Global Research & Analytics Dpt.
Valuation & Pricing Solutions
By David REGO -Paris Office- Supported by Benoit GENEST -London Office- and Ziad Fares -Paris Office-
Free Pricer Content Detail of Generic Closed Formulas Solutions
April, 2013
International Business Solutions Advisors
Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
GENESIS
PHILOSOPHY WARNING OF NO PROPERTY
The department of “Global Research & This document and all its contents, Analytic” (GRA) is a team of passionate including texts, formulas, charts and any people. One unifying criteria in the GRA other material, are not the property of remains the dominant quantitative topics, CH&Cie. including the risk modeling part. As such, each member works regularly on WARNING OF NO topics likely to be of interest to the RESPONSIBILITY financial community. The results of this work are always freely downloadable and The information, formulas and codes fully shared with anyone interested. contained in this document are merely Because we consider “risk modeling” as a informative. hobby, we try to share ideas or researches There is no guarantee of any kind, express that we found useful within our day to day or implied, about the completeness or practice. accuracy of the information provided via this document. Any reliance you place on INTRODUCTION the descriptions, mathematical formulas or related graphs is therefore strictly at The following document is in response to your own risk. repeated requests from various players in the market and asking for quick access to a conventional financial pricing library. Formerly available on the internet, it is now more difficult to find on the web.
Our approach is to bring up to date all the work done by Espen Gaarder HAUG1 and to complete it with a summary document to assist the reader. This document is based on his great work. Moreover, we would like to thank him for his significant contribution in options pricing field and to share it with the financial community.
In an initiative to promote knowledge and expertise sharing, Chappuis Halder & Cie decided to put this Options Pricer on free access. It contains a charts generator and the detail sheets of each type of options.
1 The pricing formulas and codes are from his book: “The complete guide to option pricing formulas”, edited by McGraw- Hill (second edition).
TABLE OF CONTENTS
1.A. The Generalized Black & Scholes Formula ...... 1 1.B. The generalized Black and scholes options sensitivities ...... 2 2. European option on a stock with cash dividends ...... 8 3. The Black-Scholes model adjusted for trading day volatility (French) ...... 9 4. The merton’s Jump Diffusion Model option pricing...... 10 5. American Calls on stocks with known dividends ...... 11 6.A. American approximations: The Barone-Adesi and Whaley approximation ...... 12 6.B. American approximations: The Bjerksund and Stensland approximation ...... 14 7. The Miltersen and Schwartz commodity option model ...... 16 8. Executive stock options ...... 18 9. Forward start options ...... 19 10. Time switch options ...... 20 11.A. Simple chooser options ...... 21 11.B. Complex chooser optionS ...... 22 12. Options on options ...... 24 13. Writer extendible options ...... 26 14. Two assets correlation options ...... 28 15. Option to exchange one asset for another ...... 29 16. Exchange options on exchange options ...... 31 17. Options on the maximum or the minimum of two risky assets ...... 34 18. Spread option approximation ...... 36 19. Floating strike lookback options ...... 38 20. Fixed strike lookback options ...... 40 21. Partial-Time Floating-Strike Lookback Options ...... 42 22. Partial-Time Fixed-Strike Lookback Options ...... 44 23. Extreme-spread options ...... 46 24. Standard barrier options ...... 48 25. Double barrier options ...... 52 26. Partial-time single asset barrier options ...... 55 27. Two asset barrier options ...... 60 28. Partial time two asset barrier options ...... 63 29. Look-barrier options ...... 66 30. Soft-barrier options ...... 68
Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
31. Gap options ...... 70 32. Cash-or-nothing options ...... 71 33. Two asset cash-or-nothing options ...... 72 34. Asset-or-nothing options ...... 74 35. Supershare options ...... 75 36. Binary barrier options...... 76 37. Asian Options 1: Geometric average rate options ...... 86 38. Asian Options 2: The Turnbull and Wakeman arithmetic average approximation ...... 87 39. Asian Options 3: Levy's arithmetic average approximation ...... 88 40. Foreign equity options struck in domestic currency (Value in domestic currency) ...... 90 41. Fixed exchange rate foreign equity options - Quantos (Value in domestic currency) ...... 92 42. Equity linked foreign exchange options (Value in domestic currency) ...... 94 43. Takeover foreign exchange options ...... 96 44. European swaptions in the Black-76 model ...... 97 45. The Vasicek model for european options on zero coupon bonds ...... 98
Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
1.A. THE GENERALIZED BLACK & SCHOLES FORMULA
DESCRIPTION PAYOFFS
This function allows to price plain vanilla European call and put options, The payoffs of this model can be represented as follows (for 2 positions: using the Generalized Black and Scholes formula. buying a call in the left side and buying a put in the right side)
MATHEMATICAL FORMULA
The Generalized Black & Scholes formulas for a call and put are
()b r T rT Call S.().() e CND d12 X e CND d
rT() b r T Put X.().() e CND d21 S e CND d
Where d1 and d2 are defined by the following formulas NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas
S 2 ln bT INSTRUMENT PRICE X 2 d d21 d T 1 T The prices of this model according to the price of the underlying asset And and the time to maturity can be represented as follows (for 2 positions: S = Forward Asset price buying a call in the left side and buying a put in the right side) X = Strike price r = Risk-free rate T = Time to maturity (Years) b = Cost of carry = Volatility CND(x)= The Cumulative Normal Distribution Function
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
1.B. THE GENERALIZED BLACK AND SCHOLES OPTIONS SENSITIVITIES
DELTA
DESCRIPTION DELTA VALUE NEAR MATURITY IN DEPENDENCE OF THE SPOT PRICE
The parameter Delta, noted , is the sensitivity of the plain vanilla option’s Buying a call position is in the left side while buying a put position is in the price to the underlying asset price. right side.
MATHEMATICAL FORMULA
e(b r ). T .() CND d Call 1 e(b r ). T .( CND ( d ) 1) Put 1
S 2 logbT . X 2 With: d and d21 d T 1 . T DELTA VALUE EVOLUTION IN DEPENDENCE OF BOTH THE SPOT PRICE AND THE TIME TO MATURITY
Buying a call position is in the left side while buying a put position is in the right side.
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
GAMMA
DESCRIPTION GAMMA VALUE EVOLUTION IN DEPENDENCE OF BOTH THE SPOT PRICE AND THE TIME TO MATURITY The parameter Gamma, noted , is the sensitivity of the plain vanilla option’s delta to the underlying asset price. It measures the acceleration and The gamma is the same for a call or a put. curvature of the option’s price evolution. 0 MATHEMATICAL FORMULA 0 0 0 e(b r ). T .() CND d 0 1 0 option 0 ST. 0,98 0,82 0 0,66 0 2 0,5 0 S 0 logbT . Time to 0,34 X 2 Maturity 140150 With: d and d21 d T 0,18 120130 1 . T 100 110 80 90 0,02 70 60 50 Spot GAMMA VALUE NEAR MATURITY IN DEPENDENCE OF THE SPOT PRICE
The gamma is the same for a call or a put.
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
VEGA
DESCRIPTION VEGA VALUE EVOLUTION IN DEPENDENCE OF BOTH THE SPOT PRICE AND THE TIME TO MATURITY The parameter Vega, noted , is the sensitivity of the plain vanilla option’s price to the underlying asset volatility. The vega is the same for a call or a put.
MATHEMATICAL FORMULA 40 35 vega S. e(b r ). T . CND ( d ). T 30 option 1 25 20 2 15 0,98 S 0,82 logbT . 10 0,66 X 2 5 0,5 With: d and d21 d T 1 0 0,34 . T Time to 50 60 70 80 0,18 Maturity 90 100 110 120 VEGA VALUE NEAR MATURITY IN DEPENDENCE OF THE SPOT PRICE 130 0,02 140 Spot 150 The vega is the same for a call or a put.
10 9 8 7 6 5 4 3 2 1 0 50 60 70 80 90 100 110 120 130 140 150 Spot
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
THETA
DESCRIPTION THETA VALUE NEAR MATURITY IN DEPENDENCE OF THE SPOT PRICE
The parameter Theta, noted , is the sensitivity of the plain vanilla option’s Buying a call position is in the left side while buying a put position is in the price to the time to maturity. right side.
5 5 MATHEMATICAL FORMULA 0 0 50 60 70 80 90 100 110 120 130 140 150 -5 -5 50 60 70 80 90 100 110 120 130 140 150 S. e(b r ). T CND ( d ). -10 -10 1 (br ). -15 -15 Call -20 -20 2 T -25 -25 S..()...() e(b r ). T CND d r X e r . T CND d -30 -30 12 -35 -35 -40 -40 (b r ). T -45 -45 S. e CND ( d1 ). (b r ). T -50 -50 Put (b r ). S . e . CND ( d1 ) 2 T Spot Spot rT. r...() X e CND d2 THETA VALUE EVOLUTION IN DEPENDENCE OF BOTH THE SPOT PRICE AND THE TIME TO MATURITY S 2 logbT . X 2 With: d and d21 d T Buying a call position is in the left side while buying a put position is in the 1 . T right side.
5 5 0 0 -5 -5 -10 -10 -15 -15 -20 -20 -25 -25 0,98 -30 -30 0,98 0,82 -35 -35 0,82 0,66 -40 -40 0,66 0,5 -45 -45 0,5 -50 0,34 -50 0,34 Time to Time to 150 50 60 Maturity 0,18 130140 70 80 0,18 Maturity 110120 90 100 100 110 80 90 120 0,02 70 130 0,02 60 140 50 Spot Spot 150
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
RHO
DESCRIPTION RHO VALUE EVOLUTION IN DEPENDENCE OF BOTH THE SPOT PRICE AND THE TIME TO MATURITY The parameter Rho, noted , is the sensitivity of the plain vanilla option’s price to the interest rate. Buying a call position is in the left side while buying a put position is in the right side.
MATHEMATICAL FORMULA 100 0 rT 90 -10 if b 0:call T . X . e . CND ( d2 ) 80 -20 70 -30 else T .Call (S,X,T,r,b, ) 60 -40 call Generalized BS 50 -50 -60 0,98 40 0,98 0,82 30 -70 0,82 rT 0,66 20 -80 0,66 if b 0: put T . X . e . CND ( d2 ) 0,5 10 -90 0,5 -100 0,34 0 0,34 Time to Time to else T .Put (S,X,T,r,b, ) 150 50 60 put Generalized BS Maturity 0,18 130140 70 0,18 Maturity 120 80 90 100110 100 80 90 110 120 0,02 70 130 0,02 2 60 140 S 50 Spot Spot 150 logbT . X 2 With: d and d21 d T 1 . T
RHO VALUE NEAR MATURITY IN DEPENDENCE OF THE SPOT PRICE
Buying a call position is in the left side while buying a put position is in the right side. 2,2 0,2 2 0 1,8 -0,2 1,6 -0,4 1,4 -0,6 1,2 -0,8 1 -1 0,8 -1,2 0,6 -1,4 0,4 -1,6 0,2 -1,8 0 -2 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 Spot Spot
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
COST OF CARRY
DESCRIPTION CARRY SENSITIVITY VALUE NEAR MATURITY IN DEPENDENCE OF THE SPOT PRICE The parameter Rho, noted b , is the sensitivity of the plain vanilla option’s price to the cost of carry. Buying a call position is in the left side while buying a put position is in the right side. 3 0,2 MATHEMATICAL FORMULA 2,8 0 2,6 -0,2 50 60 70 80 90 100 110 120 130 140 150 2,4 -0,4 2,2 -0,6 (b r ). T 2 -0,8 1,8 -1 bCall T...() S e CND d1 1,6 -1,2 1,4 -1,4 1,2 -1,6 -1,8 (b r ). T 1 -2 b T...() S e CND d 0,8 -2,2 Put 1 0,6 -2,4 0,4 -2,6 0,2 -2,8 0 2 -3 S 50 60 70 80 90 100 110 120 130 140 150 logbT . Spot Spot X 2 With: d and d21 d T 1 . T CARRY SENSITIVITY VALUE EVOLUTION IN DEPENDENCE OF BOTH THE SPOT PRICE AND THE TIME TO MATURITY
Buying a call position is in the left side while buying a put position is in the right side.
