M35R Introduction to Asset Liability Management for Actuarial Science
Delta: The degree to which an option price will move given a small change in the underlying stock price. For example, an option with a delta of 0.5 will move half a cent for every full cent movement in the underlying stock.
?c?pCall deltas are positive, . Put deltas are negative, (put option price ，0，0?S?S
and the underlying stock price are inversely related)
deeply out-of-the-money call : delta very close to zero
at-the-money call : delta close to 0.5
deeply in-the-money call : delta very close to 1.
Delta of a European call on a non-dividend paying stock is:
Delta = N(d) (see Black-Scholes formula for d1) 1
？r~pScXe？！？Also from Put-Call-Parity: . The first derivative gives us:
？！1. Therefore, put delta = call delta - 1. ??SS
The delta is often called the hedge ratio:
If you have a portfolio short n options (say, you have written n calls) then n multiplied by the delta gives you the number of shares (ie units of the underlying asset) you would need to create a riskless position - ie a portfolio which would be worth the same whether the stock price rose by a very small amount or fell by a very small amount. In such a "delta neutral" portfolio any gain in the value of the shares held due to a rise in the share price would be exactly offset by a loss on the value of the calls written, and vice versa. Note that above we are creating a risk-free portfolio: for two state option valuation methods in class we utilized n=1 option and delta () shares. (
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Note that as the delta changes with the stock price and time to expiration the number of shares would need to be continually adjusted to maintain the hedge. How quickly the delta changes with the stock price is given by gamma (see "Greeks" below).
Gamma: It measures how fast the delta changes for small changes in the underlying
stock price. ie the delta of the delta. The smaller it is the less often you will have to adjust the hedge to maintain a delta neutral position. If gamma is too large a small change in stock price could wreck your hedge. Adjusting gamma, however, can be tricky and is generally done using options -- unlike delta, it can't be done by buying or selling the underlying asset as the gamma of the underlying asset is, by definition, always zero so more or less of it won't affect the gamma of the total portfolio. Vega: The change in option price given a one percentage point change in volatility. Like delta and gamma, vega is also used for hedging.
Theta: The change in option price given a one one day decrease in time to expiration Rho: The change in option price given a one percentage point change in the risk-free interest rate.
Option Pricing Models
Introduction to Binomial Trees: we consider the two-state tree i.e. the value
in two different economic scenarios)
Portfolio Replication (no-arbitrage arguments) vs. Risk Neutral Valuation
In the no-arbitrage approach, we set up a riskless portfolio consisting of a position in the option and a position in the stock. By setting the return on the portfolio equal to the risk-free interest rate, we are able to value the option. When we use risk-neutral valuation, we first choose probabilities for the branches of the tree so that the expected return on the stock equals the risk-free interest rate. We then value the option by calculating its expected payoff and discounting this expected payoff at the risk-free rate.
A stock price is currently $40. It is known that at the end of 3 months it will be either $45 or $35. The risk-free rate of interest with quarterly compounding is 8% per annum. Calculate the value of a three month European put option on the stock with an exercise price of $40. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.
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I: No –arbitrage arguments (Create a risk-free portfolio):
At the end of 3 months, the value of the option is either $5 (if the stock price is $35) or $0 (if the stock price is $45).
The stock The put option
Consider a portfolio consisting of : - : shares (
+1 : option We have constructed the portfolio so that it is +1 options and - shares (rather (
than -1 options and + shares) so that the initial investment is positive. (
The future value of the portfolio is either -35+5 or -45. Whenever the value ((
of the portfolio in the upstate is the same as the value in the downstate, our portfolio is riskless. -35+5 = -45. Therefore, =-0.5. For this value of (((
delta, the future value of the portfolio is certain to be 22.5. It is riskless, i.e. it must earn the risk-free rate of interest. Hence the value of the portfolio today is 22.5/1.02.
The current value of the portfolio is also -40+p where p is the value of the (
option. Hence (-40*0.5 + p) = 22.5 /1.02.
The value of the option p = $2.06.
4;；??i4;；i！？！8%;11.02. NB: (：(：4)，
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II: Risk-Neutral Valuation:
Let q = the risk-neutral probability of an upward stock price movement i.e.
q = probability of an upward stock price movement in a risk-neutral world. At the end of 3 months, the value of the option is either $5 (if the stock price is $35) or $0 (if the stock price is $45).
$45 $0 q
The stock The put option
The expected future stock price = 45q+35(1-q).
We must have, the price of the security = the present value of the expected future value: 40=[ 45q+35(1-q)]/(1.02) ; 40(1.02) = 45q+35(1-q) ; 40.8=10q+35
; 10q=5.8 ; q=0.58
In a risk-neutral world, the expected future value of the option = 0q+5(1-q)=2.1. Price of the option = present value of expected future value = 2.1/1.02 = 2.06. This is consistent with the no-arbitrage answer.
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III Using State Price Vectors
The stock The risk-free asset
?The state price vector (psi), , combines probability and interest i.e. the risk-neutral probability and the discount function.
