Currency Options (2): Hedging and Valuation

Transcription

1 Overview Chapter 9 (2): Hedging and

2 Overview Overview The Replication Approach The Hedging Approach The Risk-adjusted Probabilities Notation Discussion Binomial Option Pricing Backward Pricing, Dynamic Hedging What can go wrong? American-style Options Black-Merton-Scholes STP-ing of European Options the Black-Merton-Scholes Equation The Delta of an Option

6 Binomial Models What & Why? Binomial Model given S t, there only two possible values for S t+1, called up and down. Restrictive? Yes, but... the distribution of the total return, after many of these binomial price changes, becomes bell-shaped the binomial option price converges to the BMS price the binomial math is much more accessible than the BMS math BinMod can be used to value more complex derivatives that have no closed-form Black-Scholes type solution. Ways to explain the model all very similar: via hedging via replication in spot market (not here) (not here) forward yes yes

9 Outline The Replication Approach The Hedging Approach The Risk-adjusted Probabilities The Replication Approach The Hedging Approach The Risk-adjusted Probabilities Notation Discussion Binomial Option Pricing Backward Pricing, Dynamic Hedging What can go wrong? American-style Options Black-Merton-Scholes STP-ing of European Options the Black-Merton-Scholes Equation The Delta of an Option

10 Our Example The Replication Approach The Hedging Approach The Risk-adjusted Probabilities Data S 0 = INR/MTL 100, r = 5%p.p.; r = %. Hence: F 0,1 = S r 0,1 1 + r 0, = = 101. S 1 is either S 1,u = 110 ( up ) or S 1,d = 95 ( down ). 1-period European-style call with X=INR/MTL S 1 C C 1 slope of exposure line (exposure): O exposure = C1,u C 1,d = 5 0 S 1,i S 1,d = 1/3 S 3,3=S 0uuu S =S uu exposure line S1

11 Our Example The Replication Approach The Hedging Approach The Risk-adjusted Probabilities Data S 0 = INR/MTL 100, r = 5%p.p.; r = %. Hence: F 0,1 = S r 0,1 1 + r 0, = = 101. S 1 is either S 1,u = 110 ( up ) or S 1,d = 95 ( down ). 1-period European-style call with X=INR/MTL S 1 C C 1 slope of exposure line (exposure): O exposure = C1,u C 1,d = 5 0 S 1,i S 1,d = 1/3 S 3,3=S 0uuu S =S uu exposure line S1

12 The Replication Approach The Replication Approach The Hedging Approach The Risk-adjusted Probabilities Step 1 Replicate the payoff from the call [5 if u] and [0 if d]: (a) = forward contract (b) = deposit, (buy MTL 1/3 at 101) V1=20 (a)+(b) S 1 = 95 1/3 ( ) = S 1 = 110 1/3 ( ) = Step 2 Time-0 cost of the replicating portfolio: forward contract is free deposit will cost INR 2/1.05 = INR Step 3 Law of One Price: option price = value portfolio C 0 = INR 1.905

15 The Hedging Approach The Replication Approach The Hedging Approach The Risk-adjusted Probabilities Replication: Hedging: Step 1 Hedge the call call = forward position + riskfree deposit call forward position = riskfree deposit (a) = forward hdege (sell MTL 1/3 at 101) (b) = call (a)+(b) S 1 = 95 1/3 (101 95) = S 1 = 110 1/3 ( ) = Step 2 time-0 value of the riskfree portfolio value = INR 2/1.05 = INR Step 3 Law of one price: option price = value portfolio C 0 + [time-0 value of hedge] = INR C 0 = INR otherwise there are arbitrage possibilities.

18 The Risk-adjusted Probabilities The Replication Approach The Hedging Approach The Risk-adjusted Probabilities Overview: Implicitly, the replication/hedging story... extracts a risk-adjusted probability up from the forward market, uses this probability to compute the call s risk-adjusted expected payoff, CEQ 0( C 1); and discounts this risk-adjusted expectation at the riskfree rate.

