1
00:00:01,030 --> 00:00:05,800
So the last Greek that I want to go
through the calculation through is theta,

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00:00:08,070 --> 00:00:09,790
and so theta is the rate of change of

3
00:00:09,790 --> 00:00:13,290
the value of the portfolio with respect to
time t.

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00:00:15,580 --> 00:00:17,700
And again, our portfolio that we're
considering

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just contains a single call option, so the

6
00:00:19,900 --> 00:00:21,460
price of our portfolio is going to be

7
00:00:21,460 --> 00:00:24,140
the Black-Scholes price for a European
call option.

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00:00:27,970 --> 00:00:31,460
And we just start by taking he partial
derivative of the Black Scholes

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00:00:31,460 --> 00:00:35,070
price formula with respect to t and that's
what we're going to call theta.

10
00:00:40,260 --> 00:00:43,900
And so here its going to be a little bit
more complicated.

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00:00:43,900 --> 00:00:47,600
Because now have this e to the minus qt
minus t.

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00:00:47,600 --> 00:00:48,900
That terms has a t in it.

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00:00:51,490 --> 00:00:57,555
E to the minus or rt minus t, that has
term in it, sorry has a t in it.

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00:00:57,555 --> 00:01:03,140
Phi of d plus is a function of d plus and
d plus depends on t.

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00:01:03,140 --> 00:01:04,570
And so does d minus.

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00:01:04,570 --> 00:01:07,220
So I have to use the product rule on each
term here.

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00:01:09,990 --> 00:01:15,220
So what I'll, I'll get is the derivative
of the first term.

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00:01:15,220 --> 00:01:18,140
So here is where the minus signs cancel
out, that

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I confused myself with in the last set of
slides.

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00:01:20,160 --> 00:01:23,270
So here, it's minus.

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00:01:26,980 --> 00:01:28,880
I really should have done this with a
substitution.

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00:01:28,880 --> 00:01:31,440
I'm even confusing myself now.

23
00:01:31,440 --> 00:01:34,910
So, what's going to happen, I have a minus
sign here, that's going to pop

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00:01:34,910 --> 00:01:40,760
out in front, and a minus sign here,
that's going to also pop out in front.

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00:01:42,920 --> 00:01:45,590
And I think maybe the easiest way to look
at this

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00:01:45,590 --> 00:01:50,540
is I have e to the minus q times capital
t.

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00:01:50,540 --> 00:01:52,180
That doesn't depend on little t.

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00:01:52,180 --> 00:01:56,290
And then, I have e to the minus q times
minus t.

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00:01:56,290 --> 00:02:01,920
So, when I take the derivative, it's e to
the qt that's kind of.

30
00:02:01,920 --> 00:02:06,410
Playing the important part of that.
So I just get a q popping out in front.

31
00:02:06,410 --> 00:02:07,800
So it's q times t.

32
00:02:08,890 --> 00:02:10,670
And there really is no minus sign here.

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00:02:10,670 --> 00:02:12,960
It's just because of the actual way the
formula's

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00:02:12,960 --> 00:02:15,190
written down that it looks like there's
two minus signs.

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00:02:15,190 --> 00:02:17,550
But with respect to the variable I'm
taking

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00:02:17,550 --> 00:02:21,880
the derivative of, there's, it's e to the
qt.

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00:02:24,180 --> 00:02:26,130
So what I'm going to end up with is

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00:02:29,670 --> 00:02:35,350
derivative of the first function times the
second function.

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00:02:35,350 --> 00:02:38,180
Plus derivative of the second function.

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00:02:38,180 --> 00:02:40,080
So that's going to be the partial
derivative with

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00:02:40,080 --> 00:02:42,039
respect to t of phi of d plus.

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00:02:43,550 --> 00:02:45,650
Times the first function, that's just the
product rule.

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00:02:47,200 --> 00:02:50,240
And then this subtraction is coming from
here just because it's

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00:02:50,240 --> 00:02:53,310
the, the first term minus the second term
in the pricing formula.

45
00:02:53,310 --> 00:02:53,810
And

46
00:02:55,500 --> 00:02:56,820
then the same thing is going to happen.

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00:02:56,820 --> 00:03:03,830
This really is e to the rt times e to the
minus r capitol t.

48
00:03:04,870 --> 00:03:07,880
So when I take the derivative, it's just
going to be one little r that pops out.

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00:03:10,410 --> 00:03:16,540
So I have derivative of the first term,
times the second term,

50
00:03:16,540 --> 00:03:21,980
plus derivative of the second term, sorry,
second function, times the first function.

51
00:03:21,980 --> 00:03:25,340
So it's just the product rule again, on
the second term.

52
00:03:25,340 --> 00:03:26,600
Of the pricing formula.

53
00:03:29,650 --> 00:03:33,650
And so now I'm going to, look at this and,
maybe we can squeeze

54
00:03:33,650 --> 00:03:37,730
just a little bit more mileage out of our,
our, kind of intermediate result.

55
00:03:39,830 --> 00:03:42,240
but what I want to do is, right now I

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00:03:42,240 --> 00:03:46,655
have d minuses in the square brackets, and
d pluses.

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00:03:46,655 --> 00:03:48,310
In, in the first line.

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00:03:50,780 --> 00:03:54,240
And that result ended up working out well,
when I had

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00:03:54,240 --> 00:03:57,620
an s in one term, and a k in the second
term.

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00:03:57,620 --> 00:04:05,630
So, what I'm going to do is just rewrite
these so that I have the, the two q's.

61
00:04:05,630 --> 00:04:08,700
Now which ones are together, plus, plus,
minus, minus.

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00:04:08,700 --> 00:04:10,760
Phi to d t.
So when I have the two

63
00:04:13,300 --> 00:04:15,740
functions that have the partial derivative
with respect to

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00:04:15,740 --> 00:04:18,320
t of the phi, those will go in one group.

