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Chapter 6

Chapter 6. The Analytic Approach to Black-Scholes. Δ V. §1. Black-Scholes Differential Equation. Let V (S, t ) be the price of an option on a stock. S is the stock price at time t , then. Taylor expansion. Recall d S = μ S d t + σ S dB( t ). Let’s look at.

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Chapter 6

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  1. Chapter 6 The Analytic Approach to Black-Scholes

  2. ΔV §1. Black-Scholes Differential Equation Let V(S,t) be the price of an option on a stock. S is the stock price at time t, then Taylor expansion

  3. Recall dS = μS dt + σS dB(t)

  4. Let’s look at Integration by parts Thus we have

  5. Hence

  6. or This is still a random term So

  7. To get rid of the stochastic term: Form a portfolio replicating the option as follows

  8. Thus The portfolio is self-financing: If we need to buy some additional stock we must sell some bonds, and vice versa. Thus

  9. Recall Choose

  10. We get Black-Scholes differential equation Note: The Black-Scholes equation is valid for any option.

  11. Initial and boundary conditions:

  12. The Black-Scholes Formula Here This is a solution to the Black-Scholes equation

  13. §6. Options on Futures • A futures contract is an agreement between the buyer and the seller to consummate a transaction on a future date. • Usually no money changed hands at beginning; • Futures are traded in the open markets. Forward contracts: traded over-the-counter. Futures contracts: traded in exchanges.

  14. How does the futures market work? • When an investor buys a futures: • no money exchanged at the beginning • but deposits funds in a margin account • At the end of each trading day: • Pay to the writer if futures price is down, or • Receive from the writer if price is up • Amount of the change in futures price through the margin account. (Marking to market) Usually 5% – 15%

  15. Example: Suppose you buy 5,000oz June futures of silver at $18. You need to deposit into the margin account 5,000x18x0.15 =13,500. Suppose next day the futures price of silver is $19, the seller need to deposit into your account $5,000, so your account will be $18,500. If the next day futures price is down to $16, you have to pay the seller $10,000 from your account, its balance is now $3,500. You will get a margin call.

  16. Suppose • Borrow S0 to buy one share of stock; • Write a futures. • Net profit at expiration time T: • Suppose the delivery date is T; • How to determine the strike price X now (at t = 0)? • Replicating investment consists of a futures and some cash amount Impossible!

  17. Thus we must have Hence the strike price is The futures price at time t is • Suppose • Short one share of stock to get S0; • Buy a futures. • Net profit at expiration time T: Impossible!

  18. Options on Futures • The underlying asset is a futures contract. • When a futures call option is exercised, the holder will get • A futures contract, and • Cash = futures price F – strike price X. • If the futures price < strike price X, will the futures call option be exercised? • What about a futures put option?

  19. Example: Suppose you hold a call option on June futures of silver (5,000oz), its strike price is $17 with expiration day May 15. Suppose the futures price of silver is $18 on May 15, you will exercise the futures option (of course!). You will get a June futures contract and cash amount 5,000x($18 – $17) = $5,000. If the futures price is $16.5, you will not exercise the option.

  20. Call Options on Futures of Stock Consider a futures contract of stock with strike price X and time to expiration τ. We first look at a call option on stock with the same strike price and expiration date. By Black-Scholes formula

  21. We have the price of a call option on futures

  22. Example: Consider a futures contract on a stock index. The futures will pay the index amount. Suppose the current futures price on index is 715, which expires in 3 months. If at the expiration day the stock index is 740, then the holder will receive 740 – 715 = $25. What is the current index? Consider a call on this futures with the same expiration date and with the strike price $740. Assume the volatility of the index is σ= 0.40 and the interest rate is 7%.

  23. Partial Differential Equation A PDE on Futures of Stock Suppose the option on future has a price G(F,t). But F = Ser(T – t) We can think of G(Ser(T – t),t) as the price of some financial derivative of the stock. Thus it satisfies the Black-Scholes equation

  24. PDE for Options on Futures

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