21 The Risk-adjusted Probabilities The Replication Approach The Hedging Approach The Risk-adjusted Probabilities Step 1 Extract risk-adjusted probability from F: Ordinary expectation: E 0( S 1) = p (1 p) 95 Risk-adjusted expectation: CEQ 0 ( S 1) = q (1 q) 95 We do not know how/why the market selects q, but q is revealed by F 0,1 (= 101): 101 = 95 + q (110 95) q = Step 2 CEQ of the call s payoff: = 6 15 = 0.4 CEQ 0( C 1) = (0.4 5) + (1 0.4) 0 = 2 Step 3 Discount at r: C 0 = CEQ0( C 1) 1 + r 0,1 = = 1.905

26 Outline Notation Discussion The Replication Approach The Hedging Approach The Risk-adjusted Probabilities Notation Discussion Binomial Option Pricing Backward Pricing, Dynamic Hedging What can go wrong? American-style Options Black-Merton-Scholes STP-ing of European Options the Black-Merton-Scholes Equation The Delta of an Option

27 Notation Notation Discussion Subscripts: n,j where n says how many jumps have been made since time 0 j says how many of these jumps were up General pricing equation: C t,j = Ct+1,u qt + C t+1,d (1 q t) 1 + r t,1period, where q t = Ft,t+1 S t+1,d S t+1,u S t+1,d, = = d t = S t+1,d S t 1+r S t,t+1 t 1+r t,t+1 S td t, S tu t S td t 1+r t,t+1 1+r t,t+1 d t, u t d t, u t = St+1,u S t. (1)

28 Notation Notation Discussion Subscripts: n,j where n says how many jumps have been made since time 0 j says how many of these jumps were up General pricing equation: C t,j = Ct+1,u qt + C t+1,d (1 q t) 1 + r t,1period, where q t = Ft,t+1 S t+1,d S t+1,u S t+1,d, = = d t = S t+1,d S t 1+r S t,t+1 t 1+r t,t+1 S td t, S tu t S td t 1+r t,t+1 1+r t,t+1 d t, u t d t, u t = St+1,u S t. (1)

29 Notation Discussion A1 (r and r ) : The risk-free one-period rates of return on both currencies are constant denoted by unsubscripted r and r Also assumed in Black-Scholes. A2 (u and d) : The multiplicative change factors, u and d, are constant. Also assumed in Black-Scholes: no jumps (sudden de/revaluations) in the exchange rate process, and a constant variance of the period-by-period percentage changes in S. Implication of A1-A2: q t is a constant. A2.01 (no free lunch in F): d < 1 + r 1 + r < u S t+1,d < F t < S t+1,u 0 < q < 1 Q: what would you do if S 1 =[95 or 110] while F=90? 115?

33 95 0 How such a tree works O Notation Discussion S 0 S =S u 1,1 S =S d 1,0 0 0 S =S uu 2,2 S =S ud 2,1 S =S dd 2, S =S uuu 3,3 S =S uud 3,2 S =S udd 3,1 S =S ddd

34 The Emerging Bellshape Notation Discussion 4.4 How to model (near-)impredictable spot rates (1) 1/16 n=4, j=4 let p=1/2 1/8 1/4 4/16 n=4, j=3 1/2 3/8 1 2/4 6/16 n=4, j=2 1/2 3/8 1/4 4/16 n=4, j=1 1/8 1/16 n=4, j=0 The emerging bell-shape 4. Time-series properties C = 4!/4!0! = 1 C = 4!/3!1! = 24/6 = 4 C = 4!/2!2! = 24/6 = 6 C = 4!/1!3! = 24/6 = 4 C = 4!/4!0! = 1

