3. Los teoremas de Markov y de Birkhoff-Kakutani 29
3.3. Metrizabilidad
– In calendar spreads, the options have the same strike price and different expiration dates
– A calendar spread can be created by selling a call option with a certain strike price and buying a longer-maturity call option with the same strike price
∗ The longer the maturity, the more expensive it is ⇒ Calendar spread requires initial investment
Short call option (K, short maturity)
Long call option (K, longer maturity)
K Calendar spread using two call options
Profit
T
S
Figure 8: Profit from calendar spread using calls
Long put option (K, longer maturity) Short put option (K, short maturity)
Calendar spread using two put options
K
Profit
T
S
Figure 9: Profit from calendar spread using puts – Profit diagrams for calendar spreads are usually produced showing profit when short-maturity
option expires on the assumption that long-maturity option is sold at that time (Fig. 8) ∗ The pattern is similar to the profit from the butterfly spread in Fig. 6
∗ The investor makes a profit if the stock price at the expiration of the short-maturity option is close to the strike price of the short-maturity option. However, a loss is incurred when the stock price is significantly above or significantly below this strike price
– Qualitative explanation of the profit pattern
∗ If the stock price is very low when the short-maturity option expires, the short-maturity option is worthless and the value of the long-maturity option is close to zero
⇒ The investor incurs a loss that is close to the cost of setting up the spread initially
∗ If the stock price ST is very high when the short-maturity option expires, the short-maturity option costs the investor ST − K, and the long-maturity option is worth close to ST − K ⇒ Again, the investor makes a net loss that is close to the cost of setting up the spread initially
∗ If ST is close to K, the short-maturity option costs the investor either a small amount or
nothing at all. However, the long-maturity option is still quite valuable ⇒ In this case a significant net profit is made
– Calendar spreads can be created with put options as well as call options:
∗ The investor buys a long-maturity put option and sells a short-maturity put option ∗ As shown in Fig. 9, the profit pattern is similar to that obtained from using calls – Types of calendar spreads
∗ Neutral calendar spread: A strike price close to the current stock price is chosen ∗ Bullish calendar spread: Involves a higher strike price
∗ Bearish calendar spread: Involves a lower strike price ∗ Reverse calendar spread: Opposite to that in Figs. 8 and 9:
· The investor buys a short-maturity option and sells a long-maturity option
· A small profit arises if the stock price at the expiration of the short-maturity option is well above or well below the strike price of the short-maturity option
· However, a significant loss results if it is close to the strike price • Diagonal spreads
– Both expiration date/strike price of calls are different ⇒ Increases range of possible profit patterns Combinations
• Combination: Strategy that involves taking a position in both calls and puts on the same stock • Straddle
– A straddle involves buying a call and put with the same strike price and expiration date ∗ If stock price close to strike price at expiration of the options, straddle leads to a loss ∗ However, if there is a sufficiently large move in either direction, significant profit results
Long call option (K, maturity T)
Long put option (K, maturity T) Bottom straddle (straddle purchase)
K
Profit
T
S
Figure 10: Profit from a straddle – Payoff from a straddle
Stock price Call payoff Put payoff Total payoff
ST ≤ K 0 K − ST K − ST
ST > K ST − K 0 ST− K
– Straddle appropriate when investor expects large move in stock price but direction unknown – Carefully consider whether the anticipated jump is already reflected in option prices
– The straddle in Fig. 10 is sometimes referred to as a bottom straddle or straddle purchase – A top straddle or straddle write is the reverse position:
∗ It is created by selling a call and a put with the same exercise price and expiration date ∗ It is a highly risky strategy:
