Binomial Model Introduction Binomial only two outcomes Binomial

Binomial Model

Introduction • Binomial – only two outcomes • Binomial model assumes that the underlying price moves only in one of the two possible new prices • Assumes discrete time--

Pricing option – No arbitrage approach • A current stock price is Rs 450 • At the end of 3 months it may rise to 490 or fall to 410 • The exercise price is 460. What will be the call option price? • The option is expected to have one of the two values Rs 490 or Rs 410 at the end of 3 months

Pricing option – No arbitrage approach • If the stock price is Rs 490 the value of the call option will be Rs 30 at the end of 3 months • If the stock price is Rs 410 the value of the option is 0

Pricing option – No arbitrage approach • To arrive at the option price in this situation under no arbitrage assumption • Set up a portfolio of a call option and a stock • In such a way that there is no uncertainty about the value of the portfolio at the end of 3 months • Because there are only two securities and only two possible outcomes. • Because the portfolio has no risk , the return on this portfolio must equal the risk free rate of return • Based on this , the option price is worked out

Pricing option – No arbitrage approach • Consider a portfolio with a long position in Δ shares of a stock and a short position in one call option • If the stock price moves from Rs 450 to Rs 490 the value of the shares is Δ 490 and the value of the option is Rs 30 so the total value of the portfolio(stock+call) is Δ 490 – 30. • If the stock price moves down to Rs 410 then the value of the portfolio is Rs 410 Δ

Pricing option – No arbitrage approach • If the Δ is chosen such that the final value of the two outcomes is the same then the portfolio is riskless • 490 Δ -30 = 410 Δ • Δ = 0. 375 • The risk less portfolio is • Long : 0. 375 shares • Short : 1 call option • Regardless of whether the stock price moves up or down the value of the portfolio is always = Rs 153. 75 • At the end of 3 months

Pricing option – No arbitrage approach • This riskless portfolio must earn the risk free rate of interest • So if Rs. 153. 75 is the value of the riskless portfolio at the end of the period 3 months • Its value at the beginning should be present value of Rs. 153. 75 discounted at the risk free rate • 153. 75 e – r. T given r = 10% T = 3/12, • 153. 75 e – 0. 1*3/12 = 149. 95

Pricing option – No arbitrage approach • The value of the portfolio today is • 450 * 0. 375 – f 0= 168. 75 - f 0 • 168. 75 - f 0 = 149. 9539 • f 0 = 18. 7961

Pricing option – Risk Neutral Approach One period Binomial Model • • • Notation S 0 = Current underlying price S 0 u = stock price increased one period later % increase in stock price = u-1 S 0 d= price decreased one period later % decrease in stock price = 1 -d f 0 = Current option price When stock price moves up to S 0 u, option pay off is f u X = exercise price r = risk free interest rate

Binomial stock price movements and Delta • If there is an up movement in the stock price, the value of the portfolio at the end of the life of the option is : • S 0 uΔ – fu • If there is down movement in the stock price the value of the portfolio at the expiration of option is: • S 0 dΔ – fd • Riskless portfolio when the two are equal will be: • S 0 uΔ – fu = S 0 dΔ – fd • Δ = fu –fd /S 0 u - S 0 d OR

Derivation of the One period Binomial Equation • Step 1: At riskless interest rate r , the present value of the portfolio is : • (S 0 uΔ – fu)e-r. T • Step 2: The cost of setting up this portfolio is: • S 0 Δ – f 0 • Hence, • S 0 Δ – f 0 = (S 0 uΔ – fu)e-r. T

Derivation of the One period Binomial Equation • S 0 Δ – f 0 = (S 0 uΔ – fu)e-r. T (contd from earlier slide) • = f 0 =S 0 Δ –(S 0 uΔ – fu)e-r. T • = f 0 = S 0 Δ(1 -ue-r. T )+ fue-r. T • Substitute for Δ , • f 0 = S 0 [fu –fd /S 0 u - S 0 d](1 -ue-r. T )+ fue-r. T • f 0 = fu (1 -de-r. T )+ fd (ue-r. T - 1) • ( u- d)

One period binomial formula(risk Neutral Approach) • f 0 = fu (1 -de-r. T )+ fd (ue-r. T - 1) • ( u- d) • f 0 = e-r Δt [p fu + (1 –p) fd] • where, • p = e r. T - d • u-d

Example pricing call option with risk neutral approach– binomial formula • • In the earlier example , S 0=450 u = 1. 0888 d = 0. 9111 r = 10% T = 0. 25 p = e 0. 1*0. 25 - 0. 9111 = 0. 642685 1. 088 -0. 91111

• f 0 = e-r Δt [p fu + (1 –p) fd] • = = 0. 97531[0. 642685 * 30 +(0)] • = 0. 97531*19. 28055 • = 18. 804

Practice Example • A stock price is currently $40. It is known that at the end of the month it will be either $42 or $ 38. • The risk free interest rate is 8% pa with continuous compounding. What is the value of one month European call option with a strike price of $ 39? Calculate with No arbitrage approach as well as risk neutral approach.

Two period Binomial • Initial stock price S 0 moves either up or down in each time step • The length of the time step is now Δt years • C 0 = e-r Δt [p Cu + (1 –p) Cd • p = e r Δt - d • u-d •

Two period binomial equation • For two period binomial model , the equation is • fu = e-r Δt [p fuu + (1 –p) fud] --(1) • fd = e-r Δt [p fud + (1 –p) fdd] ---(2) • f 0 = e-r Δt [p fu + (1 –p) fd] --- (3) • Substituting (1) and (2) in (3): • f 0 = e-2 r Δt [p 2 fuu + 2 p (1 –p) fud +(1 -p)2 fdd]

Put example • Consider 2 year European put with a strike price of $52 on a stock whose current price is $50. Suppose there are two time steps of one year and in each step the stock price either moves up by 20% or moves down by 20%. r = 5% • u= 1. 2 d= 0. 8 r= 0. 05 Δt = 1 year • p = e 0. 05*1 - 0. 08 = 0. 6282 • 1. 2 -0. 8 • Possible final stock prices are : $72, $ 48, $ 32 • fuu= 0 fud = 4 fdd = 20 • f 0 = e-2 r Δt [p 2 fuu + 2 p (1 –p) fud +(1 -p)2 fdd] • = e-20. 05*1 [(0. 6282)2 *0 + 2* 0. 6282 (1 – 0. 6282) *4 + • (1 -0. 6282)2 *20] = 4. 1923

American Option • For American option at each nod we have to compare the option value given by the above equation with the payoff from early exercise • The value of the option is greater of these two