35 What Emerging Bellshape?! lns N is normal ($S lognormal) Why multiplicative? cents vs percents no zero, negative S inverse of S Notation Discussion Constant u and d: corresponds to constant % in BMS Chosing between additive two oversimplifications: additive multiplicative multiplicative ! cents v percent: we prefer a constant distribution of percentage price changes over a constant distribution of dollar price changes. non-negative prices: with a multiplicative, the exchange rate can never quite reach zero even if it happens to go down all the time. invertible: we get a similar multiplicative process for the exchange rate as viewed abroad, S = 1/S (with d = 1/u, u = 1/d). Corresponding Limiting Distributions: additive: S n = S 0 + P n t=1 t where = {+10, 10} S n is normal if n is large (CLT) multiplicative: S n = S 0 Q n t=1 (1 + rt) where r = {+10%, -10%} ln S n = ln S 0 + P n t=1 ρt where ρ = ln(1 + r) = {+0.095, 0.095} ln S n is normal if n is large S n is lognormal.

42 Outline Backward Pricing, Dynamic Hedging What can go wrong? American-style Options The Replication Approach The Hedging Approach The Risk-adjusted Probabilities Notation Discussion Binomial Option Pricing Backward Pricing, Dynamic Hedging What can go wrong? American-style Options Black-Merton-Scholes STP-ing of European Options the Black-Merton-Scholes Equation The Delta of an Option

43 Backward Pricing, Dynamic Hedging What can go wrong? American-style Options An N-period European Call: The Problem 3.2. binomial pricing of European Options 3. pricing European S0 = 100 S 1,1 = 110 S1,0 = 90 S 2,0 = 81 u = 1.1; d =.9; 1+r = 1.05; 1+r* = ; 1+r forward factor = r* q = = 0.60 payoff S 2,2 = S = ,1 n [cashflow"at"t"if"j""up"s]"! "[risk-adj"prob"of"j""up"s] Call: C 0 A4. =! At any discrete moment (1+r)n in the model, investors j=0 can = 26"! trade "(0.62 and adjust their portfolios of HC-FC )"+"4"! "(2"! "0.6"! "0.4)"+"0"! "(0.4 2 loans. Black-Scholes: trading is continuous )

44 Backward Pricing, Dynamic Hedging Backward Pricing, Dynamic Hedging What can go wrong? American-style Options 3.2. binomial pricing of European Options 3. pricing European options S0 = 100 S 1,1 = 110 S1,0 = 90 S 2,0 = 81 u = 1.1; d =.9; 1+r = 1.05; 1+r* = ; 1+r forward factor = r* q = = 0.60 payoff S 2,2 = S = ,1 n [cashflow"at"t"if"j""up"s]"! "[risk-adj"prob"of"j""up"s] Call: C0 = if we land in node! (1+r)n j=0 (1,1): = 26"! "(0.62 )"+"4"! "(2"! "0.6"! "0.4)"+"0"! "(0.4 2 ) b 1,1 = P. Sercu and R. Uppal The Finance Workbook page 8.16 C 1,1 = if we land in node (1,0): b 1,0 = C 1,0 = = 1 (26 0.6) + (4 0.4) = =.222 (4 0.6) + (0 0.4) =

45 Backward Pricing, Dynamic Hedging Backward Pricing, Dynamic Hedging What can go wrong? American-style Options 3.2. binomial pricing of European Options 3. pricing European options S0 = 100 S 1,1 = 110 S1,0 = 90 S 2,0 = 81 u = 1.1; d =.9; 1+r = 1.05; 1+r* = ; 1+r forward factor = r* q = = 0.60 payoff S 2,2 = S = ,1 n [cashflow"at"t"if"j""up"s]"! "[risk-adj"prob"of"j""up"s] Call: C0 = if we land in node! (1+r)n j=0 (1,1): = 26"! "(0.62 )"+"4"! "(2"! "0.6"! "0.4)"+"0"! "(0.4 2 ) b 1,1 = P. Sercu and R. Uppal The Finance Workbook page 8.16 C 1,1 = if we land in node (1,0): b 1,0 = C 1,0 = = 1 (26 0.6) + (4 0.4) = =.222 (4 0.6) + (0 0.4) =