· If stock price at expiration is close to strike price, significant profits · However, loss arising from a large move is unlimited
• Strips and straps
– Strip consists of long position in one call/two puts with same strike price/expiration date – Strap consists of long position in two calls/one put with same strike price/expiration date
– In a strip the investor is betting that there will be a big stock price move and considers a decrease in the stock price to be more likely than an increase
Long put (K, T) 2 long calls (K, T) T S K Strap Profit 2 long puts (K, T) Long call (K, T) Strip K Profit T S
Figure 11: Profit from a strip and a strap
– In a strap the investor is also betting that there will be a big stock price move. However, in this case, an increase in the stock price is considered to be more likely than a decrease
• Strangles
– In a strangle, sometimes called a bottom vertical combination, an investor buys a put and a call with the same expiration date and different strike prices
) 2 Long call (K ) 1 Long put (K 2 K 1 K T S Profit
Strangle (bottom vertical combination)
Figure 12: Profit from a strangle ∗ The call strike price K2 is higher than the put strike price K1
∗ Profit pattern from strangle depends on how close the strike prices are: The farther they are apart, the less the downside risk/the farther the stock price has to move for a profit
∗ Payoff from a strangle
Stock price Call payoff Put payoff Total payoff
ST ≤ K1 0 K1− ST K1− ST
K1< ST < K2 0 0 0
ST > K2 ST − K2 0 ST − K2 – A strangle is a similar strategy to a straddle:
∗ Investor betting on large price move, but uncertain direction
∗ Stock price has to move farther in strangle than in straddle for investor to profit ∗ However, downside risk if stock price ends up at a central value is less with a strangle – The sale of a strangle is sometimes referred to as a top vertical combination:
∗ Appropriate for an investor who feels that large stock price moves are unlikely
∗ However, as with sale of straddle, it is a risky strategy involving unlimited potential loss Other payoffs
• If European options expiring at time T were available with every single possible strike price, any payoff function at time T could in theory be obtained
• Through the judicious combination of a large number of very small “spikes” (payoff from a butterfly spread), any payoff function can be approximated
Hull - Ch. 11: Binomial trees
Binomial tree
• Diagram representing different possible paths of a stock price over the life of an option • Useful and very popular technique for pricing an option
– The underlying assumption is that the stock price follows a random walk
– In each time step, it has certain probabilities of moving up/down by a certain % amount • As time step ← 0, model leads to lognormal assumption for stock prices (Black-Scholes model) A one-step binomial model and a no-arbitrage argument
• A very simple situation
– Stock price currently $20, and at the end of 3 months it will be either $22 or $18
– We are interested in valuing a European call option to buy the stock for $21 in 3 months – The only assumption needed is that arbitrage opportunities do not exist
– We set up a portfolio of the stock and the option in such a way that there is no uncertainty about the value of the portfolio at the end of the 3 months
∗ We then argue that, because portfolio has no risk, the return it earns must equal rf ∗ This enables us to work out cost of setting up the portfolio and thus the option’s price – Because there are two securities (the stock and the stock option) and only two possible outcomes,
it is always possible to set up the riskless portfolio
– Consider portfolio: Long position in ∆ shares of stock and short position in one call option ∗ We calculate the value of ∆ that makes the portfolio riskless
∗ If the stock price moves up from $20 to $22, the value of the shares is 22∆ and the value of the option is 1, so that the total value of the portfolio is 22∆ − 1
∗ If the stock price moves down from $20 to $18, the value of the shares is 18∆ and the value of the option is zero, so that the total value of the portfolio is 18∆
