(金融工程-理论与实践)Financial Engineering--Theory and PracticeGSM_04

Binomial Option Pricing ModelsTopic 4

Guanghua School of Management

4.1

K.C. John Wei

Learning Objectives Binomial Option Pricing: Replication Approach Binomial Option Pricing: Risk-free Hedge Approach Risk-Neutral Valuation Principle Two-period Binomial Option Pricing American Call and Put: Exercise Early? Binomial Model: Limiting Case

Guanghua School of Management

4.2

K.C. John Wei

Binomial Model: Valuation of Call S= Stock price today=$100 Exercise price (X)=$100 rf= 6%= risk-free rate real probability to go up (q)= 0.70 real probability to go down (1-q)= 0.3 Su= Su= up price=$125 Sd= Sd= down price=$75 Assumption: u (1.25)> 1+rf (1.06)> d (0.75) C= Call value today=?

Guanghua School of Management

4.3

K.C. John Wei

Binomial Model: Valuation of Call The binomial tree is as follows:Stock Price Su= 125 with a probability q Cu= Max(Su-X, 0)= Max(125-100, 0)= 25

S=100 C=?

Sd= 75 with a probability (1– q) Cd= Max(Sd-X, 0)= Max(75-100, 0)= 0 Time

0

1

Guanghua School of Management

4.4

K.C. John Wei

Binomial Model: Replication of Call Use a right combination of bond and stock to replicate call Strategy 1: Buyδ shares by borrowing B Payoff in period 1 will be:δSu - B(1+rf)δS - BδSd - B(1+rf) today 1 period laterδ=# of shares, B= amount of borrowing Strategy 2: Buy one call option Payoff will be: Cu C Cd today 1 period laterGuanghua School of Management 4.5 K.C. John Wei

Binomial Model: Replication of Call If these cash flows are the same thenδSu - B(1+rf)= CuδSd - B(1+rf)= Cd Let R= 1+rf and solve for the above equations:Cu Cd C 25 0 25δ= Hedge Ratio===== 0.5 Su Sd S 125 75 50S d Cu S u C d 75× 25 125× 0 1875==$35.38 B== (Su Sd )(1+ rf ) (125 75)(1.06) 53= dCu uC d 0.75× 25==$35.38 R× (u d ) 1.06× (1.25 0.75)

Guanghua School of Management

4.6

K.C. John Wei

Binomial Model: Replication of Call Buy 0.5 shares of stocks and borrow B=$35.38 0.5*125 - 35.38*1.06= 25 50 - 35.38= 14.62 0.5*75 - 35.38*1.06= 0 Buy one call option 25 C 0 To avoid arbitrage, today C= 14.62, i.e., each call option is worth$14.62. That is, if we borrow$35.38 to buy 0.5 share of stock (need additional$14.62 from our own pocket), then we can replicate the call with a price of$14.62.Guanghua School of Management 4.7 K.C. John Wei

Binomial Model: Replication of Call Redefine R= 1+ rfCu Cd Cδ== Su Sd S dCu uCd B= R(u d )

The cost of call: C=δS– B By substitutingδ and B into C and rearranging, we obtain

π C u+ (1 π )C d ( R d )C u+ ( u d )C d C== R (u d ) R R d u R whereπ=; 1 π= u d u d π= risk-neutral probability with up movement (0≤π≤1)Guanghua School of Management 4.8 K.C. John Wei

Binomial Model: Risk-free Hedge

Approach We create a risk-free hedge portfolio by buyingδ shares of stock and short selling one call. The cost of this risk-free hedge portfolio=δS– CValueδSu– Cu

δS– C

δSd– Cd Time 0 Guanghua School of Management 1 K.C. John Wei

4.9

Binomial Model: Risk-free Hedge Approach If this portfolio is risk-free, its payoffs at t= 1 will not be affected by the stock price movement and this portfolio should earn a risk-free rate. That is,δSd– Cd=δSu– Cu= (δS– C)×R. Solve forδ and C:πCu+ (1 π )Cd Cu Cd R d Cδ==,C=, whereπ= R u d S(u d ) S

This strategy is called the delta-neutral hedging strategy and is used every often by investment banks when they issue call warrants. That is, when investment banks short sell every one call warrant, they need to buyδ shares of the underlying stock to hedge.Guanghua School of Management 4.10 K.C. John Wei

Replication vs Risk-free Hedge Approaches In sum, both replication and risk-free hedging approaches obtain the same binomial option price:C=

π C u+ (1 π )C dR

,

R d whereπ= u d

Meaning ofδ for call (δ> 0): Delta (δ) is the ratio of the change in the price of stock call option to the change in the price of the underlying stock ( C/ S).δ is also a hedge ratio. That is, shorting one call needs to buyδ (δ> 0) shares of stock to hedge the risk.Guanghua School of Management K.C. John Wei

4.11

Binomial Model: Call Example Example: S= 100, u= 1.25, d= 0.75, R= 1.06, Cu= 25, Cd= 0

πC

(R d )= (1.06 0.75 )= 0.62= (u d ) (1.25 0.75 )πC u+ (1 π )C d 0.62× 25+ (1 0.62 )× 0===R 1.06

14 .62

δ= C/ S= (25-0)/(125-75)= 0.5: That is, shorting one call needs to buy 0.5 shares of stock to hedge the risk. Or shorting one call and at the same time buying 0.5 shares of stock can create a risk-free hedge portfolio.Guanghua School of Management 4.12 K.C. John Wei

Binomial Model: Valuation of Put Data: Same as before. What is the price of put (P)? The binomial tree is as follows:Stock Price Su= 125 with a probability q Pu= Max(X-Su, 0)= Max(100-125, 0)= 0

S=100 P=?

Sd= 75 with a probability (1– q) Pd= Max(X-Sd, 0)= Max(100-75, 0)= 25 Time 0 Guanghua School of Management 1 K.C. John Wei

4.13

Binomial Model: Replication of Put Use a right combination of bond and stock to replicate put Strategy 1: Buyδ shares by borrowing B Payoff in period 1 will be:δSu - B(1+rf)δS - BδSd - B(1+rf) today 1 period laterδ=# of shares, B= amount of borrowing Strategy 2: Buy one put option Payoff will be: Pu P Pd today 1 period laterGuanghua School of Management 4.14 K.C. John Wei

Binomial Model: Replication of Put If these cash flows are the same thenδSu - B(1+rf)= PuδSd - B(1+rf)= Pd Let R= 1+rf and solve for the above equations:

δ= HedgeRatio=

Pu Pd P 0 25 25

=== 0.5= Su Sd S 125 75 50

S d Pu S u Pd 75× 0 125× 25= B== $ 58 .96 (Su Sd )(1+ r f ) (125 75 )(1.06 )= dPu uP d 1 .25× 25== $ 58 …… 此处隐藏：3982字，全部文档内容请下载后查看。喜欢就下载吧 ……