140 0 130 120 -10 110 100 90 -20 80 70 -30 60 0,98 50 -40 0,98 0,82 40 0,82 0,66 30 0,66 20 -50 0,5 10 0,5 -60 0,34 0 0,34 Time to Time to 150 50 60 Maturity 0,18 130140 70 0,18 120 80 90 Maturity 100110 100 80 90 110 120 0,02 70 130 0,02 60 140 50 Spot Spot 150
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
2. EUROPEAN OPTION ON A STOCK WITH CASH DIVIDENDS
DESCRIPTION PAYOFFS
This function allows to price plain vanilla European call and put options with The payoffs of this model can be represented as follows (for 2 positions: cash dividend, using the original Black Scholes formula. Although simple, this buying a call in the left side and buying a put in the right side) approach can lead to significant mispricing and arbitrage opportunities. In particular, it will underprice options where the dividend is close to the 60 60 option's expiration date. 50 50 40 40
30 30
MATHEMATICAL FORMULA 20 20
10 10 Call S.().() CND d X erT CND d 12 0 0 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 rT Spot Spot Put X.().() e CND d21 S CND d NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas
2 S ln rT X 2 Where d ; d d T INSTRUMENT PRICE 1 T 2 1 The prices of this model according to the price of the underlying asset rt12 rt rt3 With S stock price NPVDividends s D1 . e D 2 . e D 3 . e and the time to maturity can be represented as follows (for 2 positions: Where buying a call in the left side and buying a put in the right side) s is the Stock price
60 60 DD12, and D3 are dividends for t12 , t and t3 . 50 50
X = Strike price 40 40 r = Risk-free rate 30 30 0,98 20 0,98 0,82 20 0,82 T = Time to maturity (Years) 0,66 10 10 0,66 0,5 0,5 0,34 0 0 0,34 Time to Time to = Volatility 140150 50 60 Maturity 0,18 130 70 80 0,18 Maturity 110120 90 90 100 100 CND(x)= The Cumulative Normal Distribution Function (CND) 80 110 120 0,02 70 130 0,02 60 140 50 Spot Spot 150
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
3. THE BLACK-SCHOLES MODEL ADJUSTED FOR TRADING DAY VOLATILITY (FRENCH)
DESCRIPTION PAYOFFS
This function allows to price plain vanilla European call and put options, The payoffs of this model can be represented as follows (for 2 positions: using the adjusted Generalized Black and Scholes formula. This adjustment buying a call in the left side and buying a put in the right side) was done by French in 1984 to take into consideration that the volatility is usually higher on trading days than on non-trading days. If trading days to 60 60 maturity are equals to calendar days to maturity, the output theoretical price 50 50 would be the same as the one generated by the Generalized Black Scholes 40 40 30 formula. 30 20 20
10 10
MATHEMATICAL FORMULA 0 0 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 Spot Spot ()b r T rT Call S..().() e CND d12 X e CND d NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas Put X.()..() erT CND d S e() b r T CND d 21INSTRUMENT PRICE S 2 lnbT . t X 2 d21 d t The prices of this model according to the price of the underlying asset Where : d1 and With: t and the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side) S = Stock Price
X = Strike Price 60 60 r = Risk-Free Rate 50 50 t = Trading time= Trading days until maturity / Trading days per year 40 40 30 30 T = Calendar Time = Calendar days until maturity / Calendar days per 0,98 20 0,98 0,82 20 0,82 0,66 0,66 10 10 year 0,5 0,5 0 0,34 0 0,34 Time to Time to CND(x)= The Cumulative Normal Distribution Function (CND) 150 50 60 Maturity 0,18 130140 70 80 0,18 Maturity 110120 90 100 100 110 80 90 120 = Standard Deviation 0,02 70 130 0,02 60 140 50 Spot Spot 150
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
4. THE MERTON’S JUMP DIFFUSION MODEL OPTION PRICING
DESCRIPTION PAYOFFS
This Model allows to price plain vanilla European call and put options, using The payoffs of this model can be represented as follows (for 2 positions: the Merton’s Jump Diffusion formula. This alternative model supposes a non- buying a call in the left side and buying a put in the right side) correlated Brownian motion and jumps. 60 60
50 50
MATHEMATICAL FORMULA 40 40
30 30 Ti eT() 20 20 Call Call(;;;;) S X T r ii 10 10 i0 i! 0 0 Ti eT() 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 Put Put(;;;;) S X T r Spot Spot iiNB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas i0 i!
2 22i 22 INSTRUMENT PRICE With : i z ; z T and The prices of this model according to the price of the underlying asset NB: Call and Put are calculated with the Generalized Black Scholes i i and the time to maturity can be represented as follows (for 2 positions: Function. buying a call in the left side and buying a put in the right side) With : 60 60 S = Stock Price 50 50 40 40 X = Strike Price 30 30 0,98 20 0,98 0,82 20 0,82 0,66 r = Risk-Free Rate 0,66 10 10 0,5 0,5 0 0,34 0 0,34 T = Calendar Time (time to Expiration on years) Time to Time to 150 50 60 Maturity 0,18 130140 70 80 0,18 Maturity 110120 90 100 100 110 CND(x)= The Cumulative Normal Distribution Function (CND) 80 90 120 0,02 70 130 0,02 60 140 = Standard Deviation 50 Spot Spot 150
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
5. AMERICAN CALLS ON STOCKS WITH KNOWN DIVIDENDS
DESCRIPTION WhereCall Sc ,, X T21 t = the price of European call with stock
This Model allows to price American Calls on stocks with known dividends, price of I and time to maturity Tt21 using the Roll-Geske-Whaley approximation formula. We consider here that the stock is paying a single discrete dividend yield. The method can be PAYOFFS extended to a multiple dividends. The payoff of this model can be represented as follows (for buying a call) 60
MATHEMATICAL FORMULA 50
40
t 30 Call( S Dert ). CND ( b ) ( S De rt ). M a , b ; 1 1 1 T 20 t 10 Xe.rT . Ma , b ; ( XDeCNDb ). rt ( ) 2 2T 2 0 50 60 70 80 90 100 110 120 130 140 150 Spot S Dert 2 ln rT X 2 NB: "Payoff" Chart represents prices seven days before expiry, not payoffs formulas With a ; a a T 1 T 2 1 S Dert 2 INSTRUMENT PRICE ln rT Sc 2 b1 ; b21 b T T The price of this model according to the price of the underlying asset and With: the time to maturity can be represented as follows (for buying a call) S = Stock Price; X = Strike Price; = Standard Deviation; r = Risk-Free Rate; 60 50 D = Cash Div.; T = Time to option expiration; t = time to dividend payout 40 30 CND(x)= The Cumulative Normal Distribution Function; M(a,b ; ρ) = The 0,98 0,82 20 0,66 10 Cumulative Bivariate Normal Distribution Function with upper integral limits 0,5 0,34 0 Time to a and b and correlation coefficient ρ. 140150 Maturity 0,18 120130 100110 80 90 S is the critical ex-dividend stock price that solves: 0,02 70 c 60 50 Spot CallSXT cc,,21 t S D X
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
6.A. AMERICAN APPROXIMATIONS: THE BARONE-ADESI AND WHALEY APPROXIMATION
S ** DESCRIPTION (b r ) T ** A11 1 e CND d ( S ) Q1 This quadratic approximation method by Barone-Adesi and Whaley (1987) S * allows to price American call and put options on an underlying asset with ()*b r T A211 e CND d ( S ) cost-of-carry rate b. When b > r, the American call value is equal to the Q2 European call value and can then be found by using the generalized Black- Scholes-Merton (BSM) formula. This model is fast and accurate for most MM (1)(1)4NNNN 22 (1)(1)4 practical input values. KK QQ ; 1222 MATHEMATICAL FORMULA 22rb M ; N= ; K 1 erT Q 22 With: S 2 Call( S , X , T ) A when S S * Call(,,) S X T GBS 2 * S S = Stock Price SX else b = cost of carry rate X = Strike Price Q1 S ** r = Risk-Free Rate PutGBS ( S , X , T ) A1 ** when S S Put(,,) S X T S T = Time to option expiration XS else CND(x)= The Cumulative Normal Distribution Function = Standard Deviation Where: S ** = the critical commodity price for put options S * = the critical commodity price for call options Call and Put are respectively the values of Europeans Call GBS GBS and put options computed by General Black Scholes formula. ** and S are determined by using the Newton-Raphson algorithm.
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
PAYOFFS INSTRUMENT PRICE
The payoffs of this model can be represented as follows (for 2 positions: The prices of this model according to the price of the underlying asset buying a call in the left side and buying a put in the right side) and the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side)
60 60 60 60 50 50 50 50
40 40 40 40 30 30 30 30 0,98 20 0,98 0,82 20 0,82 20 20 0,66 10 10 0,66 0,5 0,5 0 0 0,34 10 10 Time to 0,34 150 50 Time to 140 60 70 Maturity 0,18 120130 80 0,18 Maturity 0 110 90 0 90 100 100 80 110 120 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 0,02 70 130 0,02 60 140 Spot Spot 50 Spot Spot 150 NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
6.B. AMERICAN APPROXIMATIONS: THE BJERKSUND AND STENSLAND APPROXIMATION
2b k (2 1) DESCRIPTION 2
The Bjerksund and Stensland (1993) approximation can be used to price And the trigger price I is defined as American options on stocks, futures, and currencies. The method is analytical and extremely computer-efficient. Bjerksund and Stensland's approximation hT() B0 is based on an exercise strategy corresponding to a flat boundary / (trigger I B00 ( B B )(1 e ) and h ( T ) ( bT 2 T ) BB 0 price). It is demonstrated that the Bjerksund and Stensland approximation is somewhat more accurate for long-term options than the Barone-Adesi and r BXBXX and 0 max , Whaley approximation. 1 rb
If SI , it is optimal to exercise the option immediately, and the value MATHEMATICAL FORMULA must be equal to the intrinsic value of S-X. on the other hand, if br , it will never be optimal to exercice the American call option before expiration, and Call(X,S,T,r,b, ) = S (S, T, ,I, I) + (S, T, 1, I, I) - (S, T, 1, X, I) the value can be found using the generalized black-scholes formula. The - X (S, T, 0, I, I) + X (S,T, 0, X, I) value of the American put is given by the Bjerksund and Stensland put-call 2 transformation: 11bb r Where (IXI ) and 22 2 2 22 Put (S,X,T,r,b, )Call ( X , S ,T,r-b, b, ) The function (S, T, ,H, I) is given by Where Call(.) is the value of an American call with risk-free rate r-b and drift k –b. With the use of this transformation, it is not necessary to develop a IIS2ln( / ) (S, T, ,H, I)=eS CND ( d ) N d separate formula for an American put option. S T
1 r b ( 1) 2 T 2
1 2 ln(S / H ) b ( ) T 2 d T 14
Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
PAYOFFS INSTRUMENT PRICE
The payoffs of this model can be represented as follows (for 2 positions: The prices of this model according to the price of the underlying asset buying a call in the left side and buying a put in the right side) and the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side)
60 60 60 60 50 50 50 50
40 40 40 40 30 30 30 30 0,98 20 0,98 0,82 20 0,82 20 20 0,66 10 10 0,66 0,5 0,5 0,34 0 0 0,34 10 10 Time to Time to 140150 50 60 Maturity 0,18 130 70 80 0,18 Maturity 110120 90 0 0 90 100 100 80 110 120 50 60 70 80 90 100 110 120 130 140 150 0,02 70 130 0,02 50 60 70 80 90 100 110 120 130 140 150 60 140 Spot Spot 50 Spot Spot 150 NB : "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
7. THE MILTERSEN AND SCHWARTZ COMMODITY OPTION MODEL
t T2 t 2 DESCRIPTION 2 ()(,)(,)()u u s u s ds du u du z s f e FT 00u Miltersen and Schwartz (1998) developed an advanced model for pricing t t T options on commodity futures. The model is a three-factor model with (,).()(,)(,)u s ds u u s u s ds du xz f s f e stochastic futures price, a term structure of convenience yields and interest 0 uu rates. The model assumes commodity prices are log-normally distributed t and that continuously compounded forward interest rates and future (u ). ( u ) du . PFtt convenience yields are normally distributed (aka Gaussian). 0
Investigations using this option pricing model show that the time lag Where between the expiration on the option and the underlying futures will have a T significant effect on the option value. Even with three stochastic variables, ()(,)t t s ds Miltersen and Schwartz manage to derive a closed-form solution similar to a Pft t BSM-type formula. The model can be used to price European options on T commodity futures. ()()(,)(,)t t t s t s ds Ft s f e t MATHEMATICAL FORMULA This is an extremely flexible model where the variances and covariances admits several specifications. Call P F e xz CND()() d XCND d tT12
Where t is the time to maturity of the option, FT is a futures price with time to expiration T, and Pt is a zero coupon bond that expires on the option’s maturity.
ln(FX / )2 / 2 dT xz z , d d 1 2 1 z z
And the variances and covariance can be calculated as 16
Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
PAYOFFS INSTRUMENT PRICE
The payoffs of this model can be represented as follows (for 2 positions: The prices of this model according to the price of the underlying asset buying a call in the left side and buying a put in the right side) and the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side)
60 60 60 60 50 50 50 50
40 40 40 40 30 30 30 30 0,98 20 0,98 0,82 20 0,82 0,66 20 20 0,66 10 10 0,5 0,5 0,34 0 0 0,34 10 10 Time to Time to 140150 50 60 Maturity 0,18 130 70 80 0,18 Maturity 110120 90 0 90 100 100 0 80 110 120 0,02 70 130 0,02 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 60 140 Spot Spot 50 Spot Spot 150 NB : "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
8. EXECUTIVE STOCK OPTIONS
DESCRIPTION PAYOFFS
Executive stock options are priced by the Jennergren and Naslund (1993) The payoffs of this model can be represented as follows (for 2 positions: formula which takes into account that an employee or executive often loses buying a call in the left side and buying a put in the right side) his options if he has to leave the company before the option's expiration. 60 60
50 50
MATHEMATICAL FORMULA 40 40
30 30 T() b r T rT Call e Se CND()() d Xe CND d 20 20 12 10 10 Put eT Xe rT CND()() d Se() b r T CND d 0 0 21 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 Spot Spot NB : "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas Where:
ln(S / X ) ( b 2 / 2) T INSTRUMENT PRICE d d d T 1 T 2 1 The prices of this model according to the price of the underlying asset is the jump rate per year. The value of the executive option equals the and the time to maturity can be represented as follows (for 2 positions: T buying a call in the left side and buying a put in the right side) ordinary Black-Scholes option price multiplied by the probability e that the executives will stay with the firm until the option expires. 60 60 50 50 40 40 30 30 0,98 20 0,98 0,82 20 0,82 0,66 0,66 10 10 0,5 0,5 0 0,34 0 0,34 Time to Time to 150 50 60 Maturity 0,18 130140 70 80 0,18 Maturity 110120 90 100 100 110 80 90 120 0,02 70 130 0,02 60 Spot 140 50 Spot 150
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
9. FORWARD START OPTIONS
DESCRIPTION PAYOFFS
Forward start options with time to maturity T starts at-the-money or The payoffs of this model can be represented as follows (for 2 positions: proportionally in- or out-of-the-money after a known time t in the future. buying a call in the left side and buying a put in the right side) The strike is set equal to a positive constant times the asset price S after 60 60 50 50 the known time t. If is less than unity, the call (put) will start 1 - 40 40 percent in-the-money (out-of-the money); if is unity, the option will start 30 30 at-the-money; and if is larger than unity, the call (put) will start - 1 20 20 percentage out-of-the money (in-the-money). A forward start option can be 10 10 priced using the Rubinstein (1990) formula. 0 0 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 Spot Spot NB : "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas
MATHEMATICAL FORMULA INSTRUMENT PRICE
(b r ) t ( b r )( T t ) r ( T t ) Call Se e CND()() d12 e CND d The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (for 2 positions: (b r ) t r ( T t ) ( b r )( T t ) buying a call in the left side and buying a put in the right side) Put Se e CND()() d21 e CND d
Where: 60 60 50 50 ln(1/ ) (b 2 / 2)( T t ) 40 40 d ; d d T t 30 30 1 2 1 0,98 20 0,98 Tt 0,82 20 0,82 0,66 10 10 0,66 0,5 0,5 0,34 0 0 0,34 Time to Time to 140150 50 60 With: t= t1= Starting time of the option Maturity 0,18 130 70 80 0,18 Maturity 110120 90 90 100 100 80 110 120 0,02 70 130 0,02 60 140 50 Spot Spot 150
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
10. TIME SWITCH OPTIONS
DESCRIPTION PAYOFFS
A discrete time-switch call option, introduced by Pechtl (1995), pays an The payoffs of this model can be represented as follows (for 2 positions: amount At at maturity T for each time interval t the corresponding buying a call in the left side and buying a put in the right side) S asset price it has exceeded the strike price X. The discrete time-switch put option gives a similar payoff at maturity T for each time interval the asset price has been below the strike price X.
