FINS2624 Lecture Notes - Lecture 11: Arbitrage, Standard Deviation, Log-Normal Distribution

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18 May 2018

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11 – Option Valuation

Pricing Options:

• Options priced using no-arbitrage arguments

• Form portfolio that combines positions in option and its underlying asset, to

replicate risk-free asset

• Options are derivatives whose payoffs depend on the value of the underlying assets

One-Period Binomial Model:

• Setup

 Assets:

(a) Underlying asset (e.g. stock) with price S

i Stock can either have high return (price changed by u) or low return

(price changed by factor d)

ii Price of stock at time T, ST, will be either S0u or S0d

(b) Option with premium (i.e. price) f

i Payoff depends on price of stock at time T

ii Payoff either fu (if high return) or fd (if low return)

 Two periods (0 & T)

(a) Want to find arbitrage-free price of option at time 0, f0

Replicating the Risk-Free Asset:

• Payoff in two nodes should be certain (known):

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Document Summary

Pricing options: options priced using no-arbitrage arguments, form portfolio that combines positions in option and its underlying asset, to replicate risk-free asset, options are derivatives whose payoffs depend on the value of the underlying assets. Assets: (a) underlying asset (e. g. stock) with price s i. Two periods (0 & t) (a) want to find arbitrage-free price of option at time 0, f0. Replicating the risk-free asset: payoff in two nodes should be certain (known): Change in option price resulting from a change in stock price. Measures sensitivity of option value to underlying asset price. Cha(cid:374)ge i(cid:374) (cid:448)alue of shorti(cid:374)g optio(cid:374) e(cid:454)a(cid:272)tl(cid:455) offset (cid:271)(cid:455) (cid:272)ha(cid:374)ge i(cid:374) (cid:448)alue of shares of stock portfolio value will not change (i. e. risk free) Also (cid:272)alled (cid:858)hedge ratio(cid:859) of option, value change in one option position (long or short(cid:895) (cid:449)ill (cid:271)e hedged (cid:271)(cid:455) holdi(cid:374)g shares of sto(cid:272)k i(cid:374) opposite dire(cid:272)tio(cid:374: portfolio must earn risk-free rate in absence of arbitrage.

Related Questions

Consider a stock currently worth $80 (S = 80) that can go up or down by 25 percent per period. The exercise price is $80 (X = $80) and the risk-free rate is 5 percent (r = .05). The option will expire at the end of the third period (t = 3). You try to find a theoretically fair value of the put option using a three-period binomial option pricing model. Answer the following questions.

t = 0	t = 1	t = 2


		Suu = (c)
	Su = (a)
S = $80		Sud = (d)
	Sd = (b)
		Sdd = (e)



		P_uu = (f)
	P_u = (i)
P = (k)		P_ud = (g)
	P_d = (j)
		P_dd = (h)

a) What are three possible stock prices at time t = 2 (c, d, and e)?

b) Compute three values at time t = 2 of European put options. (P_uu, P_ud, and P_dd)

c) Compute two values at time t = 1 of a European put options (P_u and P_d).

d) Find the theoretical fair value of the put option today (P).

e) Compute the riskless hedge ratio (h).

f) Construct a riskless hedge position with stocks and 1,000 put options at the theoretically fair value using the riskless hedge ratio (h).

taupecamel175

6. Institutional portfolios can take advantage of what is called "portfolio margining" under the rules of the CBOE and other options exchanges. What IS portfolio margining? Choose only one answer.

A. You only pay attention to the bottom margin of the portfolio.

B. Great leverage is possible, as the intercorrelations and natural offsets (or what we call in English a "wash") are taken into consideration.

C. No margin is required, because these are not individual investors.

7. What does the symbol ^VVIX mean?

A. Apple's volatility.

B. An exchange-traded (ETF) volatility fund.

C. The implied volatility of the VIX options themselves.

8. An example of purchasing a "real" option is a company's opening of a Shanghai office, in order to have the capacity to pursue business in China.

True

False

9. A hedge fund is "short" a portfolio of VIX call options. It's "theta" is $16,788 per day. This means, barring some big market or political or terrorist event, the fund's net asset value will increase by about $16,788 per day.

True

False

10. Imagine an equity-oriented state pension fund (such as the State of Maryland Pension funds). Suppose their equity portfolio is NOT identical to the S&P 500 index. Further, suppose that due to a state budget shortfall, the year's additions to the pension fund are likely to be less than actuarially sound. Thus, the fund is becoming very risk-averse. And it needs to implement this feeling by hedging. CHOOSE THE BEST ANSWER, even though there might be more than one true answer. This is actually a question about hedge ratios. A. If the equity portfolio's beta is close to 0, then they could decide to match the overall exposure, via S&P Index futures contracts, to the fund's value. That is, # of contracts times the contract multiplier times the index would approximately equal the value of the fund.

B. If the equity portfolio's beta is close to .4, then they would have a tough time using S&P 500 contracts, so they would instead use foreign exchange contracts.

C. If the equity portfolio's beta is close to 1, then they could decide to match the overall exposure, via S&P Index futures contracts, to the fund's value. That is, # of contracts times the contract multiplier times the index would approximately equal the value of the fund.

11. Go to CMEGROUP.COM We are going to hedge ULSD https://en.wikipedia.org/wiki/Ultra-low-sulfur_diesel for a railroad, that uses ultra-light diesel fuel. Choose June 2017 as the underlying future. Let's pretend we are buying ten futures contracts. What would ten contracts cost you, using the last trade as our price? Show your math, i.e., #gallons times price per futures contract. Is this contract in contango, or normal backwardation?

12. Long options positions(long does not mean long-dated) have positive gamma, and short positions have negative gamma.

True

False

13. In valuing credit default swaps, one must estimate default probabilities. A start would be to obtain historical data for default rates from rating agencies such as Standard & Poor's. But a more complete risk profile, or risk prediction, limiting ourselves to the US market (US-based companies) would be to [pick only one choice, the best one, in your opinion]:

A. Also consider the risk of another earthquake in Japan.

B. Also consider the systematic risk associated with an economic downturn in the US, accompanied by a downturn in US stocks.

C. Also consider the possibility that Facebook options are trading at an implied volatility that is not as high as it should be, considering the lofty level of its P/E ratio.

14. A total return swap can be a form of financing vehicle to allow an investor to buy a specific bond, because: (there are two correct answers, you must pick two)

A. The payer (a financial institution) could agree to pay the bond's capital appreciation or depreciation, plus the interest payments, in return for receiving LIBOR plus some basis points.

B. The payer (a financial institution) could agree to receive the bond's capital appreciation or depreciation, plus the interest payments, in return for receiving LIBOR plus some basis points.

C. The payer retains ownership of the bond for the life of the contract.

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FINS2624 Lecture Notes - Lecture 11: Arbitrage, Standard Deviation, Log-Normal Distribution

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