look like if you want to create delta neutral portfolio using options.
Calculate the value of the european call option for the S&P500 futures contract with the following characteristics: \(S = 920\), \(K = 960\), \(r = 3\%\), \(T = 0.25\), \(σ = 29\%\). Investor has the portfolio with delta (\(\Delta_p\)) equals \(-300\) and gamma (\(\Gamma_p\)) equals \(2.9\). The investor wants to bring the value of the gamma and delta ratios of the above portfolio to zero. Suggest a strategy.