MATHEMATICAL FORMULA
n ln(S / X ) ( b 2 / 2) i t Call AerT N t NB : "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas i1 it
n 2 rT ln(S / X ) ( b / 2) i t INSTRUMENT PRICE Put Ae N t i1 it The prices of this model according to the price of the underlying asset With: and the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side) A: accumulated amount n T/ t If some of the option's total lifetime has already passed, it is necessary to add a fixed amount At Ae -rT m to the option pricing formula, where m is the number of time units where the option already has fulfilled its condition.
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
11.A. SIMPLE CHOOSER OPTIONS
DESCRIPTION PAYOFFS
A simple chooser option gives the right to choose whether the option is to be The payoff of this model can be represented as follows a standard call or put after a time t1, with strike X and time to maturity T2. The payoff from a simple chooser option at time t1 (t1 < T2) is 60
50 wSXtT(,,,)max1 2 CallGBS (,,), SXT 2 Put GBS (,, SXT 2 40
30 CallGBS (,,) S X T2 PutGBS (,,) S X T2 Where and are the general Black- 20
Scholes call and put formulas. 10
0 Spot50 60 70 80 90 100 110 120 130 140 150 NB : "Payoff" Chart represents prices seven days before expiry, not payoffs formulas
MATHEMATICAL FORMULA INSTRUMENT PRICE
A simple chooser option can be priced using the formula originally published The price of this model according to the price of the underlying asset and by Rubinstein (1991c): the time to maturity can be represented as follows
()b r T22 rT Payoff w Se CND()() d Xe CND d T2 60 ()b r T22 rT 50 Se CND()() y Xe CND y t1 40 30 Where 0,82 20 0,62 10 220,42 0 ln(S / X ) ( b / 2) T ln( S / X ) bT t / 2 Time to 2 2 1 0,22 140150 d ; y = Maturity 120130 100110 Tt80 90 210,02 70 60 50 Spot
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
11.B. COMPLEX CHOOSER OPTIONS
DESCRIPTION ln(S / X ) ( b 2 / 2) T ln(S / X ) ( b 2 / 2) T A Complex chooser option gives the right to choose whether the option is to y CC y PP 12TT T CP be a standard call option after a time t, with time to expiration C and strike t/ T t / T X T X 12CP C , or a put option with time to maturity P and strike P . The difference with regard to simple chooser options is that the calls and the puts will have S = The spot of the underlying asset different strikes ( and ) and maturities ( and ). b = The cost of carry r = The risk free rate X = The strike price The payoff from a complex chooser option at time t (t < , T) is t 1 = Time to when the holder must choose call or put wSX(,CPCP , XtTT ,,,)max Call GBS (, SXT CC ,), Put GBS (, SXT PP , T 2 = Time to maturity Call(,,) S X T Put(,,) S X T Where GBS C C and GBS P P are the general Black- = The time to maturity of the call. Scholes call and put formulas. = The time to maturity of the put. M(a,d; ρ) = The cumative bivariate normal distribution function. MATHEMATICAL FORMULA N(x) = The normal distribution function
A Complex chooser option can be priced using the formula originally And I is the solution to
(b r )( TCC t ) r ( T t ) (b r )( TP t ) r() Tp t published by Rubinstein (1991c): Ie NzXe(1 )C Nz ( 1 TtIe C ) NzXe ( 2 ) P Nz ( 2 Tt p ) 0 ln(I / X ) ( b 2 / 2)( T t ) ln(I / X ) ( b 2 / 2)( T t ) ()b r TCC rT CC PP wSe Mdy(,,)(,,)1 1 1 XeMdyCC 2 1 T 1 With z12 and z TCP t T t ()b r TPP rT Se Mdy( 1 , 2 , 2 )+ XeMdyPP ( 2 , 2 T , 2 )
ln(S / I ) ( b 2 / 2) t Where d1 d 2 d 1 t t
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
PAYOFFS INSTRUMENT PRICE
The payoff of this model can be represented as follows (for buying the The price of this model according to the price of the underlying asset and option): the time to maturity can be represented as follows (for buying the option)
60 60 50 50 40 40 30 30 20 0,82 20 10 0,62 0 0,42 10 50 Time to 60 70 0,22 80 90 Maturity 0 100 110 120 130 0,02 50 60 70 80 90 100 110 120 130 140 150 140 Spot Spot 150 NB : "Payoff" Chart represent prices seven days before expiry, not payoffs formulas
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
12. OPTIONS ON OPTIONS
DESCRIPTION PUT ON CALL This pricer allows to price options on options, namely, call on call, put on call, call on put and put on put. The pricing of such options is based on works of Payoff Max X2 CallGBS ( S , X 1 , T 2 );0 Geske(1979), Hodges and selby (1987) and Rubinstein (1999). Put XeMzyrT2(,,)(,,)() Se() b r T 2 Mzy XeNy rt 1 Call 1 2 2 1 1 2 2 MATHEMATICAL FORMULA Where the value I is found by solving the equation CALL ON CALL Call(,,) I X T t X GBS 1 2 1 2
Payoff Max CallGBS ( S , X1 , T 2 ) X 2 ;0 CALL ON PUT ()b r T2 rT 2 rt 1 Callcall Se M(,,)(,,)() z1 y 1 X 1 e M z 2 y 2 X 2 e N y 2 Payoff Max PutGBS ( S , X1 , T 2 ) X 2 ;0 2 rT() b r T rt ln(S / I ) ( b / 2) t1 Call X e2 M(,,)(,,)() z y Se 2 M z y X e 1 N y y1 y 2 y 1 t 1 put 1 2 2 1 1 2 2 t1 2 ln(S / X12 ) ( b / 2) T z1 z 2 z 1 T 2 PUT ON PUT T2
tT/ Payoff Max X2 PutGBS ( S , X 1 , T 2 );0 12 ()b r T2 rT 2 rt 1 Putput Se Mzy(,,)(,,)() 1 1 XeMzy 1 2 2 XeNy 2 2 X : strike price of the underlying option 1 Where the value I is found by solving the equation Put(,,) I X T t X GBS 1 2 1 2 X 2 : strike price of the option on the option
T2 : time to maturity of the underlying option
t1 : time to maturity of the option on option Call(,,) S X T GBS 12: the black-scholes generalized formula with T strike and time to maturity 2 M(a,d; ρ) = The cumative bivariate normal distribution function 24
Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
PAYOFFS INSTRUMENT PRICE
The payoffs of this model can be represented as follows: The prices of this model according to the price of the underlying asset and (for 4 positions: buying a call on call, buying a call on put, buying a put on the time to maturity could be represented as follows (for 4 positions: buying call, buying a put on put ) a call on call, buying a call on put, buying a put on call, buying a put on put )
60 60
50 50
40 40 30 30 20 0,82 20 0,82 0,62 10 10 0,62 0,42 0 0 0,42 Time to 50 Time to 0,22 140150 60 0,22 Maturity 130 70 80 Maturity 110120 90 100 100 110 80 90 120 0,02 70 130 0,02 60 140 50 Spot Spot 150 Call on Call Call on Put Call on Call Call on Put
60 60
50 50
40 40 30 30 20 0,82 20 0,82 0,62 10 10 0,62 0,42 0 0 0,42 Time to 50 Time to 0,22 140150 60 0,22 Maturity 130 70 80 Maturity 110120 90 100 100 110 80 90 120 0,02 70 130 0,02 60 140 50 Spot Spot 150 Put on Call Put on Put Put on Call Put on Put NB : "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
13. WRITER EXTENDIBLE OPTIONS
DESCRIPTION EXTENDIBLE PUT In general, extendible options are options where maturity can be extended. Such options can be found embedded in several financial contracts. For Payoff example, corporate warrants have frequently given the corporate issuer the (XSSX ) if Put(,,,,) S X X t T 11 right to extend the life of the warrants. Another example is options on real 1 2 1 2 PutGBS (S,X2 ,T 2 -t 1 ) else estate where the holder can extend the expiration by paying an additional fee. Pricing of such extendible options was introduced by Longstaff (1990). In Value particular, Writer extendible options can be exercised at their initial maturity t T rT date 1 but are extended to 2 if the option is out-of-the-money at . PutPut(,,)(,;) SXt Xe2 Mz Tz t GBS 1 1 2 1 2 2 1 ()b r T2 Se M(,;) z12 z MATHEMATICAL FORMULA Where EXTENDIBLE CALL 22 ln(S / X2 ) ( b / 2) T 2 ln( S / X 1 ) ( b / 2) t 1 z1 ; z 2 ; t 1 / T 2 Payoff Tt21 All formulas with (SXSX11 ) if X Call(,,,,) S X X t T 1 : strike price of the original maturity 1 2 1 2 CallGBS (S,X2 ,T 2 -t 1 ) else X 2 : strike price of the extendible maturity T : time to maturity of the extendible maturity Value 2
t1 : time to maturity of the extendible option Call Call(,,)(,;) S X t Se()b r T2 M z z GBS 1 1 1 2 Call(,,) S X T GBS 12: the black-scholes generalized formula with X erT2 M(,;) z T z t 2 1 2 2 1 T strike and time to maturity 2 M(a,d; ρ) = The cumative bivariate normal distribution function.
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
PAYOFFS INSTRUMENT PRICE
The payoffs of this model can be represented as follows (for 2 positions: The prices of this model according to the price of the underlying asset buying a call in the left side and buying a put in the right side) and the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side)
60 60
50 50
40 40 30 30 20 0,82 20 0,82 0,62 10 10 0,62 0,42 0 0 0,42 Time to 50 Time to 0,22 140150 60 0,22 Maturity 130 70 80 Maturity 110120 90 100 100 110 80 90 120 0,02 70 130 0,02 60 140 50 Spot Spot 150 NB : "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
14. TWO ASSETS CORRELATION OPTIONS
DESCRIPTION PAYOFFS
This call option pays off max(S2 - X2; 0) if S1 > X1 and 0 otherwise. The put The payoffs of this model can be represented as follows (for 2 positions: pays off max(X2 - S2) if S1 < X1 and 0 otherwise. These options are priced buying a call in the left side and buying a put in the right side) using the formulas of Zhang (1995).
MATHEMATICAL FORMULA
()b2 r T rT CallSe2 My(,;)(,;) 2 2 Ty 1 2 T XeMyy 2 2 1
rT ()b r T PutXeMyy(,;)(,;) Se2 My Ty T 2 2 1 2 2 2 1 2 NB : "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas Where
ln(S / X ) ( b 22 / 2) T ln( S / X ) ( b / 2) T INSTRUMENT PRICE yy1 1 1 1 ; 2 2 2 2 12TT 12The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (for 2 positions: With buying a call in the left side and buying a put in the right side)
S1 = The spot of the asset 1; S2 = The spot of the asset 2
X1 = Strike of asset 1; X 2 = Strike of asset 2
b1 = The cost of carry of asset 1 ; b2 = The cost of carry of asset 2;
1 = The volatility of the asset 1; 2 = The volatility of the asset 2; r = The risk free rate; = Correlation between assets 1 and 2; T = Time to expiry of the option M(a,d; ρ) = The cumative bivariate normal distribution function.
