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Option Valuation: basic concepts. Value boundaries Simple arbitrage relationships Intuition for the role of volatility. S. Mann, 2010. Option Value. Option value must be within this region. Call Option Valuation "Boundaries". Intrinsic Value - Value of Immediate exercise: S - K. 0. - PowerPoint PPT Presentation
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Option Valuation: basic concepts
S. Mann, 2010
Value boundariesSimple arbitrage relationshipsIntuition for the role of volatility
Call Option Valuation "Boundaries"
OptionValue
Define: C[S(0),T;K] =Value of American call option with strike K, expiration date T, and current underlying asset value S(0)
Result proof1) C[0,T; K] = 0 (trivial)
2) C[S(0),T;K] >= max(0, S(0) -K) (limited liability)
3) C[S(0),T;K] <= S(0) (trivial)
Intrinsic Value -Value of Immediate exercise: S - K
K S (asset price)
0
Option value must bewithin this region
European Call lower bound (asset pays no dividend)
OptionValue
Define: c[S(0),T;K] =Value of European call (can be exercised only at expiration)
value at expirationPosition cost now S(T) < K S(T) >KA)long call + T-bill c[S(0),T;K] + KB(0,T) K S(T) B)long stock S(0) S(T) S(T)
position A dominates, so c[S(0),T;K] + KB(0,T) >= S(0)
thus 4) c[S(0),T;K] >= Max(0, S(0) - KB(0,T)
Intrinsic value: S - K
KB(0,T) K S (asset price)0
Option value must bewithin this region
“Pure time value”: K - B(0,T)K
Example: Lower bound on European Call
OptionValue
Example: S(0) =$55. K=$50. T= 3 months. 3-month simple rate=4.0%. B(0,3) = 1/(1+.04(3/12)) = 0.99. KB(0,3) = 49.50.
Lower bound is S(0) - KB(0,T) = 55 – 49.50 = $5.50.What if C55 = $5.25?
Value at expirationPosition cash flow now S(T) <= $50 S(T) > $50buy call - $ 5.25 0 S(T) - $50buy bill paying K - 49.50 50 50short stock + 55.00 -S(T) -S(T)
Total + $0.25 50 - S(T) >= 0 0
Intrinsic value: 55 - 50
48.91 50 55 =S(0) S (asset price)0
Option value must bewithin this region
“Pure time value”: 50 - 48.91 = $1.09
American and European calls on assets without dividends
5) American call is worth at least as much as European Call
C[S(0),T;K] >= c[S(0),T;K] (proof trivial)
6) American call on asset without dividends will not be exercised early. C[S(0),T;K] = c[S(0),T;K]
proof: C[S(0),T;K] >= c[S(0),T;K] >= S(0) - KB(0,T)
so C[S(0),T;K] >= S(0) - KB(0,T) >= S(0) - K
and C[S(0),T;K] >= S(0) - K
Call is: worth more alive than deadEarly exercise forfeits time value
7) longer maturity cannot have negative value: for T1 > T2:
C(S(0),T1;K) >= C(S(0),T2;K)
Call Option Value
OptionValue
0
Intrinsic Value: max (0, S-K)
lower bound
No-arbitrage boundary: C >= max (0, S - PV(K))
0 K S
Volatility Value : Call option
Call payoff
Range of Asset prices at Option expiration
Pro
babi
lity
K S(T) (asset value)
Low volatility asset
High volatility asset
Volatility Value : Call option
Range of Possible Asset prices at Option expirat
Pro
ba
bil
ity
Example: Equally Likely "States of World"
"State of World" Expected Position Bad Avg Good ValueStock A 24 30 36 30Stock B 0 30 60 30
Calls w/ strike=30:Call on A: 0 0 6 2Call on B: 0 0 30 10
0
20
40
60
80
100
120
140
160
180
1 11 21 31 41 51 61 71 81 91 101111 121131141151161171181191201211 221231241251
Sto
ck P
rice
Day
lognormal evolution: mu = 1.5%, sigma=30.0%
)1,0(~)(
)](exp[)()1(
1,0
1,0
NtWwhere
tWhhtStS
Discrete-time lognormal evolution:
Put Option Valuation "Boundaries"
OptionValue
Define: P[S(0),T;K] =Value of American put option with strike K, expiration date T, and current underlying asset value S(0)
Result proof8) P[0,T; K] = K (trivial)
9) P[S(0),T;K] >= max(0, K - S(0)) (limited liability)
10) P[S(0),T;K] <= K (trivial)
Intrinsic Value -Value of Immediate exercise: K - S
K S (asset price)
0
Option value must bewithin this region
K
European Put lower bound (asset pays no dividend)
OptionValue
Define: p[S(0),T;K] =Value of European put (can be exercised only at expiration)
value at expirationposition cost now S(T) < K S(T) >K
A) long put + stock p[S(0),T;K] + S(0) K S(T)B) long T-bill KB(0,T) K K
position A dominates, so p[S(0),T;K] + S(0) >= KB(0,T)
thus 11) p[S(0),T;K] >= max (0, KB(0,T)- S(0))
Intrinsic value: K - S
KB(0,T) K S(0)
0
Option value must bewithin this region
Negative “Pure time value”: KB(0,T) - K
KB(0,T)
American puts and early exercise
OptionValue
Define: P[S(0),T;K] =Value of American put (can be exercised at any time)
12) P[S(0),T;K] >= p[S(0),T;K] (proof trivial)
However, it may be optimal to exercise a put prior to expiration (time value of money), hence American put price is not equal to European put price.
Example: K=$25, S(0) = $1, six-month simple rate is 9.5%.Immediate exercise provides $24 (1+ 0.095(6/12)) = $25.14 > $25
Intrinsic value: K - S
KB(0,T) K S(0)
0
Option value must bewithin this region
Negative “Pure time value”: KB(0,T) - K
KB(0,T)