Lecture 8: Pricing Measures And Applications To Exotic Options

Lecture 8: Pricing Measures and Applications to First-Generation Exotic Options

Pricing Measures Pricing measures, a.k.a. pricing kernels, or pricing models, are probability measures of future market scenarios which are used to pricing derivatives by discounting expected cash-flows. Example: Consider an index, e.g. the S&P500. We wish to price derivatives based on the index. The pricing measure will be such that the index satisfies St t t t rt t qt t , St where nu t are i.i.d. N(0,1) deviates, and sigma t, r t, q t are respectively the volatility, the funding rate and the dividend yield over the period (t,t t ).

Justification of the previous formula Assume that the volatility, funding rate and the dividend yield of the index are and that we know the price of the index as well. The previous formula implies that St t St St t t t St rt t St qt t so E St t St , t , rt , qt St Srt t St qt t St 1 rt t qt t Ft ,t t The conditional mean of the spot price is the forward price for the corresponding interval, ensuring that put-call parity will hold at any future time period.

Term-structures of interest rates, volatilities and dividends How do we compute t , rt , qt in practice? Derivative markets contain information about forward interest rates, dividends and volatilities. For example, consider the following hypothetical table of quantities extracted from market data: Maturity 30 days 60 days 90 days 180 days 360 days Interest rate 0.1 0.2 0.4 0.4 0.5 Dividend yield 2.1 2.2 2.2 1.9 1.8 Implied Volatility 25 27 29 30 32 Zero-coupon bond rates Implied divs. from options or actual divs. ATM implied vols.

Finding forward rates from term rates 1 1 n RT rt dt rt t j t j 1 T 0 n t j 1 j T rt d TRT dT T t The formula for backing out forward rates from term rates is rt ,t t t t Rt t tRt t

Concretely 0.5 Zero-coupon (or term) Rates (RT ) 0.4 0.4 90 180 0.2 0.1 0 30 60 360 0.8 0.6 Forward rates ( rt ) 0.4 0.3 0.1 0 30 60 90 180 360

Finding forward dividends from (implied) term dividends Dividends scale like rates, so 1 qT T T 0 1 n 1 qt dt qti ti 1 ti n t j 0 t Ti , Ti 1 qt Ti 1 q Ti 1 Ti q Ti Ti 1 Ti

Finding forward volatilities from (implied) term volatilities Variances scale linearly, like rates, so 1 T 2 T 1 n 1 2 ti ti 1 ti 0 dt n t j 0 T 2 t t Ti , Ti 1 2 t2 2 Ti 1 Ti 1 Ti Ti Ti 1 Ti These three formulas, for rates, dividends and volatilities allow us to generate forward quantities to use in the model.

Passing from term-structure to forward quantities Data (term rates) Computed (forward rates) MAT 0.0 RATE 0 DIV 0 VOL 0 F RATE 0 F DIV 0 F VOL 0 30.0 60.0 90.0 180.0 360.0 0.1 0.2 0.4 0.4 0.5 2.1 2.2 2.2 1.9 1.8 25 27 29 30 32 0.1 0.3 0.8 0.4 0.6 2.1 2.3 2.2 1.6 1.7 25 28.9 32.6 30.97 33.88

Term rates & forward rates (formal calculation) ST n 1 1 j j t rj t q j t S 0 j 0 T n t n 1 2 ln(1 ) - o( ) exp ln 1 j j t rj t q j t 2 j 0 n 1 n 1 n 1 1 n 1 2 2 exp j j t rj t q j t j j t o 1 2 j 0 j 0 j 0 j 0 n 1 n 1 n 1 1 n 1 2 exp j j t rj t q j t j t o 1 (LLN) 2 j 0 j 0 j 0 j 0 1 n 1 1 n 1 1 n 1 1 n 1 2 exp j j T rjT q jT j T n j 0 n j 0 2n j 0 n j 0 1 2 exp T T ( RT q T )T T T N(0,1) (CLT) 2 The construction gives rise to lognormal random variables with the appropriate term rates and volatilities.

