Beyond Black- Scholes: Smile & Exo6c op6ons. Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles

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1 Beyond Black- Scholes: Smile & Exo6c op6ons Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles 1

2 What is a Vola6lity Smile? Rela6onship between implied vola6lity and strike price for op6ons with a certain maturity The vola6lity smile for European call op6ons should be exactly the same as that for European put op6ons The same is at least approximately true for American op6ons 2

3 The Vola6lity Smile Foreign Currency Op6ons Implied Vola6lity Equity Op6ons Implied Vola6lity Strike Price Strike Price 3

4 Other Vola6lity Smiles? What is the vola6lity smile if True distribu6on has a less heavy lep tail and heavier right tail True distribu6on has both a less heavy lep tail and a less heavy right tail 4

5 Ways of Characterizing the Vola6lity Smiles Plot implied vola6lity against K/S 0 (The vola6lity smile is then more stable) Plot implied vola6lity against K/F 0 (Traders usually define an op6on as at- the- money when K equals the forward price, F 0, not when it equals the spot price S 0 ) Plot implied vola6lity against delta of the op6on (This approach allows the vola6lity smile to be applied to some non- standard op6ons. At- the money is defined as a call with a delta of 0.5 or a put with a delta of 0.5. These are referred to as 50- delta op6ons) 5

6 Possible Causes of Vola6lity Smile Asset price exhibits jumps rather than con6nuous changes Vola6lity for asset price is stochas6c In the case of an exchange rate vola6lity is not heavily correlated with the exchange rate. The effect of a stochas6c vola6lity is to create a symmetrical smile In the case of equi6es vola6lity is nega6vely related to stock prices because of the impact of leverage. This is consistent with the skew that is observed in prac6ce 6

7 Vola6lity Term Structure In addi6on to calcula6ng a vola6lity smile, traders also calculate a vola6lity term structure This shows the varia6on of implied vola6lity with the 6me to maturity of the op6on The vola6lity term structure tends to be downward sloping when vola6lity is high and upward sloping when it is low 7

8 Example of a Vola6lity Surface (Hull Table 18.2, page 397) K/S mnth mnth mnth year year year

9 Vola6lity surface 9

10 Exo6c Op6ons 10

11 In DerivaGem Asian Barrier: Up and in Up and out Down and in Down and out Binary Cash or nothing Asset or nothing Chooser Compound Op6on on call Op6on on put Lookback Floa6ng Fixed 11

12 Other Exo6cs Packages Nonstandard American op6ons Forward start op6ons Shout op6ons Op6ons to exchange one asset for another Op6ons involving several assets Vola6lity and Variance swaps 12

13 Binary Op6ons Cash- or- nothing: Pays Q if S T > K, otherwise pays nothing. Value = e rt Q N(d 2 ) Asset- or- nothing: Pays S T if S T > K, otherwise pays nothing. Value = S 0 e -qt N(d 1 ) 13

14 Decomposi6on of a Call Op6on Long Asset- or- Nothing op6on Short Cash- or- Nothing op6on where payoff is K Value = S 0 e -qt N(d 1 ) e rt KN(d 2 ) 14

15 Binary Op6ons - Example Index Level: Volatility: 30.00% Risk-Free Rate: 5.00% Dividend Yield: 0.00% Option Type: Analytic: European Time to Exercise: Exercise Price: Option Type: Binary: Cash or Nothing Time to Exercise: Exercise Price: Cash Amount: Option Type: Binary: Asset or Nothing Time to Exercise: Exercise Price: Price: Delta (per $): 0.62 Gamma (per $ per $): 0.01 Vega (per %): 0.38 Theta (per day): Rho (per %): 0.48 Price: Delta (per $): 1.26 Gamma (per $ per $): Vega (per %): Theta (per day): 0.01 Rho (per %): 0.77 Price: Delta (per $): 1.89 Gamma (per $ per $): 0.00 Vega (per %): Theta (per day): Rho (per %):

16 Chooser Op6on ( As You Like It ) Op6on starts at 6me 0, matures at T 2 At T 1 (0 < T 1 < T 2 ) buyer chooses whether it is a put or call This is a package! Value at 6me T 1 : Max(c,p) Put- call parity: S e r( T2 T1 ) q( T2 T1 ) 1 Chooser op6on is a package consis6ng of: One call op6on maturing at 6me T 2 p = c + Ke q( T 1) ( )( 2 1) max(, ) 2 T rq T T c p = c + e Max(0, Ke S 1) e - q(t2- T1) put op6ons maturing at 6me T 1 16

17 Chooser: example Index Level: Volatility: 30.00% Risk-Free Rate: 5.00% Dividend Yield: 0.00% Option Type: Chooser Option Data Imply Volatility Time to Exercise: Exercise Price: Decision Date: Put Call Price: Delta (per $): 0.26 Gamma (per $ per $): 0.03 Vega (per %): 0.65 Theta (per day): Rho (per %):

18 Barrier Op6ons Op6on may be a put or a call In op6ons Op6on comes into existence only if stock price hits barrier before op6on maturity Out op6ons Op6on dies if stock price hits barrier before op6on maturity Up op6ons Stock price must hit barrier from below Down op6ons Stock price must hit barrier from above Eight possible combina6ons 18

