Risk-Neutral Probabilities Lecture

Risk-Neutral Probabilities Lecture - Debt Instruments and...

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Unformatted text preview: Debt Instruments and Markets Professor Carpenter Risk-Neutral Probabilities Concepts  Risk-neutral probabilities  Risk-neutral pricing  Expected returns  True probabilities Reading  Veronesi, Chapter 9  Tuckman, Chapter 9 Risk-Neutral Probabilities 1 Debt Instruments and Markets Professor Carpenter No Arbitrage Derivative Pricing  Last lecture, we priced a derivative by constructing a replicating portfolio from the underlying zeroes: – We started with a derivative with a payoff at time 0.5. The payoff depended on the time 0.5 price of the zero maturing at time 1. – We modeled the random future price of the zero and the future payoff of the derivative. – We constructed a portfolio of 0.5-year and 1year zeroes with the same payoff of the derivative by solving simultaneous equations. – We then set the price of the derivative equal to the value of the replicating portfolio. General Bond Derivative • Any security whose time 0.5 payoff is a function of the time 0.5 price of the zero maturing at time 1 can be priced by no arbitrage. • Suppose its payoff is Ku in the up state, and Kd in the down state: Time 0 0.5-year zero 0.973047 1-year zero 0.947649 General portfolio 0.973047 N0.5 + 0.947649 N1 General derivative ? Time 0.5 1 0.972290 1N0.5 + 0.97229N1 Ku 1 0.976086 1N0.5 + 0.976086N1 Kd Risk-Neutral Probabilities 2 Debt Instruments and Markets Professor Carpenter Replicating and Pricing the General Derivative 1) Determine the replicating portfolio by solving the equations 1N0.5 + 0.97229N1 = Ku 1N0.5 + 0.96086N1 = Kd for the unknown N's. (The two possible K's are known.) 2) Price the replicating portfolio as 0.973047N0.5 + 0.947649N1 This is the no arbitrage price of the derivative. Risk-Neutral Probabilities • Finance: The no arbitrage price of the derivative is its replication cost. • We know that’s some function of the prices and payoffs of the basic underlying assets. • Math: We can use a mathematical device, risk-neutral probabilities, to compute that replication cost more directly. • That’s useful when we only need to know the price of the replicating portfolio, but not the holdings. Risk-Neutral Probabilities 3 Debt Instruments and Markets Professor Carpenter Start with the Prices and Payoffs of the Underlying Assets • In our example, the derivative payoffs were functions of the time 0.5 price the zero maturing at time 1. • So the underlying asset is the zero maturing at time 1 and the riskless asset is the zero maturing at time 0.5. • The prices and payoffs are, in general terms: Time 0 Time 0.5 1 u 0.5 1 d0.5 d d1 1 d 0.5 1 d € € Find the “Probabilities” that “Risk-Neutrally” Price the Underlying Risky Asset • Find the “probabilities” of the up and down states, p and 1-p, that make the price of the underlying asset equal to its “expected” future payoff, discounted back at the riskless rate. • I.e., find the p that solves “Risk-Neutral Pricing Equation” (RNPE) Price = discounted “expected” future payoff for the underlying risky asset. • In our example, this is the zero maturing at time 1, so d1 = d0.5 [ p × 0.5 d1u + (1− p) × 0.5 d1d ] d1 − 0.5 d1d d0.5 ⇒ p= u d 0.5 d1 − 0.5 d1 € Risk-Neutral Probabilities 4 Debt Instruments and Markets Professor Carpenter Example of p In our example, 0.947649 − 0.976086 0.973047 p= = 0.576 0.972290 − 0.976086 1− p = 0.424 € Result • The same p prices all the derivatives of the underlying risk-neutrally. • I.e., if a derivative has payoffs Ku in the up state and Kd in the down state, its replication cost turns out to be equal to Derivative price = d0.5 [ p × K u + (1− p) × K d ] € using the same p that made this risk-neutral pricing equation (RNPE) hold for the underlying asset. • That is, a single p makes the RNPE price = discounted “expected” future payoff hold for the underlying and all its derivatives. Risk-Neutral Probabilities 5 Debt Instruments and Markets Professor Carpenter Examples of Risk-Neutral Pricing With the risk-neutral probabilities, the price of an asset is its expected payoff multiplied by the riskless zero price, i.e., discounted at the riskless rate: call option: (0.576 × 0 + 0.424 ×1.086) × 0.9730 = 0.448 or, 0.576 × 0 + 0.424 ×1.086 = 0.448 1.0277 Class Problem: Price the put option with payoffs Ku=2.71 and Kd=0 € using the risk-neutral probabilities. Examples of Risk-Neutral Pricing... 1-year zero: (0.576 × 0.9723 + 0.424 × 0.9761) × 0.9730 = 0.9476 or, € 0.576 × 0.9723 + 0.424 × 0.9761 = 0.9476 1.0277 0.5-year zero (riskless asset): (0.576 ×1+ 0.424 ×1) × 0.9730 = 0.9730 or, 0.576 ×1+ 0.424 ×1 = 0.9730 1.0277 € Risk-Neutral Probabilities 6 Debt Instruments and Markets Professor Carpenter Why does the p that makes the RNP Equation hold for the underlying also make the RNPE work for all its derivatives? 1) By construction, p makes price of underlying risky asset = discount factor x [p x underlying’s up payoff + (1-p) x underlying’s down payoff]. 