CS476/676 Numeric Computation for Financial Modelling
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Financial Options:
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A financial option/derivative is a financial contract stipulated today at t = 0. The value of the contract at the future expiry T is determined exactly by the market price of an underlying asset at T .
St, T, payoff(ST )
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underlying asset St: stock, commodity, market index, interest rate/bond, exchange rate.
Let St, or S(t), denote the underling price at time t, a stochastic process
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Knowing the future value of the contract in relation to the underlying allows it to be used as an insurance.
Holder bought insurance. How much should the holder pay today?
Writer bought risk (uncertainty). How much should the writer receive today?
premium V0
VT =payoff(ST)
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European calls and puts
A European call option is the right to buy an underlying asset at a preset strike price K. The right can only be exercised at the expiry T.
Note: asymmetry: holder has the option to exercise. Writer has the obligation.
A European put option is the right to sell an underlying asset at a preset strike price K. The right can only be exercised at the expiry T.
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An American put option is the right to sell an underlying asset at a preset strike price K. The right can be exercised any time from now to the expiry T.
Holder: buyer of the option, enters a long position Writer: Seller of the option, enters a short position. Let V (S(t), t), or Vt, denote the option value at time t.
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Central Question: what is the fair value V0 of the option today? How should a writer hedge risk?
What are payoff functions VT = payoff(ST ) for calls/puts?
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Call value at the expiry T
If ST ≤ K, holder should not exercise the call. VT = 0
If ST ≥ K, holder exercises the right: gets ST and pays K.
⇒VT =payoff(ST)=max(ST −K,0)=
ST−K ifST≥K 0 otherwise
Put value at the expiry T
If ST ≤ K, holder should exercise. VT = K − ST If ST ≥ K, holder should not exercise
⇒VT =payoff(ST)=max(K−ST,0)=
ifST ≤K otherwise
Many payoff functions have been created. Straddle: max(ST −K,0)+max(K −ST,0)
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Bulb wholesaler can purchase a call to have the option of buying
tulip $.50/dozen at a fixed price in 3 months.
Bulb growers can purchase a put to allowing selling tulip $ 1/dozen at a fixed price in 3 months.
A bet on the underlying price can be done by trading either St or Vt.
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Note: Option is more risky compared to the underlying (leverage effect):
ST −S0 VT −V0 VT −V0 S << V , VT = payoff = 0 ⇒ V
When option expires out of money, 100% loss for the option holder
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What do we know about the pricing problem?
We know at expiry VT = payoff(ST ). We don’t know S0 → ST .
We know S0.
What is V0?
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We focus on stock option with expiry T ≤ 1 and interest rate randomness is reasonably ignored.
Stock: a share in ownership of a company.
Dividend: payment to shareholders from the profits.
Note: when stock pays dividend to the shareholder, holder of option on the stock receives nothing. Option is said to be dividend protected.