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
15. OPTION TO EXCHANGE ONE ASSET FOR ANOTHER
DESCRIPTION r = The risk free rate An exchange-one-asset-for-another option gives the holder the right, as its name indicates, to exchange one asset S for another S at expiration. The T = Time to expiry of the option 2 1 = Correlation between assets 1 and 2 payoff from an exchange-one-asset-for-another option is Q1 Max( Q1 S 1 Q 2 S 2 ;0) . = Quantity of asset 1 Q 2 = Quantity of asset 2 MATHEMATICAL FORMULA CND = The cumulative normal distribution function
EUROPEAN CALL AMERICAN CALL
()()b12 r T b r T Bjerksund and Stensland (1993) showed that an American Exchange one asset Call Q1 S 1 e CND()() d 1 Q 2 S 2 e CND d 2 for another option (S2 for S1) can be priced using a formula for pricing a plain where vanilla American option, with the underlying asset S1 with a risk-adjusted drift equal ln(Q S / Q S ) ( b b 2 / 2) T d1 1 2 2 1 2 ; d d T to b1-b2, the strike price equal to S2 , time to maturity T, risk free rate equal to r-b2, 1 T 2 1 and volatilityequal to (defined in the same way as it is for the European option). 22 1 2 2 1 2 and where
= The spot of the underlying asset 1
S2 = The spot of the underlying asset 2
b1 = The cost of carry of asset 1; b2 = The cost of carry of asset 2 1 = The volatility of the asset 1; 2 = The volatility of the asset 2
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
PAYOFFS INSTRUMENT PRICE
The payoffs of this model can be represented as follows (for 2 positions: The prices of this model according to the price of the underlying asset and buying a European call in the left side and buying an American call in the the time to maturity can be represented as follows for 2 positions: buying a right side) European call in the left side and buying an American call in the right side)
60 60
50 50
40 40 30 30 20 0,82 20 0,82 0,62 10 10 0,62 0,42 0,42 0 0 Time to Time to 0,22 150 50 60 0,22 Maturity 130140 70 80 Maturity 110120 90 100 100 110 80 90 120 0,02 70 130 0,02 60 140 50 Spot Spot 150 NB : "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
16. EXCHANGE OPTIONS ON EXCHANGE OPTIONS
DESCRIPTION ln(S / S ) ( b b 2 / 2) T An Exchange options on exchange options can be found embedded in y1 2 1 2 2 ; y y T 1 T 2 1 2 sequential exchange opportunities. An example described by Carr (1988) is a 2 bond holder converting into a stock and later exchanging the shares received ln(S / S ) ( b b 2 / 2) T y2 1 2 1 2 ; y y T for stocks of an acquiring firm. Those options can be priced analytically using 3 4 3 2 T2 a model introduced by Carr (1988).
22 1 2 2 1 2 MATHEMATICAL FORMULA [2] Option to exchange the option to exchange S2 for S1 in return for Q*S2 [1] Option to exchange Q*S2 for the option to exchange S2 for S1 The value of the option to exchange asset for in return for a The value of the option to exchange the option to exchange a fixed fixed quantity of asset is : quantity Q of asset S2 for the option to exchange asset for S1 is :
()()b2 r T 2 b 1 r T 2 Call S2 e M(,;/)(,;/) d 3 y 2 t 1 T 2 S 1 e M d 4 y 1 t 1 T 2 Call S e()()b1 r T 2 M(,;/)(,;/ d y t T S e b 2 r T 2 M d y t T 1 1 1 1 2 2 2 2 1 2 ()b21 r t QS23 e CND() d ()b21 r t QS22 e CND() d (b1 r )( T 2 t 1 ) Se1 where I is the unique critical price ratio I solving 1 (b2 r )( T 2 t 1 ) Se2 ln(S / IS ) ( b b 2 / 2) t 1 2 1 2 1 I1 N()() z 1 N z 2 Q d1 ; d 2 d 1 t 1 t 2 1 ln(I1 ) ( T 2 t 1 ) / 2 z1; z 2 z 1 T 2 t 1 2 ln(IS / S ) ( b b / 2) t Tt21 d2 1 2 1 1 ; d d t 3 t 4 3 1 1
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
[3] Option to exchange Q*S2 for the option to exchange S1 for S2 where
The value of the option to exchange a fixed quantity Q of asset S2 for the = The spot of the underlying asset 1 option to exchange asset S1 for is: = The spot of the underlying asset 2
b1 = The cost of carry of the asset 1 Call S e()()b2 r T 2 M(,;/)(,;/) d y t T S e b 1 r T 2 M d y t T 2 3 3 1 2 1 4 4 1 2 b2 = The cost of carry of the asset 2 ()b21 r t QS23 e CND() d r = The risk free rate 1 = Volatility of asset 1 [4] Option to exchange the option to exchange S1 for S2 in return for Q*S2 2 = Volatility of asset 2 t The value of the option to exchange the option to exchange asset for 1 = Time to expiration of the "original" option. in return for a fixed quantity Q of asset is : T 2 = Time to expiration of the underlying option (T2 > t1) = Correlation between assets 1 and 2. Call S e()()b1 r T 2 M(,;/)(,;/) d y t T S e b 2 r T 2 M d y t T 1 1 4 1 2 2 2 3 1 2 Q = Quantity of asset delivered if option is exercised ()b21 r t QS22 e CND() d CND = The cumulative normal distribution function
Se(b2 r )( T 2 t 1 ) where I is now the unique critical price ratio I 2 that solves 2 (b1 r )( T 2 t 1 ) Se1
N()() z1 I 2 N z 2 Q 2 ln(I2 ) ( T 2 t 1 ) / 2 z1; z 2 z 1 T 2 t 1 Tt21
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
PAYOFFS INSTRUMENT PRICE
The payoffs of this model can be represented as follows: The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows:
[1] Q2S2 for Option (S2 for S1) [2] Q2S2 for Option (S1 for S2) [1] Q2S2 for Option (S2 for S1) [2] Q2S2 for Option (S1 for S2)
[3] Option (S2 for S1) for Q2S2 [4] Option (S1 for S2) for Q2S2 [3] Option (S2 for S1) for Q2S2 [4] Option (S1 for S2) for Q2S2 NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
17. OPTIONS ON THE MAXIMUM OR THE MINIMUM OF TWO RISKY ASSETS
DESCRIPTION [2] CALL ON THE MAXIMUM OF TWO ASSETS These options on the minimum or maximum of two risky assets are priced by using the formula of Stulz (1982) witch have later been extended and Payoff: Max max( S12 , S ) X ,0 discussed by Johnson (1987), Rubinstein (1991) and others.
()()b12 r T b r T Callmin12(,,,),;(,;) S S X T S 1 e M y 112 d S e M y 2 d T 2 MATHEMATICAL FORMULA XerT 1 M ( y T , y T ; ) 1 1 2 2
[1] CALL ON THE MAXIMUM OF TWO ASSETS [3] PUT ON THE MINIMUM OF TWO ASSETS
Payoff: Max min( S , S ) X ,0 Payoff: Max X min( S , S ),0 12 12
()b1 r T rT Callmin(,,,),; S 1 S 2 X T S 1 e M y 1 d 1 Putmin(,,,) S 1 S 2 X T Xe Call min (,,0,) S 1 S 2 T Call min (,,,) S 1 S 2 X T ()()()b r T b r T b r ()b2 r T rT 1 1 2 Se2 Myd(,;)(,;) 2 T 2 XeMy 1 1 Ty 2 2 T Where Callmin ( S 1 , S 2 ,0, T ) S 1 e S 1 e CND ( d ) S 2 e CND ( d T )
22 ln(S1 / S 2 ) ( b 1 b 2 / 2) T ln( S 1 / X ) ( b 1 1 / 2) T Where dy ; 1 TT [4] PUT ON THE MAXIMUM OF TWO ASSETS 1 2 ln(S2 / X ) ( b 2 2 / 2) T y2 Payoff: Max X max( S12 , S ),0 2 T
rT 22 1 2 2 1 Putmax(,,,) S 1 S 2 X T Xe Call max (,,0,) S 1 S 2 T Call max (,,,) S 1 S 2 X T 1 2 2 1 2 ; 1 ; 2 ()()()b2 r T b 1 r T b 2 r Where Callmax ( S 1 , S 2 ,0, T ) S 2 e S 1 e CND ( d ) S 2 e CND ( d T )
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
PAYOFFS INSTRUMENT PRICE
The payoffs of this model can be represented as follows: The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows:
[1] Call on Minimum [2] Call on Maximum [1] Call on Minimum [2] Call on Maximum
60 60
60 60 50 50 50 50
40 40 40 40
30 30 30 30 0,98 20 20 0,98 0,82 0,82 20 20 0,66 10 10 0,66 0,5 0,5 Time to 0 0,34 0 0,34 Time to 10 10 Maturity 150 50 60 Maturity 130140 70 0,18 0,18 120 80 90 110 100 90 100 110 0 0 80 120 0,02 70 130 0,02 Spot Spot 60 140 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 50 Spot Spot 150 [3] Put on Minimum [4] Put on Maximum [3] Put on Minimum [4] Put on Maximum NB : "Payoffs" Charts represent prices seven days before expiry, not payoffs formula
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
18. SPREAD OPTION APPROXIMATION
DESCRIPTION Where 2 A European spread option is constructed by buying and selling equal number ln(ST ) ( / 2) d1 ; d 2 d 1 T of options of the same class on the same underlying asset but with different T strike prices or expiration dates. They can be valued using the standard Black Scholes (1973) model by performing the following transformation, as Q S e()b1 r T S 11 originally shown by Kirk(1995). ()b2 r T rT Q22 S e Xe
MATHEMATICAL FORMULA And the volatility can be approximated by
22 1 ( 2FF ) 2 1 2
CALL SPREAD Q S e()b2 r T Where F 22 Q S e()b2 r T XerT S1 22 Payoff: Max ( S1 S 2 X ,0) max 1,0 ( S 2 X ) SX 2 where ()b r T rT 2 S1 = The spot of the underlying asset 1 Call()()() Q2 S 2 e Xe SN d 1 N d 2
S2 = The spot of the underlying asset 2
Q1 = Quantity of asset 1 PUT SPREAD Q2 = Quantity of asset 2
b1 = The cost of carry of asset 1; b2 = The cost of carry of asset 2 S1 Payoff: Max ( X S1 S 2 ,0)max1 ,0( S 2 X ) 1 = The volatility of the asset 1; 2 = The volatility of the asset 2 SX2 = Correlation between assets 1 and 2 ()b2 r T rT Put()()() Q2 S 2 e Xe N d 2 SN d 1 r = The risk free rate T = Time to expiry of the option
CND = Cumulative Normal Distribution
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
PAYOFFS INSTRUMENT PRICE
The payoffs of this model can be represented as follows (for 2 positions: The prices of this model according to the price of the underlying asset and the buying a call in the left side and buying a put in the right side) time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side)
60 60 60 60
50 50 50 50
40 40 40 40
30 30 30 30 0,98 20 20 0,98 0,82 0,82 0,66 0,66 20 20 10 10 0,5 0,5 Time to 0 0,34 0 0,34 Time to Maturity 10 10 140150 50 60 Maturity 0,18 130 70 0,18 110 120 80 90 90 100 100 80 110 0,02 70 120 0,02 0 0 60 130 Spot Spot Spot 140 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 50 Spot 150 NB : "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
19. FLOATING STRIKE LOOKBACK OPTIONS
DESCRIPTION Floating Strike Lookback Put A floating strike lookback call gives the holder the right to buy the underlying asset at the lowest price observed, Smin , during the option’s lifetime. Payoff: Max ( Smax S ;0) Similarly, a floating-strike put gives the holder the right to sell the underlying if b 0 then asset at the higher price observed, Smax , during the option’s lifetime. rT() b r T PutSeNbmax()() 2 Se Nb 1 2b MATHEMATICAL FORMULA 2 2 Sb 2 SerT N b T ebT N() b 11 2bSmax Floating Strike Lookback Call
Payoff: Max ( S Smin ;0) And if b=0 we have
rT() b r T rT if b 0 then PutSeNbmax()()()()) 2 Se Nb 1 Se Tnb 1 Nbb 1 1 ()b r T rT Call Se N()() a1 S min e N a 2 Where 2b 2 2 2 rT Sb2 bT ln(S / S ) ( b / 2) T Se N a T e N() a max 11 b1 b 2 b 1 T 2bSmin T
And if b=0 we have
rT rT rT Call Se N() a1 S min e N () a 2 Se T n ()(()1) a 1 a 1 N a 1
Where PAYOFFS ln(S / S ) ( b 2 / 2) T amin a a T 1 2 1 The payoffs of this model can be represented as follows (for 2 positions: T buying a call in the left side and buying a put in the right side) 38
Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side)
NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas
INSTRUMENT PRICE
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
20. FIXED STRIKE LOOKBACK OPTIONS
When X S : Call erT ( S X ) Se() b r T N ( e ) S e rT N ( e ) DESCRIPTION max max 1 max 2 2b 2 Sb 2 2 In a fixed-strike lookback call, the strike is fixed in advance. At expiration, the SerT N e T ebT N() e 11 option pays out the maximum of the difference between the highest 2bSmax S observed price during the option's lifetime, max and the strike X, and 0. Similarly, a put at expiration pays out the maximum of the difference ln(S / S ) ( b 2 / 2) T S Where emax and e e T between the fixed-strike X and the minimum observed price min , and 0. 1 T 2 1 Fixed-strike lookback options can be priced using the Conze and Viswanathan (1991) formula. FIXED STRIKE LOOKBACK PUT
MATHEMATICAL FORMULA Payoff: Max ( X Smin ;0)
rT() b r T Put Xe N()() d21 Se N d FIXED-STRIKE LOOKBACK CALL 2b 2 2 rT Sb 2 bT Payoff: Max ( S X ;0) Se N d11 T e N() d max 2bX Call Se()b r T N()() d Xe rT N d 12 rT() b r T rT 2b When X Smin : Pute ( XS min ) Se Nf ( 1 ) SeNf min ( 2 ) 2 2 rT Sb 2 bT 2b Se N d11 T e N() d 2 2 2bX rT Sb2 bT Se N f11 T e N() f 2bS min ln(S / X ) ( b 2 / 2) T Where d1 ; d 2 d 1 T 2 ln(S / Smin ) ( b / 2) T T Where f1 and f 2 f 1 T T
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
PAYOFFS INSTRUMENT PRICE
The payoffs of this model can be represented as follows (for 2 positions: The prices of this model according to the price of the underlying asset and buying a call in the left side and buying a put in the right side) the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side)
NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
21. PARTIAL-TIME FLOATING-STRIKE LOOKBACK OPTIONS