19 Parity Rela6ons c = c ui + c uo p = p ui + p uo Example Stock pric 100 DivYield 0% Strike 100 Maturity 1 Int.Rate 5% Volatility 30% c = c di + c do p = p di + p do Standard Down: barrier = 90 Up: barrier = 110 In Out In Out Call Put

20 Barrier Op6on Valua6on (wonkish!) ) ( ) ( T y N S H Ke y N S H Se C rt qt di σ λ λ = [ ] [ ] ) ( ) ( ) ( ) ( ) ( ) ( T y N T y N S H Ke y N y N S H Se T x N Ke x N Se P rt qt rt qt di σ σ σ λ λ = / σ σ λ + = q r T T K S H y λσ σ + = / ln 2 T T S H y λσ σ + = ln 1 Formulas exit but not really user friendly!!) Example: down and in Payoff depends on two events: - Hiong the barrier - Being in the money at maturity 20

21 Compound Op6on Op6on to buy or sell an op6on Call on call Put on call Call on put Put on put Can be valued analy6cally Price is quite low compared with a regular op6on 21

22 Example Index Level: Volatility (% per year): 30.00% Risk-Free Rate (% per year): 5.00% Dividend Yield (% per yer): 0.00% At each node: Upper value = Underlying Asset Price Lower value = Option Price Values in red are a result of early exercise Strike price = Discount factor per step = Time step, dt = years, days Growth factor per step, a = Probability of up move, p = Up step size, u = Down step size, d = Node Time:

23 Option Type: Option Data Compound: Option on Call Time to First Exercise: First Exercise Price: Time to Final Exercise: Final Exercise Price: Imply Volatility Put Call Price: Delta (per $): Gamma (per $ per $): Vega (per %): Theta (per day): Rho (per %):

24 Lookback Op6ons Floa6ng lookback call pays S T S min at 6me T (Allows buyer to buy stock at lowest observed price in some interval of 6me) Floa6ng lookback put pays S max S T at 6me T (Allows buyer to sell stock at highest observed price in some interval of 6me) Fixed lookback call pays max(s max K, 0) Fixed lookback put pays max(k S min, 0) Analy6c valua6on for all types 24

25 Lookback Example Option Type: Option Type: Option Type: Analytic: European Imply Volatility Floating Lookback Fixed Lookback Time to Exercise: Exercise Price: Put Call Time to Exercise: Maximum to Date: Minimum to Date: Time to Exercise: Maximum to Date: Minimum to Date: Exercise Price: Price: Delta (per $): 0.62 Gamma (per $ per $): 0.01 Vega (per %): 0.38 Theta (per day): Rho (per %): 0.48 Price: Delta (per $): 0.24 Gamma (per $ per $): 0.03 Vega (per %): 0.64 Theta (per day): Rho (per %): 0.42 Price: Delta (per $): 1.23 Gamma (per $ per $): 0.02 Vega (per %): 0.93 Theta (per day): Rho (per %):

26 Asian Op6ons Payoff related to average stock price Average Price op6ons pay: Call: max(s ave K, 0) Put: max(k S ave, 0) Average Strike op6ons pay: Call: max(s T S ave, 0) Put: max(s ave S T, 0) 26

27 Asian Op6ons No exact analy6c valua6on Can be approximately valued by assuming that the average stock price is lognormally distributed Option Type: Asian Time to Exercise: Exercise Price: Time since Inception: Current Average: Imply Volatility Put Call Price: 2.03 Delta (per $): 0.18 Gamma (per $ per $): 0.01 Vega (per %): 0.11 Theta (per day): Rho (per %):

28 Packages Porpolios of standard op6ons Examples from Chapter 10: bull spreads, bear spreads, straddles, etc OPen structured to have zero cost One popular package is a range forward contract 28

29 Non- Standard American Op6ons Exercisable only on specific dates (Bermudans) Early exercise allowed during only part of life (ini6al lock out period) Strike price changes over the life (warrants, conver6bles) 29

30 Forward Start Op6ons (page 556) Op6on starts at a future 6me, T 1 Implicit in employee stock op6on plans OPen structured so that strike price equals asset price at 6me T 1 30

31 Shout Op6ons Buyer can shout once during op6on life Final payoff is either Usual op6on payoff, max(s T K, 0), or Intrinsic value at 6me of shout, S τ K Payoff: max(s T S τ, 0) + S τ K Similar to lookback op6on but cheaper How can a binomial tree be used to value a shout op6on? 31

32 Exchange Op6ons Op6on to exchange one asset for another For example, an op6on to exchange one unit of U for one unit of V Payoff is max(v T U T, 0) 32

33 Basket Op6ons (page 567) A basket op6on is an op6on to buy or sell a porpolio of assets This can be valued by calcula6ng the first two moments of the value of the basket and then assuming it is lognormal 33

34 Vola6lity and Variance Swaps Agreement to exchange the realized vola6lity between 6me 0 and 6me T for a prespecified fixed vola6lity with both being mul6plied by a prespecified principal Variance swap is agreement to exchange the realized variance rate between 6me 0 and 6me T for a prespecified fixed variance rate with both being mul6plied by a prespecified principal Daily return is assumed to be zero in calcula6ng the vola6lity or variance rate 34

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