2) It’s also always true for any p that price of riskless asset = discount factor = discount factor x [p x 1 + (1-p) x 1] so the p that works for the underlying also works for the riskless asset, because any p does. 3) Therefore, this p also works for any portfolio of these two assets. I.e., for any portfolio with holdings N0.5 and N1: N0.5 x price of riskless asset + N1 x price of underlying = disc.factor x [p x (N0.5 x 1 + N1 x underlying’s up payoff ) + (1-p) x (N0.5 x 1 + N1 x underlying’s down payoff ) ] i.e., portfolio price = disc.factor x [p x portfolio’s up payoff + (1-p) x portfolio’s down payoff] Since every derivative of the underlying is one of these portfolios, the RNPE, using the same p, holds for all of them too. Class Problem • Suppose the time 0 price of the zero maturing at time 1 is slightly lower: Time 0 0.973047 0.947007 Time 0.5 1 0.972290 1 0.976086 • What would be the risk-neutral probabilities p and 1-p of the up and down states? Risk-Neutral Probabilities 7 Debt Instruments and Markets Professor Carpenter Class Problem 1) Price the call using these new RN probs: Time 0 Time 0.5 1 0.972290 0.973047 0 0.947007 Call? 1 0.976086 1.086 2) Calculate the replication cost of the call the old way and verify that it matches the price above. Hint: the payoffs are unchanged so the replicating portfolio is still N0.5 = -278.163 and N1 = 286.091. Expected Returns with RN Probs • Note that we can rearrange the risk-neutral pricing equation, price = discounted “expected” payoff, as V = d0.5 [ p × K u + (1− p) × K d ], or p × K u + (1− p) × K d V= 1+ r0.5 /2 p × K u + (1− p) × K d ⇔ = 1+ r0.5 /2 V • I.e., “expected” return = the riskless rate. € (Here return is un-annualized. ) • Thus, with the risk-neutral probabilities, all assets have the same expected return--equal to the riskless rate. • This is why we call them "risk-neutral" probabilities. Risk-Neutral Probabilities 8 Debt Instruments and Markets Professor Carpenter True Probabilities  The risk-neutral probabilities are not the same as the true probabilities of the future states.  Notice that pricing contingent claims did not involve the true probabilities of the up or down state actually occurring.  Let's suppose that the true probabilities are 0.5 chance the up state occurs and 0.5 chance the down state occurs.  What could we do with this information?  For one, we could compute the true expected returns of the different securities over the next 6 months. True Expected Returns Recall that the unannualized return on an asset over a given horizon is future value −1 initial value For the 6-month zero the unannualized return over the next 6 months is € 1 −1 = 2.77% 0.973047 with certainty. This will be the return regardless of which state occurs. That's why € this asset is riskless for this horizon. Of course, the annualized semi-annually compounded ROR is 5.54%, the quoted zero rate. Risk-Neutral Probabilities 9 Debt Instruments and Markets Professor Carpenter True Expected Returns... The return on the 1-year zero over the next 6 months will be either 0.972290 −1 = 2.60% with probability 0.5, or 0.947649 0.976086 −1 = 3.00% with probability 0.5. 0.947649 € The expected return on the 1-year zero over the next 6 months is 2.80%. Notice that it is higher than the return of 2.77% on the riskless asset. True Expected Returns...  Why might the longer zero have a higher expected return? – Investors have short-term horizons, and dislike the price risk of the longer zero. – Investors require a premium to hold securities that covary positively with long bonds (bullish securities) because government bonds are in positive net supply.  Sometimes the reverse could be true.  In general, assets with different risk characteristics have different expected returns. Their expected returns also depend on how their payoffs covary with other assets. Risk-Neutral Probabilities 10 Debt Instruments and Markets Professor Carpenter True Expected Returns... What is the expected rate of return on the call over the next 6 months? The possible returns are: 0 −1 = −100% with probability 0.5, or 0.448 1.086 −1 = 142% with probability 0.5. 0.448 € The expected return on the call is 21%. True Expected Returns... What is the expected rate of return on the put over the next 6 months? The possible returns are: 2.71 −1 = 78% with probability 0.5, or 1.52 0 −1 = −100% with probability 0.5. 1.52 € Risk-Neutral Probabilities The expected return on the put is -11%. The put is bearish--it insures (hedges) the risk of bullish positions. By no arbitrage, if bullish assets have positive risk premia, bearish assets must have negative risk premia. Intuitively, investors must pay up for this insurance. 11 Debt Instruments and Markets Professor Carpenter Risk-Neutral Expected Returns Using the risk-neutral probabilities to compute expected (unannualized) returns sets all expected returns equal to the riskless rate. Asset 0.5-Year Zero 1-Year Zero Call Put Unannualized Up Return ("prob"=0.576) Unannualized Down Payoff ("prob"=0.424) 1/0.9730 - 1 1/0.9730 - 1 = 2.77% = 2.77% 0.97229/0.947649 - 1 0.976086/0.947649 - 1 = 2.60% = 3.00% 0/0.448 - 1 = -100% 2.7103/1.519 - 1 = 78.42% 1.0859/0.448 - 1 = 142.39% 0/1.519 - 1 = -100% "Expected" Unannualized Return 2.77% 2.77% 2.77% 2.77% Why Does the p that Works for the Underlying Asset Also Work for All Its Derivatives? Reprise: 1)  The expected return on a portfolio is the average of the expected returns of the individual assets. 2)  The risk-neutral probabilities are constructed to make the expected return on the underlying risky asset equal to the riskless asset return. 3)  So under the risk-neutral probabilities, the expected return on every portfolio of the underlying and riskless assets is also that same riskless return. 4)  Every derivative of the underlying can be viewed as a portfolio of the underlying asset and the riskless asset. 5)  So the derivative’s expected return must also equal the riskless return under the risk-neutral probabilities. 6)  So the derivative’s price must equal its expected payoff, using the risk-neutral probabilities, discounted back at the riskless rate. Risk-Neutral Probabilities 12 ...
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