DESCRIPTION The factor enables the creation of so called “fractional” lookback options In the partial-time floating-strike lookback options, the lookback period is at where the strike is fixed at some percentage above or below the actual the beginning of the option's lifetime. Time to expiration is T2, and time to extreme, 1for calls and 01 for puts. the end of the lookback period is t1 (t1 < T2). Except for the partial lookback period, the partial-time floating-strike lookback option is similar to a Where standard floating-strike lookback option. However, a partial lookback option must naturally be cheaper than a similar standard floating-strike lookback 2 ln(S / M02 ) ( b / 2) T option. Heynen and Kat (1994) have developed formulas for pricing these d1 d 2 d 1 T 2 T options. 2 2 (b / 2)( T21 t ) e1 e 2 e 1 T 2 t 1 MATHEMATICAL FORMULA Tt21 2 ln(S / M01 ) ( b / 2) t PARTIAL TIME FLOATING-STRIKE LOOKBACK CALL f1 f 2 f 1 t 1 t1
()b r T22 rT Call Se N()() d1 g 1 S min e N d 2 g 1 ln( ) ln( ) g12 g 2b T T t 2 2 2 1 S 22b t12 b T 2 M f1 d 1 g 1;/ t 1 T 2 rT Where 2 Smin Se 2b 2b 2 bT2 S if call e M d1 g 1, e 1 g 2 ; 1 t 1 / T 2 min M 0 Smax if put ()b r T2 Se M d1 g 1, e 1 g 2 ; 1 t 1 / T 2
rT2 Smin e M(,;/) f 2 d 2 g 1 t 1 T 2 2 b()() T2 t 1 b r T 2 e1 Se N ( e2 g 2 ) N ( f 1 ) 2b
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
PARTIAL TIME FLOATING-STRIKE LOOKBACK PUT PAYOFFS
rT22() b r T The payoffs of this model can be represented as follows (for 2 positions: Put Smax e N()() d 2 g 1 Se N d 1 g 1 buying a call in the left side and buying a put in the right side) 2b 2 S 22b t12 b T 2 M f1 d 1 g 1;/ t 1 T 2 rT Se 2 Smax 2b 2b 2 ebT2 M d g, e g ; 1 t / T 1 1 1 2 1 2 Se()b r T2 M d g, e g ; 1 t / T 1 1 1 2 1 2
rT2 Smax e M(, f 2 d2 g 1;/) t 1 T 2
2 NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas b()() T2 t 1 b r T 2 e1 Se N ( e2 g 2 ) N ( f 1 ) 2b INSTRUMENT PRICE
The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side)
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
22. PARTIAL-TIME FIXED-STRIKE LOOKBACK OPTIONS
DESCRIPTION PARTIAL TIME FIXED-STRIKE LOOKBACK PUT For Partial-Time Fixed-Strike Lookback option, the lookback period starts at a predetermined date t1 after the option contract is initiated. The partial time rT22() b r T fixed-strike lookback call payoff is given by the maximum of the highest Put Xe N()() d21 Se N d observed price of the underlying asset in the lookback period, in excess of 2b 2 S 22b T21 b t the strike price X, and 0. The put pays off the maximum of the fixed-strike 2 M d1 ,;/ f 1 t 1 T 2 rT2 price X minus the minimum observed asset price in the lookback period Se X 2b ()Tt21 Smin , and 0. This option is naturally cheaper than a similar standard bT2 e M e1, d 1 ; 1 t 1 / T 2 fixed-strike lookback option. Partial-time fixed strike lookback options can be priced analytically using a model introduced by Heynen and Kat (1994). ()b r T2 Se M e1, d 1 ; 1 t 1 / T 2
XerT2 M(,;/) f d t T MATHEMATICAL FORMULA 2 2 1 2 2 b()( T21 t b rT) 2 e1 Se N()() e21 N f PARTIAL TIME FIXED-STRIKE LOOKBACK CALL 2b
()b r T22 rT Call Se N()() d12 Xe N d 2b Where d1, e 1 and f 1 are defined under the floating-strike Lookback options. 2 S 22b T21 b t 2 M d1 f 1 ;/ t 1 T 2 rT2 Se X 2b ebT2 M e, d ; 1 t / T 1 1 1 2
()b r T22 rT Se M e1, d 1 ; 1 t 1 / T 2 Xe M ( f 2 , d 2 ; t 1 / T 2 ) 2 b()() T21 t b r T2 e1 Se N()() f12 N e 2b
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
PAYOFFS INSTRUMENT PRICE
The payoffs of this model can be represented as follows (for 2 positions: The prices of this model according to the price of the underlying asset and buying a call in the left side and buying a put in the right side) the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side)
NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
23. EXTREME-SPREAD OPTIONS
DESCRIPTION 2 2 These options are closer to lookback options than spread options, due to the rT2 And where e M ; k 1 ; = 2 way the time to maturity is divided. It is divided into two periods: one period 2(rD ) starting today and ending at time t1 , and another period starting at and 22 ln(M / S ) ; 12 r D 0.5 ; r D 0.5 ending at the maturity of the option T2 . Extreme spread options can be priced analytically using a model introduced by Bermin (1996). . 1 if Call 1 if extreme spread MaximumValue if 1 ; and = ;M -1 if Put 1 if reverse extreme spreadMinimumValue if 1 MATHEMATICAL FORMULA
EXTREME-SPREAD OPTIONS REVERSE EXTREME-SPREAD OPTIONS
(t1 , T 2 ) (0, t 1 ) (t , T ) (0, t ) Payoff( Call ) : Max ( Smax S max ;0) Payoff( Call ) : Max ( S1 2 S 1 ;0) min min (0,t1 ) ( t 1 , T 2 ) Payoff( Put ) : Max ( S S ;0) (0,t1 ) ( t 1 , T 2 ) min min Payoff( Put ) : Max ( Smax S max ;0)
DT2( D r )( T 2 t 1 ) DT 2 Se KN() A e Se SeDT2 KN()() A N B Spread KN( B ) N ( C ) ( k 1) e N ( D ) DT2 extreme SpreadReverseextreme ( k 1) e N ( C ) Se KN ( G ) N( E ) ( k 1) e N ( F ) (D r )( T2 t 1 ) DT 2 e Se( k 1) N ( H ) m 2 T 2 m 2 t 1 m 1 T 2 ()()T t T t Where ABC ; ; Where GH2 2 1 ; 1 2 1 T2 t 1 T 2 T t T t 2 1 2 1 m T m t m t DEF1 2 ; 1 1 ; 1 1 T t t 2 1 1
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
PAYOFFS INSTRUMENT PRICE
The payoffs of this model can be represented as follows (for 2 positions: The prices of this model according to the price of the underlying asset and buying a call in the left sides and buying a put in the right sides) the time to maturity can be represented as follows (for 2 positions: buying a call in the left sides and buying a put in the right sides)
Extreme Spread options Extreme Spread options
Reverse Extreme Spread options Reverse Extreme Spread options
NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
24. STANDARD BARRIER OPTIONS
DESCRIPTION MATHEMATICAL FORMULA
There are four types of single barrier options. The type flag "cdi" denotes a The different formulas use a common set of factors: down-and-in call, "cui" denotes an up-and-in call, "cdo" denotes a down-and- ()b r T rT out call, and "cuo" denotes an up-and-out call. Similarly, the type flags for A Se N()() x11 Xe N x T the corresponding puts are pdi, pui, pdo, and puo. A down-and-in option B Se()b r T N()() x Xe rT N x T comes into existence if the asset price, S, falls to the barrier level, H. An up- 22 (b r ) T 2( 1) rT 2 and-in option comes into existence if the asset price rises to the barrier level. C Se(/)()(/)() H S N y11 Xe H S N y T A down-and-out option becomes worthless if the asset price falls to the D Se(b r ) T(/)()(/)() H S 2( 1) N y Xe rT H S 2 N y T barrier level. An up-and-out option becomes worthless if the asset price rises 22 to the barrier level. In general a prespecified cash rebate K is included. It is E KerT N()(/)( x T H S2 N yT ) 2 2 paid out at option expiration if the option has not been knocked in during its lifetime for «in» barriers or if the option is knocked out before expiration for FKHSNzHSNz(/) ()(/) ( 2 T ) «out » barriers. Where European single barrier options can be priced analytically using a model introduced by Reiner and Rubinstein (1991). ln(SXSH / ) ln( / ) x12 (1 ) T ; x (1 ) T TT ln(H2 / SX ) ln( H / S ) y (1 ) T ; y (1 ) T 12TT 2 ln(H / S ) b / 22 2 r zT ; 22 ; T
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Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt.
”IN” BARRIERS “OUT” BARRIERS
Down-and-in Call S>H Down-and-out Call S>H
Payoff: Max ( S X ;0) if S H before T else K at expiration Payoff: Max ( S X ;0) if S> H before T else K at hit
Cdi (XHCE ) =1, 1 Cdo (XHACF ) =1, =1
Cdi (XHABDE ) =1, 1 Cdo (XHBDF ) =1, =1
Up-and-in Call S Payoff: Max ( S X ;0) if S H before T else K at expiration Payoff: Max ( S X ;0) if S< H before T else K at hit Cui (XHAE ) =-1, =1 Cuo (XHF ) =-1, =1 Cui (XHBCDE ) =-1, =1 Cuo (XHABCDF ) =-1, =1 Down-and-in put S>H Down-and-out put S>H Payoff: Max ( X S ;0) if S H before T else K at expiration Payoff: Max ( X S ;0) if S> H before T else K at hit Pdi (XHBCDE ) =1, = -1 PXHABCDFdo ( ) =1, =-1 Pdi (XHAE ) =1, = -1 PXHFdo ( ) =1, =-1 Up-and-in Put S Payoff: Max ( X S ;0) if S H before T else K at expiration Payoff: Max ( X S ;0) if S< H before T else K at hit Pui (XHABDE ) =-1, = -1 PXHBDFuo ( ) =-1, =-1 Pui (XHCE ) =-1, = -1 PXHACFuo ( ) =-1, =-1 49 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS The payoffs of this model can be represented as follows (Buying positions, Rebate = 3): Call Up and In Call Up and Out Put Up and In Put Up and Out Call Down and In Call Down and Out Put Down and In Put Down and Out NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 50 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. INSTRUMENT PRICE The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions, Rebate = 3): Call Up and In Call Up and Out Put Up and In Put Up and Out Call Down and In Call Down and Out Put Down and In Put Down and Out 51 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 25. DOUBLE BARRIER OPTIONS Where DESCRIPTION ln(SU2n / ( XL 2 n )) ( b 2 / 2) T ln( SU 2 n / ( FL 2 n )) ( b 2 / 2) T A double-barrier option is knocked either in or out if the underlying price d ; d 12TT touches the lower boundary L or the upper boundary U prior to expiration. ln(L2n 2 / ( XSU 2 n )) ( b 2 / 2) T ln( L 2 n 2 / ( FSU 2 n )) ( b 2 / 2) T The formulas below pertain only to double knock-out options. The price of a d ; d double knock-in call is equal to the portfolio of a long standard call and a 34TT short double knock-out call, with identical strikes and time to expiration. 22bn21 2 Similarly, a double knock-in put is equal to a long standard put and a short 1 ; 2n 1 1222 double knock-out put. Doublebarrier options can be priced using the Ikeda 22bn and Kuintomo (1992.) 21 T 1 ; F Ue 1 3 2 MATHEMATICAL FORMULA Where 1 and 2 determine the curvature of L and U. CALL UP-AND-OUT-DOWN-AND-OUT PUT UP-AND-OUT-DOWN-AND-OUT Payoff: Call ( S , U , L , T ) Max ( S k ;0) if L 13 2 nn2 1 n 1 2 ()b r T ULL UL Call Se N()()()() d N d N d N d nn 1 2 3 4 n N()() y12 T N y T LSUS LS n rT Put Xe 2 2 n1 3 n 1 2 n UL L N()() d T N d T n N()() y34 T N y T n 12 US LS XerT 1 n1 3 2 n 2 n L UL N()() d T N d T n N()() y12 N y n 34 LS US ()b r T Se n1 3 n L n N()() y34 N y US 52 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. Where ln(SU2n / ( EL 2 n )) ( b 2 / 2) T ln( SU 2 n / ( XL 2 n )) ( b 2 / 2) T y;; y E Le2 2T 12TT ln(L2n 2 / ( ESU 2 n )) ( b 2 / 2) T ln( L 2 n 2 / ( XSU 2 n )) ( b 2 / 2) T yy; 34TT CALL UP-AND-IN-DOWN-AND-IN Call CallGBS CallUp-and-Out-Down-and-Out PUT UP-AND-IN-DOWN-AND-IN Put PutGBS PutUp-and-Out-Down-and-Out 53 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS INSTRUMENT PRICE The payoffs of this model can be represented as follows (Buying positions): The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions): Call Out Call In Call out Call In 100 100 100 100 90 90 90 90 80 80 80 80 70 70 70 70 60 60 60 60 50 50 50 50 40 40 0,98 0,98 40 40 0,82 30 30 0,82 0,66 20 20 0,66 30 30 0,5 10 10 0,5 Time to 0,34 0 0 0,34 Time to 20 20 Maturity Maturity 140150 50 60 0,18 130 70 0,18 10 10 110 120 80 90 90 100 100 110 80 120 0 0 0,02 70 130 0,02 Spot Spot 60 140 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 50 Spot Spot 150 Put Out Put in Put Out Put in 100 100 100 100 90 90 90 90 80 80 80 80 70 70 70 70 60 60 60 60 50 50 50 50 40 40 0,98 30 0,98 40 40 0,82 30 0,82 0,66 20 20 0,66 30 30 0,5 10 10 0,5 Time to 0 0,34 0 0,34 Time to 20 20 Maturity 140150 50 Maturity 0,18 130 60 70 10 10 120 80 0,18 100 110 90 90 100 110 80 120 0 0 0,02 70 130 0,02 Spot Spot 60 140 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 50 Spot Spot 150 NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 54 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 26. PARTIAL-TIME SINGLE ASSET BARRIER OPTIONS DESCRIPTION MATHEMATICAL FORMULA For single asset partial-time barrier options, the monitoring period for a PARTIAL-TIME-START-OUT OPTIONS: UP-AND-OUT & DOWN-AND-OUT CALLS barrier crossing is confined to only a fraction of the option's lifetime. There TYPE A are two types of partial-time barrier options: partial-time-start (type A) and partial-time-end (type B). Partial-time-start barrier options (type A) have the 2( 1) monitoring period start at time zero and end at an arbitrary date before H Call Se()b r T2 M(,;)(,;) d e M f e expiration. Partial-time-end barrier options (Type B) have the monitoring A 1 1 1 3 S period start at an arbitrary date before expiration and end at expiration. 2 Partial-time-end barrier options (type B) are then broken down again into H XerT2 M(,;)(,;) d e M f e two categories: B1 and B2. Type B1 is defined such that only a barrier hit or 2 2 2 4 S crossed causes the option to be knocked out. There is no difference between Where up and down options. Type B2 options are defined such that a down-and-out call is knocked out as soon as the underlying price is below the barrier. 1 for an up-and-out call (CuoA ) Similarly, an up-and-out call is knocked out as soon as the underlying price is above the barrier. Partial-time barrier options can be priced analytically 1 for a down-and-out call (CdoA ) using a model introduced by Heynen and Kat (1994). 