dstefan
06-14-2008, 04:11 PM
Chapter 3, Problem 14
(i) The vega of a plain vanilla European call or put is positive, since
\mbox{vega}(C) ~=~ \mbox{vega}(P) ~=~ S e^{-q(T-t)} \sqrt{T-t}~\frac{1}{\sqrt{2 \pi}} e^{-\frac{d_1^2}{2}}
Can you give a financial explanation for this?
(ii) Compute the vega of ATM Call options with maturities of fifteen days, three months, and one year, respectively, on a non--dividend--paying underlying asset with spot price 50 and volatility 30%. For simplicity, assume zero interest rates, i.e., r=0.
(iii) If r=q=0, the vega of ATM call and put options is \mbox{vega}(C) ~=~ \mbox{vega}(P) ~=~ S \sqrt{T-t}~\frac{1}{\sqrt{2 \pi}} e^{-\frac{d_1^2}{2}}, where d_1 = \frac{\sigma \sqrt{T-t}}{2}.
Compute the dependence of vega(C) on time to maturity T-t, i.e.,
\frac{\partial~(\mbox{vega}(C))}{\partial (T-t)}
and explain the results from part (ii) of the problem.
Solution:
(i) The fact the vega of a plain vanilla European call or put is positive means that, all other things being equal, options on underlying assets with higher volatility are more valuable (or more expensive, depending on whether you have a long or short options position).
This could be understood as follows: the higher the volatility of the underlying asset, the higher the risk associated with writing options on the asset, and therefore, the premium charged for selling the option will be higher.
If you have a long position in either put or call options you are essentially ``long vol".
(ii) The input in the Black--Scholes formula for the Gamma of the call is S=K=50, \sigma=0.3, r=q=0. For T = 15/252, T = 1/4, and T = 1, the following values of the vega of the ATM call are obtained:
\mbox{vega}(15 \mbox{days}) = 4.863340; \mbox{vega}(3 \mbox{months}) = 9.945546; \mbox{vega}(1 \mbox{year}) = 19.723967
(iii) For clarity, let \tau = T - t. Then, for r=q=0, we know that
\mbox{vega}(C) ~=~ \frac{S \sqrt{\tau}}{\sqrt{2 \pi}} e^{-\frac{\sigma^2 \tau}{8}}
By direct computation, we find that
\frac{\partial~(\mbox{vega}(C))}{\partial \tau} = \frac{S}{2\sqrt{2 \pi \tau}} e^{-\frac{\sigma^2 \tau}{8}} ~-~ \frac{\sigma^2 S \sqrt{\tau}}{8 \sqrt{2 \pi}} e^{-\frac{\sigma^2 \tau}{8}} = \frac{S}{2\sqrt{2 \pi \tau}} \left( 1 - \frac{\sigma^2 \tau^2}{4} \right) e^{-\frac{\sigma^2 \tau}{8}}
For \sigma = 0.3 and time to maturity \tau less than one year, we find that 1 - \frac{\sigma^2 \tau^2}{4} ~\geq~ 0.9775 and therefore \frac{\partial~(\mbox{vega}(C))}{\partial \tau}~\geq~ 0
We conclude that, for options with moderately large time to maturity, the vega is increasing as time to maturity increases. Therefore we expect that
\mbox{vega}(1 \mbox{year})~>~ \mbox{vega}(3 \mbox{months}) ~>~ \mbox{vega}(15 \mbox{days}),
which is what we previously obtained by direct computation.
(i) The vega of a plain vanilla European call or put is positive, since
\mbox{vega}(C) ~=~ \mbox{vega}(P) ~=~ S e^{-q(T-t)} \sqrt{T-t}~\frac{1}{\sqrt{2 \pi}} e^{-\frac{d_1^2}{2}}
Can you give a financial explanation for this?
(ii) Compute the vega of ATM Call options with maturities of fifteen days, three months, and one year, respectively, on a non--dividend--paying underlying asset with spot price 50 and volatility 30%. For simplicity, assume zero interest rates, i.e., r=0.
(iii) If r=q=0, the vega of ATM call and put options is \mbox{vega}(C) ~=~ \mbox{vega}(P) ~=~ S \sqrt{T-t}~\frac{1}{\sqrt{2 \pi}} e^{-\frac{d_1^2}{2}}, where d_1 = \frac{\sigma \sqrt{T-t}}{2}.
Compute the dependence of vega(C) on time to maturity T-t, i.e.,
\frac{\partial~(\mbox{vega}(C))}{\partial (T-t)}
and explain the results from part (ii) of the problem.
Solution:
(i) The fact the vega of a plain vanilla European call or put is positive means that, all other things being equal, options on underlying assets with higher volatility are more valuable (or more expensive, depending on whether you have a long or short options position).
This could be understood as follows: the higher the volatility of the underlying asset, the higher the risk associated with writing options on the asset, and therefore, the premium charged for selling the option will be higher.
If you have a long position in either put or call options you are essentially ``long vol".
(ii) The input in the Black--Scholes formula for the Gamma of the call is S=K=50, \sigma=0.3, r=q=0. For T = 15/252, T = 1/4, and T = 1, the following values of the vega of the ATM call are obtained:
\mbox{vega}(15 \mbox{days}) = 4.863340; \mbox{vega}(3 \mbox{months}) = 9.945546; \mbox{vega}(1 \mbox{year}) = 19.723967
(iii) For clarity, let \tau = T - t. Then, for r=q=0, we know that
\mbox{vega}(C) ~=~ \frac{S \sqrt{\tau}}{\sqrt{2 \pi}} e^{-\frac{\sigma^2 \tau}{8}}
By direct computation, we find that
\frac{\partial~(\mbox{vega}(C))}{\partial \tau} = \frac{S}{2\sqrt{2 \pi \tau}} e^{-\frac{\sigma^2 \tau}{8}} ~-~ \frac{\sigma^2 S \sqrt{\tau}}{8 \sqrt{2 \pi}} e^{-\frac{\sigma^2 \tau}{8}} = \frac{S}{2\sqrt{2 \pi \tau}} \left( 1 - \frac{\sigma^2 \tau^2}{4} \right) e^{-\frac{\sigma^2 \tau}{8}}
For \sigma = 0.3 and time to maturity \tau less than one year, we find that 1 - \frac{\sigma^2 \tau^2}{4} ~\geq~ 0.9775 and therefore \frac{\partial~(\mbox{vega}(C))}{\partial \tau}~\geq~ 0
We conclude that, for options with moderately large time to maturity, the vega is increasing as time to maturity increases. Therefore we expect that
\mbox{vega}(1 \mbox{year})~>~ \mbox{vega}(3 \mbox{months}) ~>~ \mbox{vega}(15 \mbox{days}),
which is what we previously obtained by direct computation.