65
00:04:18,320 --> 00:04:20,660
And then the other two will go in the
second group.

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00:04:22,280 --> 00:04:25,390
So I end up with this expression here.
And

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00:04:27,440 --> 00:04:28,419
now I have this.

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00:04:31,430 --> 00:04:31,560
Oops.

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00:04:31,560 --> 00:04:33,570
So, now, it's this one that I can get rid
of.

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00:04:33,570 --> 00:04:38,520
So, I have a way to replace this with
something that depends on s.

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00:04:38,520 --> 00:04:40,850
So, actually something that's going to be
equal to this again.

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00:04:45,130 --> 00:04:45,750
So, let's see.

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00:04:48,580 --> 00:04:51,870
So, here I'm just working on the, the part
in the square brackets.

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00:04:51,870 --> 00:04:55,000
So, here I've used the chain rule so the

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00:04:55,000 --> 00:04:57,120
derivative, with respect to t of capital
phi of

76
00:04:57,120 --> 00:04:59,490
d plus, that's become lower case phi of d

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00:04:59,490 --> 00:05:04,180
plus times partial d plus with respect to
t.

78
00:05:04,180 --> 00:05:06,460
And, then, the same thing with the second
term the.

79
00:05:06,460 --> 00:05:10,060
Derivative with respect to t of capital
phi has become

80
00:05:10,060 --> 00:05:13,620
lowercase phi of d minus times the partial
of d

81
00:05:13,620 --> 00:05:14,890
minus with respect to t.
And

82
00:05:19,480 --> 00:05:23,080
now I can do the same trick that I had
done before.

83
00:05:23,080 --> 00:05:24,880
I have now partial derivative.

84
00:05:27,090 --> 00:05:29,080
So I can replace this guy here

85
00:05:32,890 --> 00:05:33,460
with this.

86
00:05:33,460 --> 00:05:36,480
That was that intermediate result that I
had before.

87
00:05:36,480 --> 00:05:40,810
And so I can rewrite this bit in the
square brackets like this.

88
00:05:40,810 --> 00:05:43,510
And now last time I had.

89
00:05:43,510 --> 00:05:46,360
The derivative with respect to sigma, this
time it's

90
00:05:46,360 --> 00:05:48,380
the derivative with respect to t, but I'm
still going

91
00:05:48,380 --> 00:05:51,190
to have this d, d plus minus d minus,

92
00:05:51,190 --> 00:05:53,320
which is going to simplify that expression
quite a bit.

93
00:05:55,800 --> 00:05:59,230
So I'm going to look at this quantity in a
little bit closer detail.

94
00:06:01,900 --> 00:06:04,040
So I can rewrite that again as the partial
derivative

95
00:06:04,040 --> 00:06:06,900
with respect to t just a d plus minus d
minus.

96
00:06:10,730 --> 00:06:14,210
And so in, in light gray here just, I tend
to confuse

97
00:06:14,210 --> 00:06:17,150
myself if I try to take a derivative of
something with a square

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00:06:17,150 --> 00:06:20,445
root just Like when I have a, a rational
function, if I have

99
00:06:20,445 --> 00:06:24,280
1 over sigma, I'd rather write that as
Sigma to the minus one.

100
00:06:24,280 --> 00:06:27,720
Here I have square root of t minus t, I'd
rather

101
00:06:27,720 --> 00:06:30,670
write that as the quantity t minus t to
the one half.

102
00:06:32,730 --> 00:06:36,210
And so now I'm set up to use the power
rule to take the derivative here.

103
00:06:37,890 --> 00:06:39,690
So what I'm going to end up with, I have
one half.

104
00:06:39,690 --> 00:06:43,290
So that's where this two comes from.

105
00:06:43,290 --> 00:06:46,830
Then I'll have the quantity, t minus t to
the negative one half.

106
00:06:46,830 --> 00:06:51,240
That's going to be where the 1 over square
root of t minus t comes from.

107
00:06:51,240 --> 00:06:54,400
And then I have this sigma left over in
front.

108
00:06:54,400 --> 00:06:56,900
And that's where the sigma in the top
comes from.

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00:06:58,560 --> 00:07:01,870
And then by the chain rule, so the
derivative

110
00:07:01,870 --> 00:07:05,670
that I'm taking here is with respect to t,
and

111
00:07:05,670 --> 00:07:09,120
in the inside function, t is negative, so
I have

112
00:07:09,120 --> 00:07:11,370
to multiply this by negative one by the
chain rule.

113
00:07:11,370 --> 00:07:16,150
So if I'd made a u substitution, and said
u is equal to capital t

114
00:07:16,150 --> 00:07:20,240
minus little t, then when I took the
derivative of u, I would get minus one.

115
00:07:24,930 --> 00:07:28,680
And so this is the expression for theta
that I had before.

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00:07:29,970 --> 00:07:34,110
So, it's the second term that I'm going to
be able to simplify using my, my trick.

117
00:07:36,080 --> 00:07:39,000
And I now have an expression for this
here, which is just

118
00:07:39,000 --> 00:07:42,340
minus sigma divided by 2 times the square
root of t minus t.

119
00:07:44,520 --> 00:07:47,270
And so that's going to finally give me my
expression for theta.

120
00:07:47,270 --> 00:07:50,810
So this one, unfortunately, doesn't
simplify much further than this.

121
00:07:52,150 --> 00:07:54,180
And so all I've done is just substitute
the

122
00:07:54,180 --> 00:07:58,900
expression we had last time into the first
term here.

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00:07:58,900 --> 00:08:00,650
And then left the other two terms alone.

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00:08:00,650 --> 00:08:00,650
[BLANK_AUDIO]

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.