2 ln(S / X ) ( b / 2) T2 d1 ; d 2 d 1 T 2 T2 2 ln(S / X ) 2ln( H / S ) ( b / 2) T2 f1 ; f 2 f 1 T 2 T2 ln(S / H ) ( b 2 / 2) t 2ln(HS / ) e1 ; e e t ; e e 1tt 2 1 1 3 1 11 b 2 /2 t e e t ; ; 1 4 3 1 2 T2 55 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 2( 1) PARTIAL-TIME-START-IN OPTIONS (TYPE A) H C Se()b r T2 M(,;)(,;) g e M g e oB1 1 1 3 3 S The price of "in" options of type A can be found using "out" options in 2 H combination with plain vanilla call options computed by the Generalized rT2 Xe M(,;)(,;) g2 e 2 M g 4 e 4 Black-Scholes formula (GBS). S 2( 1) ()b r T H Up-and-in Call CuiA Call GBS C uoA 2 Se M(,;)(,;) d1 e 1 M f 1 e 3 S Down-and-in Call C Call C diA GBS doA 2 rT2 H Xe M(,;)( d22 e M fe24,;) S PARTIAL-TIME-END-OUT OPTIONS (TYPE B) 2( 1) H Out Call Type B1: No difference between up-and-out and down-and-out ()b r T2 Se M(,;)(,;) g1 e 1 M g 3 e 3 options S H 2 When x > H, the knock-out call value is given by: rT2 Xe M(,;)(,;) g2 e 2 M g 4 e 4 2( 1) H S C Se()b r T2 M(,;)(,;) d e M f e oB1 1 1 1 3 Where S 2 ln(S / H ) ( b 2 / 2) T rT2 H 2 Xe M(,;)(,;) d2 e 2 M f 2 e 4 g1; g 2 g 1 T 2 S T2 When X < H, the knock-out call value is given by: 2ln(HS / ) g g ; g g T 3 1 T 4 3 2 2 56 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. Down-and-Out Call type B2 (case of X < H) 2( 1) H C Se()b r T2 M(,;)(,;) g e M g e doB2 1 1 3 3 S H 2 rT2 Xe M(,;)(,;) g2 e 2 M g 4 e 4 S Up-and-Out Call type B2 (case of X < H) 2( 1) H C Se()b r T2 M(,;)(,;) g e M g e uoB2 1 1 3 3 S 2 rT2 H Xe M(,;)(,;) g2 e 2 M g 4 e 4 S 2( 1) H ()b r T2 Se M(,;)(,;) d1 e 1 M e 3 f 1 S 2 rT2 H Xe M(,;) d22 e M (,;)ef42 S 57 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS Call Out (B1) Put Out (B1) 70 70 The payoffs of this model can be represented as follows (Buying positions): 60 60 50 50 Call Up and Out (A) Call Down and Out (A) 40 40 60 60 30 30 50 50 20 20 40 40 10 10 30 30 0 0 Spot Spot 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 20 20 10 10 Call Up and Out (B2) Call Down and Out (B2) 70 70 0 0 Spot Spot 60 60 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 50 50 Put Up and Out (A) Put Down and Out (A) 40 40 60 60 30 30 50 50 20 20 40 40 10 10 30 30 0 0 Spot Spot 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 20 20 10 10 Put Up and Out (B2) Put Down and Out (B2) 70 70 0 0 Spot Spot 60 60 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 50 50 40 40 30 30 20 20 10 10 0 0 Spot Spot 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 58 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. INSTRUMENT PRICE Call Out (B1) Put Out (B1) 70 70 The prices of this model according to the price of the underlying asset and 60 60 the time to maturity can be represented as follows (Buying positions): 50 50 40 40 30 30 Call Up and Out (A) Call Down and Out (A) 0,98 0,98 0,82 20 20 0,82 0,66 0,66 10 10 60 60 0,5 0,5 Time to 0,34 0 0 0,34 Time to Maturity 50 50 140150 50 60 Maturity 0,18 130 70 0,18 120 80 90 100 110 90 100 110 40 40 80 120 0,02 70 130 0,02 60 Spot Spot 140 30 30 50 150 0,98 20 20 0,98 0,82 0,82 0,66 10 10 0,66 0,5 0,5 Call Up and Out (B2) Call Down and Out (B2) Time to 0 0,34 0 0,34 Time to Maturity 50 Maturity 70 70 140150 60 0,18 130 70 0,18 120 80 90 110 100 90 100 110 60 60 80 120 0,02 70 130 0,02 60 140 50 Spot Spot 150 50 50 40 40 30 30 0,98 0,98 0,82 20 20 0,82 0,66 0,66 Put Up and Out (A) Put Down and Out (A) 10 10 0,5 0,5 Time to 0 0,34 0 0,34 Time to 60 60 Maturity 140150 50 60 Maturity 0,18 120130 70 80 0,18 100 110 90 50 50 90 100 110 80 120 0,02 70 130 0,02 60 140 40 40 50 Spot Spot 150 30 30 0,98 20 20 0,98 0,82 0,82 0,66 10 10 0,66 0,5 0,5 Put Up and Out (B2) Put Down and Out (B2) Time to 0,34 0 0 0,34 Time to Maturity 70 70 140150 50 60 Maturity 0,18 130 70 0,18 120 80 90 100 110 90 100 110 60 60 80 120 0,02 70 130 0,02 60 140 50 Spot Spot 150 50 50 40 40 30 30 0,98 0,98 0,82 20 20 0,82 0,66 0,66 10 10 0,5 0,5 Time to 0,34 0 0 0,34 Time to Maturity 140150 50 60 Maturity 0,18 130 70 0,18 120 80 90 100 110 90 100 110 80 120 0,02 70 130 0,02 60 140 50 Spot Spot 150 59 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 27. TWO ASSET BARRIER OPTIONS DESCRIPTION TWO-ASSET "OUT" BARRIERS In a two asset barrier option, the underlying asset S1 determines how much Down-and-out call (Cdo ) 1; -1 the option is in or out-of-the-money. The other asset S2 is the trigger asset Payoff:Max(SXH ;0) if S before T else 0 at hit which is linked to barrier hits. Two-asset barrier options can be priced 12 analytically using a model introduced by Heynen and Kat (1994). Up-and-out call (Cuo ) 1; 1 Payoff:Max(SXH ;0) if S before T else 0 at hit 12 MATHEMATICAL FORMULA Down-and-out put (Pdo ) 1; 1 Payoff:Max ( X S ;0) if S H before T else 0 at hit M(,;) d e 12 11 Up-and-out put (P ) 1; 1 ()b12 r T uo w S1 e 2( )ln(HS / ) exp2 1 2 2 M ( d , e ; ) Payoff:Max ( X S ;0) if S H before T else 0 at hit 2 33 12 2 TWO-ASSET "IN" BARRIERS rT 222 ln(HS / ) Xe M(,;)exp d2 e 2 2 M (,;) d 4 e 4 2 Down-and-in call (Cdi ) C di Call GBS C do Payoff:Max(SXH ;0) if S before T else 0 at expiration ln(SXT / ) (2 ) 12 d1 1 1 ; d d T 1 2 1 1 Up-and-in call (Cui ) C ui Call GBS C uo 1 T Payoff:Max(S X ;0) if S H before T else 0 at expiration 2 ln(HSHS / ) 2 ln( / ) 1 2 d d 22; d d 3 1 4 2 Down-and-in put (Pdi ) P diPut GBS P do 22TT Payoff:Max ( X S12 ;0) if S H before T else 0 at expiration ln(HSTHS /2 ) ( 2 1 2 ) 2ln( / 2 ) Up-and-in put (Pui ) P diPut GBS Puo e1;; e 2 e 1 1 T e 3 e 1 22TTPayoff:Max ( X S12 ;0) if S H before T else 0 at expiration 2ln(HS / ) e e 2 ; b 22 / 2; b / 2 4 1 T 1 1 1 2 2 2 2 60 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS The payoffs of this model can be represented as follows (Buying positions, Payoff1= 20): Call Up and In Call Up and Out Put Up and In Put Up and Out Call Down and In Call Down and Out Put Down and In Put Down and Out NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 61 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. INSTRUMENT PRICE The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions, Payoff1= 20): Call Up and In Call Up and Out Put Up and In Put Up and Out Call Down and In Call Down and Out Put Down and In Put Down and Out 62 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 28. PARTIAL TIME TWO ASSET BARRIER OPTIONS DESCRIPTION MATHEMATICAL FORMULA Partial-time two-asset barrier options are similar to standard two-asset M(,;/) d e t T barrier options, except that the barrier hits are monitored only for a fraction 1 1 1 2 w S e()b12 r T of the option's lifetime. The option is knocked in or knocked out if Asset 2 1 2(2 1 2 )ln(HS / 2 ) exp2 M ( d3 , e 3 ; t 1 / T 2 ) hits the barrier during the monitoring period. The payoff depends on Asset 1 2 and the strike price. Partial-time two-asset barrier options can be priced analytically using a model introduced by Bermin (1996). M(,;/) d2 e 2 t 1 T 2 rT Xe 2 ln(HS / ) 22 exp2 M ( d4 , e 4 ; t 1 / T 2 ) 2 2 ln(SXT1 / ) ( 1 1 ) 2 d1; d 2 d 1 1 T 2 12T 2 ln(HSHS / ) 2 ln( / ) 22 d3 d 1 ; d 4 d 2 2TT 2 2 2 ln(H / S2 ) ( 2 1 2 ) t 1 2ln( H / S 2 ) e1;; e 2 e 1 1 t 1 e 3 e 1 2tt 1 2 1 2ln(HS /2 ) 22 e4 e 2 ; 1 b 1 1 / 2; 2 b 2 2 / 2 21t TWO-ASSET "OUT" BARRIERS cf specification Pricer n°27: TWO ASSET BARRIER OPTIONS TWO-ASSET "IN" BARRIERS cf specification Pricer n°27: TWO ASSET BARRIER OPTIONS 63 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS The payoffs of this model can be represented as follows (Buying positions, Payoff1= 20): Call Up and In Call Up and Out Put Up and In Put Up and Out Call Down and In Call Down and Out Put Down and In Put Down and Out NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 64 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. INSTRUMENT PRICE The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions, Payoff1= 20): Call Up and In Call Up and Out Put Up and In Put Up and Out Call Down and In Call Down and Out Put Down and In Put Down and Out 65 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 29. LOOK-BARRIER OPTIONS 2b 2 DESCRIPTION S m1 t 1 k 1 T 2 SM ,; 2 X tT A look-barrier option is the combination of a forward starting fixed strike rT2 12 e 2b Lookback option and a partial time barrier option. The option’s barrier 2b 2 monitoring period starts at time zero and ends at an arbitrary date before H m22 h 1 t 1 h k 1 T 2 HM ,; expiration. If the barrier is not triggered during this period, the fixed strike X tT 12 Lookback option will be kick off at the end of the barrier tenor. Lookback 22 barrier options can be priced analytically using a model introduced by ()Tt b() T t 11Ne2 2 1 21 Bermin (1996). 22bb Tt21 ()b r T2 rT2 Se g12 e Xg 1()Tt 2 1 MATHEMATICAL FORMULA N Tt 21 A Where N ( x ) N ( x ) and M ( a , b ; ) M a , b ; m2 t 1 k 2 T 2 M n ,; 2 1 if up-and-out call min(hk , ) when 1 tT12 ()b r T2 ; m A Se 1 2b 1 if down-and-out put max(hk , ) when 1 2 2/2h m22 h 2 t 1 h k 2 T 2 eM ,; tT 22t1 12 hln( H / S ); k ln( X / S );12 b / 2; b / 2; T2 m t k T M 1 1,; 1 2 h t 22 h t m t m 2 h t tT 2 1222hh / 2 1 2 1 2 / 2 1 rT 12 g1 N e N N e N eX2 t t t t 1 1 1 1 2/h 2 m22 h 1 t 1 h k 1 T 2 eM ,; tT 12 h t 22 h t m t m 2 h t g N1 1 e122hh / N 1 1 N 1 1 e 1 / N 1 1 2 t1 t 1 t 1 t 1 66 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS INSTRUMENT PRICE The payoffs of this model can be represented as follows (Buying positions): The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions): Call Up and Out Call Up and In Call Up and Out Call Up and In Put Down and Out Put Down and In Put Down and Out Put Down and In NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 67 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 30. SOFT-BARRIER OPTIONS DESCRIPTION Where A soft-barrier option is similar to a standard barrier option, except that the 1 if down-and-in call barrier is no longer a single level. Rather, it is a soft range between a lower level and an upper level. Soft-barrier options are knocked in or knocked out 1 if up-and-in put proportionally. Introduced by Hart and Ross (1994), the valuation formula ln(U22 / ( SX )) ln( U / ( SX )) can be used to price soft-down-and-in call and soft-up-and-in put options. d1 T; d 2 d 1 ( 0.5) T ; d 3 ( 1) T Soft-barrier options can be priced analytically using a model introduced by TT 2 Hart and Ross (1994). ln(L / ( SX )) d4 d 3 ( 0.5) T ; e 1 T ; e 2 e 1 ( 0.5) T T ln(L2 / ( SX )) e3 ( 1) T ; e 4 e 3 ( 0.5) T T 2 0.522TT ( 0.5)( 0.5) 0.5 ( 0.5)( 1.5) b /2 MATHEMATICAL FORMULA ee; ; 12 2 SOFT "IN" BARRIERS SOFT « OUT » BARRIERS 1 A Soft down-and-out call = standard call - soft down-and-in call. UL Soft up-and-out put = standard put - soft up-and-in put 0.5 U 2 N d1 1 N() d 2 Standard Calls and puts are calculated with the generalized Black scholes ()SX 0.5 SX A Se(b r ) T S 2 formula. 