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Call Option Pricing Through Binomial Trees and Hedging Strategies

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13 diapositives · 302 Views · Feb 17, 2013

The process of pricing call options using a binomial tree model. Specifically, it focuses on a stock currently priced at $100, with two future dates where the stock can increase by 10% or decrease by 5%. By breaking down the pricing problem into simpler two-date problems, we apply standard calculation methods to determine the option's value. We also analyze hedge ratios and present values of portfolios, ruling out arbitrage opportunities through a systematic approach, culminating in an understanding of the Black-Scholes option pricing formula's foundation.

#call option pricing #binomial tree model #hedge ratio #stock market analysis #blackscholes formula #riskfree rate #future stock prices #financial derivatives
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Call Option Pricing Through Binomial Trees and Hedging Strategies
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  1. Options Pricing: addon Riccardo Colacito

  2. Setup of the problem • Stock price today is $100 • Two future dates • At each date • Stock price can go up by 10% • Stock price can go down by 5% • Call strike price is $110 • Risk free rate for each period is 5%

  3. Binomial Tree: what’s the price of the stock at each date and state? 121 110 100 104.50 95 90.25

  4. The recipe • After reading this set of slides you should be convinced that pricing calls in this exercise is just identical to what we did in the simple binomial tree with only two dates. • The only secret is that you have to decompose the problem into a bunch of simple `two dates problems’ and apply the standard recipe that is reported on the next slide.

  5. Option pricing using hedge ratio • Find hedge ratio • Find tomorrow’s value of portfolio made of H share and one written call • Find present value of portfolio (discount at risk free rate) • Set this value equal to today’s cost of portfolio and solve for call price

  6. Start from the end! 121 (call value: 11) 110 100 104.50 (call value: 0) 95 90.25

  7. Call price when stock is worth $110 • Hedge ratio: • Date 3 portfolio value: • Discounted value: • Rule out arbitrage:

  8. Start from the end! 121 110 100 104.50 (call value: 0) 95 90.25 (call value: 0)

  9. Call price when stock is worth $95 • Here you can proceed `brute force’ as we did before, or use a shortcut. • Here’s the shortcut: if today’s price is $95, the option tomorrow is going to expire out of the money no matter what happens. This option is worthless, hence its price should be zero! • That is C95=$0.

  10. Now work on the initial date We found these values before! 121 110 (call value: 6.984) 100 104.50 95 (call value: 0) 90.25

  11. Call price when stock is worth $100 • Hedge ratio: • Date 2 portfolio value: • Discounted value: • Rule out arbitrage:

  12. Value of the call option at each date and state 11 6.984 4.434 0 0 0

  13. Black and Scholes • In a nutshell, the Black and Scholes option pricing formula is based on the binomial asset pricing model, by: • letting the distance between two consecutive dates shrink (think about two consecutive nodes being one millisecond away from each other!) • Extending the tree to a large number of nodes