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[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"This spreadsheet is an educational example.  The Society of Actuaries Research Institute does not warrant this spreadsheet as fit for business use in any respect, and no warranty whatsoever should be assumed or implied by any individual of this spreadsheet or its fitness for any particular purpose."]
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[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"© 2022  Society of Actuaries Research Institute"]

{"sheet":"Calculations","index":1,"n_rows":229,"n_cols":25,"rows_kept":229,"cut":false,"hidden":false,"merged":[[2,3,2,5],[2,7,2,11],[16,0,16,1],[16,22,16,24]]}
["Valuation of a put option on equity stock"]
["One-year tenor",null,null,null,null,null,null,null,null,null,null,null,null,"Values to try:"]
[null,null,null,"Risk-Neutral",null,null,null,"Real-World"]
[null,null,null,null,null,null,null,"Market price of risk (MPR):",null,0.3," = risk spread / volatility",null,null,"<- zero or roughly 30%"]
[null,null,null,"Risk-free rate:",null,{"$f":"J5","v":0.04},null,"Risk-free rate:",null,0.04,null,null,null,"<- Reasonable range is 0% to 10%"]
[null,null,null,"Implied volatility:",null,{"$f":"impliedVolatilityBS(1+J7,1,$F$5,1,$J$15,$J$6)","v":0.15479467741755026},null,"Est. realized volatility:",null,0.15,null,null,null,"<- Broad stock market index volatility is roughly 15%"]
[null,null,null,"Strike:",null,{"$f":"J7","v":0},null,"Strike:",null,0," % difference from current",null,null,"<- Reasonable range is 0% +/-  10%"]
[null,null,null,"Expected returns on underlying equity:",null,null,null,"Expected returns on underlying equity:"]
[null,null,null,"Risk-free rate:",null,{"$f":"F5","v":0.04},null,"Risk spread:",null,{"$f":"J4*J6","v":4.4999999999999998E-2}," = volatility * MPR"]
[null,null,null,null,null,null,null,"Risk-adjusted return:",null,{"$f":"J5+J4*J6","v":8.4999999999999992E-2}," = risk free + risk spread"]
[null,null,null,"Valuation of the put option:",null,null,null,"Valuation of the put option:",null,null,null,null,null,"Real-world calculation differences from risk-neutral approach:"]
[null,null,null,"Expected present value:",null,{"$f":"F121/EXP(F5)","v":4.2877986101618998E-2},null,"Expected present value:",null,{"$f":"J121/EXP(J5)","v":2.6704346787942906E-2}]
[null,null,null,null,null,null,null,"Std dev of present value:",null,{"$f":"SQRT(L121)/EXP(J5)","v":5.3917996054546828E-2},null,null,null,"1.  The mean expected return includes a risk spread"]
[null,null,null,null,null,null,null,"Risk adjustment:",null,{"$f":"J4*J13","v":1.6175398816364046E-2}," = std dev * MPR",null,null,"2.  The volatility is the expected realized volatility, not the implied volatility"]
[null,null,null,"Price = expected value",null,{"$f":"F12","v":4.2877986101618998E-2},null,"Price",null,{"$f":"(J12+J14)","v":4.2879745604306949E-2}," = expected val + risk adj",null,null,"3.  The option price includes a risk adjustment over the expected value"]
[]
["Standard Normal Distribution",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Probability distribution of 1-year log return"]
["Std errs","Probability weight",null,"Return","Payoff","Probability weighted payoff",null,"Return","Payoff","Probability weighted payoff","Diff from mean","Probability weighted diff^2",null,null,null,null,null,null,null,null,null,null,null,"Risk neutral","Real world"]
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[-4.0000000000000036,1.3383022576488348E-5,null,-0.44631503308058673,0.44631503308058673,5.9730441639436358E-6,null,-0.40918361787851876,0.40918361787851876,5.4761135959973976E-6,0.38138944602385788,1.9466664872810087E-6,null,null,null,null,null,null,null,null,null,null,{"$f":"W28+0.005","v":-0.44999999999999996},1.7190028842238601E-2,4.5968954575189092E-3]
[-3.9000000000000035,1.9865547139276991E-5,null,-0.4376776055758147,0.4376776055758147,8.6947051053722288E-6,null,-0.40025457171949153,0.40025457171949153,7.9512760622046823E-6,0.37246039986483065,2.7558827810246854E-6,null,"If the market price of risk is not zero, the distributions peak at a different value.",null,null,null,null,null,null,null,null,{"$f":"W29+0.005","v":-0.44499999999999995},1.9030741398988787E-2,5.1743540413927355E-3]
[-3.8000000000000034,2.9194692579145637E-5,null,-0.42890543510643719,0.42890543510643719,1.2521762323457133E-5,null,-0.39119058030890608,0.39119058030890608,1.1420688731976096E-5,0.3633964084542452,3.8553620487727007E-6,null,null,null,null,null,null,null,null,null,null,{"$f":"W30+0.005","v":-0.43999999999999995},2.1046587600971004E-2,5.8178846336384114E-3]
[-3.7000000000000033,4.2478027055074647E-5,null,-0.41999641969631596,0.41999641969631596,1.7840619278894595E-5,null,-0.38198960421045625,0.38198960421045625,1.6226164742409015E-5,0.35419543235579537,5.32905558010626E-6,null,"If implied volatility is not equal to est. realized volatility, the lines don't reach ",null,null,null,null,null,null,null,null,{"$f":"W31+0.005","v":-0.43499999999999994},2.3251691694766458E-2,6.5341864085024914E-3]
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