Financial Engineering is a multidisciplinary field involving finance and economics, mathematics, statistics, engineering and computational methods. The emphasis of FE & RM Part II will be on the use of simple stochastic models to (i) solve portfolio optimization problems (ii) price derivative securities in various asset classes including equities and credit and (iii) consider some advanced applications of financial engineering including algorithmic trading and the pricing of real options. We will also consider the role that financial engineering played during the financial crisis. We hope that students who complete the course and the prerequisite course (FE & RM Part I) will have a good understanding of the "rocket science" behind financial engineering. But perhaps more importantly, we hope they will also understand the limitations of this theory in practice and why financial models should always be treated with a healthy degree of skepticism.
The Greeks: Vega and Theta
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來自 Columbia University 的課程
金融工程与风险管理，第 2 部分
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從本節課中
Equity Derivatives in Practice: Part I
Problems with mean-variance analysis; ETFs and leveraged ETFs; VaR and CVaR for asset allocation; survivorship bias, performance evaluation and other statistical pitfalls.
與講師見面
Martin HaughCo-Director, Center for Financial Engineering
Industrial Engineering & Operations Research
Garud IyengarProfessor
Industrial Engineering and Operations Research Department
In the last module, we saw Delta and Gamma.
In this module, we're going to see Vega and Theta.
Vega is the sensitivity of the option price with respective changes in the
parameter Sigma, the volatility parameter Sigma.
Whereas, Theta is the sensitivity of the option price with respective changes in
the time to maturity. So, were going to discuss Vega and Theta
in this module. We'll see how they behave as the function
of the underlying security price and indeed as the function of the underlying
time to maturity. It is very important that we understand
how all of the Greeks work. And the Vega and the Theta are very
important Greeks in practice. Again, here we have the Black-Scholes
formula. Were going to use the Black-Scholes
formula in this module to compute the Vega and Theta of an option.
So, just remind yourselves, the Black-Scholes model assumes that the
stock price follows a geometric Brownian motion, so that the stock price of time
LT is equal to S0e to the r minus c minus Sigma squared over 2 times T, plus Sigma
Wt, where Wt is a Brownian motion under the risk neutral probability distribution
Q. So, this is the formula here.
And you saw in the last module what the delta of an option is.
We also saw what the gamma of an option is.
And we can calculate the Delta and Gamma by taking the appropriate derivatives of
this expression here, for a call option. Likewise, we could the same for a put
option or if we liked, we could use put-call parity to compute those
expressions for put options. So, first let's deal with the Vega.
The Vega of an option is a partial derivative of the option price with
respect to the volatility parameter Sigma.
So, the volatility parameter is this parameter over here.
Now, if you stop and think about it for a moment, you might think that as Sigma
increases, the value of the option will increase.
And indeed that is true. For example, the payoff of a call option,
capital T, CT, is equal to the maximum of 0 and ST minus K.
While it makes sense the Sigma gets larger, the value of the security will
also increase. an easy way to see that perhaps is
imagine that S0, the initial stock price, is much less than in, than the strike of
K. Well, in that case, if Sigma is very very
small, the chances of the stock price growing enough, so that the option ends
up in the money, will be zero, or approximately zero.
On the other hand, as Sigma gets sufficiently large, the probability that
st will be greater than K will actually increase, in which case, the option value
will be non-zero. And so it makes sense that the call
price, the initial price of the option C zero, should be increasing in Sigma.
And we will see that that is indeed the case.
The Vega of an option is a partial derivative of the option price with
respect to the volatility parameter Sigma.
Vega, therefore, measures the sensitivity of the option price to Sigma.
And using the Black-Scholes formula, it can easily be calculated.
Vega is equal to Delta C Delta Sigma, which turns out to be e to the minus eT,
s square root of time to majority times phi of d1, where phi is the probability
density function of a standard normal random variable.
Now, we can also compute the Vega for a European put option by using put-call
parity. So, this is put-call parity here.
Remember, so it actually implies. So, it implies that the put price is
equal to the the call plus e to the minus rt times K minus e to the minus cT times
S. And so, therefore, we can actually
compute Delta P, Delta Sigma, we see it's equal to, well Delta C, Delta Sigma.
That's the first term here. And then these other two terms don't
depend on Sigma at all. So, it's plus zero minus zero.
And so, we see Delta P, Delta Sigma equals Delta c, Delta Sigma.
And so, the Vega of a European put option is the same as the Vega of a European
call option. Here's a question for us.
Is the concept of Vega inconsistent in any way with the Black-Scholes model?
And the answer is yes. If you recall, the Black-Scholes mo-,
model assumes that St, the stock price of any time t, is equal to S zero e to the
Mu minus Sigma squared over 2 times t, plus Sigma times Wt, where Wt is a
standard Brownian motion. Mu and Sigma are constants in this model.