46 Backward Pricing, Dynamic Hedging Backward Pricing, Dynamic Hedging What can go wrong? American-style Options C 1 = j if S1 = if S 1 = 90 at time 0 we do have a two-point problem: Summary: b 0 = C 1,1 = we hedge dynamically: = ( ) + ( ) = start the hedge at time 0 with units sold forward. The time-1 hedge will be to sell forward 1 or units of foreign currency, depending on whether the rate moves up of down. we price backward, step by step

50 Hedging Verified Backward Pricing, Dynamic Hedging What can go wrong? American-style Options 102 S0 = Delta hedging S 1,1 = S1,0 = 90 S 2,0 = 81 payoff S 2,2 = S = ,1 Example at time 0: u = 1.1; d invest =.9; r 129 = 1.05; at 5%, 1+r* buy fwd = MTL ; at = 102 value forward if up: 1+r factor = q = ( r* = 102) 0.60 = value if down: ( ) = one-period forward rates added above spot rates if in node (1,1): invest at 5%, buy fwd MTL 1 at = !!4 b 1,1 = 121!!99 = 1, V (26! 0.6)!+!(4! 0.4) value if up: ,1 = 000 ( ) = 1.05 = value if down: ( ) = !!0 b 1,0 = 99!!81 = , V 4! 0.6!+!0!0.4 1,0 = 1.05 = if in node (1,0): invest at 5%, buy fwd MTL at = 91.8 P. Sercu and R. Uppal The Finance Workbook page 8.21 value if up: ( ) = value if down: ( ) =

51 Backward Pricing, Dynamic Hedging What can go wrong? American-style Options What can go wrong? 4.4 Note: what could go wrong? Everything can and will go wrong: Suppose that, unexpectedly, the risk changes in (1,1): u = 1.20, d = 0.80 in (1,0): u = 1.05, d = Multiperiod binomial P. Sercu and R. Uppal The Finance Workbook page Change V 1,1 of risk: = ±20% 1.05 if up, = ±5% 19.36, ifnot down, 16.38; instead of the current ±10%: and V 1,0 = C 1,1 = 1.05 = 0, not 2.29 = 19.36, not 16.38, 1.05 to hedge/replicate, we should have C used ,0 = = = ?! 0.00, not 2.29, You would have mishedged: You would lose, as a writer, in the upstate (risk up) You would gain, as a writer, in the downstate (risk down)

52 Backward Pricing, Dynamic Hedging What can go wrong? American-style Options What can go wrong? 4.4 Note: what could go wrong? Everything can and will go wrong: Suppose that, unexpectedly, the risk changes in (1,1): u = 1.20, d = 0.80 in (1,0): u = 1.05, d = Multiperiod binomial P. Sercu and R. Uppal The Finance Workbook page Change V 1,1 of risk: = ±20% 1.05 if up, = ±5% 19.36, ifnot down, 16.38; instead of the current ±10%: and V 1,0 = C 1,1 = 1.05 = 0, not 2.29 = 19.36, not 16.38, 1.05 to hedge/replicate, we should have C used ,0 = = = ?! 0.00, not 2.29, You would have mishedged: You would lose, as a writer, in the upstate (risk up) You would gain, as a writer, in the downstate (risk down)

