## Risk Management

Basis: Long hedge: Effective purchase price = S2 – (F2 – F1) = F1 + (S2 – F2) = F1 + b2
Short hedge: Effective selling price = S2 + (F1 – F2) = F1 + (S2 – F2) = F1 + b2
Optimal hedge ratio:


F
S h =
Optimal number of contracts:
A
F
V
N h
V
=
Optimal number of contracts using β:
P
N
F
= 
CAPM:
( ) ( ) E r r r r = + − f m f 
Forward/futures price: F0
▪ No income: F0 = S0 e
r T
▪ Known dollar income: F0 = (S0 – I0) e
r T
▪ Known yield income: F0 = S0 e
(r – q) T
o Index: F0 = S0 e
(r – q) T
o Currency: F0 = S0 e
(r – rf)T
o Investment commodity:F0 = S0 e
(r + u) T or F0 = (S0 + U0) e
r T
Cost of carry (investment): F0 = S0 e c T
where c = r, r-q, r-rf , r+u
▪ Consumption commodity: F0 ≤ S0e
(r + u) T or F0 ≤ (S0 + U0) e
r T
Convenience yield: F0 = S0 e
(r+u – y) T or F0 = (S0 + U0) e
(r-y) T
Value of a long forward contract: ft = (Ft – F0) e
-r (T-t)
Conversion from continuous to m compounding: Rm = m(e
Rc/m
– 1)
Conversion from m to continuous compounding: Rc = m ln(1 + Rm/m)
Fixed side (swap):

=

= +

n
i
rt
B Le ke
i
r t
fix
i
n n
1
Floating side (swap):
B Le k e
r t
r t
fl
= + 
− −
1 1
1 1
Options
European option payoff: Long call = max (ST – K, 0)
Short call = –max (ST – K, 0) or min (K – ST, 0)
Long put = max (K – ST, 0)
Short put = –max (K – ST, 0) or min (ST – K, 0)
Long asset = ST – S0
Short asset = S0 – ST
European option bounds: Upper (no dividends): c ≤ S0 Lower (no div/s): c ≥ S0 – Ke
-rT
p ≤ Ke
-rT p ≥ Ke
-rT – S0
Upper (dividends): c ≤ S0 – D Lower (div/s): c ≥ (S0 – D) – Ke
-rT

p ≤ Ke
-rT
p ≥ Ke
-rT – S0 + D
Upper (yield): c ≤ S0e
-qT
Lower (yield): c ≥ S0e
-qT – Ke
-rT

p ≤ Ke
-rT
p ≥ Ke
-rT – S0e
-qT
Upper (currency): c ≤ S0e
-rfT
Lower (currency): c ≥ S0e
-rfT – Ke
-rT

p ≤ Ke
-rT
p ≥ Ke
-rT – S0e
-rfT
Upper (futures): c ≤ F0 e
-rT
Lower (futures): c ≥ (F0 – K)e
-rT

p ≤ Ke
-rT
p ≥ (K – F0)e
-rT
Put-call parity: European options: No dividends: c + Ke
-rT = p + S0
Dividends: c + Ke
-rT = p + (S0 – D)
Yield: c + Ke
-rT = p + S0e
-qT
Currency: c + Ke
-rT = p + S0e
-rfT
Futures: c + Ke
-rT = p + F0e
-rT
American: No dividends: C + Ke
-rT ≤ P + S0
Binomial Trees
Up and down parameters related to volatility:
,
t t u e d e    − 
= =
Single-step delta: No income:
0
( )
f f u d
S u d

 =

Yield
0 ( )
u d
q t
f f
u d e S

 =

Risk-neutral probability: No income:
r t d
p
u d
e

=

Yield:
( ) r q t d
p
u d
e
− 

=

Futures:
u d
d
p

=
1
Discounted expected payoff: f = e
–r t
[pfu + (1 – p)fd]
Black-Scholes option pricing
Lognormal distribution:
2
0
ln ln μ ,
2
S S T
 T T 
   
  + −            
European (no dividends): c = S0 N(d1) – Ke
-rT N(d2)
p = Ke
-rT N(–d2) – S0 N(–d1)
2
0
1
ln( / ) ( / 2) S K r T d
T

+ +
=
d 2 = d1− T
European (yield): c = S0e
-qTN(d1) – Ke
-rT N(d2)
p = Ke
-rT N(–d2) – S0e
-qTN(–d1)
2
0
1
ln( / ) ( / 2) S K r q T d
T

+ − +
=
European (currency): c = S0e
-rfTN(d1) – Ke
-rT N(d2)
p = Ke
-rT N(–d2) – S0e
-rfTN(–d1)
2
0
1
ln( / ) ( / 2) S K r rf T d
T

+ − +
=
European (futures): c = e
-rT[F0N(d1) – KN(d2)]
p = e
-rT [KN(–d2) – F0N(–d1)]

2
0
1
ln( / ) / 2 F K T d
T

+
=
Greek values
/ 2
2

1
( )
x N x e

=
Delta: ΔCALL = N(d1), ΔPUT = N(d1) – 1 with
2
0
1
ln( / ) ( / 2) S K r T d
T

+ +
= ,
ΔCALL = e-qT N(d1), ΔPUT = e-qT (N(d1)-1) with
2
0
1
ln( / ) ( / 2) S K r q T d
T

+ − +
=
Gamma:
S T
N d
0
( 1)
 = , or
0
( )1
qT N d e
S T 

 =
Vega:
 = S0 TN(d1), or 0 ( )1
qT  S T N d e− =

Hedging with futures versus hedging with spot: HF = e –rT HA, HF = e –(r-q)T HA
Table for N(x) when x ≤ 0
This table shows values for N(x) for x ≤ 0. The table should be used with interpolation. For example:
N(-0.1234)= N(-0.12) -0.34[N(-0.12)-N(-0.13)]
= 0.4522-0.34*(0.4522-0.4483)
=0.4509
x 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641
-0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
-0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
-0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483
-0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121
-0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
-0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
-0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
-0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867
-0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611
-1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379
-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
-1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
-1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
-1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681
-1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
-1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455
-1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367
-1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294
-1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233
-2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
-2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
-2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
-2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
-2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
-2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
-2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
-2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
-2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019
-2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014
-3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
-3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
-3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005
-3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
-3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
-3.5 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002
-3.6 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
-3.7 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
-3.8 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
-3.9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
-4.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Table for N(x) when x ≥ 0
This table shows values for N(x) for x ≥ 0. The table should be used with interpolation. For example:
N(0.6278)= N(0.62) + 0.78[N(0.63)-N(0.62)]
= 0.7324+0.78*(0.7357-0.7324)
=0.7350
x 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998
3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
4.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000