?represents state i. We form simultaneous equations from the stock price and i
the risk-free asset). The risk-free asset is like a bank account, i.e if we invest $1 at
the risk-free rate we receive $1.02 after 3 months, regardless of the economic scenario.
??The current price of the stock : 45+ 35= 40 12
??The current price of the risk-free asset: 1.02+1.02= 1 12
??The simultaneous equations yield: =0.5686 ; = 0.4118 . 12
，??, the current price of the option = (0) +(5) 12
= (0) 0.5686+(5) 0.4118=2.06
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IV: Create a risk-free portfolio:Buy n shares, borrow $B(buying on margin).
The stock The risk-free asset
The risk-free portfolio replicates the Call option
payoffs of the option.
The risk-free portfolio replicates the payoffs of the option:
In the upstate: 45n – 1.02B = 5
In the downstate: 35n – 1.02B= 0
Solve simultaneously for the values of n and B: n=0.5 B=$ 17.1569
Current price of the option = c= 40n –B = 40(0.5)-17.1569=2.8431 Any information on this handout that was not done in class is FYP (for your pleasure). 6
?cNB: Note that n=0.5, the delta, , of the call option . (?S
Exercise 1: Use method 4 above to determine the price of the put option. i.e. Create a risk-free portfolio ( Short n shares and invest B dollars, eg. short selling i.e. current value of portfolio is B-nS, where n is the delta of the put option ?p). ?S
Exercise 2: A stock price is currently $25. It is known that at the end of 2 months it will be either $23 or $27. The risk-free interest rate is 10% per
Sannum with continuous compounding. Suppose is the stock price at the T
2Send of 2 months. What is the value of a derivative that pays off at this T
? Used to calculate theoretical call price
? Ignores dividends paid during the life of the option
? Uses the five key determinants of an option's price: stock price, strike price,
volatility, time to expiration, and short-term (risk free) interest rate.
？？rTt()cSNdXeNd！？.()..()The theoretical call option price: 12
2??S；??lnrTt？？？;；(：(：X2)，)，d！ddTt！？？； or simply . 221Tt？；
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The variables are:
S = stock price; X = strike price;
T-t = time remaining until expiration, expressed as a percent of a year; r = current continuously compounded risk-free interest rate
； = annual volatility of stock price (the standard deviation of the short-term returns over one year).;
ln = natural logarithm
N(x) = standard normal cumulative distribution function
e = the exponential function
Share Price Delta PV(X)= Bank Loan
??Sln(：PVX()；Tt？)，d！？Exercise: Show that the following is true: . 12Tt；？
There are 3 steps to using the formula
1. Calculate d1 and d2.
2. Finds N(d1), N(d2).
In Excel use the NORMSDIST function; when using tables interpolate.
If only positive values of d are tabulated, recall that N(-d)=1-N(d)
3. Plug ‘n play.
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? normal distribution of underlying asset returns
(which is the same thing as saying that the underlying asset prices themselves are
? A lognormal distribution has a longer right tail compared with a normal, or
bell-shaped, distribution. The lognormal distribution allows for a stock price
distribution of between zero and infinity (ie no negative prices) and has an
upward bias (representing the fact that a stock price can only drop 100% but
can rise by more than 100%).
In practice underlying asset price distributions often depart significantly from the lognormal. For example historical distributions of underlying asset returns often have fatter left and right tails than a normal distribution indicating that dramatic market moves occur with greater frequency than would be predicted by a normal distribution of returns-- ie more very high returns and more very low returns.
Unlike volatility, which is all important for determining the fair value of an option, views about the future direction of an underlying asset (ie whether you think it will go up or down in the future and by how much) are completely irrelevant.
Significantly, the expected rate of return of the stock (which would incorporate risk preferences of investors as an equity risk premium) is not one of the
variables in the Black-Scholes model (or any other model for option valuation). The important implication is that the value of an option is completely independent of the expected growth of the underlying asset (and is therefore risk neutral).
Thus, while any two investors may strongly disagree on the rate of return they expect on a stock they will, given agreement to the assumptions of volatility and the risk free rate, always agree on the fair value of the option on that underlying asset.
This key concept underlying the valuation of all derivatives -- that fact that the price of an option is independent of the risk preferences of investors -- is called risk-neutral
valuation. It means that all derivatives can be valued by assuming that the return from their underlying assets is the risk free rate.
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Advantage: main advantage of the Black-Scholes model is speed (calculate a very large number of option prices in a very short time).
Limitation: one major limitation: it cannot be used to accurately price options
with an American-style exercise as it only calculates the option price at one point in time -- at expiration. It does not consider the steps along the way where there could be the possibility of early exercise of an American option. As all exchange traded equity options have American-style exercise (ie they can be exercised at any time as opposed to European options which can only be exercised at expiration) this is a significant limitation. The exception to this is an American call on a non-dividend paying asset. In this case the call is always worth the same as its European equivalent as there is never any advantage in exercising early.
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