We consider here some experiments on the sensitivity of vanilla European put prices as computed under variance gamma as σ, ν and ϑ changes.
The numerical experiments presented here can be obtained using the code in C language available in appendix B. The vanilla put case studied here can is realized by choosing the following variable values:
• callput = 0
• euroamerican = 0
• barrier switch = 0 in the C code.
Sensitivity with Respect to
σ
Let’s consider here the sensitivity of vanilla European put option prices to
σ. The parameters used in this case are the following:
• T = 1 Time to maturity in years;
• K = $100 Strike price;
• r = 0.03 Interest rate;
• q = 0.01 Dividend yield;
• ϑ =−0.1;
• ν = 0.2;
• N = 1700 Number of space intervals;
• M = 400 Number of time intervals;
• xmin = 1.60944 Min xvalue considered, corresponding to S = $5;
• xmax = 5.85793 Max x value considered, corresponding toS = $350;
• ∆x≃0.0025 Size of a space interval;
• ∆t= 0.0025 Size of a time interval;
• ∆t ∆x ≃1;
Keeping ϑ, ν and the other parameters constant, we move σ from 10% to 90%. Volatility, skewness and kurtosis are affected in the following way:
σ 0.1 0.2 0.3 0.4 0.5 0.7 0.9 r Eh(X(t)−E[X(t)])2i 0.110 0.205 0.303 0.402 0.502 0.701 0.901 E[(X(t)−E[X(t)])3] q {E[(X(t)−E[X(t)])2 ]}3 −0.517 −0.288 −0.196 −0.148 −0.119 −0.085 −0.067 E[(X(t)−E[X(t)])4] {E[(X(t)−E[X(t)])2]} 2 3.783 3.656 3.626 3.615 3.609 3.605 3.603
We can see that when σ increases, volatility increases and skewness and kurtosis are reduced. The graph below shows that the option value increases as σ increases. 0 10 20 30 40 50 60 70 80 Option Price 20 40 60 80 100 120 140 160 180 200 Stock Price
Final Payoff Sigma = 10% Sigma = 20% Sigma = 30% Sigma = 40% Sigma = 50% Sigma = 70% Sigma = 90%
Sigma Sensitivity Analysis
European Put
Sensitivity with Respect to
ν
We consider here the sensitivity of a vanilla European put option with respect to ν. The parameters used for these experiments are the following:
• T = 1 Time to maturity in years;
• K = $100 Strike price;
• r = 0.03 Interest rate;
• q = 0.01 Dividend yield;
• σ = 0.2;
• ϑ = 0;
• N = 1700 Number of space intervals;
• M = 400 Number of time intervals;
• xmin = 1.60944 Min xvalue considered, corresponding to S = $5;
• xmax = 5.85793 Max x value considered, corresponding toS = $350;
• ∆x≃0.0025 Size of a space interval;
• ∆t= 0.0025 Size of a time interval;
• ∆t ∆x ≃1;
Using these values, we modify ν from 0.01 to 2.0. We note that here, as we did for the call case, we constrained ϑ= 0. By doing this, volatility and and skewness are not affected byν and the impact of changing ν is reflected only on the kurtosis. In particular we have the following values as ν changes:
ν 0.01 0.5 1 2 q E£ (X(t)−E[X(t)])2¤ 0.2 0.2 0.2 0.2 E[(X(t)−E[X(t)])3] q {E[(X(t)−E[X(t)])2]}3 0 0 0 0 E[(X(t)−E[X(t)])4 ] {E[(X(t)−E[X(t)])2 ]}2 3.03 4.5 6 9
In the first graph we compare a case of low kurtosis with a case of higher kurtosis. We remember that when ϑ = 0 and ν = 0.01 we are quite close to the geometric Brownian motion case.