53 Backward Pricing, Dynamic Hedging What can go wrong? American-style Options American-style Options Exchange rate American Options European Put with X= (0) (0) 7.81 (10) u=1.1, d=0.9, r=5%, (1+r)/(1+r*) = 1.02, q=0.60 P. Sercu and R. Uppal The Finance Workbook page American Put with X= (0) 4.03 (0) 7.81 (10) European: Node (1,1) P 1,1 In = (0.6!0)!+!(0.4!1) this node 1.05 the choices = 0.38 are, P 1,0 = (0.6!1)!+!(0.4!19) 1.05 = 7.81 PV of later exercise P 0 = (0.6!0.381)!+!(0.4!7.81) (0 or 1): Value of immediate exercise: 0 so we wait; 1.05 = V ,1 =.381 American: Node (1,0) In every Nownode the (except choicesthe are last period, but including time 0): chose between exercising PV of (" later value exercise dead) or (0 postponing or 19): 7.81 (" Value alive). The value is given by the larger of the Value two. of immediate exercse: 10 so we exercise; V 1,0 = 10 not 7.81 Node (0) We now choose between PV of later exercise (0 or 1 at time 2, or 10 at time 1): P alive = = Value of immediate exercise: 0 so we wait; V 0 =

56 Outline STP-ing of European Options Option s Delta The Replication Approach The Hedging Approach The Risk-adjusted Probabilities Notation Discussion Binomial Option Pricing Backward Pricing, Dynamic Hedging What can go wrong? American-style Options Black-Merton-Scholes STP-ing of European Options the Black-Merton-Scholes Equation The Delta of an Option

57 Straight-Through-Pricing a 3-period Put The long way: ! ,15 16, STP-ing of European Options Option s Delta C 2,2 = C 2,1 = C 2,0 = C 1,1 = C 1,0 = C 0 = = 0.00, = 4.152, = 16.55, = 1.582, = 8, 678, , =

58 Straight-Through-Pricing a 3-period Put The long way: ! ,15 16, STP-ing of European Options Option s Delta C 2,2 = C 2,1 = C 2,0 = C 1,1 = C 1,0 = C 0 = = 0.00, = 4.152, = 16.55, = 1.582, = 8, 678, , =

59 Straight-Through-Pricing a 3-period Put STP-ing of European Options Option s Delta ! , , The fast way: pr 3 =... pr 2 =... pr 1 =... pr 0 =... The (risk-adjusted) chance of ending in the money is C 0 = = 4.21.

60 Straight-Through-Pricing: 2-period Math STP-ing of European Options Option s Delta C 1,1 = q C 2,2 + (1 q)c 2,1, 1 + r C 1,0 = q C 2,1 + (1 q)c 2,0, 1 + r C 0 = q C 1,1 + (1 q)c 1,0, 1 + r = q h q C2,2 +(1 q)c 2,1 1+r i + (1 q) 1 + r = q2 C 2,2 + 2q (1 q)c 2,1 + (1 q) 2 C 2,0 (1 + r) 2 h q C2,1 i +(1 q)c 2,0 1+r,

61 Straight-Through-Pricing: 3-period Math STP-ing of European Options Option s Delta C 1,1 = q2 C 3,3 + 2q (1 q)c 3,2 + (1 q) 2 C 3,1 (1 + r) 2 C 1,0 = q2 C 3,2 + 2q (1 q)c 3,1 + (1 q) 2 C 3,0 (1 + r) 2, C 0 = q C 1,1 + (1 q)c 1,0, 1 + r = q ˆq 2 C 3,3 + 2q (1 q) C 3,2 + (1 q) 2 C 3,1 + (1 q)ˆq 2 C 3,2 + 2q (1 q)c 3,1 + (1 q) 2 C 3,0 (1 + r) 3, = q3 C 3,3 + 3q 2 (1 q)c 3,2 + 3q(1 q) 2 C 3,1 + (1 q) 3 C 3,0 (1 + r) 3

62 Toward BMS 1: two terms STP-ing of European Options Option s Delta Let pr (Q) n,j = risk-adjusted chance of having j ups in n jumps N = q j (1 q) N j = q j (1 q) N j j n! j! (n j)! {z } # of paths with j ups {z } prob of such a path and let a : {j a} {S n,j X}; then C 0 = = = P N j=0 pr(q) n,j C n,j (1 + r) N = CEQ 0( C N ) discounted, P N j=0 pr(q) n,j (S n,j X) + (1 + r) N, P N j=a pr(q) n,j (S n,j X) (1 + r) N, = P N j=a pr(q) n,j S n,j X (1 + r) N (1 + r) N NX j=a pr (Q) n,j. (2)

