In the previous article we learned about how to map various complex positions such as coupon paying bonds, FRAs, and swaps. All these complex positions had linear payoffs and were fairly straightforward to map. However, there are other positions such as options where the payoff is non-linear. In such a case the mapping process becomes a bit more complex.
For portfolios and positions with non-linear payoff, the mapping is done with the help of first-order and second-order Taylor approximation, i.e., the Delta and Delta-Gamma approach.
Let’s look at these two approaches.
Let’s say that we have a call option with a value of c. The value of this option will depend on many variables such as the underlying stock price (S), the strike price, and the volatility. Ignoring other factors and considering the underlying stock price as the only factor, we can use the first order approximation, in which case:
Δc = δΔS
Where, δ is the option’s delta, which is readily available for actively traded options.
With this information, we can calculate the option’s VaR as follows:
VaR = δZασS
This approach however assumes that we have a short holding period and the delta remains constant. Also, it may not be very reliable for positions with high optionality or additional nonlinear features.
Our approximation can be made more accurate by using the second order Taylor approximation (Delta-Gamma), in which case the call value will be represented as follows:
Δc = δΔS +g/2(ΔS)2
Note that for a long call option, both Delta and Gamma will be positive, and for a short position both will be negative. A positive gamma favourably impacts the option value, and vice verse.
The VaR of the position can be calculated as follows:
VaR = δZασΔS – γ/2(ZασS)2
Note the minus sign in the formula. This is because positive gamma reduces VaR and vice verse.
Let’s say we have the following information about a European call option:
S = 100
r = 6%
Option maturity = 3 months
S = 25% (Annual)
D = 0.6
G = 2.2
A 10-day VaR at 99% confidence interval can be calculated as follows:
VaR = 0.6*2.33*Sqrt(10/250)*0.25 –(2.2)/2*(2.33* Sqrt(10/250)*0.25)^2
VaR = 0.0549
Note that VaR without incorporating gamma would have been higher.