0 5 10 15 20 25 30 35 40 Option Price 60 70 80 90 100 110 120 130 140 Stock Price Final Payoff Nu = 0.01 Nu = 2.0 Nu Sensitivity Analysis
European Put
Figure 4.10: European Put: ν Sensitivity Analysis
the option whose underlying has lower kurtosis is worth more, while on the two sides the option on the stock with higher ν is worth more.
In this second case we consider a zoom in the area near the money. We can see that in this area a large kurtosis reduces the value of the option.
2 4 6 8 10 12 Option Price 90 95 100 105 110 115 Stock Price Nu = 0.01 Nu = 0.5 Nu = 1.0 Nu = 2.0 Nu Sensitivity Analysis
European Put
In this last case, we consider an in the money area. We compare here a level of kurtosis close to the normal one with a larger kurtosis case and we see that the option having large kurtosis is worth more as the stock moves far from the money.
12 14 16 18 20 22 24 Option Price 75 80 85 90 Stock Price Nu = 0.01 Nu = 2.0 Nu Sensitivity Analysis
European Put
Sensitivity with Respect to
ϑ
We consider now the sensitivity of European vanilla put options with respect to ϑ. In particular let’s consider the following parameters:
• T = 1 Time to maturity in years;
• K = $100 Strike price;
• r = 0.03 Interest rate;
• q = 0.01 Dividend yield;
• σ = 0.2;
• ν = 0.2;
• N = 1700 Number of space intervals;
• M = 400 Number of time intervals;
• xmin = 1.60944 Min xvalue considered, corresponding to S = $5;
• xmax = 5.85793 Max x value considered, corresponding toS = $350;
• ∆x≃0.0025 Size of a space interval;
• ∆t= 0.0025 Size of a time interval;
• ∆t ∆x ≃1;
We consider here the impact of moving ϑ from +0.2 to -1.0. The following table shows the impact on volatility, skewness and kurtosis of a move in ϑas the other variables are unchanged
ϑ +0.2 0.0 −0.2 −0.5 −0.7 −1.0 q E£ (X(t)−E[X(t)])2¤ 0.22 0.20 0.22 0.30 0.37 0.49 E[(X(t)−E[X(t)])3] q {E[(X(t)−E[X(t)])2]}3 +0.52 0.00 −0.52 −0.81 −0.86 −0.88 E[(X(t)−E[X(t)])4 ] {E[(X(t)−E[X(t)])2]}2 3.78 3.60 3.78 4.08 4.15 4.18
In this first graph we consider only the case of a non skewed or negative skewed distribution. This case is consistent with market data and with the economic interpretation of skewness in stock price processes. From the table above, we see that as ϑbecomes increasingly negative, the negative skewness increases. Moreover we can see that volatility and kurtosis increases as ϑ
goes from 0 to -1.0. As a result of this when the put price increases as ϑ
moves from zero to larger negative values.
0 10 20 30 40 50 60 Option Price 40 60 80 100 120 140 160 180 200 Stock Price
Final Payoff Theta = 0 Theta = −0.2 Theta = −0.5 Theta = −0.7 Theta = −1.0 Theta Sensitivity Analysis
European Put
In this second graph we compare the case of ϑ = −0.2 with the case of
ϑ = +0.2. Because ϑ impacts the second and fourth moments only in the form of its square and its fourth power, the fact that we change the sign of the variable does not produce any impact on volatility and kurtosis. In this way we isolated the skewness effect on the put price. We can see that when the option is out of the money, the option with negative skew is worth more because it is more likely that it will jump in the money.
This is consistent with what we found for the call case: in that situation the out of the money option was worth more if the skewness was positive since a positive jump was required to bring the option in the money. In the put case, on the other side, we need a negative jump to move in the money, that is why the out of the money option with negative skewness is the one having the highest value.
0 5 10 15 20 25 30 Option Price 70 80 90 100 110 120 130 140 150 Stock Price
Final Payoff Theta = +0.2 Theta = −0.2
Theta Sensitivity Analysis