They are not assumed to change. And that indeed was the assumption of the
Black-Scholes model. They assumed continuous trading.
They assumed that there were no transactions, costs, and that short sales
were allowed, and that borrowing or lending at the risk free interest rate,
or was also possible. Using these assumptions, they constructed
a self-financing trading strategy that replicated the payoff of the option, and
that is indeed how they are paying the Black-Scholes formula.
Nothing in their model allowed Sigma to change, Sigma was a known constant.
And yet when we're talking about Delta C, Delta Sigma, we're implicitly recognizing
the fact that Sigma can change. And indeed, in the marketplace, Sigma
does change. So, in that sense, the, While,
mathematically, one can always define Delta C, Delta Sigma, there's no problem
with that. Within the economics of the Black-Scholes
model, it isn't consistent to talk about Sigma changing.
Because we ob-, we obtained the Black-Scholes option price under the
assumption that Sigma could not change. Here are some plots of Vega for options,
for European options, as a function of the stock price at time t equal to zero,
and as the time to maturity varies. So, we've got three different times to
maturity T equals 0.05 years, T equals 0.25 years, and T equals 0.5 years.
There are probably two things to notice first.
The first observation is as follows. Note that if I pick any one of these
options, the Vega goes to 0 as the stock price moves away from the strike, which
was $100 here. So, what is going on here.
Well, it's very simple. So, again re-, returning to what we did
in previous modules, we know the following.
Let's take a call option as our example. .
We know that the call option price[BLANK_AUDIO] will be approximately
equal to, and again, ignoring interest rate factors and so on.
It would approximately be equal to S0 minus K, for S0 being very large.
And by very large, I mean much larger than K.
And sufficiently large, that I'm almost certain I'm going to be exercising the
option. It will be equal to zero for S0 being
very small. And very small here means much smaller
than K. And indeed, small enough that the chances
of exercising the option are approximately zero.
Well, we can see here that Delta C, Delta Sigma, therefore, must be equal to 0 in
this situation because the partial derivative of S0 minus K with respect to
sigma is 0. And also 0 down in this situation as
well. And therefore, for S0 very large, which
is up here, or S0 very small, which is down here, we see that Delta C, Delta
Sigma goes to 0. And indeed, that's what we see for each
of these three options. So, that's the first observation.
The second observation, so let's call this observation 1.
The second observation here is that the Vega[BLANK_AUDIO] increases in time
maturity capital T. So, we see that the option where t equal
.05 years, the blue curve here, is larger than the Vega for the option where t
equals 0.25 years and so on. And in fact, this is not surprising
because if we go back to the Black-Scholes formula.
Over here, we can see that every place where Sigma appears, we find it together
with a square root of T. Or if you like, when Sigma squared
appears in the Black-Scholes formula, I see a T appearing.
So, I've got Sigma square root T or sigma square root T appearing here.
So, basically, every time I see a sigma, I'm multiplying it by the square root of
T. And, therefore, the impact of a change in
Sigma, i.e. the Vega, it would be amplified by the
square root of T. And it is, therefore, the case.
And, by the way, maybe I should have mentioned or was assumed to be equal to
c, was assumed to be equal to zero in these plots here.
We can see that the blue curve has, is a factor of square root of 2, which is
approximately equal to 1. 4, times higher than the green curve.
And the green curve is a factor of the square root of 5, which is approximately
equal to 2.2 something, greater than the red curve.
And that's no surprise because 0.5 years divided by 0.25 years is equal to 2.
So, the square root of 2 is approximately 1.4.
And, indeed, we see the green curve reaches a peak of 20 here.
1.4 times 20 is 28. And that's roughly the peak of the blue
curve. Likewise, down here, we have the red
curve reaching maybe a peak of approximately 8 and a half or 9, 8 and a
half or 9 multiplied by 2.2 brings us up towards approximately 20.
So in fact, this behavior is entirely predictable.
The, the change, the, the Vega for the option is magnified by the square root of
the time to maturity. Another way of saying that is, another
way of seeing this is by looking at this figure here, where we have plotted the
Vegas for three options. And not the money option.
A 10% out of the money option and a 20% out of the money option.
And in all three cases, we see that the Vega converges to zero as the time ot
maturity goes to zero. And the, this would also be true if i
showed a 10% in the money option or 20% in the money option.
The next Greek I want to talk about is the Theta of an option.
The Theta of an option is the negative of the partial derivative of the option
price with respect with time to maturity. So, therefore, mathematically speaking,
Theta equals minus Delta c, Delta t. And that's for a call option.
We can also compute it for a put option, if we actually go ahead and do the
mathematics, compute the derivatives of the Black-Scholes formula.
We will find that theta's equal to this long expression here, where, phi is the
standard normal PDF. So, if you recall, N is the standard
normal CDF and phi is the standard normal PDF.