∗ Portfolio riskless if: 22∆ − 1 = 18∆ ⇒ ∆ = 0.25, i.e. Long: 0.25 shares, Short: 1 option – Riskless portfolios must, in the absence of arbitrage opportunities, earn the risk-free rate
∗ Suppose that in this case the risk-free rate is 12% per annum
∗ Portfolio today must be worth PV of 4.5 (= 22 × 0.25 − 1 = 18 × 0.25) = 4.5e−0.12×3/12= 4.367 – The value of the stock price today is known to be $20. Suppose the option price is denoted by f
∗ The value of the portfolio today is: 20 × 0.25 − f = 5 − f = 4.367 ⇒ f = 0.633
∗ In the absence of arbitrage opportunities, the current value of the option must be 0.633 • Generalization
– Suppose that the option lasts for time T and that during the life of the option the stock price can either move up from S0 to a new level S0u (u > 1), or down from S0 to a new level S0d (d < 1) – % increase in stock price when up movement: u − 1. % decrease when down movement: 1 − d – If the stock price moves up to S0u, we suppose that the payoff from the option is fu. If the stock
price moves down to S0d, we suppose the payoff from the option is fd
– Consider portfolio consisting of long position in ∆ shares and short position in one option. We calculate the value of ∆ that makes the portfolio riskless:
S0u∆ − fu = S0d∆ − fd ⇒ ∆ =
fu− fd S0u − S0d
(1) – Eq. (1): ∆ = ratio [change in option price] ÷ [change in stock price] moving between nodes at T
– PV of portfolio = (S0u∆ − fu)e−rT, and cost of setting up portfolio = S0∆ − f . Hence:
S0∆ − f = (S0u∆ − fu)e−rT ⇒ f = S0∆(1 − ue−rT) + fue−rT
– Eqs. (2)/(3) enable option to be priced when stock moves are given by a one-step binomial tree:
f = e−rT[pfu+ (1 − p)fd] (2)
p = e
rT − d
• Irrelevance of the stock’s expected return
– Option pricing formula in Eq. (2) does not involve probabilities of stock moving up/down: ∗ This is surprising and seems counterintuitive
– The key reason is that we are not valuing the option in absolute terms: ∗ We are calculating its value in terms of the price of the underlying stock
∗ Probabilities of future up/down moves already incorporated into stock price: No need to take them into account again when valuing option in terms of stock price
Risk-neutral valuation
• Natural to interpret variable p in Eq. (2) as the probability of an up movement in stock price
– 1 − p is then probability of down move, and pfu+ (1 − p)fd= expected payoff from option
– Eq. (2) then states that the option value today is its expected payoff discounted risk-free • Expected return from the stock when the probability of an up movement is p:
– The expected stock price E[ST] at time T is:
E[ST] = pS0u + (1 − p)S0d = pS0(u − d) + S0d = S0erT (4)
– Probability of up move ≡ p ⇔ Assuming that return on stock = r • Risk neutral world
– All individuals are indifferent to risk. In such a world, investors require no compensation for risk, and the expected return on all securities is the risk-free interest rate
– Eq. (4): We assume risk-neutral world when we set probability of an up move to p – Eq. (2): Option value = its expected payoff in a risk-neutral world discounted risk-free ⇒ Principle of risk-neutral valuation:
– States that we can with complete impunity assume the world is risk neutral when pricing options – The resulting prices are correct not just in a risk-neutral world, but in other worlds as well • The one-step binomial example revisited
– Define p as the probability of an upward movement in the stock price in a risk-neutral world: ∗ We can calculate p from Eq. (3)
∗ Alternatively, argue that expected return on stock in risk-neutral world = risk-free rate of 12%
⇒ p must satisfy: 22p + 18(1 − p) = 20e0.12×3/12 ⇒ p = 0.6523
– At the end of the 3 months, the call option has a 0.6523 probability of being worth 1 and a 0.3477 probability of being worth zero. Its expected value is therefore: 0.6523 × 1 + 0.3477 × 0 = 0.6523 – In a risk-neutral world, the value of the option today is therefore: 0.6523e−0.12×3/12= 0.633 – No-arbitrage arguments and risk-neutral valuation give the same answer
• Real world vs. risk-neutral world
– p = probability of up move in risk-neutral world 6= probability of up move in real world (in general)
