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)