e-?tr(0) + e-?t µ(e?t – 1) + e-?te?s dBs
r(t) = e-?tr(0) + µ(1-e-?t)+ e-?(t-s) dBs
Call option price of a bond at t
(current time) which has an expiry at time m
such that (t
pay a price more than the actual price of the share at time T , resulting in a
loss.However, the option will not be exercised in this case, which is why the
owner pays an extra price for the option.
European put option gives the owner the
right , but not the obligation, to sell the share to the issuer on an agreed
time in the future known as the expiry date (T) at an agreed price called the
strike price (K).While,the writer of the option is obliged to buy the share.
In this case,
the payoff would be C = (K-ST)+
Similarly if K
> ST (in-the-money) then the owner will be selling the share at a
price more than the share price at time T , thereby gain the difference.
Binomial model :
Ina financial market,
a portfolio ( set of assets ,called stocks) which can be bought and sold at
discrete times (1,2,3….) are priced using the binomial model which assumes:
No minimum or maximum units of trading
Short-term interest rate is fixed over the
period of time and is represented by ?
No trading costs or tax
Allowed to borrow/short sell shares
Principle of no-arbitrage applies :
d < (1+?) < u where u/d is the amount by which the share price increase/decrease respectively. TWO-STEP BINOMIAL MODEL: The portfolio of a discrete time market is generally made up of two assets : one bond (non-risky asset) which has a fixed rate of interest (?) and one share (risky asset) with share prices at times 0,1,2 defined as S0 (initial share price) , S1= S0Y1, S2=S1Y2 ,where Yi (i= 0,1,2) are iid random variables with P(Y= u) + P(Y= d) = 1 ,meaning that at each time, the share price either goes up by 'u' or goes down by 'd' . Let Xo and X1 be the no. of bonds and no. of shares invested at time 0 respectively. The binomial model is based on pricing using arbitrage.Our portfolio is formulated in such a manner that if it is said to replicate the payoff of an option, then the price of the portfolio is nothing but the option price, assuming arbitrage-free condition in the market. That is, Wo = Xo+X1So = OP(C) Capital at time 2 : W2 = Xo(1+r)2 + X1S2 = C Under binomial model, (qu + qd ) = 1 (1) and EQY = u qu + d qd = 1+? (2) Solving these two equations gives us the martingale probabilities , qu = and qd = = (1 - qu) We want W2 = C = f(S2) C / S2 = S0u2 = Cuu = Xo(1+r)2 + X1 S0u2 C/ S2 = S0ud = Cud = Xo(1+r)2 + X1 S0ud C / S2 = S0d2 = Cdd = Xo(1+r)2 + X1 S0d2 The binomial tree representation : qu S2 = S0u2 , Cuu = (S0u2 - K )+ qu2 S1= S0u qu qd S0 = 8 S2 = S0ud, Cud = (S0ud - K) + qu 2quqd qd S1= S0d qd S2 = S0d2 , Cdd = (S0d2 -K) + qd2 The option claim of the option price is given by , OP(C) = (1+ ?)-2 EQ(C) OP(C) = (1+ ?)-2 Cuu qu2 + Cud (2quqd) + Cdd qd2 Hedging : To find the conditional ptice of the share at each level, we need to use hedging technique, which is given by: The no-arbitrage time t conditional option price of option claim C at maturity (T) is defined by: OP(C) = (1+ ?)-(T-t) EC/St = f In as two-step binomial model, we need no-arbitrage time 1 conditional option price of the option claim C where S1 can either be S0u or S0d. Hence, the tree is divided into two blocks – U block and D block with conditional option prices v(u) and v(d) respectively. v(u) = (1+ ?)-(T-1) EC/S1 = S0u should be equal to Xo(1+ ?)+X1S0u (1) v(d) = (1+ ?)-(T-1) EC/S1 = S0d should be equal to Xo(1+ ?)+X1S0d (2) Solving these equations will give us the hedging portfolio (Xo, X1): Xo = & X1 = Generally, Option price of n steps are given by: (T= 1,2,….n steps) OPf (ST) = (1+ ?)-T S0ujdT-j ) qu j qd T-j where j is given by pascal tree method. EXAMPLE TWO-STEP BINOMIAL MODEL Consider a discrete financial market having a portfolio made of one risky asset(share) and one non-risky asset(bond).The share prices at times 0,1,2 are defined by the two-period binomial model: S0 = 8 , S1 = S0Y1, S2 = S1Y2 , Yi (i= 0,1,2) are iid random variables where, P(Y=1.15) + P(Y=0.90) =1 (The share price either goes up by 15% or goes down by 10%) The fixed rate of interest (?) is given by 3% per period. (a) Write down the binomial tree representation of the arbitrage-free model. (b) Find the arbitrage-free time 0 and time 1 option price of the European call option with a strike price K = 7 (c) Identify its intial hedging portfolio. SOLUTION: u = 1.15 , d = 0.90 , ? = 0.03 , d < (1+ ?)< u for no-arbitrage condition to hold true and in this case its true. (0.90 < 1.03 < 1.15) Option claim is given by C = (S2 – 7)+ Martingale probabilities: qu = = = 0.52 qd = (1 - qu) = 1-0.52 = 0.48 Binomial tree: qu S2 = S0u2 = 10.58 , Cuu = (10.58 - 7)+ = 3.58 qu2 = 0.2704 S1= S0u = 9.2 qu = 0.52 qd S0 = 8 S2 = S0ud = 8.28 , Cud = (8.28 - 7) + = 1.28 qu 2quqd = 0.4992 qd = 0.48 S1= S0d = 7.2 qd S2 = S0d2 = 6.48 , Cdd = (6.48 - 7) + = 0 qd2 = 0.2304 OP(C) = (1+ ?)-2 EQ(C) OP(C) = (1+ ?)-2 Cuu qu2 + Cud (2quqd) + Cdd qd2 OP(C) = (1.03) -2 (3.58*0.2704) + (1.28*0.4992) + (0*0.2304) Time 0 Option price of the option claim is 1.51476 Conditional time-1 Option price: Hedging: U-BLOCK: (C') We start with S0u (time 1) which either moves up or down at time 2 S1' = S0u2 = 10.58 , Cu'= Cuu = (10.58 - 7)+ = 3.58 qu = 0.52 qu = 0.52 S0'= S0u = 9.2 qd = 0.48 S1' = S0ud = 8.28 , Cd'= Cud = (8.28 - 7) + = 1.28 qd = 0.48 OP(C') = (1+ ?)-1 Cu' qu + Cd' qd V(u) = OP(C') = (1.03) -1 3.58(0.52) + 1.28(0.48) = 2.4038835 D-BLOCK: (C'') Similarly, we start with S0d (time 1) which either moves up or down at time 2 S1'' = S0ud = 8.28 , Cu''= Cud = (8.28 - 7) + = 1.28 qu = 0.52 qu = 0.52 S0''= S0d = 7.2 qd = 0.48 S1'' = S0d2 = 6.48 , Cd''= Cdd = (6.48 - 7) + = 0 qd = 0.48 OP(C'') = (1+ ?)-1 Cu'' qu + Cd'' qd V(d ) = OP(C'') = (1.03) -1 1.28(0.52) + 0(0.48) = 0.6462136 The hedging portflios are given by: Xo = Xo = Xo = -5.516 (borrow/short sell bonds) X1 = X1 = X1 = 0.878835 (buy shares) We know that Wo = Xo+X1So Wo = -5.516 + 0.878835(8) = 1.51476 This gives us the intuitive result ,i.e, Wo = OP(C) The initial capital invested is nothing but the price paid for the option.