0.5 2( 0.5) L2 N()() e N e 1 1 2 SX 0,5 U 2 N()() d3 2 N d 4 ()SX 0.5 SX XerT S 2( 1) 0,5 2( 0.5) L2 N()() e3 2 N e 4 SX 68 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS INSTRUMENT PRICE The payoffs of this model can be represented as follows (Buying positions): The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions): Call Down and In Call Down and Out Call Down and In Call Down and Out Put Up and In Put Up and Out Put Up and In Put Up and Out NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 69 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 31. GAP OPTIONS DESCRIPTION PAYOFFS Gap options are similar to plain options, except for the payoff. The payoff is a The payoffs of this model can be represented as follows (for 2 positions: function of the exercise price. The payoff on a gap option depends on all of buying a call in the left side and buying a put in the right side) the factors of a plain option, but it is also affected by the gap amount, which can be either positive or negative. A gap call option is equivalent to being long an asset-or-nothing call and short a cash-or-nothing call. A gap put option is equivalent to being long a cash-or-nothing put and short an asset- or-nothing put. Gap options can be priced analytically using a model introduced by Reiner and Rubinstein (1991). MATHEMATICAL FORMULA NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 0 if S X1 Payoff(): Call SX if S > X1 2 INSTRUMENT PRICE ()b r T rT Call Se N()() d1 X 2 e N d 2 The prices of this model according to the price of the underlying asset 0 if S X1 and the time to maturity can be represented as follows (for 2 positions: Payoff(): Put XS if S < X buying a call in the left side and buying a put in the right side) 2 1 rT() b r T Put Xe2 Nd()() 2 Se Nd 1 Where 2 ln(S / X1 ) ( b / 2) T d1 ; d 2 d 1 T T Notice that the payoff from this option can be negative, depending on the settings of Xi and X2 70 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 32. CASH-OR-NOTHING OPTIONS DESCRIPTION PAYOFFS In a cash-or-nothing option, a predetermined amount is paid if the asset is, at The payoffs of this model can be represented as follows (for 2 positions: expiration, above for a call or below for a put some strike level, independent buying a call in the left side and buying a put in the right side) of the path taken. These options require no payment of an exercise price. Instead, the exercise price determines whether or not the option returns a payoff. The value of a cash-or-nothing call (put) option is the present value of the fixed cash payoff multiplied by the probability that the terminal price will be greater than (less than) the exercise price. Cash-or-nothing options can be priced analytically using a model introduced by Reiner and Rubinstein (1991). NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas MATHEMATICAL FORMULA INSTRUMENT PRICE 0 if S K Payoff() Call The prices of this model according to the price of the underlying asset K if S > K and the time to maturity can be represented as follows (for 2 positions: Call KerT N() d buying a call in the left side and buying a put in the right side) 0 if S K Payoff() Put K if S < K Put KerT N() d ln(S / X ) ( b 2 / 2) T Where d T 71 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 33. TWO ASSET CASH-OR-NOTHING OPTIONS DESCRIPTION MATHEMATICAL FORMULA Two-asset cash-or-nothing options can be useful building blocks for Payoffs constructing more complex exotic options. There are four types of two-asset cash-or-nothing options: KXX if S1 1 and S 2 2 Payoff 1: 0 else 1. A two-asset cash-or-nothing call pays out a fixed cash amount if Asset 1 is above Strike 1 and Asset 2 is above Strike 2 at expiration. KXX if S1 1 and S 2 2 Payoff 2: 2. A two-asset cash-or-nothing put pays out a fixed cash amount if 0 else Asset 1 is below Strike 1 and Asset 2 is below Strike 2 at expiration. KXX if S1 1 and S 2 2 3. Payoff 3: A two-asset cash-or-nothing up-down pays out a fixed cash amount 0 else if Asset 1 is above Strike 1 and Asset 2 is below Strike 2 at expiration. KXX if S1 1 and S 2 2 Payoff 4: 4. A two-asset cash-or-nothing down-up pays out a fixed cash amount 0 else if Asset 1 is below Strike 1 and Asset 2 is above Strike 2 at expiration. Values Two-asset cash-or-nothing options can be priced analytically using a model rT Value1 Ke M ( d1,1 , d 2,2 ; ) introduced by Heynen and Kat (1996). rT Value2 Ke M ( d1,1 , d 2,2 ; ) rT Value3 Ke M ( d1,1 , d 2,2 ; ) rT Value4 Ke M ( d1,1 , d 2,2 ; ) Where 2 ln(Si / X j ) ( b i i / 2) T dij, i T 72 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS INSTRUMENT PRICE The payoffs of this model can be represented as follows (Buying positions): The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions): (1)Call (asset 1) (2) Put (Asset 2) (1)Call (asset 1) (2) Put (Asset 2) (3)Up Down(Asset1) (4) Down Up(Asset2) (3)Up Down(Asset1) (4) Down Up(Asset2) NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 73 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 34. ASSET-OR-NOTHING OPTIONS DESCRIPTION PAYOFFS In an asset-or-nothing option, the asset value is paid if the asset is, at The payoffs of this model can be represented as follows (for 2 positions: expiration, above for a call or below for a put some strike level, independent buying a call in the left side and buying a put in the right side) 150 150 of the path taken. The exercise price is never paid. Instead, the value of the 140 140 130 130 asset relative to the exercise price determines whether or not the option 120 120 110 110 returns a payoff. The value of an asset-or-nothing call (put) option is the 100 100 90 90 present value of the asset multiplied by the probability that the terminal 80 80 70 70 price will be greater than (less than) the exercise price. Asset-or-nothing 60 60 50 50 options can be priced analytically using a model introduced by Cox and 40 40 30 30 Rubinstein (1985). 20 20 10 10 0 0 Spot Spot 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150 NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas MATHEMATICAL FORMULA INSTRUMENT PRICE 0 if S X The prices of this model according to the price of the underlying asset Payoff() Call S if S > K and the time to maturity can be represented as follows (for 2 positions: Call Se()b r T N() d buying a call in the left side and buying a put in the right side) 150 150 140 140 0 if S X 130 130 120 120 Payoff() Put 110 110 S if S < X 100 100 90 90 80 80 ()b r T 70 70 Put Se N() d 60 60 0,98 50 50 0,98 0,82 40 40 0,82 0,66 30 30 0,66 20 20 2 0,5 10 10 0,5 0 ln(S / X ) ( b / 2) T Time to 0,34 0 0,34 Time to Where d Maturity 150 50 60 Maturity 130140 70 0,18 0,18 120 80 90 T 110 100 90 100 110 80 120 0,02 70 130 0,02 60 140 50 Spot Spot 150 74 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 35. SUPERSHARE OPTIONS DESCRIPTION PAYOFFS A supershare is a financial instrument that represents a contingent claim on The payoff of this model can be represented as follows (buying position): a fraction of the underlying portfolio. The contingency is that the value of the 2 portfolio must lie between a lower X and an upper bound X on its L H 1,8 expiration date. If the value lies within these boundaries, the supershare is 1,6 worth a proportion of the assets underlying the portfolio, else the 1,4 1,2 supershare expires worthless. A supershare has a payoff that is basically like 1 a spread of two asset-or-nothing calls, in which the owner of a supershare 0,8 purchases an asset-or-nothing call with an strike price of Lower Strike and 0,6 0,4 sells an asset-or-nothing call with an strike price of Upper Strike. Supershare 0,2 0 options can be priced analytically using a model introduced by Hakansson Spot 50 60 70 80 90 100 110 120 130 140 150 (1976). NB: "Payoff" Chart represents prices seven days before expiry, not payoff formula MATHEMATICAL FORMULA INSTRUMENT PRICE The price of this model according to the price of the underlying asset and SXSX/LLH if X Payoff the time to maturity can be represented as follows (buying position): 0 else 2 2 ()b r T 2 w(/) Se XL N d12 N d 1 1 22 1 ln(S / XLH ) ( b / 2) T ln( S / X ) ( b / 2) T 1 Where dd; 0,98 12 0,82 1 TT0,66 0 0,5 0 Time to 0,34 0 Maturity 140150 0,18 130 110 120 90 100 80 0,02 70 60 50 Spot 75 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 36. BINARY BARRIER OPTIONS DESCRIPTION MATHEMATICAL FORMULA Binary-barrier options combine characteristics of both binary and barrier We begin by introducing 9 factors: ()b r T options. They are path dependent options with a discontinuous payoff. A11 Se N x Similar to barrier options, the payoff depends on whether or not the asset rT B Ke N x T price crosses a predetermined barrier. There are 28 different types of binary 11 ()b r T barrier options, which can be divided into two main categories: Cash-or- A22 Se N x nothing and Asset-or-nothing barrier options. rT B22 Ke N x T ()b r T 2( 1) Cash-or-nothing barrier options pay out a predetermined cash amount or A31 Se N H/ S N y nothing, depending on whether the asset price has hit the barrier. rT 2 B31 Ke H/ S N y T ()b r T 2( 1) Asset-or-nothing barrier options pay out the value of the asset or nothing, A42 Se N H/ S N y depending on whether the asset price has crossed the barrier. rT 2 B42 Ke H/ S N y T The barrier monitoring frequency can be adjusted to account for discrete AKHSNzHSNz5 / / 2 T monitoring using an approximation developed by Broadie, Glasserman, and Kou (1995). Binary-barrier options can be priced analytically using a model Where K is a prespecified cash amount. The binary variables and introduced by Reiner and Rubinstein (1991). each take the value 1 or —1. Moreover: ln(SXSH / ) ln( / ) x12 1 T ; x 1 T TT 2 ln(H / SX ) ln( H / S ) y12 1 T ; y 1 T TT ln(H / S ) b 2 / 2 2 r zT ; ; 2 T 22 By using A1 to A5 and B1 to B4 in different combinations, we can price the 28 binary barrier options described below : 76 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. case Binary Barrier Options case X>H X [15] Down-and-in asset-(at-expiry)-or- A1- nothing call (S>H) 1 1 A3 A2+A4 77 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS The payoffs of this model can be represented as follows (Buying positions): Down and In Cash (hit) or Nothing Up and In Cash (hit) or Nothing Down and In Cash (expiry) or Nothing Up and In Cash (expiry) or Nothing Down and In Asset (hit) or Nothing Up and In Asset (hit) or Nothing Down and In Asset (expiry) or Nothing Up and In Asset (expiry) or Nothing NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 78 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS The payoffs of this model can be represented as follows (Buying positions): Down and Out Cash (expiry) or Nothing Up and Out Cash (expiry) or Nothing Down and Out Asset (expiry) or Nothing Up and Out Asset (expiry) or Nothing Down and In Cash or Nothing Call Up and In Cash or Nothing Call Down and In Asset or Nothing Call Up and In Asset or Nothing Call NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 79 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS The payoffs of this model can be represented as follows (Buying positions): Down and In Cash or Nothing Put Up and In Cash or Nothing Put Down and In Asset or Nothing Put Up and In Asset or Nothing Put Down and Out Cash or Nothing Call Up and Out Cash or Nothing Call Down and Out Asset or Nothing Call Up and Out Asset or Nothing Call NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 80 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS The payoffs of this model can be represented as follows (Buying positions): Down and Out Cash or Nothing Put Up and Out Cash or Nothing Put Down and Out Asset or Nothing Put Up and Out Asset or Nothing Put NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 81 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. INSTRUMENT PRICE The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions): Down and In Cash (hit) or Nothing Up and In Cash (hit) or Nothing Down and In Cash (expiry) or Nothing Up and In Cash (expiry) or Nothing Down and In Asset (hit) or Nothing Up and In Asset (hit) or Nothing Down and In Asset (expiry) or Nothing Up and In Asset (expiry) or Nothing 82 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. INSTRUMENT PRICE The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions): Down and Out Cash (expiry) or Nothing Up and Out Cash (expiry) or Nothing Down and Out Asset (expiry) or Nothing Up and Out Asset (expiry) or Nothing Down and In Cash or Nothing Call Up and In Cash or Nothing Call Down and In Asset or Nothing Call Up and In Asset or Nothing Call 83 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. INSTRUMENT PRICE The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions): Down and In Cash or Nothing Put Up and In Cash or Nothing Put Down and In Asset or Nothing Put Up and In Asset or Nothing Put Down and Out Cash or Nothing Call Up and Out Cash or Nothing Call Down and Out Asset or Nothing Call Up and Out Asset or Nothing Call 84 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. INSTRUMENT PRICE The prices of this model according to the price of the underlying asset and the time to maturity can be represented as follows (Buying positions): Down and Out Cash or Nothing Put Up and Out Cash or Nothing Put Down and Out Asset or Nothing Put Up and Out Asset or Nothing Put 85 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 37. ASIAN OPTIONS 1: GEOMETRIC AVERAGE RATE OPTIONS DESCRIPTION PAYOFFS Asian options are path-dependent options, with payoffs that depend on the The payoffs of this model can be represented as follows (for 2 positions: average price of the underlying asset or the average exercise price. There are buying a call in the left side and buying a put in the right side) two categories or types of Asian options: average rate options (also known as average price options) and average strike options. The payoffs depend on the average price of the underlying asset over a predetermined time period. An average option is less volatile than the underlying asset, therefore making Asian options less expensive than standard European options. Asian options are commonly used in currency and commodity markets. Asian options are of interest in markets with thinly traded assets. Due to the little effect it will have on the option’s value, options based on an average, such as Asian options, have a