– Suppose, in real world, expected return is 16% and p∗ = probability of up move:
22p∗+ 18(1 − p∗) = 20e0.16×3/12 ⇒ p∗ = 0.7041
– The expected payoff from the option in the real world is then: p∗× 1 + (1 − p∗) × 0 = 0.7041 – Unfortunately, correct discount rate to apply to expected payoff in real world is unknown
∗ A position in a call option is riskier than a position in the stock
⇒ The discount rate to be applied to the payoff from a call option is greater than 16%
– Using risk-neutral valuation is convenient because in a risk-neutral world the expected return on all assets (and thus the discount rate to use for all expected payoffs) is the risk-free rate
Two-step binomial trees
• We can extend the analysis to a two-step binomial tree such as that shown in Fig. 1
– Stock price initially S0. During each time step, it moves up/down to [u/d] × [its initial value] – Suppose that the risk-free interest rate is r and the length of the time step is ∆t years
] dd f 2 ) p + (1{ ud f ) p (1{ p + 2 uu f 2 p [ t ¢ r {2 e = f ) ] d f ) p + (1{ u pf [ t ¢ r { e = f ] dd f ) p + (1{ ud pf [ t ¢ r { e = d f ] ud f ) p + (1{ uu pf [ t ¢ r { e = u f p p 1{ p 1{ p p 1{ p d { u||| | { d t ¢ r e = p t ¢ uu f ud f dd f d f f u f 2 u 0 S 2 d 0 S ud 0 S d 0 S 0 S u 0 S
Figure 1: Stock and option prices in general two-step tree • Eqs. (2) and (3) become:
f = e−r∆t[pfu+ (1 − p)fd] (5)
p = e
r∆t− d
u − d (6)
• Repeated application of Eq. (5) gives:
fu = e−r∆t[pfuu+ (1 − p)fud] (7)
fd= e−r∆t[pfud+ (1 − p)fdd] (8)
f = e−r∆t[pfu+ (1 − p)fd] (9)
• Substituting Eqs. (7) and (8) into Eq. (9), we get:
f = e−2r∆t[p2fuu+ 2p(1 − p)fud+ (1 − p)2fdd] (10)
• This is consistent with the principle of risk-neutral valuation
– p2, 2p(1 − p), (1 − p)2 are probabilities that upper/middle/lower final nodes reached – Option price = its expected payoff in risk-neutral world discounted risk-free
• As we add more steps to the binomial tree, the risk-neutral valuation principle continues to hold ⇒ Option price always = its expected payoff in risk-neutral world discounted risk-free
• Put options
– The procedures described above can be used to price puts as well as calls • American options
– The procedure is to work back through the tree from the end to the beginning, testing at each node to see whether early exercise is optimal
– The value of the option at the final nodes is the same as for the European option – At earlier nodes the value of the option is the greater of:
1. The value given by Eq. (5) 2. The payoff from early exercise • Delta
– ∆ is an important parameter in the pricing and hedging of options
∗ ∆ = ratio of [change in option price] ÷ [change in underlying stock price]
∗ ∆ = # of units of stock we should hold for each option shorted to create riskless portfolio – The construction of a riskless hedge is sometimes referred to as delta hedging
– The delta of a call option is positive, whereas the delta of a put option is negative
Matching volatility with u and d
• In practice, when constructing a binomial tree, we choose u and d to match volatility of stock price – Suppose that the expected return on a stock (in the real world) is µ and its volatility is σ
– The step is of length ∆t. Stock price starts at S0 and moves either up to S0u or down to S0d
– Probability of up move in real world is p∗ and in risk-neutral world it is p
– The expected stock price at the end of the first time step in the real world is S0eµ∆t. On the tree, the expected stock price at this time is: p∗S0u + (1 − p∗)S0d
– To match the expected return on the stock with the tree’s parameters, we must have: p∗S0u + (1 − p∗)S0d = S0eµ∆t ⇒ p∗=
eµ∆t− d
u − d (11)
• The volatility σ of a stock price is defined so that σ√∆t is the std. dev. of the return on the stock price
in a short period of time of length ∆t ⇔ Variance of return = σ2∆t
– On the tree, variance of stock price return = p∗u2+ (1 − p∗)d2− [p∗u + (1 − p∗)d]2 – To match the stock price volatility with the tree’s parameters, we must have:
p∗u2+ (1 − p∗)d2− [p∗u + (1 − p∗)d]2= σ2∆t (12)
– Substituting Eq. (11) into Eq. (12) gives: eµ∆t(u + d) − ud − e2µ∆t = σ2∆t and when terms in ∆t2 and higher powers of ∆t are ignored, one solution to this equation is:
u = eσ √