The pricing model can be implemented in a trinomial tree max t PU ,n U (n,j) M 0 PM ,n 1 f n , j , D PD ,n max t t n t 1 f n , j 1 max 2 2 2 max f n, j n2 2 , n rn qn max t n t 1 f n , j 1 max 2 2 2 max

Pricing models as probabilities on future price paths F ST1 , T1 F ST2 , T2 F ST3 , T3 F ST4 , T4 F STN , TN N RT j T j V E e F ST j , T j j 1 Once a pricing measure has been specified, the value of a derivative security is the expected value of its discounted cash-flows.

Barrier options Barrier options are puts and calls which are activated/deactivated as the underling asset crosses a given level These are sometimes called first generation exotic options’’. They are OTC contracts which are tailored to end-user expectations and usually are cheaper than regular options. From a risk-management perspective, barrier options are more difficult than plain-vanilla options because their value is not monotone in the underlying price and the deltas, gammas and vegas can change sign. Barrier options are popular in FX, Equities and Fixed-income derivatives

Down-and Out Calls A down-and-call is a standard call with the additional provision that the contract is void if the underlying asset price goes below some level. Example: 180 Day European SPY Call with strike 120 and knockout barrier 100. Last price 123.12 Payoff max(S-120) S 120 Payoff 0 S 100 T 180

Pricing a down-and-out call V(S,180) max(S-120,0) S 100 V(S 100,t) 0 0 30 60 90 180 Piecewise-constant interest, dividend, volatility to fit term-structures! Recursive algorithm: Vnj e rn t PU ,nVnj 11 PM ,nVnj 1 PD ,nVnj 11 , VNj max(S Nj 120,0), V nj0 0 j j0 , n N if S nj0 100

Closed-form solution for DO Call (from Hull) If the parameters r, q, sigma are constant, there is a closed-from solution: Cdo S , T , K , H , r , q, Se qT N d1 Ke rT N d 2 Se d1 F T ln , 2 T K e1 H2 T , ln T SK 1 1 r q 2 / 2 2 2 qT H rT H N e1 Ke S S d 2 d1 T e2 e1 T 2 2 N e2

Intuition for the pricing formula H Down and Out long Vanilla Call, short Down-and-In Call

D&O Call versus plain vanilla Call K 110, T 1, H 80, r 10bps, q 200 bps, vol 40%

1y 110 Call KO 90 K 110, T 1, H90, r 10bps, q 200 bps, vol 20%

Reverse Knock-outs Up and out calls are sometimes called reverse knock-outs. They have a discontinuity at the KO barrier which renders the Delta increasingly large as maturity approaches. Formula (Hull) for a RKO Call: K strike, H barrier, constant rates and vol. Cuo ( S , T , K , H , r , q, ) BSCall ( S , T , K , r , q, ) Cui ( S , T , K , H , r , q, ) Cui ( S , T , K , H , r , q, ) Se qT N ( f1 ) Ke rT N ( f 2 ) Se qT S H 2 N ( e1 ) N ( g ) Ke S f1 ln T , T H H2 1 T , e1 ln SK T 1 g H ln T T S 1 rT S H f 2 f1 T 2 2 N ( e , 1 T ) N ( g T ) r q 2 / 2 2

110 CALL KO 130 T 0.30Y T 0.04Y T 0.15Y T 0.01Y

One-touch Contract delivers a cash payoff (or share payoff) if a barrier is hit within a certain time-period. t 0 Receive cash payout here A two-touch receives a payoff if either a lower level or an upper level is attained by the price of the underlying asset. T T

Pricing measures, a.k.a. pricing kernels, or pricing models, are probability measures of future market scenarios which are used to pricing derivatives by discounting expected cash-flows. Example: Consider an index, e.g. the S&P500. We wish to price derivatives based on the index. The pricing measure will be such that the index satisfies