63 Toward BMS 2: two probabilities Recall: C 0 = P Nj=a pr (Q) n,j S n,j (1 + r) N We can factor out S 0, in the first term, by using We also use 1 (1 + r) N = 1 (1 + r ) N S n,j = S 0 u j d N j. X NX (1 + r) N j=a pr (Q) n,j. 1 + r «j 1 + r «N j 1 + r 1 + r STP-ing of European Options Option s Delta P N j=a pr(q) n,j S n,j (1 + r) N = = S 0 (1 + r ) N S 0 (1 + r ) N NX N q 1 + «j (1 r q) 1 + «N j r j j=a NX j=a 1 + r N j π j (1 π) N j 1 + r where π := q 1 + r 1 + r 1 π = (1 q) 1 + r 1 + r.

64 BMS 3: the limit STP-ing of European Options Option s Delta C 0 = S 0 (1 + r ) N {z } price of the underlying FC PN a j a probability-like expression z! } { NX N π j (1 π) N j j j=a Special case a = 0: a = 0 means that... so both probabilities become... and we recognize the value of... X (1 + r) N {z } discounted strike In the limit for N (and suitably adjusting u, d, r, r ) j/n becomes Gaussian, so we get Gaussian probabilities prob (Q) of j a z } { NX j=a pr (Q) n,j. (3) first prob typically denoted N(d 1 ), d 1 = ln(f t,t /X)+(1/2)σt,T 2, with σ σ t,t t,t the effective stdev of ln S T as seen at time t second prob typically denoted N(d 2 ), d 2 = ln(f t,t /X) (1/2)σ 2 t,t σ t,t

69 The Delta of an Option STP-ing of European Options Option s Delta Replication: in BMS the option formula is still based on a portfolio that replicates the option (over the short time period dt): a fraction P n j=a πj or N(d1) of a FC PN with face value unity, and a fraction P n j=a prj or N(d2) of a HC PN with face value X. Hedge: since hedging is just replication reversed, you can use the formula to hedge: version of formula hedge instrument unit price size of position C 0 = S 0 1+r 0,T N(d 1 )... FC PN expiring at T S 0 1+r 0,T N(d 1 ) N(d C 0 = S 1 ) 0 1+r 0,T N(d... FC spot deposit S 1 ) 0 1+r 0,T C 0 = F 0,T N(d 1 ) 1+r 0,T... Forward expiring at T F 0,T N(d 1 ) 1+r 0,T

70 The Delta of an Option STP-ing of European Options Option s Delta Replication: in BMS the option formula is still based on a portfolio that replicates the option (over the short time period dt): a fraction P n j=a πj or N(d1) of a FC PN with face value unity, and a fraction P n j=a prj or N(d2) of a HC PN with face value X. Hedge: since hedging is just replication reversed, you can use the formula to hedge: version of formula hedge instrument unit price size of position C 0 = S 0 1+r 0,T N(d 1 )... FC PN expiring at T S 0 1+r 0,T N(d 1 ) N(d C 0 = S 1 ) 0 1+r 0,T N(d... FC spot deposit S 1 ) 0 1+r 0,T C 0 = F 0,T N(d 1 ) 1+r 0,T... Forward expiring at T F 0,T N(d 1 ) 1+r 0,T

71 What have we learned in this chapter? STP-ing of European Options Option s Delta Why binomial? does basically the same as the BMS pde, but... is much simpler One-period problems hedging/replication gets us the price without knowing the true p and the required risk correction in the discount rate but that s because we implicitly use q instead: the price is the discounted risk-adjusted expectation Multiperiod models basic model assumes constant u, d, r, r we can hedge dynamically and price backward for American-style options, we also compare to the value dead Black-Merton-Scholes For European-style options, you can Straight-Through-Price the option This gets us a BMS-like model BMS itself is a limit case