Why do we take the negative? Well, we take the negative because in
practice, time goes forward. So, in practice the time to maturity of
an option decreases. Suppose I have an option right now which
is 200 days to maturity. Well, then tomorrow, it would have 199
days to maturity. So, therefore, the time to maturity is
always decreasing in practice. And so, it's conventional to take Theta
to be the negative of the partial derivative of the option price with
respect to time to maturity. Here are some figures.
Again, we see Theta for European call option as a function of the stock price.
K was equal to 100 in these examples. We assumed r, the interest rate, and
indeed c, the dividend yield, was equal to 0%.
We plot the Theta here for 0.05 years, 0.25 years and 0.5 years.
Notice, number one, that the Theta is negative in all cases.
Now, in general, Theta will be negative for European call and put options.
It's, it's not always the case that it's negative.
There are certain situations where theta could be positive.
But in general, most of the time, theta is negative.
In other words, when you hold a European call or put option, you lose a little
piece of money every day if the underlying stock price does not change.
That's what Theta means. Remember Theta is equal to minus the
partial derivative of the call option price with respect to time to maturity.
So, as the time to maturity decreases, I lose a little piece of the value of the
option, the Delta c decreases. Again, another observation, is that as
the stock price moves away from the current strike of K equals 100.
We see that the Theta goes toward zero. And again, we can use our earlier
examples to see why this is the case. We know, and this time we'll say in the
case of we have a call option here, so we'll stick with the call option, in the
case of a call option, we know that c0 will be approximately equal to, as zero
minus K, if S zero is much bigger than K or very large, and it's approximately
equal, if S zero is much smaller than K. In both cases, the partial derivative of
this term, with respect to capital T, the time to maturity is zero.
Likewise, the partial derivative of this term, zero, with respect to time to
maturity capital T, is also zero. And so, that's why we see, for large
values of S and for very small values of S, we see that the partial derivative
with respect to time to maturity is zero. And that's why all of these curves
approach zero as S moves away from the strike, K.
Why is the theta most negative around the strike for short times to maturity?
That is, for time to maturity of two and a half weeks of 0.05 years.
Well, one way to see that is the following.
Suppose I've just got one day to maturity.
So, I've got one day. So, T is equal to one day to maturity.
Well, then, and suppose the stock price is equal to K.
So, I'm at the money. This means I've got one day to maturity.
If the stock price increases, I'm going to exercise the option and make some
money. if the stock price decreases over the
next day, I'm not going to exercise the money.
So, the option value will be none-zero at this point because over the next day,
there is a chance that the stock price will increase and I'll be exercising and
make me some money. However, imagine rolling time forward one
day without changing the stock price. Well, in that case, this is going to go
to zero days to maturity. That's zero is still equal to K.
And now the option expires worthless. So, when there's just one day to
maturity, the Theta is larger, larger and more negative, because I have more to
lose over the next day than I would if there was one year to maturity.
If there was one year to maturity, I would have 365 days left.
Moving time forward, one day isn't really going to impact the value of the option
very much at all. However, when I've just one day left to
maturity, that one day encapsulates all of the value of the option when I'm at
the money. And if I roll time forward one day
without changing the stock price, I'm going to expire worthless and, therefore,
receive nothing. So, the Theta becomes more negative and
peaked around the strike as the time to maturity decreases towards 0.
And on this plot, we see the Theta for European put options as a function of the
time to maturity. We've plotted here three different option
curves. One for not the money option, the blue
curve, and the green and red curves for 10% out of the money and 20% out of the
money options, respectively. So, in fact, just so we're clear, a 10%
out of the money option, in this case, it's a European put option.
So, 10% out of the money option will have K being equal to 0.9 times S zero.
So, the strike is below the current stock price.
And, so, currently it is out of the money.
For the 20% case, we will have K equal to 0.8 times S0.
So, in these cases, we see this out of the money options that they're Theta is
decreasing. Let's take this green curve here.
We see Theta is decreasing for a while, that's negative and decreasing.
But beyond the certain point, it becomes, it, it turns around and moves toward
zero. And that's because it's becoming
increasingly unlikely that the option will be exercised.
It's value is moving towards 0, and so its Theta will be 0.
The partial derivative of 0 with respect to t is equal to 0.
So, at this point, the value is moving towards 0 because it's becoming less and
less likely to be exercised. The red curve, corresponding to a 20% out
of the money option, has actually be turned earlier than the green curve
because its 20% out of the money is further away.
And so, it's becoming less and less likely to be exercised at an earlier
point than this green curve here. And, in case you're wondering, we can
easily create these plots, just by using this expression here.
This is an expression, so for those of you who are comfortable with Code in R or
Matlab or Python. Or indeed in Excel, you could create a
table of values for t. Create the, Thetas for these different T
values and create a plot. You can easily create these kinds of
curves that I'm showing you.