∆t and d = e−σ√∆t (13)
– These are values of u/d proposed by Cox, Ross, Rubinstein (1979) for matching volatility • Risk-neutral analysis
– The variable p is given by Eq. (6) as:
p = a − d
u − d where a = e
r∆t (14)
– The expected stock price at the end of the time step is S0er∆t – The variance of the stock price return in the risk-neutral world is:
pu2+ (1 − p)d2− [pu + (1 − p)d]2= [er∆t(u + d) − ud − e2r∆t]
– Substituting u/d from Eq. (13), this equals σ2∆t when terms in ∆t2 are ignored
⇒ When we move from the real world to the risk-neutral world the expected return on the stock changes, but its volatility remains the same (at least in the limit as ∆t tends to zero)
• Girsanov’s theorem
– When we move from a world with one set of risk preferences to a world with another set of risk preferences, the expected growth rates in variables change, but their volatilities remain the same – Moving from one set of risk preferences to another is sometimes referred to as changing the measure – Real-world measure: aka, P -measure. Risk-neutral world measure: aka, Q-measure
Increasing the number of steps
• When binomial trees are used in practice, option life is typically divided into 30+ time steps • In each time step there is a binomial stock price movement
• With 30 time steps, 31 terminal stock prices and 230≈ 109 stock price paths implicitly considered Options on other assets
• We can construct and use binomial trees for options on indices, currencies, and futures contracts in exactly the same way as for options on stocks except that the equations for p change
• Options on stocks paying a continuous dividend yield – Consider a stock paying a known dividend yield at rate q
∗ The total return from dividends and capital gains in a risk-neutral world is r ∗ The dividends provide a return of q ⇒ Capital gains must provide a return of r − q
– If stock starts at S0, its expected value after one time step ∆t must be S0e(r−q)∆t. Hence,
pS0u + (1 − p)S0d = S0e(r−q)∆t ⇒ p = e
(r−q)∆t− d u − d – We match volatility by setting u = eσ
√
∆t and d = 1/u
⇒ We can use Eqs. (13) to (14), except that we set a = e(r−q)∆t instead of a = er∆t • Options on stock indices
– We assume that the stocks underlying the index provided a dividend yield at rate q – Valuation of option on stock index ⇔ Option on stock paying known dividend yield • Options on currencies
– Foreign currency ∼ asset providing a yield at foreign risk-free rate of interest rf
– Analogy with stock index: For options on a currency, use Eqs. (13)/(14) and set a = e(r−rf)∆t
• Options on futures
– It costs nothing to take a long or a short position in a futures contract
⇒ In a risk-neutral world a futures price should have an expected growth rate of zero – If F0 = initial futures price, expected futures price at end of time step ∆t is also F0. Thus,
pF0u + (1 − p)F0d = F0 ⇒ p =
1 − d u − d – Therefore, we can use Eqs. (13)/(14) with a = 1
Fabozzi: Valuation of bonds with embedded options
Introduction
• The complication in building a model to value bonds with embedded options and option-type derivatives is that cash flows will depend on interest rates in the future
• The first step is to move from the yield curve to a valuation lattice
• Lattice holds all the info required to value certain option-like interest-rate products – First, the lattice is used to generate the CFs across the life of the security – Next, the interest rates on the lattice are used to compute the PV of those CFs • Binomial model: Lattice model where only two rates are possible in next period
• Regardless of underlying assumptions, models share common restriction: Interest-rate tree generated must produce a value for an on-the-run optionless issue consistent with current par yield curve
– Output from the model must be equal to observed market price for optionless instrument – Under these conditions, the model is said to be “arbitrage free”
– A lattice that produces an arbitrage-free valuation is said to be “fair” – The lattice is used for valuation only when it has been calibrated to be fair The interest rate lattice
• Fig. 1 provides an example of a binomial interest-rate tree: Consists of number of “nodes”/“legs”