Option Pricing: basic principles Definitions Value boundaries simple arbitrage relationships put-call parity

Transcription

1 Option Pricing: basic principles Definitions Value boundaries simple arbitrage relationships put-call parity Finance 7523 Spring 1999 M.J. Neeley School of Business Texas Christian University Assistant professor Steven C. Mann

2 "Moneyness" Call moneyness Call value asset price (S) Put moneyness Out of the money in the money (S < ) (S > ) Put value asset price (S) in the money out of the money (S < ) (S >)

3 Call Option Valuation "Boundaries" Option Value Option value must be within this region Intrinsic Value - Value of Immediate exercise: S - Define: C[S(),T;] =Value of American call option with strike, expiration date T, and current underlying asset value S() Result proof 1) C[,T; ] = (trivial) 2) C[S(),T;] max(, S() -) (limited liability) 3) C[S(),T;] S() (trivial) S (asset price)

4 European Call lower bound (asset pays no dividend) Option Value Option value must be within this region Pure time value : - B(,T) Intrinsic value: S - B(,T) S (asset price) Define: c[s(),t;] =Value of European call (can be exercised only at expiration) value at expiration position cost now S(T) < S(T) > A) long call + T-bill c[s(),t;] + B(,T) S(T) B) long stock S() S(T) S(T) position A dominates, so c[s(),t;] + B(,T) S() thus 4) c[s(),t;] Max(, S() - B(,T)

5 Example: Lower bound on European Call Option Value Option value must be within this region Pure time value : = $1.9 Intrinsic value: =S() S (asset price) Example: S() =$55. =$5. T= 3 months. 3-month simple rate=8.9%. B(,3) = 1/(1+.89(3/12)) = B(,3) = Lower bound is S() - B(,T) = = $6.9. What if c = $4.? Value at expiration position cash flow now S(T) $5 S(T) > $5 buy call - $ 4. S(T) - $5 buy bill paying short stock S(T) -S(T) Total + $ S(T)

6 Forward derivation of lower bound (asset pays no dividend) European call c[s(),t;], forward written on S, maturity T, delivery price : current market forward price f (,T)B(,T) = S() ; f (,T) = S()/B(,T) what is value of long forward with delivery price? Value = PV [ f (,T) - ] = B(,T) [ S()/B(,T) -] = S() - B(,T) consider buying call and selling forward with delivery. outcomes: S(T) > : use call to buy asset for, deliver against forward, receive (net = ) S(T) : buy spot asset at S(T), deliver against forward, receive (net > ) Example: S() =$55. =$5. T= 3 months. 3-month simple rate=8.9%. B(,3) = 1/(1+.89(3/12)) = B(,3) = Value of forward, delivery is: S() - B(,T) = $ = $6.9 Lower bound is S() - B(,T) = = $6.9. What if c = $5.? Value at expiration position cash flow now S(T) $5 S(T) > $5 buy call - $ 5. S(T) - $5 sell forward Total + $ S(T)

7 American and European calls on assets without dividends 5) American call is worth at least as much as European Call C[S(),T;] c[s(),t;] (proof trivial) 6) American call on asset without dividends will not be exercised early. C[S(),T;] = c[s(),t;] proof: C[S(),T;] c[s(),t;] S() - B(,T) so C[S(),T;] S() - B(,T) S() - and C[S(),T;] S() - Call is: worth more alive than dead Early exercise forfeits time value 7) longer maturity cannot have negative value: for T 1 > T 2: C(S(),T 1 ;) C(S(),T 2 ;)

8 Call Option Value Option Value No-arbitrage boundary: C >= max (, S - PV()) lower bound Intrinsic Value: max (, S-) S

9 Volatility Value : Call option Low volatility asset Probability High volatility asset Call payoff S(T) (asset value) Range of Asset prices at Option expiration

10 Volatility Value : Call option Example: Equally Likely "States of World" "State of World" Expected Position Bad Avg Good Value Stock A Stock B Probabili Range of Possible Asset prices at Option expirat Calls w/ strike=3: Call on A: 6 2 Call on B: 3 1

11 Put Option Valuation "Boundaries" Option Value Option value must be within this region Intrinsic Value - Value of Immediate exercise: - S Define: P[S(),T;] =Value of American put option with strike, expiration date T, and current underlying asset value S() Result proof 8) P[,T; ] = (trivial) 9) P[S(),T;] max(, - S()) (limited liability) 1) P[S(),T;] (trivial) S (asset price)

12 European Put lower bound (asset pays no dividend) B(,T) Option Value Intrinsic value: - S Option value must be within this region Negative Pure time value : B(,T) - B(,T) S() Define: p[s(),t;] =Value of European put (can be exercised only at expiration) value at expiration position cost now S(T) < S(T) > A) long put + stock p[s(),t;] + S() S(T) B) long T-bill B(,T) position A dominates, so p[s(),t;] + S() B(,T) thus 11) p[s(),t;] max (, B(,T)- S())

13 American puts and early exercise B(,T) Option Value Intrinsic value: - S Option value must be within this region Negative Pure time value : B(,T) - Define: P[S(),T;] =Value of American put (can be exercised at any time) 12) P[S(),T;] p[s(),t;] (proof trivial) B(,T) S() 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: =$25, S() = $1, six-month simple rate is 9.5%. Immediate exercise provides $24 (1+.95(6/12)) = $25.14 > $25

14 European Put-Call parity: Asset plus Put Asset Put S(T) Asset plus European put: S() + p[s(),t;]

15 European Put-Call parity: Bond plus Call Bond Call S(T) Bond + European Call: c[s(),t;] + B(,T)

16 European Put-Call parity Value at expiration Position cost now S(T) S(T) > Portfolio A: Stock S() S(T) S(T) put p[s(),t;] - S(T) total A: S + P S (T) Portfolio B: Call c[s(),t;] S(T) - Bill B(,T) total B: C + B(,T) S(T) European Put-Call parity: S() + p[s(),t;] = c[s(),t;] + B(,T)

17 Bull Spread: value at maturity ( J&T problem 3.1) S() = $5 value at maturity position: S(T) S(T) 5 S(T) > 5 Long call with strike at $45 S(T) - 45 S(T) -45 Short call w/ strike at $5 - [ S(T) - 5] net: S(T) Position value at T S(T)

18 Bear Spread: value at maturity ( J&T problem 3.2) S() = $3 value at maturity position: S(T) S(T) 35 S(T) >35 Long call with strike at $35 S(T) -35 Short call w/ strike at $25 -[S(T) - 25] - [ S(T) -25] net: 25 - S(T) -1 Position value at T S(T)

19 Butterfly Spread: value at maturity ( J&T problem 3.3) S() = $5 value at maturity position: S(T) S(T) 5 5 S(T) 55 S(T) > 55 Long call, = $45 S(T) - 45 S(T) - 45 S(T) - 45 Short 2 calls, = $5-2 [S(T) - 5] -2[S(T) - 5] Long call, = $55 S(T) - 55 net: S(T) S(T) 1 Position value at T S(T)

20 Straddle value at maturity ( J&T problem 3.4) S() = $25 value at maturity position: S(T) 25 S(T) > 25 Long call, = $25 S(T) - 45 Long put, = $ S(T) net: 25 - S(T) S(T) - 25 Position value at T 1 5 straddle Bottom straddle S(T) Bottom straddle: call strike > put strike: put = 23; call = 27