reduced incentive to manipulate the underlying price at expiration. NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas MATHEMATICAL FORMULA INSTRUMENT PRICE Payoff( call ) Max ( S X ;0) Average The prices of this model according to the price of the underlying asset rT ()bAdjusted r T Call Se N()() d12 Xe N d and the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side) Payoff( Put ) Max ( X SAverage ;0) rT ()bAdjusted r T Put Xe N()() d21 Se N d 2 ln(S / X ) ( bAdjusted Adjusted / 2) T d1; d 2 d 1 Adjusted T T Where Adjusted 1 2 Adjusted ; b Adjusted b 3 26 86 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 38. ASIAN OPTIONS 2: THE TURNBULL AND WAKEMAN ARITHMETIC AVERAGE APPROXIMATION DESCRIPTION PAYOFFS It is not possible (or very hard) to find a closed-form solution for the value of The payoffs of this model can be represented as follows (for 2 positions: options on an arithmetic average. The main reason is that when the asset is buying a call in the left side and buying a put in the right side) assumed to be lognormally distributed, the arithmetic average will not itself have a lognormal distribution. Arithmetic average rate options can be priced by the analytical approximation of Turnbull and Wakeman (1991). This approximation adjusts the mean and variance so that they are consistent with the exact moments of the arithmetic average. The adjusted mean, bA , and 2 variance, A , are then used as input in the generalized BSM formula. MATHEMATICAL FORMULA NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas Call Se()bA r T N()() d XerT N d INSTRUMENT PRICE 12 rT ()bA r T Put Xe N()() d21 Se N d The prices of this model according to the price of the underlying asset ln(S / X ) ( b 2 / 2) T and the time to maturity can be represented as follows (for 2 positions: Where dAA ; d d T 1 2 1 A buying a call in the left side and buying a put in the right side) A T where T is the time to maturity in years. ln(MM ) ln( ) 21 2b ; b AAATT 2 2 ebT ebt12 e(2 b ) T 2 e(2 b ) t 1 1 e b ( T t 1 ) With M12; M 2 2 2 2 2 2 b() T t1 (b )(2 b )( T t11 ) b ( T t ) 2 b b Where t1 is the time to the beginning of the average period. 87 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 39. ASIAN OPTIONS 3: LEVY'S ARITHMETIC AVERAGE APPROXIMATION DESCRIPTION Where Levy's arithmetic average approximation (1992) is an alternative to the Turnbull and Wakeman formula described below. S A = Arithmetic average of the known asset price fixings. S = Asset price. MATHEMATICAL FORMULA X = Strike price. r = Risk-free interest rate. * rT2 Call SNdE ()()12 Xe Nd b = Cost-of-carry rate. T2 = Remaining time to maturity. Where T = Original time to maturity. S a = Volatility of natural logarithms of return of the underlying asset. S() e()b r T22 e rT E T b NB : The formula does not allow for b = 0 1 ln(D ) * d1 ln( X ) ; d 2 d 1 V V 2 TT M X* X 2 S; V ln( D ) 2 rT ln( S ) ; D= T AE2 T 2 2 2 (2b ) T22 bT 2S e 1 e 1 M 22 bb2 b The Asian put value can be found by using the following put-call parity: Put Call S X* erT2 E 88 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS INSTRUMENT PRICE The payoffs of this model can be represented as follows (for 2 positions: The prices of this model according to the price of the underlying asset buying a call in the left side and buying a put in the right side) and the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side) NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 89 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 40. FOREIGN EQUITY OPTIONS STRUCK IN DOMESTIC CURRENCY (VALUE IN DOMESTIC CURRENCY) DESCRIPTION Where As the name indicates, these are options on foreign equity where the strike is ln(ES*2 / X ) ( r q / 2) T denominated in domestic currency. At expiration, the foreign equity is ES* d1; d 2 d 1 ES * T translated into the domestic currency. Valuation of these options is achieved T ES* using the formula attributed to Reiner (1992). 22 ES**** E S 2 ES E S MATHEMATICAL FORMULA S* = Underlying asset price in foreign currency. X = Delivery price in domestic currency. The payoff to a European investor for an option linked to the DowJones index is : r = Domestic interest rate. Payoff( Call ) Max ( E S* X ;0) q = Instantaneous proportional dividend payout rate of the EUR/(/)(/)(/) share EUR USD USD share EUR share * underlying asset. Payoff( PutEUR/(/)(/)(/) share ) Max ( X EUR share E EUR USD S USD share ;0) E = Spot exchange rate specified in units of the domestic currency per Call ES* eqT N()() d Xe rT N d unit of the foreign currency. 12 * rT* qT S = Volatility of the underlying asset. Put Xe N()() d21 ES e N d E = Volatility of the domestic exchange rate. * = ES Correlation between asset and domestic exchange rate. 90 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS INSTRUMENT PRICE The payoffs of this model can be represented as follows (for 2 positions: The prices of this model according to the price of the underlying asset buying a call in the left side and buying a put in the right side) and the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side) NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 91 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 41. FIXED EXCHANGE RATE FOREIGN EQUITY OPTIONS - QUANTOS (VALUE IN DOMESTIC CURRENCY) DESCRIPTION Where A fixed exchange-rate foreign-equity option (Quanto) is denominated in another currency than that of the underlying equity exposure. The face value ln(S* / X * ) ( r q 2 / 2) T fESS** of the currency protection expands or contracts to cover changes in the d1; d 2 d 1 S * T T foreign currency value of the underlying asset. Quanto options can be priced S* analytically using a model published by Dravid, Richardson, and Sun (1993) S* = Underlying asset price in foreign currency. X* = Delivery price in foreign currency. MATHEMATICAL FORMULA r = Domestic interest rate. rf = Foreign interest rate. q = Instantaneous proportional dividend payout rate of the * underlying asset. Payoff( CallEUR/(/)(/)(/) share ) E P EUR USD Max ( S USD share X USD share ;0) Payoff( Put ) E Max ( X S* ;0) Ep = Predetermined exchange rate specified in units of domestic EUR/(/)(/)(/) share P EUR USD USD share USD share currency per unit of foreign currency. ()r r q T E* = Spot exchange rate specified in units of foreign currency per Call E S** ef S* E N()() d X erT N d p 12unit of domestic currency. PutEXeNd**rT ()() Se()rf r q S* E T Nd * p 21 S = Volatility of the underlying asset. E = = Volatility of the domestic exchange rate. = Correlation between asset and domestic exchange rate. 92 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS INSTRUMENT PRICE The payoffs of this model can be represented as follows (for 2 positions: The prices of this model according to the price of the underlying asset buying a call in the left side and buying a put in the right side) and the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side) NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 93 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 42. EQUITY LINKED FOREIGN EXCHANGE OPTIONS (VALUE IN DOMESTIC CURRENCY) DESCRIPTION Where An equity-linked foreign-exchange option is an option on the foreign ln(E / X ) ( r r 2 / 2) T exchange rate and is linked to the forward price of a stock or equity index. fS* E E d1; d 2 d 1 E T This option can be priced analytically using a model introduced by Reiner E T (1992). S* = Underlying asset price in foreign currency. MATHEMATICAL FORMULA X = Currency strike price in domestic currency. r = Domestic interest rate. * rf = Foreign interest rate. Payoff( CallEUR/(/)(/)(/) share ) S USD share Max ( E EUR USD X EUR USD ;0) * q = Instantaneous proportional dividend payout rate of the Payoff( Put ) S Max ( X E ;0) EUR/(/)(/)(/) share USD share EUR USD EUR USD underlying asset. **qT ()rf r q S* E T E = Spot exchange rate specified in units of the domestic currency Call ES e N()() d XS e N d 12 per unit of the foreign currency. Put XS** e()rf r q S* E T N()() d ES eqT N d 21 E* = Spot exchange rate specified in units of the foreign currency per unit of the domestic currency. * S = Volatility of the underlying asset. E =- Volatility of the domestic exchange rate. = Correlation between asset and the domestic exchange rate. 94 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS INSTRUMENT PRICE The payoffs of this model can be represented as follows (for 2 positions: The prices of this model according to the price of the underlying asset buying a call in the left side and buying a put in the right side) and the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side) NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 95 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 43. TAKEOVER FOREIGN EXCHANGE OPTIONS DESCRIPTION PAYOFFS A takeover foreign exchange call option gives the buyer the right to purchase The payoff of this model can be represented as follows (buying position): a specified number of units of foreign currency at a strike price if the corporate takeover is successful. This option can be priced analytically using a model introduced by Schnabel and Wei (1994). NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas MATHEMATICAL FORMULA INSTRUMENT PRICE CallNEeMarTf (,;)(,;) Ta T XeMaarT 2EE 1 1 2 The price of this model according to the price of the underlying asset and the time to maturity can be represented as follows (buying position): Where 22 ln(V / N ) ( rf E V V / 2) T ln( E / X ) ( r r f E / 2) T aa12; VETT Both the strike price X and the currency price E are quoted in units of the domestic currency per unit of the foreign currency. 96 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 44. EUROPEAN SWAPTIONS IN THE BLACK-76 MODEL DESCRIPTION PAYOFFS A Swaption (option on Interest Rate Swap IRS) reserves the right for its The payoffs of this model can be represented as follows (for buying position): holder to purchase a swap at a prescribed time and interest rate in the future (European Option). The holder of such a call option has the right, but not the obligation to pay a fixed interest rate in exchange of a variable interest rate. This option is also known as “Payer Swaption”. The holder of the equivalent put option has the right, but not the obligation to receive the fixed interest rate (Receiver Swaption) and pay the variable interest rate. NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas MATHEMATICAL FORMULA 1 INSTRUMENT PRICE 1 tm (1 Fm / ) 1 Call( Payer swaption ) erT FN ( d ) XN ( d ) F 12The price of this model according to the price of the underlying asset and 1 the time to maturity can be represented as follows (for buying positions): 1 tm (1 Fm / ) 1 Put( Receiver swaption ) erT XN ( d ) FN ( d ) F 21 ln(FXT / ) ( 2 / 2) Where d1; d 2 d 1 T T t1= Tenor of swap in years; m= compounding per year F= Underlying Swap Rate; X =Strike Price T= Time to Maturity; r =Risk-free Rate; = Swap Rate Volatility 97 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. 45. THE VASICEK MODEL FOR EUROPEAN OPTIONS ON ZERO COUPON BONDS DESCRIPTION MATHEMATICAL FORMULA The Vasicek (1977) model is a yield-based one-factor equilibrium model that ZERO COUPON BOND VALUE assumes that the short rate is normally distributed. The model incorporates mean reversion and is popular in the academic community—mainly due to The price at time t of a discount bond maturing at time T is : its analytic tractability. The model is not used much by market participants B(,)() t T r t because it is not ensured to be arbitrage-free relative to the underlying P(,)(,) t T A t T e securities already in the marketplace. This model is given by the following general formula : Where ()Tt dr k() r dt d 1 e z B(,) t T K is the speed of the mean reversion, and is the mean reversion level. ((,)B t T T t )(2 2 /2) 2 B (,) t T 2 A( t , T ) exp 2 4 OPTION VALUE The value of a European option maturing at time T on a zero-coupon bond that matures at time is: Call Pt(,)()(,)() Nh XPtTNh p Put XPtTN(,)()(,)() h p Pt N h Where r(t) is the rate at time t and 1Pt ( , ) h ln P P( t , T ) X 2 P 2(1 e 2 (Tt ) ) Bt(,) P 2 98 Chappuis Halder & Cie Valuation & Pricing Solutions Global Research & Analytics Dpt. PAYOFFS INSTRUMENT PRICE The payoffs of this model can be represented as follows (for 2 positions: The prices of this model according to the price of the underlying asset buying a call in the left side and buying a put in the right side) and the time to maturity can be represented as follows (for 2 positions: buying a call in the left side and buying a put in the right side) NB: "Payoffs" Charts represent prices seven days before expiry, not payoffs formulas 99